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[ "Cherenkov Telescope Array: The next-generation ground-based gamma-ray obser- vatory", "Cherenkov Telescope Array: The next-generation ground-based gamma-ray obser- vatory" ]	[ "G Hermann \nMax Planck Institute for Nuclear Physics\nSaupfercheckweg 169117HeidelbergGermany\n", "W Hofmann \nMax Planck Institute for Nuclear Physics\nSaupfercheckweg 169117HeidelbergGermany\n", "T Schweizer [email protected] \nFOR THE CTA CONSORTIUM\n\n\nMax Planck Institute for Physics (Werner-Heisenberg-Institut)\nFöhringer Ring 680805Munich, Ger-many\n", "M Teshima " ]	[ "Max Planck Institute for Nuclear Physics\nSaupfercheckweg 169117HeidelbergGermany", "Max Planck Institute for Nuclear Physics\nSaupfercheckweg 169117HeidelbergGermany", "FOR THE CTA CONSORTIUM\n", "Max Planck Institute for Physics (Werner-Heisenberg-Institut)\nFöhringer Ring 680805Munich, Ger-many" ]	[]	High energy gamma-ray astronomy is a newly emerging and very successful branch of astronomy and astrophysics. Exciting results have been obtained by the current generation Cherenkov telescope systems such as H.E.S.S., MAGIC, VERITAS and CANGAROO. The H.E.S.S. survey of the galactic plane has revealed a large number of sources and addresses issues such as the question about the origin of cosmic rays. The detection of very high energy emission from extragalactic sources at large distances has provided insights in the star formation during the history of the universe and in the understanding of active galactic nuclei. The development of the very large Cherenkov telescope array system (CTA) with a sensitivity about an order of magnitude better than current instruments and significantly improved sensitivity is under intense discussion. This observatory will reveal an order of magnitude more sources and due to its higher sensitivity and angular resolution it will be able to detect new classes of objects and phenomena that have not been visible until now. A combination of different telescope types will provide the sensitivity needed in different energy ranges.	null	[ "https://arxiv.org/pdf/0709.2048v1.pdf" ]	18,314,795	0709.2048	81a83200c2ada4727bcc906b8975d89795b333a5	Cherenkov Telescope Array: The next-generation ground-based gamma-ray obser- vatory G Hermann Max Planck Institute for Nuclear Physics Saupfercheckweg 169117HeidelbergGermany W Hofmann Max Planck Institute for Nuclear Physics Saupfercheckweg 169117HeidelbergGermany T Schweizer [email protected] FOR THE CTA CONSORTIUM Max Planck Institute for Physics (Werner-Heisenberg-Institut) Föhringer Ring 680805Munich, Ger-many M Teshima Cherenkov Telescope Array: The next-generation ground-based gamma-ray obser- vatory 30TH INTERNATIONAL COSMIC RAY CONFERENCE High energy gamma-ray astronomy is a newly emerging and very successful branch of astronomy and astrophysics. Exciting results have been obtained by the current generation Cherenkov telescope systems such as H.E.S.S., MAGIC, VERITAS and CANGAROO. The H.E.S.S. survey of the galactic plane has revealed a large number of sources and addresses issues such as the question about the origin of cosmic rays. The detection of very high energy emission from extragalactic sources at large distances has provided insights in the star formation during the history of the universe and in the understanding of active galactic nuclei. The development of the very large Cherenkov telescope array system (CTA) with a sensitivity about an order of magnitude better than current instruments and significantly improved sensitivity is under intense discussion. This observatory will reveal an order of magnitude more sources and due to its higher sensitivity and angular resolution it will be able to detect new classes of objects and phenomena that have not been visible until now. A combination of different telescope types will provide the sensitivity needed in different energy ranges. Introduction The Cherenkov Telescope Array (CTA) [1] is a proposed advanced facility for ground based highenergy gamma ray astronomy, based on the observation of Cherenkov radiation. This approach has proven to be extremely successful for gamma rays of energies above a 100 GeV. The facility will consist of an array of telescopes enhancing the all sky monitoring capability. CTA is designed to be the next generation observatory after currently running Cherenkov telescope projects such as the H.E.S.S. experiment [2], MAGIC [3], VERITAS [4] and CANGAROO [5]. In order to cover the full sky, it is planned to build two stations, one in the northern hemisphere and the other one in the southern hemisphere. CTA will be about a factor of ten more sensitive than current experiments (see Fig.1). It will allow in-depth studies of known classes of gamma emitters and also detect new source classes that are below the sensitivity of current instruments. In its core energy range from about 100 GeV up to 10 TeV, CTA will reach millicrab sensitivity. The full energy range will be three to four orders of magnitude from some 10 GeV up to 100 TeV, which is crucial for the understanding of the physical processes. Together with multiwavelength observations from radio wavelengths to optical, Xray and MeV energy range, CTA will reveal deep insight in the most violent and energetic physical processes in the universe. For selected sets of gamma-ray events, CTA will have an angular resolution by a factor of 5 better than current instruments because of the large amount of telescopes within the array. This angular resolution will be necessary for resolving the high density of galactic objects within the milky way. It allows also detailed morphological studies of extended objects. Due to its large detection area and lower threshold it will collect a much larger event statistics and improve the time resolution of transient objects down to the sub-minute scale. Some few telescopes can be used for monitoring transient objects. This will deliver interesting details about the acceleration process of particles in cosmic accelerators. All in all, CTA is expected to detect and observe about 1000 sources. In March 2007 a European FP7 design study was applied for to examine the exact system layout and technical details of the CTA. 34 institutes in 15 countries (France, Germany, Italy, Spain, United Kingdom, Poland, Finnland, Switzerland, Netherlands, Czech Republic, Armenia, Ireland, United States, Republic of South Africa) are participating in the CTA design study. Physics prospects of CTA The large list of the astrophysics cases for CTA can be roughly split into two parts, the galactic sources and the extragalactic sources. In this short overview we will only mention the most promising science cases. In the galaxy many TeV gamma sources have already been discovered, mainly in the galactic scan by H.E.S.S. (see Fig.2). The sources are supernova remnants (SNRs), pulsar wind nebulae (PWNs), binary systems, the galactic center itself and diffuse gamma radiation from our galaxy. The galactic astrophysical subjects are manyfold. One main subject is the question about the origin of cosmic rays (CR). The origin of the bulk of CR observed at earth remains unknown even almost one century after their discovery by Victor Hess in 1912. Several arguments hint to supernova remnants (SNRs) being the main candidate sources, but a solid evidence is still missing. The detection of TeV radiation from 30-100 SNRs that will be detected by an instrument with a higher sensitivity would be the ultimate proof of these sources to be the bulk producers of CR arriving at earth. The measurement of their spectra at 100 GeV up to 100 TeV region will reveal the physics of the particle acceleration mechanism and probe the 'knee' in the hadronic spectrum of CR at 3 · 10 15 eV using the gamma emission radiated during the acceleration process of particles at the shock front of the SNR with the interstellar medium. The diffuse gamma radiation from the region around the galactic center seems to originate from the interaction of cosmic rays with molecular clouds. Molecular clouds are a tool that allow to probe the cosmic ray spectrum in different locations in the galaxy. A new instrument like CTA has the necessary sensitivity to improve these measurements and even separate additional contributions to diffuse radiation due to more exotic physics. In addition, also pulsar wind nebulae as well as binary systems might accelerate not only leptons but significantly also hadrons to multihundred TeV radiation and therefore contribute to the CRs. Purely leptonic acceleration models as well as hadronic acceleration models have been sug-gested for both cases. Precise multiwavelength measurements from radio, optical, X-ray, MeV up to 100 TeV gamma radiation enable phycisists to fit their models to the full wavelength range and verify or falsify one or another of these models. It is therefore crucial to measure an energy range as wide as possible from several tens GeV up to 100 TeV. CTA will provide this energy range. Very interesting source classes are the various types of binary systems, Wolf-Rayet-Star binaries, Be-Star-Pulsar binary, Star-black hole binary, micro quasars etc. which are transient TeV gamma emitters. They are ideal objects to study various types of particle acceleration and Gamma ray absorption. The low energy threshold together with a good sensitivity will allow to detect many extragalactic objects at high redshifts that are not visible at higher energies due to the so-called gamma ray horizon. High energy gamma rays are absorbed due to electron-positron pair creation with extragalactic background light in the universe. The study of the spectra and the energy cut-offs of active galactic nuclei (AGN) enables us to calculate and measure the spectrum of the so-called extragalactic background light (EBL). The main contributions to the EBL is redshifted starlight and its reemission from dust in the universe. This measurement not only reflects the history of star formation in the universe but also contains information about the evolution of the universe expressed in the Hubble constant and cosmic constant. Simultaneous multi wavelength campains of the transient AGN with instruments at other wavelengths will supply more information about leptonic and hadronic acceleration models in AGN jets by observing these objects in the multiwavelength band, same as for the galactic sources. It is not understood if the highest energy CR have been accelerated by these objects. The final proof of hadronic acceleration can be only given by the detection of a neutrino signal from these sources. Therefore, multimessenger observations are important in gamma astronomy. The northern CTA observatory will observe the same hemisphere of the sky as the ICECUBE neutrino telescope. The future KM3NET neutrino telescope will observe the southern sky and will probably start operation around 2015. AGN exhibit very fast flux changes at all wavelengths that are especially difficult to explain for the TeV radiation. The large scales of the emission regions in the jet imply large gamma factors for the bulk motion. Another type of very interesting transient objects are the so-called gamma ray bursts. These objects exhibit ultrafast X-ray and gamma-ray flux variability on second and subsecond time scale. As of today, no radiation at hundreds of GeV energies could be detected, it is believed mainly due to the gamma ray horizon. Very distant transient objects with ultrafast flux variability can be used to test Lorentz invariance violation. Furthermore, CTA has considerable discovery potential e.g. for the detection additional non-blazar AGN of pulsed gamma emission from pulsars, clusters of galaxies and the detection of a dark matter candidate, the neutralino. Design of the CTA system The exact design of CTA is being studied and not precisely defined as of now. First MC studies indicate some possible scenarios in which the sensitivity and angular resolution aimed for can be achieved. Fig. 3 illustrates the possible design, a combination of arrays of 2-3 different telescope sizes. A large number (several tens) of mid-size telscopes will provide for the millicrab sensitivity and high angular resolution in the core energy range from about 100 GeV up to 10 TeV. An extension of a few (4-9) very large diameter telescopes and many (up to 100) very small telescopes distributed over a large area enlarge the energy range from several 10 GeV up to 100 TeV. In this way one achieves an energy coverage of 3-4 orders of magnitude while keeping the high sensitivity. This is crucial for detailed studies of the shape of the spectrum which contains a lot of information about the mechanism of particle acceleration and physics in the object. In order to achieve full sky coverage it is planned to install two stations, one in the southern hemisphere and one in the northern hemisphere. The design of the telescope camera and the performance predictions of CTA will be based on classical photomultipliers but an eye is kept also on the technical development of future photosensors (such as SiPM) with higher photon detection efficiency to lower the energy threshold of the system. Detailed MC simulations will define the details such as FOV, optimal telescope sizes, camera pixel sizes and telescope spacing in the array. The real challenge will be to reach the perfection and reliability of the hardware that is needed to run a large number of telescopes without major interruptions or failure in a quasi robotic modus since it is simply impossible to do frequent repairs on every telescope. The design should start in 2008 and continue until the end of 2011. We aim to begin the construction of the full array around 2012. It is intented to have an overlap with the GLAST satellite project. Conclusions We present the new Cherenkov Telescope Array (CTA) project that unifies for the first time the research groups working in gamma ray astronomy in a common strategy, resulting in a new facility that is well beyond possible upgrades of existing instruments like H.E.S.S., MAGIC, VERITAS and CANGAROO. The new facility will be run as an observatory open to external astronomers. Its sensitivity will be better by about one order of magnitude combined with a larger energy range coverage. It is expected that at the order of 1000 new sources and also new classes of sources will be discovered which have been not visible with today's instruments. The physics of most objects can only be understood, if the whole multi-wavelength picture is generated. One of the main topics of CTA will be therefore the multi-wavelength and multimessenger observation together with X-ray and gamma-ray satellites, optical and radio telescopes as well as with the forthcoming neutrino experiments. Figure 1 : 1The figure illustrates the sensitivity that is aimed at with CTA. The exact values will finally depend on the real layout of the system. arXiv:0709.2048v1 [astro-ph] 13 Sep 2007 Figure 2 : 2The milky way seen in multiwavelength observations.The H.E.S.S. observations at TeV energies reveal a plethora of sources. Most of them are supernova remnants and pulsar wind nebulae. Figure 3 : 3This figure illustrates possible telescope configurations which achieve the aimed sensitivity. AcknowledgementsWe wish to thank all the scientists in 34 institutes in 15 countries who are contributing and participating in the Cherenkov Telescope Array (CTA) project. (See http://www.mpi-hd.mpg.de/hfm/CTA/) The Cherenkov Telescope Project. "The Cherenkov Telescope Project". http://www.mpi-hd.mpg.de/hfm/CTA/. [2] "The H.E.S.S. project". http://www.mpi-hd.mpg.de/hfm/HESS/. . " The, Magic Telescope, "The MAGIC Telescope". http://wwwmagic.mppmu.mpg.de/. The VERITAS project. "The VERITAS project". http://veritas.sao.arizona.edu/. The CANGAROO telescope system. "The CANGAROO telescope system". http://icrhp9.icrr.u-tokyo.ac.jp/.	[]
[ "A quantitative analysis of stellar activity based on CoRoT ⋆ photometric data", "A quantitative analysis of stellar activity based on CoRoT ⋆ photometric data" ]	[ "J C Hulot \nInstitut d'Astrophysique Spatiale\nUMR 8617\nCNRS\nUniversité Paris XI\nF-91405OrsayFrance\n", "F Baudin \nInstitut d'Astrophysique Spatiale\nUMR 8617\nCNRS\nUniversité Paris XI\nF-91405OrsayFrance\n", "R Samadi \nLESIA\nUMR8109\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\nObservatoire de Paris\n92195MeudonFrance\n", "M J Goupil \nLESIA\nUMR8109\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\nObservatoire de Paris\n92195MeudonFrance\n" ]	[ "Institut d'Astrophysique Spatiale\nUMR 8617\nCNRS\nUniversité Paris XI\nF-91405OrsayFrance", "Institut d'Astrophysique Spatiale\nUMR 8617\nCNRS\nUniversité Paris XI\nF-91405OrsayFrance", "LESIA\nUMR8109\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\nObservatoire de Paris\n92195MeudonFrance", "LESIA\nUMR8109\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\nObservatoire de Paris\n92195MeudonFrance" ]	[]	Context. The CoRoT satellite has made available high precision photometric observations of a large number of stars of different spectral types. Continuous photometric time series allow the characterization of stellar microvariability in a systematic way. Aims. We determine an index indicating the level of activity, derived from photometric data, for a large sample of stars with different color temperatures. We also assess to what extent this index can be related to an estimated Rossby number for stars whose rotation period can be estimated from the light curve. We also estimate a characteristic lifetime of the surface heterogeneities causing the variability of selected light curves. Methods. Most work on the photometric impact of stellar activity has been based either on measuring the variance of stellar microvariability, or on fitting the light curve. Following similar research for the Sun, our work is based on the Fourier analysis of stellar light curves. We have analyzed the Fourier power spectra of 430 selected light curves obtained by CoRoT during three observation runs. The low-frequency contribution of the stellar variability is modelled by a "generalized semi-lorentzian" profile. An activity index is derived from the fitted amplitude and width of the semi-lorentzian model. Some of the Fourier spectra exhibit a rotational modulation which enables the determination of the rotation period. In addition, a convective turnover time is derived from a grid of stellar models, so that a Rossby number can be estimated for this subsample of stars. A characteristic lifetime of the phenomena causing the observed power at low frequency is assessed from the fitted model of the power spectrum and is compared to the rotation period. Results. Higher values of the microvariability index are observed among the coolest stars from our sample. 28 light curves show a clear rotational modulation. The rotation periods derived from the observed low-frequency peaks in the Fourier spectrum decrease with the color temperature, in accordance with previous observations. The estimated Rossby number of most of the observed stars with a rotational modulation is less than 1, generally accepted as a critical value under which stars are expected to be active. The activity index, computed from the Fourier spectrum, decreases with increasing Rossby number. The quality of the CoRoT data enables the determination of the characteristic lifetime of active structures. It is shown to increase with the rotation period, but some scatter could arise from different surface heterogeneities (spot and faculae for example).	null	[ "https://arxiv.org/pdf/1104.2185v2.pdf" ]	119,181,413	1104.2185	185a118349d5555b01b54e2360d277d9aa81d63c	A quantitative analysis of stellar activity based on CoRoT ⋆ photometric data 18 Apr 2011 January 19, 2013 J C Hulot Institut d'Astrophysique Spatiale UMR 8617 CNRS Université Paris XI F-91405OrsayFrance F Baudin Institut d'Astrophysique Spatiale UMR 8617 CNRS Université Paris XI F-91405OrsayFrance R Samadi LESIA UMR8109 Université Pierre et Marie Curie Université Denis Diderot Observatoire de Paris 92195MeudonFrance M J Goupil LESIA UMR8109 Université Pierre et Marie Curie Université Denis Diderot Observatoire de Paris 92195MeudonFrance A quantitative analysis of stellar activity based on CoRoT ⋆ photometric data 18 Apr 2011 January 19, 2013Received ; AcceptedarXiv:1104.2185v2 [astro-ph.SR] Astronomy & Astrophysics manuscript no. projetJCH c ESO 2013stellar microvariability, stellar activity Context. The CoRoT satellite has made available high precision photometric observations of a large number of stars of different spectral types. Continuous photometric time series allow the characterization of stellar microvariability in a systematic way. Aims. We determine an index indicating the level of activity, derived from photometric data, for a large sample of stars with different color temperatures. We also assess to what extent this index can be related to an estimated Rossby number for stars whose rotation period can be estimated from the light curve. We also estimate a characteristic lifetime of the surface heterogeneities causing the variability of selected light curves. Methods. Most work on the photometric impact of stellar activity has been based either on measuring the variance of stellar microvariability, or on fitting the light curve. Following similar research for the Sun, our work is based on the Fourier analysis of stellar light curves. We have analyzed the Fourier power spectra of 430 selected light curves obtained by CoRoT during three observation runs. The low-frequency contribution of the stellar variability is modelled by a "generalized semi-lorentzian" profile. An activity index is derived from the fitted amplitude and width of the semi-lorentzian model. Some of the Fourier spectra exhibit a rotational modulation which enables the determination of the rotation period. In addition, a convective turnover time is derived from a grid of stellar models, so that a Rossby number can be estimated for this subsample of stars. A characteristic lifetime of the phenomena causing the observed power at low frequency is assessed from the fitted model of the power spectrum and is compared to the rotation period. Results. Higher values of the microvariability index are observed among the coolest stars from our sample. 28 light curves show a clear rotational modulation. The rotation periods derived from the observed low-frequency peaks in the Fourier spectrum decrease with the color temperature, in accordance with previous observations. The estimated Rossby number of most of the observed stars with a rotational modulation is less than 1, generally accepted as a critical value under which stars are expected to be active. The activity index, computed from the Fourier spectrum, decreases with increasing Rossby number. The quality of the CoRoT data enables the determination of the characteristic lifetime of active structures. It is shown to increase with the rotation period, but some scatter could arise from different surface heterogeneities (spot and faculae for example). lar spots, can also result in broad-band photometric microvariability. Large enough spots may be detected through the rotational modulation they induce in the star light curve. Due to their intrinsic temporal evolution, the surface heterogeneities are also expected to produce a power excess at low frequency in a Fourier spectrum of a velocity or photometric time series (Harvey 1985). These heterogeneities can be considered as a random process with some memory. They give rise to an exponentially decaying autocorrelation in time, yielding a Lorentzian shape in the power spectrum. In solar observations, at low frequency, several processes contribute to the power spectrum: supergranulation (in velocity series, Jimenez et al. 1988), bright points (in photometry, Harvey et al. 1993) or granulation (in both). As it is the process with the longest time scales, magnetic activity seen through the signature of transiting spots on the solar surface in photometric series is detected at the lowest frequencies in the power spectrum. In solar-type stars, we expect the active structures to be the phenomena with the longest time scales. Mosser et al. (2009) have shown that, in the case of several solar-type stars, the microvariability at the lowest frequencies is well modelled by the transit of stellar spots. In addition, simulation of the granulation and com-parison with observations in the case of one of these solartype stars (HD49933) shows that granulation has time scales shorter than that of spots (Ludwig et al. 2009), as for the Sun. Modeling of the spectrum with Harvey profiles is commonly used in the seismic analysis and shows a similar shape to that of the Sun (Michel et al. 2009;Benomar et al. 2009). It is not excluded that other physical processes, such as large convective cells for instance, which are suspected to exist in giant stars, could have the longest times. However, we selected our targets among stars that are identified as main-sequence. Stellar activity has been raised as an obstacle to the detection of exo-planets around cool stars (e.g. Aigrain et al. (2004), Lagrange et al. (2010), Meunier et al. (2010)). The impact of stellar activity on exoplanet detectability has been estimated based on the solar case and on scaling laws (Aigrain et al. 2004). Simulations have given estimates of the "stellar activity noise" that strongly depend on spectral types. Actual light curves collected by the CoRoT (Convection, Rotation and planetary Transits) now provide us new insights on the microvariability that can be attributed to stellar activity. As stressed by Ribárik et al. (2003), CoRoT gives the opportunity to analyze a large amount of photometric data related to spotted stars. While ground-based observations have a limited precision and a strongly constrained observational window, the space-based CoRoT photometric instrument has collected numerous high-precision and continuous light curves. It has been designed to study several bright seismic targets and to search for exoplanets around a large amount of fainter stars. Seismic and exoplanet targets belong to different spectral types and luminosity classes. Exoplanet targets have however been mainly selected from F to M spectral types. The observed noise from an exoplanet search point of view has been measured and has been shown to be from two to three times the expected photon noise (Aigrain et al. 2009). We have performed a more detailed analysis of the stellar microvariability based on the Fourier power spectrum, which allows us to discriminate several phenomena with different time scales. The purpose of this work is to show, based on a still limited but homogenous sample, that the available light curves can deepen our knowledge of stellar activity as the time scales of magnetic structures provide hints to understand the underlying mechanisms. There are at least two approaches to process the thousands of available light curves. The first is based on directly modeling a time series, as suggested by Ribárik et al. (2003) and applied by Lanza et al. (2009), Mosser et al. (2009) and Fröhlich et al. (2009. Models of light curves are based on various assumptions regarding the number of spots, their physical properties, differential rotation and limb darkening. There is no unique solution, but some "reasonable" hypotheses can give interesting results for a case by case analysis. A second approach is based on the Fourier power spectrum of a time series (Harvey 1985). It can be more reliable in identifying different features in a given signal as it can discriminate different time scales and quantify the lowest frequency contributions to the stellar microvariability. In addition, it allows the identification, in some cases, of a rotational modulation. This second approach is adopted in the present work. Our work is based on a 430-star sample, which is large enough to obtain several interesting results regarding stellar activity. Here we develop and test an analysis method on the present sample in order to determine physical constraints on stellar activity. In Section 2, we present our initial sample of 430 stars and describe our analysis method of the Fourier power spectra of the selected light curves. In Section 3, we first summarize some useful results related to the solar microvariability. We then define our stellar activity index computed from the Fourier spectrum. The distribution of our activity index against the color temperature is analyzed. In Section 4, we present our findings related to 28 light curves, out of the 430-star sample, showing a rotational modulation. In Section 5, the Rossby number for stars showing a rotational modulation is derived. The correlation between the observed activity index and the estimated Rossby number is investigated. In Section 6, we analyze our results regarding the characteristic evolution times of microvariability. In Section 7, we conclude and discuss the opportunity to go further into a wider analysis of the CoRoT data. The sample of CoRoT light curves Selection criteria Our sample 1 was initially made of the first 30 seismic targets observed by CoRoT during three observation "runs" (IRa01 -January/April 2007-SRa01 -April/May 2007-and LRc01 -May/October 2007) and of the 400 brightest exoplanet targets observed during the initial run (IRa01). The apparent V magnitudes of the seismic targets range from 5.45 to 9.48, and between 12 and 13 for the selected exoplanet targets. Our sample includes stars with spectral types from B to M, but most of the exoplanet targets are redder than A. Most of the selected light curves are 60-days long. However, some of the seismic targets light curves are limited to 30 days and a few are 150 days long. Seismic targets are observed in a broad-band white-light flux. For exoplanet targets, the flux is the sum of three broad-band chromatic fluxes, equivalent to a white-light flux. There is no bolometric correction. Broadband photometric data are collected with two sampling times (32 or 512 s) depending on the kind of target. The available photometric time series have a very high precision that allows, after correcting for an instrumental linear trend with time, a systematic study of stellar microvariability. More details on the CoRoT time series are given by Auvergne et al. (2009). Available stellar information Apparent B, V, R and I magnitudes were determined during ground-based photometric campaigns in preparation for the CoRoT mission (Deleuil et al. 2009). A color temperature was derived from those magnitudes, which were not corrected for interstellar extinction (Deleuil et al. 2009). We checked that there is a monotonic relationship between this color temperature and the observed B-V index of the selected stars. In the next sections, we use this temperature as a proxy for the stellar spectral type. Analysis procedure Each light curve has been analyzed through a 4-step process. The first step starts with a visual inspection of the light curve. It is performed in order to identify any significant discontinuities of the recorded flux which could be caused by a high-energy particle on the photometric detector. As the resulting discontinuities do not show any systematic time scale, no automatic procedure worked well enough to remove their impact in the time series or in the Fourier power spectrum. Consequently, a visual inspection has been performed (for more details Fig. 1. The distribution of the selected stars by spectral type. on these discontinuities, see Auvergne et al. 2009). The light curves showing such discontinuities followed by a slow recovery are excluded from the next steps of our analysis. Exceptions are accepted if the discontinuity occurs close to the extremities of the time-series, so that the truncated time-series, excluding the discontinuity, is long enough to be included in the further analysis. All of the seismic targets went through the first step, but 67 exoplanet targets had to be excluded. A linear variation was removed with an ordinary least square linear fit. The observed temporal drift is mainly attributed to instrumental aging. We checked that a model with a second or third order polynomial drift does not significantly change the Fourier power spectrum. Missing observations are replaced by linearly interpolated values. They account for close to 10 % and mainly result from the crossing of the South-Atlantic Anomaly by the CoRoT spacecraft. Their signature appears in the Fourier spectrum at the orbital frequency and harmonics. The seismic targets include 4 stars that were found out to be binaries. They were excluded from our analysis. At the end of this first step, our sample includes 359 stars (Table 1) which are mainly distributed within the A to G spectral types (Figure 1). In a second step, we derive the Fourier power spectrum from a uniformly sampled time series. The third step consists in computing several indexes to characterize the low-frequency power spectrum. Narrow spectral peaks are searched for and their frequencies are compared in order to identify the rotation period (see Section 4). The fourth step is designed to fit a generalized semi-lorentzian model to the low frequency spectrum. The power spectra of 58 stars, among the hottest ones from our sample, were however too flat to be fitted to our model. The corresponding stars were probably not active or had too low activity to be detected. The number of stars surviving the different steps of our analysis is shown in Table 1. Low-frequency microvariability The solar case The solar microvariability has been extensively analyzed based on the SOHO/VIRGO photometric data ( Aigrain et al. 2004;Seleznyov et al. 2003) before being extrapolated to other stars (using scaling laws to describe the rotation, the amplitude and characteristic evolution time of activity, in order to simulate an "activity noise" impact on exo-planet detectability by Aigrain et al. (2004). VIRGO data can be compared to CoRoT data, as they are also intensity measurements. However, the effects of the instruments (bandwith) are different and should be taken into account. VIRGO data are based on 3 narrow bandwidths, whereas CoRoT relies on one wide band. As shown by Aigrain et al. (2004), VIRGO power spectra show several contri-butions which are attributed to solar activity, super-granulation and granulation. The first contribution is observed at the lowest frequencies and shows a clear variability through the solar cycle. A lorentzian profile was first suggested by Harvey (1985) to model the low-frequency background in such a power spectrum. Each contribution to the power spectrum is actually better described by a 'generalized' lorentzian profile, the total profile being the sum of each contribution: P( f ) = N i=1 A i 1 + f ∆ f i α i + B (1) P( f ) is the power density in ppm 2 µHz −1 , with ppm for partper-million. α i equals 2 when the phenomenon causing the flux variation exponentially decays, but different values can result from more complex temporal evolution. A i is the amplitude of the i-th contribution and 1/∆ f i is its characteristic time. B corresponds to an additional white noise. It is important to stress that, in the solar case, the low frequency contribution has a 10-day characteristic time ( Aigrain et al. 2004). It is then significantly shorter than the evolution time of a solar active region, generally estimated around 2 months, and should be interpreted as a characteristic evolution time of individual spots and faculae forming an active region. A simplified model of the low-frequency stellar microvariability A variance derived directly from a light curve could capture different superposed phenomena. On the contrary, a Fourier analysis allows the identification and quantification of the lowfrequency background contribution. What is observed in the solar case is expected to be seen in some stellar light curves if the photometric precision is high enough. The low-frequency part (below 100 µHz) of the power spectrum may result from several contributions that must be disentangled. A first contribution to the power excess at low frequency results from instrumental effects. All our light curves are corrected for a linear trend and the CoRoT stability is confirmed by a significant number of flat power spectra, mainly observed for A stars. A second contribution is from the so-called "hot pixels" that result from incident high-energy particles on the photometric detector. We recall that the light curves concerned are excluded from our analysis through a visual inspection, so that this contribution should not be observed in our sample. A third contribution to the power spectrum relates to intense and narrow peaks that are due either to long-period oscillations or to a rotational modulation. The fourth contribution is the one that we try to identify and to model. It corresponds to a low-frequency background and is well described by a continuous profile centered on zero frequency (see Eq. 1). It varies significantly from one light curve to another. Our sample shows that an A-star spectrum is more often flat, while a G-star spectrum shows a higher power at low frequencies. ). However, we cannot exclude that a giant star is included in our sample and that its low-frequency microvariability comes from another phenomenon than magnetic activity. Such a case is likely to happen within some red giants whose surface could show large convective cells with long evolution time scales. Our sample of stars has been selected from main-sequence stars from the CoRoT database but misclassifications cannot be excluded. Considering the power spectrum at frequencies lower than 100 µHz, we exclude the peaks that may result from a rotational modulation or from long-period oscillations. This 'cleaned' power spectrum is then fitted to a single generalized semilorentzian distribution to which we add a constant free parameter: P( f ) = A 1 + f ∆ f α +B (2) A is the amplitude of the lower-frequency contribution to the spectrum and ∆ f is related to its characteristic time. It captures as shown in the solar case (Harvey 1985), the stellar activity.B captures all the other contributions, including granulation, and not only the additional white noise as in Eq. 1. The low-frequency background index We derive a microvariability index from the generalized semilorentzian model (Eq. 2). We call it a "low-frequency background index" (LFBI). It is defined as I LF = A∆ f and it is expressed in ppm 2 . It represents the integral of the power density of the model which describes the microvariability disentangled from other contributions. A possible bias in the computation of the index corresponds to the case of a slowly rotating star whose light curve has an unresolved rotational modulation. In such a case, the rotational modulation peak cannot be removed from the power spectrum: it affects the very first bins of the spectrum and may lead to an over-estimated LFBI. In order to use the solar case as a reference when considering CoRoT light curves, we have analyzed some VIRGO data. These data need to be corrected for the instrument bandwith (see Section 3.1), following Michel et al. (2009).The correction factor to VIRGO data to be compared with CoRoT data is estimated to be 1.4. The results obtained from the VIRGO solar data at different times of the solar cycle indicates that A∆ f is a representative index for activity. In the solar case, while I LF is close to 1.4 10 5 ppm 2 at the maximum of activity, it is nearly 100 times lower at its minimum. Correspondingly, A varies from 1.5 10 5 ppm 2 µHz −1 at maximum to nearly 100 times lower at minimum, while ∆ f remains close to 1, which corresponds to nearly 10 days. In order to check the robustness of our index I LF , two additional indices were computed, consisting of the direct integration of the Fourier power spectrum, respectively from 0 up to 10 µHz and from 0 up to 100 µHz. They were found to be highly correlated with I LF , but we recall that these latter indices take into account all of the power at low frequency, whereas I LF does not take into account the power due to pulsation peaks, peaks due to rotational modulation or the constant term capturing other phenomena and noise. A study case: HD 49933 HD 49933 is a bright solar-like star for which several parameters are now well known from ground-based observations. It is an F5 star with a visual magnitude 5.77. Its effective temperature is between 6500 K and 6780 K (Ryabchikova et al. 2009;Bruntt et al. 2008;Bruntt 2009). HD 49933 was observed by CoRoT during its 60-day initial observation run with a 32s sampling time. The detrended stellar intensity shows some significant variations with a ∼1000 ppm amplitude (see Fig. 2). We note that the VIRGO photometric data show a peak-to-peak variation which is also close to 1000 ppm at the maximum of solar activity. The power spectrum of HD 49933 light curve confirms a power excess at low frequency (see Fig. 3). Following the method presented above, we obtained the following fitted parameters (see Fig. 4), using a MLE fitting algorithm described by Appourchaux et al. (2008): A = 1.0 10 4 ppm 2 µHz −1 , B = 1.7 ppm 2 µHz −1 , ∆ f = 2µHz and α = 2.4. A is about 15 times lower than the observed value for the Sun at its activity maximum, but less than 10 times larger than the same parameter at the Sun's minimum of activity. The HD 49933 activity index is 2. 10 4 ppm 2 . Observations on a long time scale would be necessary to get a better knowledge of its activity cycle (García et al. 2010;Wilson 1978). Fig. 3. Power spectrum of HD 49933 computed from a 60-day time series after it was corrected for a linear trend which captures some instrumental effects. The power is presented here on a logarithmic scale that allows us to see the rotational modulation peak at 3.5 ± 0.2 µHz and its third harmonics. . Color temperature and microvariability The LFBI was computed for the 301 stars of the sample for which a fit could be performed. As was noted in section 2.3, 58 power spectra could not be correctly fit. All of them were "flat" spectra with a low power density at low frequency. Such spectra confirm the instrument stability. The corresponding stars are among the hottest and do not show a photometric signature of surface heterogeneities. The activity indexes do not show a clear dependence on color temperature. However, our sample is not uniformly distributed in terms of temperature: F stars are over-represented. In order to avoid this bias, we sort our sample by increasing temperatures, and divide it into nine sub-samples, each of them including nearly the same number of stars. We determine a threshold value of the LFBI defined so that 20% of the full sample has an index higher than the threshold. We then determine, within each temperature sub-sample, the fraction of stars with a LFBI higher than the threshold and labelled by the fraction of most active stars. The fraction of most active stars appears to be clearly related to the Fig. 5. The fraction of most active stars, defined as stars with a LFBI higher than a given threshold (see text). Here, the threshold was defined so that the most active stars account for 20 % of the total sample. The fraction of stars with an index higher than the threshold is larger for the coolest stars. This general trend is not sensitive to the threshold value. temperature, being higher in the low temperature sub-samples (see Fig. 5). As the 20% threshold is arbitrary, we checked that the same result is observed when the threshold is determined so that we select respectively the 10%, 30%, 40% or 50% most active stars from the whole sample. The higher fraction of active stars among the G to K spectral types is consistent with previous observations (Berdyugina 2005). According to our results, cool stars are more likely to show an enhanced microvariability. Rotational signature Rotational narrow peaks Some of the analyzed power spectra show super-imposed narrow peaks in addition to the low frequency background. Such peaks are attributed to the transit of surface heterogeneities which are large and contrasted enough to cause a periodic intensity modulation over a significant fraction of the observation (see Mosser et al. 2009). A Fourier analysis shows that a single corotating permanent spot results in a strong peak and weaker even harmonics in the Fourier power spectrum of its photometric flux (Clarke 2003). Indeed, we often observed harmonics above the peak corresponding to the rotation period. We checked with simulated light curves that a more complicated distribution of spots or, more generally of active areas, gives the expected peaks, with even and odd harmonics. These peaks are thus considered as the signature of activity and rotation. Because the observation time is limited, the frequency resolution might be too low to detect such peaks in slowly-rotating stars. As previously noted, the LFBI of such stars could be over-estimated. An example of rotational modulation : HD 49933 In order to identify a rotational modulation in the CoRoT light curves, we first focused on the brightest stars which were selected as seismological targets, and chose HD 49933 among them. While p-modes were detected in this star, a rotational modulation of the observed spectral lines was attributed to one or more spots (Mosser et al. 2005 Table 2. Number of selected light curves with a detected rotational modulation signature. The spectral type is derived from a color temperature determined from ground-based photometric observations. carried out our analysis on the full sample of stars, we present in more detail our results for this star. The power spectrum in the range [0,20] µHz shows a strong peak and weaker ones (see Fig. 3). The main peak is centered at 3.5 ± 0.2 µHz which corresponds to a rotation period of 3.4 ± 0.2 days and is confirmed by the observed p modes oscillations. (Appourchaux et al. 2008). A second peak is observed at twice this frequency and a third one at a frequency four times larger. The second harmonic is ten times weaker than the fundamental but twice as strong as the fourth harmonic. We see neither a third harmonic, nor a fifth. Such a result would be obtained with a single and long-lived co-rotating active region (Clarke 2003) and is also in agreement with the analysis of the light curve by Mosser et al. (2009). Rotational modulation within our sample There may be active stars whose power spectra do not show rotational peaks. A first reason would be an axis of rotation too close to the line of sight. A second reason is a rotation period that is too long to be identified in a too low-frequency resolution Fourier power spectrum. A third reason could be that stars having spots evolving on time scales shorter than the rotation period do not show a clear rotational modulation. Indeed, our analysis in terms of LFBI shows cool stars with a high excess power at low frequency, but without any clear rotational modulation. However, 6 stars (including HD 49933) out of 30 from the seismological sub-sample and 22 stars from the exo-planets sub-sample show a rotational modulation. The fractions of identified rotational modulations in the two sub-samples may be biased by their magnitude distributions. The asteroseismology targets are significantly brighter than the exoplanet ones, so that we detect weakly active stars within the former, but only strongly active ones within the latter. We identified two G stars with a period too long (respectively more than 23 days and more than 29 days) to be confirmed with a 60-day time series. We decided not to include these two stars in our sub-sample of stars with a rotational modulation. The rotational modulation appears to be significantly more frequent among G and, to a lesser extent, F spectral types (see Table 2). The low proportion of K and M stars with an observed rotational modulation seems to be at variance with the expected activity of K and M stars (see for instance Berdyugina 2005). However, our sample of K and M stars is too limited and their rotation periods are likely to be too long for a rotational modulation to be observed in a 60-day photometric time-series. Indeed, Kiraga & Stepien (2007) identify several M stars with 30-day and longer rotation periods. The rotation periods of stars with an activity signature are derived from the rotational peaks. In most of the observed cases, the frequency resolution is too low to determine the width of the peak that is attributed to the rotational modulation. In two cases, we can only determine a lower estimate of the rotation period, so that the corresponding stars are excluded from the next steps of our analysis. When plotted against the color temperature (see Fig. 6), the rotation period shows a clearly decreasing trend versus the color temperature. This result is consistent with previous observations (Gray 1982;Lockwood et al. 1984). A significant scatter results from additional parameters which are thought to impact the rotation period (the age for example). Rossby number A determinant of stellar activity? To exhibit solar-like activity, a star needs to have a convective envelope. According to our present understanding of stellar interiors, stars redder (cooler) than F2 have a convective envelope whose thickness decreases withan increasing surface temperature. A dynamo effect is expected to take place when the dynamo number, which compares advective to diffusive effects on the magnetic fields, is larger than 1. Under some very simplified assumptions, a dynamo number can be expressed as a decreasing function of the Rossby number (Parker 1979). The Rossby number is the ratio between the surface rotation period and the convective turnover time at the bottom of the convective envelope. Gilliland (1985) actually observed that a chromospheric activity index, based on the Ca II H and K emission fluxes, is correlated with an estimated Rossby number within a 41-star sample. The convective turnover time is expected to be a decreasing function of the effective temperature. A rough relationship between the B-V index and the rotation period can be inferred for mainsequence stars from a statistical study of a large population of stars (Gray 1982). On average, a cooler star rotates more slowly than a hotter one. However, the rotation period and the convective turnover time are likely to show significant scatter within a given spectral type, so that the B-V index is not enough to determine if a star is active or not. The Rossby number is expected to be relevant as an activity index. From a 277-star sample for which ground-based photometric observations are available, Hall (1991) concluded that the Rossby number of active stars must be lower than 2/3. Stepien (1994) showed that an activity index is well correlated with the Rossby number for a limited color class of main-sequence stars, but he also stressed that, for the other stars, this dimensionless number does not explain activity better than the rotation period. As noted by Hall (1991), the conditions for a star to be active have generally been addressed focusing on samples in which (very) active stars are over-weighted. A more systematic study of a larger sample of stars, as made possible with CoRoT, spread over the HR diagram could avoid such bias. Estimating a Rossby number The Rossby number is defined as the ratio of the rotation period to the convective turnover time at the bottom of the convective envelope (τ CV ). The rotation period is derived from the frequency of the main peak in the Fourier power spectrum. Looking at M stars, Kiraga & Stepien (2007) suggested an empirical determination of the convective turnover time based on a linear relation between the logarithm of an X ray-flux index, not available for our targets, and the rotation period. Noyes et al. (1984) used a scaling relation between a color index and a convective turnover time. Here, we prefer to estimate this turnover time using a grid of stellar models with 1 to 2 solar masses, for a star whose effective temperature is presumed to be known ± 100 K. The global 1D models are obtained with the CESAM code (version 4, see Morel & Lebreton 2008) assuming standard physics. Convection is described according to Böhm-Vitense (1958)'s local mixing-length theory of convection (MLT) with a mixinglength Λ = α c H p , where H p is the pressure scale height and α c is the mixing-length parameter. The calibration of the associated solar model gives α c = 1.62. Turbulent pressure and microscopic diffusion are not included. All models have a solar iron-to-hydrogen abundance and the chemical mixture of the heavy elements of Grevesse & Noels (1993). We use OPAL opacities (Iglesias & Rogers 1996) extended with the Alexander & Ferguson (1994) data for T 10 4 K, both sets of data being given for the Grevesse & Noels (1993) solar mixture. Finally, the CEFF (Christensen-Dalsgaard & Däppen 1992) equation of state is assumed. The convective time τ CV is then directly deduced from the mixing length Λ and the convective velocity u at one height scale from the base of the convective zone. There is indeed a difficulty in estimating a turnover time at the bottom of the convective envelope where the velocity is zero, which is why we estimate it at a height H p above the bottom. The Rossby number and the low-frequency background index of the Sun are useful references. The solar convective turnover time is computed with the same grid of stellar interior model. We obtain a Rossby number close to 1. The LFBI for the Sun is derived from VIRGO data for a 180-day active period. We obtain log(LFBI) ∼ 5, which places it among very moderately active stars. Rossby number and microvariability index In our sample, one star was expected to be fully convective so that our estimation concerns 27 stars from our 28 stars with a rotational modulation. Fig. 7 shows a clear anti-correlation between the activity index and the Rossby number: the most active stars having the lower estimated Rossby number. The observed scatter could result from various parameters, one of which is the stars metallicity. A similar relation was searched for between the rotation period and the activity index. No trend was observed. The Rossby number seems then to be a more useful predictor of the low-frequency microvariability than the rotation period. The Rossby numbers are limited by the observed rotational periods and by the color temperature of the selected stars. Based on the CoRoT photometric data from the initial run, our analysis method allows the determination of the rotation period over an interval from a few to nearly 30 days. The upper limit is quite restrictive for the coolest stars. It will be enhanced by using the CoRoT data from longer observation periods. On the other hand, the spectral range studied corresponds to a wide range of convective turnover times (from 0.05 day to 170 days). Characteristic evolution time of the surface heterogeneities 6.1. Low frequency power density and characteristic evolution time While stellar activity has been mainly addressed in terms of activity index (in Ca II H and K flux, X-ray flux or broadband photometric flux) and activity cycle, based on long observations, there is much less available data about the characteristic evolution time of the surface heterogeneities. The parameter τ = 1/∆ f determined when fitting the Fourier power spectrum to a generalized semi-Lorentzian has been previously used to compute an activity index. The parameter itself gives interesting information about the characteristic evolution time of the surface heterogeneities. When there are two different phenomena, for instance spots and faculae, or when spots evolve on a time scale that is shorter than the evolution time of active areas, the lorentzian contribution may be significantly wider than the peaks resulting from the rotational modulation. Such a result is observed in most of the identified stars with rotational modulation. For instance, in the case of HD 49933, ∆ f = 2µHz corresponds to a 5.5-day characteristic time. This value is slightly less than two rotation periods, whereas the width of the rotational modulation peaks is com- parable to the frequency resolution of a 60-day time series. The rotational modulation could be caused by a phenomenon whose evolution time is quite long compared to the rotation period, for example faculae. The observed characteristic evolution times The characteristic time (τ) is plotted against the rotation period (P) in Fig. 8, showing a clear increase of τ with P. Several stars show a rotational modulation caused by a phenomenon whose characteristic time is very large compared to the rotation period. In our sample, very few stars show a rotational modulation caused by phenomena with rapid evolution, on a time-scale shorter than the rotation period. However, there is an interesting exception, a M1 star with a 11.6-day period and a 2.5-day characteristic evolution time. M stars have a deep convective envelope or are fully convective. Even if they are generally slow rotators, they are expected to be very active, as confirmed by Xray flux observations (Kiraga & Stepien 2007). A more extensive sample might allow us to show several different relationships between rotation period and characteristic evolution time. It might be explained by different physical mechanisms linked to surface heterogeneities. Conclusion Summary and discussion of our results Our first goal was to test an analysis method designed to identify and quantify an activity signature in the photometric data collected by CoRoT. Based on a homogeneous sample of more than 350 stars, we obtain original results regarding a contribution to stellar microvariability from stellar activity. CoRoT, thanks to its sensitivity and stability, allows the investigation of weaker activity stars than from ground-based observations. Our work shows that CoRoT data allow the detection and measurement in many cases of a low-frequency power excess, which is interpreted as a photometric signature of stellar activity. After removing several contributions to the low-frequency power excess, we are able to fit the low-frequency background to a generalized semi-lorentzian model. A low-frequency back-ground index (LFBI) is then determined in a systematic way. Such an index is more reliable than a microvariability index directly derived from the light curve, because it captures only the low-frequency contribution which is likely to result from stellar activity and excludes other possible sources of variability, such as low frequency pulsations. Despite some scatter, our activity index is correlated to the color temperature, the highest values being obtained for the coolest stars. This result is consistent with previous observations, as reviewed by Berdyugina (2005) for example. Our sample allows us to quantify more precisely this relation between the color temperature (or any parameter locating a star on the mainsequence) and its level of activity. However, a single-parameter analysis is likely to show significant scatter. Several factors, such as rotation or age and possibly metallicity through its influence on convection, may induce a scatter of an activity index against the temperature. Another possible cause for the observed scatter could be the cyclicity of stellar activity: the duration of our observations is likely to be shorter than the activity cycle. Our method allows us to quantify an activity index even if there is no rotational modulation. However a detected rotational modulation permits us to go further by estimating the Rossby number. We have seen a clear rotational modulation in the light curves of 28 out of the 430 stars initially included in our sample. There is a higher proportion of such stars within the G, and to a less extent, F spectral types. It could be explained by two biases: hotter stars are less likely to be active and the cooler ones are expected to be slow rotators, so that their rotational modulation is not detected in a 60-day light curve. The most active stars have a low Rossby number, sometimes greater than unity but still anti-correlated with the activity index of the star. This result is consistent with previous work (for instance Hall 1991) but our sample allows the exploration of larger values of the Rossby number. Some scatter is also observed, again as expected from a single-parameter analysis. The Rossby number is only a proxy for a dynamo number. The scaling relation between the two dimensionless numbers is based on the strong assumption regarding the differential rotation in the convective envelope that may be too approximate for some spectral classes. Moreover, the scatter of the activity index at a given Rossby number can be explained by adverse observing conditions (low activity during a cycle or axis of rotation close to the line of sight). In addition, no clear relation is found between activity and rotation period in our sample, confirming the relevance of the Rossby number. However, we also investigated the variation of the characteristic evolution time, which is clearly correlated with the rotation period. Identifying active stars and sorting them according to their physical conditions is an interesting goal in itself. Perspectives Our sample covers a large part of the HR diagram, but is not unbiased. The exoplanet targets have been mainly selected among the cooler main-sequence stars. The available data however cover A to M spectral types. In any case, the study of a larger sample must take into account the non-uniform distribution in terms of color temperature. In order to study the correlation between an activity index and an estimated Rossby number, we need to determine the rotation period. Some power spectra do not show any low-frequency peaks, but have a high-amplitude low-frequency power excess. The corresponding stars are likely to be active. As was ex-plained for M stars, a power excess at very low frequency can result from an unresolved modulation peak which corresponds to a slow rotation. When an activity index is plotted against the color temperature, the non-rotationally modulated but active stars are taken into account. Unresolved modulation peaks may over-estimate the corresponding activity index. To study the K and M stars more precisely, longer time-series are needed. More generally, a larger sample of stars is needed to describe more precisely the relationship between the characteristic evolution time, the rotation period and the color temperature (or any parameter that locates a main-sequence star in the HR diagram). M stars are again particularly interesting as they are supposed to be fully convective and thus cannot be described with the same definition of the Rossby number. In slightly hotter stars, the convective turnover time is strongly sensitive to the color temperature, since we consider stars with a shallow convective envelope, which are expected to show steep variations of the convective turnover (Kiraga & Stepien 2007). CoRoT has now collected numerous longer time series and Kepler is collecting even longer time-series, both missions are thus very promising for the understanding of stellar magnetic activity. Fig. 2 . 2HD 49933 light curve. The observed signal has been corrected for a linear trend and converted into a relative flux variation and presented in ppm. Fig. 4 . 4A 'generalized' lorentzian model (with a uniform background) is fitted (solid line) to the power spectrum of HD 49933 (crosses), from which the rotational modulation peaks have been removed. Fig. 6 . 6Rotation periods versus the color temperature of the 28 stars whose light curves show a rotational modulation. Fig. 7 . 7The low-frequency background, or microvariability, index versus the Rossby number. The active Sun, i.e. at its maximum, corresponds to the square, while the quiet Sun would have an index one hundred times lower (see section 3.3). Fig. 8 . 8The characteristic time τ against the rotation period. showing a clear increasing trend of τ with P rot . The slow rotator with a rapid evolution time, indicated by a square, is a M1 star. We suppose that late-type main sequence stars show surface heterogeneities with time scales that follow the same hierarchy as in the solar case, that is activity has a longer time scale compared to supergranulation and granulation. This relies on previous observations: first, as already mentioned, the results of Mosser et al. (2009) about spot modeling; second, in the case of HD49933, the power spectrum expected from a granulation model has been compared to the actual power spectrum observed with CoRoT and show the expected hierarchy of time scales (Ludwig et al.Table 1. Number of selected light curves through the different steps4 J.C. Hulot et al.: Stellar activity from CoRoT data step number of number of total number seismic targets exoplanet targets of stars initial selection 30 400 430 after excluding 'hot pixels' and binaries 26 333 359 fitted Fourier power spectrum 25 276 301 with rotational modulation 6 22 28 with estimated Ro 6 21 27 2009 ). In order to illustrate how wespectral number out of which with proportion with type of stars detected rotational detected rotational step 1 modulation modulation O/B 9 0 0% A 129 2 2 % F 140 13 9 % G 57 12 21 % K/M 24 1 4 % Total 359 28 8 % All data are available from the public CoRot archive at: idoc-corot.ias.u-psud.fr . 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[ "On Generalizations of (k 1 , k 2 )-runs", "On Generalizations of (k 1 , k 2 )-runs" ]	[ "A N Kumar \nDepartment of Mathematics\nIndian Institute of Technology Madras\nChennai-600036India\n", "N S Upadhye \nDepartment of Mathematics\nIndian Institute of Technology Madras\nChennai-600036India\n" ]	[ "Department of Mathematics\nIndian Institute of Technology Madras\nChennai-600036India", "Department of Mathematics\nIndian Institute of Technology Madras\nChennai-600036India" ]	[]	The paper deals with three generalized dependent setups arising from an independent sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, an application to demonstrate the relevance of the results is given.	null	[ "https://arxiv.org/pdf/1707.08367v1.pdf" ]	119,147,354	1707.08367	24a09e5eb0728296d23f402f3d47dfe58e898853	On Generalizations of (k 1 , k 2 )-runs 26 Jul 2017 A N Kumar Department of Mathematics Indian Institute of Technology Madras Chennai-600036India N S Upadhye Department of Mathematics Indian Institute of Technology Madras Chennai-600036India On Generalizations of (k 1 , k 2 )-runs 26 Jul 2017(k 1 , k 2 )-runswaiting timeprobability generating functionprobability mass functionmomentsFibonacci words MSC 2010 Subject Classifications : Primary : 60E05, 62E15Secondary : 60C05, 60E10 The paper deals with three generalized dependent setups arising from an independent sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, an application to demonstrate the relevance of the results is given. Introduction Runs and patterns play a crucial role in applied statistics and has numerous applications, for example, reliability theory (see Fu [9] and Fu and Hu [11]), nonparametric hypothesis testing (Balakrishnan and Koutras [5]), DNA sequence analysis (Fu et al. [10]), statistical testing (Balakrishnan et al. [7]), computer science (Sinha et al. [21]) and quality control (Moore [18]) among many others. A run can be defined as an occurrence of specific patterns of failures or successes or both in a sequence of Bernoulli trials. In particular, a pattern of consecutive successes of length k is considered by Philippou et al. [19] and described geometric and negative binomial distribution of order k. Also, Philippou and Makri [20] discussed binomial distribution of order k. Later, Huang and Tsai [13] extended the pattern by observing at least k 1 consecutive failures followed by at least k 2 consecutive successes and studied a modified binomial distribution of order k or (k 1 , k 2 )-runs. Recently, Dafnis et al. [8] also considered three types of (k 1 , k 2 )-runs which include the pattern discussed in Huang and Tsai [13]. Though there have been several studies on this topic, still there are many problems which can not be studied based on the available literature. For example, (i) let us consider the quality control problem in which the system is said to be in control, whenever, (on the control chart) not more than two consecutive points exceed the control limits and at least three succeeding points are inside the control limits (see (T1) below with ℓ 1 = 1, k 1 = 2 and ℓ 2 = 3). Similarly, (ii) consider a climatology problem, in which, climatologist is interested in knowing the distribution of at least two consecutive rainy days followed by exactly five consecutive dry days (see (T2) below with ℓ 1 = 2 and ℓ 2 = k 2 = 5). Also, there are several such problems that occur in brand switching, learning, reliability and queuing models. Hence, there is a need to generalize the results related to (k 1 , k 2 )-runs. In this paper, we generalize (k 1 , k 2 )-runs to include the following patterns, for 1 ≤ ℓ 1 ≤ k 1 and 1 ≤ ℓ 2 ≤ k 2 , (T1) at least ℓ 1 and at most k 1 consecutive 0's followed by at least ℓ 2 consecutive 1's. (T2) at least ℓ 1 consecutive 0's followed by at least ℓ 2 and at most k 2 consecutive 1's. (T3) at least ℓ 1 and at most k 1 consecutive 0's followed by at least ℓ 2 and at most k 2 consecutive 1's. Note that (T1), (T2) and (T3) contain various (k 1 , k 2 )-runs. For example, 1. if ℓ 1 = k 1 then (T1) leads to, exactly ℓ 1 consecutive 0's followed by at least ℓ 2 consecutive 1's, 2. if ℓ 2 = k 2 then (T2) leads to, at least ℓ 1 consecutive 0's followed by exactly ℓ 2 consecutive 1's, 3. if ℓ 1 = ℓ 2 = 1 then (T3) leads to, at most k 1 consecutive 0's followed by at most k 2 consecutive 1's, 4. if ℓ 1 = k 1 and ℓ 2 = k 2 then (T3) leads to, exactly k 1 consecutive 0's followed by exactly k 2 consecutive 1's and similarly, other special cases can be seen by choosing the values for ℓ 1 and ℓ 2 , k 1 and k 2 appropriately. Dafnis et al. [8] considered two special cases of (T3), namely, (i) ℓ 1 = 1 = ℓ 2 and (ii) ℓ 1 = k 1 and ℓ 2 = k 2 . Next, let us define I (m) s :=      (1 − ζ m ) · · · (1 − ζ m+ℓ1−1 )(1 − ζ m+ℓ1 ) · · · (1 − ζ m+s+ℓ1−1 )ζ m+s+ℓ1 · · · ζ m+s+ℓ1+ℓ2−1 , m = 1, ζ m (1 − ζ m+1 ) · · · (1 − ζ m+ℓ1 )(1 − ζ m+ℓ1+1 ) · · · (1 − ζ m+s+ℓ1 )ζ m+s+ℓ1+1 · · · ζ m+s+ℓ1+ℓ2 , m ≥ 2, J (m) t := (1 − ζ m ) · · · (1 − ζ m+ℓ1−1 )ζ m+ℓ1 · · · ζ m+ℓ1+ℓ2−1 ζ m+ℓ1+ℓ2 · · · ζ m+t+ℓ1+ℓ2−1 (1 − ζ m+t+ℓ1+ℓ2 ), K (m) s,t :=                  (1 − ζ m ) · · · (1 − ζ m+ℓ1−1 )(1 − ζ m+ℓ1 ) · · · (1 − ζ m+s+ℓ1−1 )ζ m+s+ℓ1 · · · ζ m+s+ℓ1+ℓ2−1 ζ m+s+ℓ1+ℓ2 · · · ζ m+s+t+ℓ1+ℓ2−1 (1 − ζ m+s+t+ℓ1+ℓ2 ), m = 1, ζ m (1 − ζ m+1 ) · · · (1 − ζ m+ℓ1 )(1 − ζ m+ℓ1+1 ) · · · (1 − ζ m+s+ℓ1 )ζ m+s+ℓ1+1 · · · ζ m+s+ℓ1+ℓ2 ζ m+s+ℓ1+ℓ2+1 · · · ζ m+s+t+ℓ1+ℓ2 (1 − ζ m+s+t+ℓ1+ℓ2+1 ), m ≥ 2,I m : = max 0≤s≤k1−ℓ1 I (m) s , J m := max 0≤t≤k2−ℓ2 J (m) t , K m := max 0≤s≤k1−ℓ1 0≤t≤k2−ℓ2 K (m) s,t , where ζ 1 , ζ 2 , . . . , ζ n be a finite sequence of independent Bernoulli trials with success (denoted by 1) probability p and failure (denoted by 0) probability q = 1 − p. This paper is organized as follows. In Section 2, we obtain the double probability generating function (PGF) and waiting time for H n ℓ1,k1,ℓ2 , H n ℓ1,ℓ2,k2 and H n ℓ1,k1,ℓ2,k2 . Next, using double PGF, we derive recursive relation in PGF, probability mass function (PMF) and moments and also derive an explicit form of PGF and PMF. Finally, using double PGF for waiting time, we obtain the PGF, recursive relations in PMF and moments. In Section 3, we demonstrate the relevance of the results with an interesting application to Fibonacci words. 2 Distributions Related to H n ℓ 1 ,k 1 ,ℓ 2 , H n ℓ 1 ,ℓ 2 ,k 2 and H n ℓ 1 ,k 1 ,ℓ 2 ,k 2 In this section, we discuss various distributional properties such as PGF, PMF and moments for H n ℓ1,k1,ℓ2 , H n ℓ1,ℓ2,k2 and H n ℓ1,k1,ℓ2,k2 and their waiting time. The method used, to derive the results is general, can be formulated in the following way. Let Y n be a random variable related to (k 1 , k 2 )-runs. Then, we can define a Markov chain {Z t , t ≥ 0} on discrete space Ω (which can be partitioned into discrete subspaces {0, 1, 2, . . . , r} of maximum length ε n and containing one and only one (k 1 , k 2 )-event) such that (k 1 , k 2 )-runs has occurred v times if and only if Markov chain is in v-th discrete subspace (say E v = {E v,0 , E v,1 , . . . , E v,r } such that Ω = ∪ v≥0 E v ) . Now, assume A and B be (r + 1) × (r + 1) matrices when (k 1 , k 2 )-runs are observed from v to v and v to v + 1 times, respectively. Let φ n (·) and Φ(·, ·) be the single and double generating function of Y n and H j (·) and H(·, ·) be the single and double generating function of j-th waiting time for Y n . Then, the double generating function for Y n and its waiting time is given by Φ(t, z) = ∞ j=0 φ j (t)z j = κ 0 (ϑ(z, t)) −1 1 t(1) and H(t, z) = ∞ j=0 H j (t)z j = 1 + tzκ 0 (ϑ(t, z)) −1 B1 t (2) respectively, where κ 0 is the initial distribution, ϑ(z, t) = I − z(A + tB) be (r + 1) × (r + 1) matrix, 1 t is the transpose of row matrix (1, 1, . . . , 1) with (r + 1) entries and I is (r + 1) × (r + 1) identity matrix. For more details, we refer the reader to Antzoulakos et al. [3] and Dafnis et al. [8]. Let us define some notations as a(p) := q ℓ1 p ℓ2 , ℓ := ℓ 1 + ℓ 2 , m 1 := k 1 − ℓ 1 + 1, m 2 := k 2 − ℓ 2 + 1, ρ r is the r-th waiting time for (k 1 , k 2 )-runs, p ·,n and g r (·) be the PMF of (k 1 , k 2 )-runs and ρ r , respectively. Also, define µ n,j andμ r,j be the j-th (non-central) moment of (k 1 , k 2 )-runs and ρ r , respectively, where n denotes the number of Bernoulli trials. Distribution of H n ℓ 1 ,k 1 ,ℓ 2 and its Waiting Time Recall that H n ℓ1,k1,ℓ2 is the number of occurrences of (at least ℓ 1 ) at most k 1 consecutive 0's followed by at least ℓ 2 consecutive 1's. Here, r = k 1 + ℓ 2 + 1 and k + 1 is the element after k 1 consecutive 0's (if failures occur) in {0, 1, . . . , k 1 , k + 1 = k 1 + 1, k 1 + 2, . . . , k 1 + ℓ 1 + 1}. It is easy to see that P H 0 ℓ1,k1,ℓ2 = 0 = 1 and ε n := sup x : P H n ℓ1,k1,ℓ2 = x > 0 = ⌊n/ℓ⌋. Therefore, κ 0 = (1, 0, . . . , 0) 1×(k1+ℓ2+2) , A = [a i,j ] (k1+ℓ2+2)×(k1+ℓ2+2) with non-zero entries • a i,1 = p and a i,i+1 = q for 1 ≤ i ≤ ℓ 1 , • a i,k1+3 = p and a i,i+1 = q for ℓ 1 + 1 ≤ i ≤ k 1 + 1, • a k1+2,1 = p and a k1+2,k1+2 = q, • a i,2 = q for k 1 + 3 ≤ i ≤ k 1 + ℓ 2 + 2 and a i,i+1 = p for k 1 + 3 ≤ i ≤ k 1 + ℓ 2 , • a k1+ℓ2+2,k1+ℓ2+2 = p and B = [b i,j ] (k1+ℓ2+2)×(k1+ℓ2+2) is the matrix of non-zero entry b k1+ℓ2+1,k1+ℓ2+2 = p. Hence, using (1), it can be easily verified that Φ(t, z) = ∞ n=0 φ n (t)z n = 1 1 − z − (qz) ℓ1 (pz) ℓ2 (1 − (qz) k1−ℓ1+1 ) = 1 1 − z − a(p)z ℓ (t − 1) (1 − (qz) m1 ) . (3) Now, using (3), we have the following results. Theorem 2.1. The recursive relation in PGF, PMF and moments of H n ℓ1,k1,ℓ2 , for n ≥ ℓ, are given by (i) φ n (t) = φ n−1 (t) + a(p)(t − 1) [φ n−ℓ (t) − q m1 φ n−ℓ−m1 (t)] with initial condition φ n (t) = 1, for n ≤ ℓ − 1. (ii) p m,n = p m,n−1 + a(p) [p m−1,n−ℓ − p m,n−ℓ − q m1 (p m−1,n−ℓ−m1 − p m,n−ℓ−m1 )] with initial conditions p 0,n = 1 and p m,n = 0, m > 0 for n ≤ ℓ − 1. (iii) µ n,j = µ n−1,j + a(p) j−1 k=0 j k [µ n−ℓ,k − q m1 µ n−ℓ−m1,k ], for j ≥ 1 with initial conditions µ n,0 = 1 and µ n,j = 0 for all j ≥ 1 and n ≤ ℓ − 1. Proof. From (3), (i) follows and using the definition of PGF, (ii) follows. Substituting t = e x = ∞ m=0 x m /m! in (i) and comparing the coefficient of x m /m!, (iii) follows. Next, we obtain an explicit form of PGF and PMF using Theorem 2.1. Theorem 2.2. Assume the conditions of Theorem 2.1 hold, then PGF and PMF of H n ℓ1,k1,ℓ2 are given by (3) can be written as (i) φ n (t) = ⌊ n ℓ ⌋ u=0 n−uℓ ℓ+m 1 v=0 n − u(ℓ − 1) − v(ℓ + m 1 − 1) n − uℓ − v(ℓ + m 1 ), u, v (−1) v q vm1 (a(p)(t − 1)) u+v . (ii) p m,n = ⌊ n ℓ ⌋ u=0 n−uℓ ℓ+m 1 v=0 n − u(ℓ − 1) − v(ℓ + m 1 − 1) n − uℓ − v(ℓ + m 1 ), u, v u + v m (−1) u−m q vm1 a(p) u+v , where n u1,u2,...,us = n! u1!u2!···us! . Proof. (i) For (t, z) ∈ \|t\| ≤ 1, \|z\| < 1 and \|z + a(p)z ℓ (t − 1)(1 − (qz) m1 )\| < 1 ,Φ(t, z) = ∞ n=0 z + a(p)z ℓ (t − 1)(1 − (qz) m1 ) n . Now, using binomial expansion and interchanging summations, we get the required result. (ii) Following the steps similar to (i) with recursive relation (ii) of Theorem 2.1, the proof follows. Next, using (2) with some algebraic manipulations, it can be easily verified that H(t, z) = 1 + ∞ r=1 a(p)t ℓ (1 − (qt) m1 ) 1 − t + a(p)t ℓ (1 − (qt) m1 ) r z r .(4) Hence, using (4), we have the following theorem. Theorem 2.3. Let δ i,j denote Kronecker delta function. The PGF, PMF and moments of ρ r , for r ≥ 1, are given by (i) H r (t) = a(p)t ℓ (1 − (qt) m1 ) 1 − t + a(p)t ℓ (1 − (qt) m1 ) r . (ii) g r (m) = g r (m − 1) + a(p) [g r−1 (m − ℓ) − g r (m − ℓ) − q m1 (g r−1 (m − ℓ − m 1 ) − g r (m − ℓ − m 1 ))] , for m ≥ ℓr with initial condition g 0 (m) = δ m,0 , g r (m) = 0 for m ≤ ℓr − 1. (iii)μ r,j = j k=0 j k [μ r,k + a(p)(ℓ j−k − q m1 (ℓ + m 1 ) j−k )(μ r−1,k −μ r,k )], j ≥ 1 with initial conditionμ 0,i = δ i,0 . Proof. Following the steps similar to the proof of Theorem 2.1, the results follow. 2.2 Distribution of H n ℓ 1 ,ℓ 2 ,k 2 and its Waiting Time Recall that H n ℓ1,ℓ2,k2 is the number of occurrences of at least ℓ 1 consecutive 0's followed by (at least ℓ 2 ) at most k 2 consecutive 1's. Here, r = ℓ 1 + k 2 , P H 0 ℓ1,ℓ2,k2 = 0 = 1 and ε n := sup x : P H n ℓ1,ℓ2,k2 = x > 0 = ⌊n/ℓ⌋. Also, if 0 occurs after at least ℓ 1 consecutive 0's followed by (at least ℓ 2 ) at most k 2 consecutive 1's then H n ℓ1,ℓ2,k2 moves v (any) to v + 1 times. Therefore, κ 0 = (1, 0, . . . , 0) 1×(ℓ1+k2+1) , A = [a i,j ] (ℓ1+k2+1)×(ℓ1+k2+1) with non-zero entries • a i,1 = p and a i,i+1 = q for 1 ≤ i ≤ ℓ 1 , • a ℓ1+1,ℓ1+1 = q and a ℓ1+1,ℓ1+2 = p, • a i,2 = q for ℓ 1 + 2 ≤ i ≤ ℓ 1 + ℓ 2 and a i,i+1 = p for ℓ 1 + 2 ≤ i ≤ ℓ 1 + k 2 , • a ℓ1+k2+1,1 = p and B = [b i,j ] (ℓ1+k2+1)×(ℓ1+k2+1) is the matrix of non-zero entries b i,2 = q for ℓ 1 + ℓ 2 + 1 ≤ i ≤ ℓ 1 + k 2 + 1. Hence, using (1), it can be easily verified that Φ(t, z) = 1 − a(p)z ℓ (t − 1) m2 i=1 (pz) i−1 1 − z − a(p)z ℓ (t − 1) (1 − (pz) m2 ) .(5) Now, using (5), the following theorem can be easily derived. Theorem 2.4. The recursive relation in PGF, PMF and moments of H n ℓ1,ℓ2,k2 , for n ≥ ℓ + 1, are given by (i) φ n (t) = φ n−1 (t) + a(p)(t − 1) [φ n−ℓ (t) − p m2 φ n−ℓ−m2 (t)] − a(p)(t − 1)p n−ℓ 1(ℓ + 1 ≤ n ≤ ℓ + m 2 − 1) with initial condition φ n (t) = 1, for n ≤ ℓ, where 1(A) denotes the indicator function of set A. (ii) p m,n = p m,n−1 + a(p) [p m−1,n−ℓ − p m,n−ℓ − p m2 (p m−1,n−ℓ−m2 − p m,n−ℓ−m2 )] − a(p) p n−ℓ [1(m = 1, ℓ + 1 ≤ n ≤ ℓ + m 2 − 1) − 1(m = 0, ℓ + 1 ≤ n ≤ ℓ + m 2 − 1)] with initial conditions p 0,n = 1, p m,n = 0, m > 0 for n ≤ ℓ. (iii) µ n,j = µ n−1,j + a(p) j−1 k=0 j k [µ n−ℓ,k − p m2 µ n−ℓ−m2,k ] − a(p) p n−ℓ 1(ℓ + 1 ≤ n ≤ ℓ + m 2 − 1), for j ≥ 1 with initial conditions µ n,0 = 1 and µ n,j = 0 for all j ≥ 1 and n ≤ ℓ. Next, we obtain an explicit form for PGF and PMF using Theorem 2.4. Theorem 2.5. Assume the conditions of Theorem 2.4 hold, then PGF and PMF of H n ℓ1,ℓ2,k2 are given by (i) φ n (t) = χ n (t) − a(p)(t − 1) ℓ+m2−1 i=ℓ p i−ℓ χ n−i (t) (ii) p m,n = V m,n − a(p) ℓ+m2−1 i=ℓ p i−ℓ (V m−1,n−i − V m,n−i ), where χ n (t) = ⌊ n ℓ ⌋ u=0 n−uℓ ℓ+m 2 v=0 n − u(ℓ − 1) − v(ℓ + m 2 − 1) n − uℓ − v(ℓ + m 2 ), u, v (−1) v p vm2 (a(p)(t − 1)) u+v and V m,n = ⌊ n ℓ ⌋ u=0 n−uℓ ℓ+m 2 v=0 n − u(ℓ − 1) − v(ℓ + m 2 − 1) n − uℓ − v(ℓ + m 2 ), u, v u + v m (−1) u−m p vm2 a(p) u+v . Next, using (2), it can be easily verified that H(t, z) = 1 + qt 1 − pt ∞ r=1 a(p)t ℓ (1 − (pt) m2 ) 1 − t + a(p)t ℓ (1 − (pt) m2 ) r z r .(6) Hence, using (6), the following theorem can be easily derived. Theorem 2.6. The PGF, PMF and moments of ρ r , for r ≥ 1, are given by (i) H r (t) = qt 1 − pt a(p)t ℓ (1 − (pt) m2 ) 1 − t + a(p)t ℓ (1 − (pt) m2 ) r . (ii) g r (m) = g r (m − 1) + a(p)[g r−1 (m − ℓ) − g r (m − ℓ) − p m2 (g r−1 (m − ℓ − m 2 ) − g r (m − ℓ − m 2 ))], r ≥ 2 with initial condition g 0 (m) = δ m,0 and g 1 (m) = g 1 (m − 1) − a(p)[g 1 (m − ℓ) − p m2 g 1 (m − ℓ − m 2 )] + qa(p)p m−ℓ−1 1(ℓ + 1 ≤ m ≤ ℓ + m 2 ), for m ≥ ℓr + 1, g r (m) = 0 whenever m ≤ ℓr and r ≥ 1. (iii)μ r,j = j k=0 j k μ r,k + a(p) ℓ j−k − p m2 (ℓ + m 2 ) j−k (μ r−1,k −μ r,k ) , j ≥ 1 and r ≥ 2 with initial conditionμ 0,i = δ i,0 and µ 1,j = j k=0 j k μ 1,k 1 − a(p) ℓ j−k − p m2 (ℓ + m 2 ) j−k + qa(p) ℓ+m2 k=ℓ+1 k j p k−ℓ−1 . The proofs of Theorems 2.4 -2.6 follow using steps similar to the proofs of Theorems 2.1 -2.3. 2.3 Distribution of H n ℓ 1 ,k 1 ,ℓ 2 ,k 2 and its Waiting Time Recall that H n ℓ1,k1,ℓ2,k2 is the number of occurrences of (at least ℓ 1 ) at most k 1 consecutive 0's followed by (at least ℓ 2 ) at most k 2 consecutive 1's. Here, r = k 1 + k 2 + 1 and k + 1 is the element after k 1 consecutive 0's (if failures occur) in {0, 1, . . . , k 1 , k + 1 = k 1 + 1, k 1 + 2, . . . , k 1 + k 2 + 1}. It is easy to see that P H 0 ℓ1,k1,ℓ2,k2 = 0 = 1 and ε n := sup x : P H n ℓ1,k1,ℓ2,k2 = x > 0 = ⌊n/ℓ⌋. Also, if 0 occurs after (at least ℓ 1 ) at moat k 1 consecutive 0's followed by (at least ℓ 2 ) at most k 2 consecutive 1's then H n ℓ1,k1,ℓ2,k2 moves v (any) to v + 1 times. Therefore, κ 0 = (1, 0, . . . , 0) 1×(k1+k2+2) , A = [a i,j ] (k1+k2+2)×(k1+k2+2) with non-zero entries • a i,1 = p and a i,i+1 = q for 1 ≤ i ≤ ℓ 1 , • a i,k1+3 = p and a i,i+1 = q for ℓ 1 + 1 ≤ i ≤ k 1 + 1, • a k1+2,1 = p and a k1+2,k1+2 = q, • a i,2 = q for k 1 + 3 ≤ i ≤ k 1 + ℓ 2 + 1 and a i,i+1 = p for k 1 + 3 ≤ i ≤ k 1 + k 2 + 1, • a k1+k2+2,1 = p and B = [b i,j ] (k1+k2+2)×(k1+k2+2) is the matrix of non-zero entries b i,2 = q for k 1 + ℓ 2 + 2 ≤ i ≤ k 1 + k 2 + 2. Hence, using (1), it can be easily verified that Φ(t, z) = ∞ n=0 φ n (t)z n = 1 − a(p)z ℓ (t − 1) (1 − (qz) m1 ) m2 i=1 (pz) i−1 1 − z − a(p)z ℓ (t − 1) (1 − (qz) m1 ) (1 − (pz) m2 ) .(7) Now, using (7), the following theorem can be easily derived. Theorem 2.7. The recursive relations in PGF, PMF and moments of H n ℓ1,k1,ℓ2,k2 , for n ≥ ℓ + 1, are given by (i) φ n (t) = φ n−1 (t) + a(p)(t − 1) [φ n−ℓ (t) − q m1 φ n−ℓ−m1 (t) − p m2 φ n−ℓ−m2 (t) + q m1 p m2 φ n−ℓ−m1−m2 (t)] − a(p)(t − 1)p n−ℓ 1(ℓ + 1 ≤ n ≤ ℓ + m 2 − 1) − q p m1 1(ℓ + m 1 ≤ n ≤ ℓ + m 1 + m 2 − 1) with initial condition φ n (t) = 1, for n ≤ ℓ. (ii) p m,n = p m,n−1 − a(p) p n−ℓ 1(m = 1, ℓ + 1 ≤ n ≤ ℓ + m 2 − 1) − 1(m = 0, ℓ + 1 ≤ n ≤ ℓ + m 2 − 1) − (q/p) m1 1(m = 1, ℓ + m 1 ≤ n ≤ ℓ + m 1 + m 2 − 1)−1(m = 0, ℓ + m 1 ≤ n ≤ ℓ + m 1 + m 2 − 1) + a(p) [p m−1,n−ℓ − p m,n−ℓ − q m1 (p m−1,n−ℓ−m1 − p m,n−ℓ−m1 ) − p m2 (p m−1,n−ℓ−m2 − p m,n−ℓ−m2 ) +q m1 p m2 (p m−1,n−ℓ−m1−m2 − p m,n−ℓ−m1−m2 )] with initial conditions p 0,n = 1 and p m,n = 0, m > 0 for n ≤ ℓ. (iii) µ n,j = µ n−1,j + a(p) j−1 k=0 j k [µ n−ℓ,k − q m1 µ n−ℓ−m1,k − p m2 µ n−ℓ−m2,k + q m1 p m2 µ n−ℓ−m1−m2,k ] − a(p) p n−ℓ 1(ℓ + 1 ≤ n ≤ ℓ + m 2 − 1) − (q/p) m1 1(ℓ + m 1 ≤ n ≤ ℓ + m 1 + m 2 − 1) , j ≥ 1 with initial conditions µ n,0 = 1 and µ n,j = 0 for all j ≥ 1 and n ≤ ℓ. Next, we obtain an explicit form for PGF and PMF using Theorem 2.7. Theorem 2.8. Assume the conditions of Theorem 2.7 hold, then PGF and PMF of H n ℓ1,k1,ℓ2,k2 are given by (i) φ n (t) = ϕ n (t) − a(p)(t − 1) ℓ+m2−1 i=ℓ p i−ℓ ϕ n−i (t) − q p m1 ℓ+m1+m2−1 i=ℓ+m1 p i−ℓ ϕ n−i (t) (ii) p m,n = κ m,n − a(p) ℓ+m2−1 i=ℓ p i−ℓ (κ m−1,n−i − κ m,n−i ) − q p m1 ℓ+m1+m2−1 i=ℓ+m1 p i−ℓ (κ m−1,n−i − κ m,n−i ) , where ϕ n (t)= ⌊ n ℓ ⌋ u=0 n−uℓ ℓ+m 1 w=0 f (n,u,w,0,0) ℓ+m 2 r=0 f (n,u,w,r,0) ℓ+m 1 +m 2 v=0 (−1) w+r f (n, u, w, r, v)+u+v+r+w f (n, u, w, r, v), u, w, r, v q (v+w)m1 p (v+r)m2 (a(p)(t−1)) u+w+r+v κ m,n = ⌊ n ℓ ⌋ u=0 n−uℓ ℓ+m 1 w=0 f (n,u,w,0,0) ℓ+m 2 r=0 f (n,u,w,r,0) ℓ+m 1 +m 2 v=0 (−1) u+v−m f (n, u, w, r, v)+u+v+r+w f (n, u, w, r, v), u, w, r, v u + w + r + v m q (v+w)m1 p (v+r)m2 a(p) u+w+r+v and f (n, u, w, r, v) = n − uℓ − w(ℓ + m 1 ) − r(ℓ + m 2 ) − v(ℓ + m 1 + m 2 ). Next, using (2), it can be easily verified that H(t, z) = 1 + qt 1 − pt ∞ r=1 a(p)t ℓ (1 − (qt) m1 )(1 − (pt) m2 ) 1 − t + a(p)t ℓ (1 − (qt) m1 )(1 − (pt) m2 ) r z r .(8) Hence, using (8), the following theorem can be easily derived. Theorem 2.9. The PGF, PMF and moments of ρ r , for r ≥ 1, are given by (i) H r (t) = qt 1 − pt a(p)t ℓ (1 − (qt) m1 )(1 − (pt) m2 ) 1 − t + a(p)t ℓ (1 − (qt) m1 )(1 − (pt) m2 ) r . (ii) g r (m)=g r (m − 1) + a(p)[g r−1 (m − ℓ) − g r (m − ℓ) − q m1 (g r−1 (m − ℓ − m 1 ) − g r (m − ℓ − m 1 )) − p m2 (g r−1 (m−ℓ−m 2 )−g r (m−ℓ−m 2 ))+q m1 p m2 (g r−1 (m−ℓ−m 1 −m 2 )−g r (m−ℓ−m 1 −m 2 ))], for r ≥ 2 with initial condition g 0 (m) = δ m,0 and g 1 (m) = g 1 (m − 1) + qa(p)p m−ℓ−1 1(ℓ + 1 ≤ m ≤ ℓ + m 2 ) − q p m1 1(ℓ + m 1 + 1 ≤ m ≤ ℓ + m 1 + m 2 ) − a(p)[g 1 (m − ℓ) − q m1 g 1 (m − ℓ − m 1 ) − p m2 g 1 (m − ℓ − m 2 ) + q m1 p m2 g 1 (m − ℓ − m 1 − m 2 )], for m ≥ ℓr + 1, g r (m) = 0 whenever m ≤ ℓr and r ≥ 1. (iii)μ r,j = j k=0 j k μ r,k + a(p) ℓ j−k − q m1 (ℓ + m 1 ) j−k − p m2 (ℓ + m 2 ) j−k + q m1 p m2 (ℓ + m 1 + m 2 ) j−k (μ r−1,k −μ r,k ), j ≥ 1, and r ≥ 2 with initial conditionμ 0,i = δ i,0 and µ 1,j = j k=0 j k μ 1,k 1 − a(p) ℓ j−k − q m1 (ℓ + m 1 ) j−k − p m2 (ℓ + m 2 ) j−k + q m1 p m2 (ℓ + m 1 + m 2 ) j−k + qa(p) ℓ+m2 k=ℓ+1 k j p k−ℓ−1 − q p m1 ℓ+m1+m2 k=ℓ+m1+1 k j p k−ℓ−1 . The proofs of Theorems 2.7 -2.9 follow using steps similar to the proofs of Theorems 2.1 -2.3. Remarks 2.1. (i) It is important to note that the expression m2 i=1 (pz) i−1 = k2−ℓ1+1 i=1 (pz) i−1 appears in (5) and (7), as expected, since the pattern can be completed if a failure occurs after ℓ 2 +1 (up to k 2 ) consecutive successes. Also, with the same justification, the expressions (4) and (8) have the term qt/(1−pt). However, (3) and (4) are in easy form as the pattern is completed just after ℓ 2 consecutive successes. (ii) The explicit form of PGF and PMF in Theorems 2.2, 2.5 and 2.8 can also be expressed in different forms as the binomial expansion can be written (a + b) n = n u=0 n u a u b n−u = n u=0 n u a n−u b u . It is up to the end-user to choose an appropriate form and modify the results. (iii) The results derived in Section 2, are based on Markov chain approach (see Fu and Koutras [12] and Dafnis et al. [8]). However, the results can also be derived using combinatorial method similar to Huang and Tsai [13]. (iv) It can be easily verified that for ℓ 1 = k 1 and ℓ 2 = k 2 , Theorems 2.7 -2.9 are same as Theorems 3.1 -3.8 of Kumar and Upadhye [16], as expected. (v) For ℓ 1 = 1 = ℓ 2 , H n 1,1,k1,k2 = X (3) n of Dafnis et al. [8] (in their notation). Also, Dafnis et al. [8] in Theorem 4.7 proved that for r ≥ 1, the PGF for waiting time for X (3) n is given by H r (z) = (qz)(pz)(1 − (qz) k1 )(1 − (pz) k2 ) 1 − z + (qz)(pz)(1 − (qz) k1 )(1 − (pz) k2 ) r (1 − (pz) k2 ) −1 .(9) But, observe that H r (1) = 1/(1 − p k2 ) = 1 unless p = 0. Therefore, the expression (9) is incorrect and hence Theorems 4.8 and 4.9 of Dafnis et al. [8] are also incorrect. We correct and generalize these erroneous results in Theorem 2.9. An Application to Fibonacci Words Fibonacci words are particular sequences of binary numbers 0 and 1 (or two alphabets) and it is used to model physical systems with the aperiodic order such as quasi-crystals. Also, Fibonacci word studied widely in the field of combinatorics on words. Fibonacci words are formed in a similar way as Fibonacci numbers (repeated addition) and, in this process, n-th Fibonacci word depends on (n − 1)-th and (n − 2)-th Fibonacci words of 0's and 1's. The construction can be explained as follows: C 0 = 0 and C 1 = 01 then n-th Fibonacci word is given by C n = C n−1 C n−2 . For example, 10-th element of Fibonacci words is given by (1 − ζ 1 )ζ 2 (1 − ζ 3 )(1 − ζ 4 )ζ 5 (1 − ζ 6 )ζ 7 (1 − ζ 8 )(1 − ζ 9 )ζ 10 (1 − ζ 11 )(1 − ζ 12 )ζ 13 (1 − ζ 14 )ζ 15 (1 − ζ 16 )(1 − ζ 17 )ζ 18 (1 − ζ 19 )ζ 20 (1 − ζ 21 )(1 − ζ 22 )ζ 23 (1 − ζ 24 )(1 − ζ 25 )ζ 26 (1 − ζ 27 )ζ 28 (1 − ζ 29 )(1 − ζ 30 ) . . . . Also, the sub-words "11" and "000" never occur in Fibonacci words and last two digits are "01" and "10", alternately. For more details on Fibonacci words, we refer the reader to Berstel [6]. Now, observe that Fibonacci words can be seen as a pattern of either exactly one 1 followed by (at least one) at most two consecutive 0's or (at least one) at most two consecutive 0's followed by exactly one 1 and hence the distribution of patterns adopted the distribution of either H n 1,1,1,2 or H n 1,2,1,1 respectively, for n-th Fibonacci word. For large values of n, the probabilities and moments of the distribution of these patterns can be calculated from the distribution of either H n 1,1,1,2 or H n 1,2,1,1 . Next, we compute some probabilities and mean for H n 1,2,1,1 and its waiting time for various values of p and n = 60. Observe that the upper range of m is ⌊n/ℓ⌋ = ⌊60/2⌋ = 30, while we obtain the probabilities up to m = 5 and others can be computed in a similar way. Also, for waiting time distribution, it is known that m ≥ ℓr + 1 = 3. So, we obtain probabilities by taking m up to 10 in Table 2. Moment for H 60 1,2,1,1 and ρ 1 are obtained in Table 1 and Table 2, respectively. For more details about runs and patterns, we refer the reader to Aki[1], Aki et al.[2], Antzoulakos et al.[3], Koutras[14,15] and Makri et al.[17] and references therein.Also, Let H n ℓ1,k1,ℓ2 , H n ℓ1,ℓ2,k2 and H n ℓ1,k1,ℓ2,k2 be the number of occurrences for (T1), (T2) and (T3) type event, respectively. Then, random variable representation of H n ℓ1,k1,ℓ2 , H n ℓ1,ℓ2,k2 and H n ℓ1,k1,ℓ2,k2 can be seen as follows: H n ℓ1,k1,ℓ2 = n−ℓ1−ℓ2+1 m=1 I m , H n ℓ1,ℓ2,k2 = n−ℓ1−ℓ2 m=1 J m and H n ℓ1,k1,ℓ2,k2 = n−ℓ1−ℓ2 m=1 K m . Now, let us consider a particular realization in a sequence of 20 Bernoulli trials given by 0 0 1 1 1 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1. Here, note that (T1) H 20 1,1,1 = 2, H 20 1,2,2 = 2, H 20 2,2,3 = 1 and H 20 1,2,1 = 3. (T2) H 20 1,1,2 = 3, H 20 3,1,2 = 1, H 20 2,2,2 = 0 and H 20 1,4,4 = 1. (T3) H 20 1,1,1,1 = 1, H 20 1,2,2,2 = 1, H 20 1,1,1,2 = 2 and H 20 1,2,1,2 = 2. Antzoulakos and Chadjiconstantinidis [4], Balakrishnan and Koutras [5], Dafnis et al. [8], Fu and Koutras [12], Table 1 : 1Distribution and moments of H 60 1,2,1,1 . Table 2 : 2Distribution and moments of waiting time for H 60 1,2,1,1 . On sooner and later problems between success and failure runs. S Aki, Advances in Combinatorial Methods and Applications to Probability and Statistics. Borkhäuser, BostonAki, S. (1997). On sooner and later problems between success and failure runs. Advances in Combinatorial Methods and Applications to Probability and Statistics (Ed., N. Balakrishnan), 385-400, Borkhäuser, Boston. On discrete distributions of order k. S Aki, H Kuboki, K Hirano, Ann. Inst. Statist. Math. 36Aki, S., Kuboki, H. and Hirano, K. (1984). On discrete distributions of order k. Ann. Inst. Statist. Math., 36, 431-440. Waiting times associated with the sum of success run lengths. D L Antzoulakos, S Bersimis, M V Koutras, Mathematical and Statistical Methods in Reliability. Lindqvist, B., Doksum, K.SingaporeWorld ScientificAntzoulakos, D. L., Bersimis, S., Koutras, M. V. (2003). Waiting times associated with the sum of success run lengths. In: Lindqvist, B., Doksum, K. (Eds.), Mathematical and Statistical Methods in Reliability, World Scientific, Singapore, 141-157. Distributions of numbers of success runs of fixed length in Markov dependent trials. D L Antzoulakos, S Chadjiconstantinidis, Ann. Inst. Statist. Math. 53Antzoulakos, D. L. and Chadjiconstantinidis, S. (2001). Distributions of numbers of success runs of fixed length in Markov dependent trials. Ann. Inst. Statist. Math., 53, 599-619. Runs and Scans with Applications. N Balakrishnan, M V Koutras, John Wiley & SonsNew YorkBalakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications, John Wiley & Sons, New York. Fibonacci words -a survey. J Berstel, Rozenberg, G., Salomaa, A.SpringerBerlinThe Book of LBerstel, J. (1986). Fibonacci words -a survey, in: Rozenberg, G., Salomaa, A. (Eds.), The Book of L, Springer, Berlin. Start-up demonstration tests under Markov dependence model with corrective actions. N Balakrishnan, S G Mohanty, S Aki, Ann. Inst. Statist. Math. 49Balakrishnan, N., Mohanty, S. G. and Aki, S. (1997). Start-up demonstration tests under Markov dependence model with corrective actions. Ann. Inst. Statist. Math., 49, 155-169. 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[ "Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials", "Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials" ]	[ "Tian-Xiao He \nDepartment of Mathematics Illinois\nWesleyan University Bloomington\n61702-2900ILUSA\n", "Jinze Zheng \nDepartment of Mathematics Illinois\nWesleyan University Bloomington\n61702-2900ILUSA\n" ]	[ "Department of Mathematics Illinois\nWesleyan University Bloomington\n61702-2900ILUSA", "Department of Mathematics Illinois\nWesleyan University Bloomington\n61702-2900ILUSA" ]	[]	A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If the pair of matrices are the same, they define a dual relationship. Here presented is a unified approach to construct dual relationships via pseudo-involution of Riordan arrays. Then we give four dual relationships for Bernoulli numbers and Euler numbers, from which the corresponding dual sequences of Bernoulli polynomials and Euler polynomials are constructed. Some applications in the construction of identities of Bernoulli numbers and polynomials and Euler numbers and polynomials are discussed based on the dual relationships.	null	[ "https://arxiv.org/pdf/1507.03055v1.pdf" ]	119,706,269	1507.03055	5691c16eed75091a8aa71417db92ffffeeced8bb	Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials 11 Jul 2015 Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington 61702-2900ILUSA Jinze Zheng Department of Mathematics Illinois Wesleyan University Bloomington 61702-2900ILUSA Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials 11 Jul 2015arXiv:1507.03055v1 [math.CO]AMS Subject Classification: 05A1505A0511B3911B7315B3615A0605A1911B83 Key Words and Phrases: inverse matricesdualBernoulli numbersBernoulli poly- nomialsEuler numbersEuler polynomialsRiordan arrayspseudo-involution A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If the pair of matrices are the same, they define a dual relationship. Here presented is a unified approach to construct dual relationships via pseudo-involution of Riordan arrays. Then we give four dual relationships for Bernoulli numbers and Euler numbers, from which the corresponding dual sequences of Bernoulli polynomials and Euler polynomials are constructed. Some applications in the construction of identities of Bernoulli numbers and polynomials and Euler numbers and polynomials are discussed based on the dual relationships. Introduction Krattenthaler defines a class of matrix inverses in [26], which has numerous famous special cases presented by Gould and Hsu [13], Carlitz [4], Bressoud [2], Chu and Hsu [9], Ma [29], etc. Hsu, Shiue, and one of the authors study Riordan matrix inverses in [22]. Let infinite low triangle matrices (d n,k ) 0≤k≤n and (d n,k ) 0≤k≤n be matrix inverses. Then a sequence inverse relationship can be defined as f n = n k=0 d n,k g k ⇐⇒ g n = n k=0d n,k f k , where sequences {f n } n≥0 and {g n } n≥0 are called the inverse sequences with respect to inverse matrices (d n,k ) 0≤k≤n and (d n,k ) 0≤k≤n . If d n,k =d n,k , i.e., {d n,k } 0≤k≤n is selfinverse, then {f n } n≥0 and {g n } n≥0 are said to be a pair of dual sequences (or they are dual each other) with respect to self-inverse matrix (d n,k ) 0≤k≤n , i.e., they satisfy f n = n k=0 d n,k g k ⇐⇒ g n = n k=0 d n,k f k , If there exists a sequence {f n } n≥0 that is dual to itself with respect to a self-inverse matrix (d n,k ) 0≤k≤n , i.e., f n = n k=0 d n,k f k ,(3) then, {f n } n≥0 is called self-dual sequence with respect to the matrix (d n,k ) 0≤k≤n . Let d n,k = n k (−1) k . Then (d n,k ) 0≤k≤n is a self-inverse matrix because n k=0 k j=0 n k k j (−1) k+j = δ n,j , where δ n,j is the Kronecker symbol. Let {a n } n≥0 be a sequence of complex numbers. Then {a * n } n≥0 defined by a * n = n k=0 n k (−1) k a k(4) is the dual sequence of {a n } n≥0 (see, for example, Graham, Knuth, and Patashnik [14]) with respect to ( n k (−1) k ). Hence, a n = n k=0 n k (−1) k a * k . {a n } n≥0 and {a * n } n≥0 are a pair of inverse sequences with (a * ) * n = a n . If a * n = a n , then {a n } n≥0 is called the self-dual sequence (see Z. W. Sun [48]). For instance, the following number sequences are self-dual sequences with respect to the dual relationship (4) (see, for example, Z. H. Sun [47]): 1 2 n , Bernoulli numbers B n and Euler numbers E n for n = 0, 1, . . . are defined by B n := B n (0) and E n = 2 n E n 1 2 . A large literature scatters widely in books and journals on Bernoulli numbers B n , and Bernoulli polynomials B n (x) and Euler numbers E n and Euler polynomials E n (x). They can be studied by means of the binomial expressions connecting them, B n (x) = n k=0 n k B k x n−k , n ≥ 0,(8)E n (x) = n k=0 n k x − 1 2 n−k E k 2 k , n ≥ 0,(9) where E k = 2 k E k (1/2). The study brings consistent attention of researchers working in combinatorics, number theory, etc. This paper will study the duals of Bernoulli number sequence and Euler number sequence with respect to ( n k (−1) k ) and other self-inverse matrices. The dual sequence of {B n } n≥0 with respect to ( n k (−1) k ) is denoted by {B * n } n≥0 , and the corresponding dual Bernoulli polynomials denoted by {B * n (x)} n≥0 is defined similarly to (8) by using the the duals of sequences {B n } n≥0 , i.e., B * n (x) = n k=0 n k B * k x n−k , n ≥ 0,(10) where B * n are the duals of the Bernoulli numbers {B n } n≥0 , i.e., B * n = n k=0 n k (−1) k B k , n ≥ 0.(11) Hence, there are several questions raised: (1) With respect to which self-inverse matrix the Bernoulli number sequence {B n } is self-dual? (2) What is the relationship between the self-inverse matrix in (1) and the self-inverse matrix for {(−1) n B n } n≥0 , ( n k (−1) k )? And (3) does there exist a unified approach to construct self-inverse matrices and self-duals with respect to a self-inverse matrix? We will answer those questions by using the Riordan array theory. Riordan arrays are infinite, lower triangular matrices defined by the generating function of their columns. They form a group, called the Riordan group (see Shapiro et al. [44]). Some of the main results on the Riordan group and its application to combinatorial sums and identities can be found in [5]- [8], [10]- [12], [15]- [25], [27]- [28], [31]- [32], [34], [36]- [37], [39]- [43], [45]- [46], and [50]- [53]. More formally, let us consider the set of formal power series (f.p.s. ) F = R[[t]]; the order of f (t) ∈ F , f (t) = ∞ k=0 f k t k (f k ∈ R) , is the minimal number r ∈ N such that f r = 0; F r is the set of formal power series of order r. It is known that F 0 is the set of invertible f.p.s. and F 1 is the set of compositionally invertible f.p.s., that is, the f.p.s. f (t) for which the compositional inversef (t) exists such that f (f (t)) =f (f (t)) = t. Let d(t) ∈ F 0 and h(t) ∈ F 1 ; the pair (d(t), h(t)) defines the (proper) Riordan array D = (d n,k ) n,k∈N = (d(t), h(t)) having d n,k = [t n ]d(t)h(t) k(12) or, in other words, having d(t)h(t) k as the generating function whose coefficients make-up the entries of column k. It is immediately to be known that the usual row-by-column product of two Riordan arrays is also a Riordan array: (d 1 (t), h 1 (t)) * (d 2 (t), h 2 (t)) = (d 1 (t)d 2 (h 1 (t)), h 2 (h 1 (t))),(13) where the fundamental theorem of Riordan arrays (d(t), h(t))f (t) = d(t)f (h(t))(14) is applied to obtain (13). The Riordan array I = (1, t) is everywhere 0 except that it contains all 1's on the main diagonal; it can be easily proved that I acts as an identity for this product, that is, (1, t) * (d(t), h(t)) = (d(t), h(t)) * (1, t) = (d(t), h(t)) . From these facts, we deduce a formula for the inverse Riordan array: (d(t), h(t)) −1 = 1 d(h(t)) ,h(t) (15) whereh(t) is the compositional inverse of h(t). In this way, the set R of proper Riordan arrays forms a group. From [40], an infinite lower triangular array [d n,k ] n,k∈N = (d(t), h(t)) is a Riordan array if and only if a sequence A = (a 0 = 0, a 1 , a 2 , . . .) exists such that for every n, k ∈ N there holds d n+1,k+1 = a 0 d n,k + a 1 d n,k+1 + · · · + a n d n,n , which is shown in [24] to be equivalent to h(t) = tA(h(t)).(17) Here, A(t) is the generating function of the A-sequence. In [24,31] it is also shown that a unique sequence Z = (z 0 , z 1 , z 2 , . . .) exists such that every element in column 0 can be expressed as the linear combination d n+1,0 = z 0 d n,0 + z 1 d n,1 + · · · + z n d n,n , or equivalently, d(t) = d 0,0 1 − tZ(h(t)) . We call sequences A and Z the A-(characterization) sequence and Z−(characterization) sequence of the Riordan array (d(t), h(t)), respectively, and the generating functions of called A-sequence and Z−sequence the A (generating) function and Z (generating) function, respectively. In next section, we present a unified approach to construct a class of self-inverse matrices and their application for the construction of dual number sequences and dual polynomial sequences. In Section 3 we will discuss the algebraic structure of dual number sequences with respect to the dual relationships established in Section 2. Some identities of self-dual number sequences are found accordingly. In Section 4, more identities of dual number sequences will be constructed by using the dual relationships. 2 Construction of self-inverse matrices and the corresponding self-duals It is calear that a Riordan array (d(t), h(t)) and its inverse (d(t), h(t)) −1 = 1 d(h(t)) ,h(t) , whereh(h(t)) = h(h(t)) = t, is a pair of inverse matrices. If (d(t), h(t)) −1 = (d(t), h(t)), i.e., (d(t), h(t)) is an involution, i.e., it has an order of 2, then (d(t), h(t)) is a self-inverse matrix. Here, if g is an element of a group G, then the smallest positive integer n such that g n = e, the identity of the group, if it exists, is called the order of g. If there is no such integer, then g is said to have infinite order. It is well-known that (see Shapiro [42]) if we restrict all entries of a Riordan array to be integers, then any element of finite order in the Riordan group must have order 1 or 2, and each element of order 2 generates a subgroup of order 2. Sprugnoli and one of the author find the characterization of Riordan arrays of order 2 in [24]. In combinatorial situations, a Riordan array often has nonnegative integer entries and hence it can not have order 2. Therefore, we consider (see definition, for example, in [6]) an element R ∈ R to have pseudo-order 2, i.e., RM has order 2, where M = (1, −t). Those R are called pseudo-Riordan involutions or briefly pseudo-involutions (see Cameron and Nkwanta [3] and [42]). We now present some sufficient and necessary conditions to identify pseudo-involutions. Theorem 2.1 Let (d(t), h(t) ) be a pseudo-involution, and let A(t) and Z(t) be the generating functions of the A-sequence and Z-sequence of (d(t), h(t)), i.e., A-function and Z−function. Then the following statements are equivalent to the statement that the Riordan array (d(t), h(t)) is a pseudo-involution: ( 1) ±(d(t), h(t))(1, −t) = (±d(t), −h(t)) are involutions. (2) (1, −t)(d(t), h(t))(1, −t) = (d(t), h(t)) −1 , the inverse of (d(t), h(t)). (3) ±(1, −t)(d(t), h(t)) = (±d(−t), h(−t)) are involutions. (4) A(t) = −t h(−t) and Z(t) = d(−t)−1 h(−t) . Proof. Let (d(t), h(t)) be a pseudo-involution. Then (d(t), h(t))(1, −t) = (d(t), −h(t)) is an involution, i.e., (d(t), −h(t))(d(t), −h(t)) = (d(t)d(−h(t)), −h(−h(t)) = (1, t). Additionally, −(d(t), h(t))(1, −t) = (−d(t), −h(t)) satisfies (−d(t), −h(t))(−d(t), −h(t)) = (d(t), −h(t))(d(t), −h(t)) = (1, t), which implies −(d(t), h(t))(1, −t) = (−d(t), −h(t)) is also an involution. Hence, we have proved the assumption that (d(t), h(t)) is a pseudo-involution implies (1). (2) follows from (1) because I = ((d(t), h(t))(1, −t)) ((d(t), h(t))(1, −t)) = (d(t), h(t)) ((1, −t)(d(t), h(t))(1, −t)) . From the above equations, we have (±(1, −t)(d(t), h(t))) (±(1, −t)(d(t), h(t))) = ((1, −t)(d(t), h(t))(1, −t)) (d(t), h(t)) = (1, t) = I. Hence, we obtain (3) from (2). Similarly, (1) follows from (3). To find the A-function and Z-function of pseudo-involution (d(t), h(t)), we recall (see Theorem 3.3 of [24]) the A-sequence of (d 3 (t), h 3 (t)) = (d 1 (t), h 1 (t))(d 2 (t), h 2 (t)) has the A-(generating) function A 3 (t) = A 2 (t)A 1 t A 2 (t) , where A 1 (t) and A 2 (t) are the A-(generating) functions of (d 1 (t), h 1 (t)) and (d 2 (t), h 2 (t)), respectively. Since the A-functions of (d(t), h(t)) and (1, −t) are respectively A(t) and −1, the A-function of (d(t), h(t))(1, −t) is A 3 (t) = −A (−t) .(20) On the other hand, from Theorem 4.3 of [24] we know the A-(generating) function of the involution (d(t), h(t))(1, −t) = (d(t), −h(t)) is A 3 (t) = t −h(t) .(21) Comparing (20) and (21) we have A(t) = −t h(−t) . From Theorem 3.4 of [24], the Z-(generating) function of (d 3 (t), h 3 (t)) = (d(t), h(t))(1, −t) = (d(t), −h(t)) is Z 3 (t) = Z(−t) because the Z-functions of (d(t), h(t)) and (1, −t) are Z(t) and 0, respectively. Since (d 3 (t), h 3 (t)) is an involution, from Theorem 4.3 of [24] we also have Z 3 (t) = 1 − d(t) −h(t) . Comparing the last two equations, we immediately know Z(t) = d(−t) − 1 h(−t) . From the definition of A-sequence and Z-sequence, a Riordan array (d(t), h(t)) that possesses the above A-function and Z-function is a pseudo-involution. Corollary 2.2 Let nonzero d(t) ∈ F 0 and h(t) ∈ F 1 . Then the infinite lower triangle matrix (d(t), h(t)) is a pseudo-involution if and only if h(t) = −h(−t) and d(t) = h(−t) h(−t) − td(−t) + t , where the denominator is assumed not being zero. Proof.h(t) = −h(−t) if and only if t h(t) = −t h(−t) , or equivalently, there exists the function A(t) such that A(t) = t h(t) and A(t) = −t h(−t) , which implies that (d(t), h(t)) has an A-function (see, for example, [24]) satisfying (4) of Theorem 2.1. d(t) = h(−t)/(h(−t) − td(−t) + t) if and only if d(−t) − 1 h(−t) = d(t) − 1 td(t) , or equivalently, there exists the function Z(t) such that Z(t) = d(t) − 1 td(t) and Z(t) = d(−t) − 1 h(−t) , which implies that (d(t), h(t)) has a Z-function (see, for example, [24]) which satisfies (4) of Theorem 2.1. Combining all above statements together we know (d(t), h(t)) is a pseudoinvolution, completing the proof of Corollary 2.2. Corollary 2.2 also presents an algorithm to find a pseudo-involution. Generally one can accomplish this by carrying through the procedure demonstrated by the following example as suggested by the corollary. For instance, it is clear that h(t) = t/(1 − t) ( or t/(1 + t)) has the compositional inverseh(t) = t/(1 + t) (or t/(1 − t)) and satisfiesh(t) = −h(−t). From Corollary 2.2, we may also find that d(t) = 1/(1 − t) (or 1/(1 + t)) satisfies d(t) = h(−t)/(h(−t) − td(−t) + t). Therefore (1/(1 − t), t/(1 − t)) ((1/(1 + t), t/(1 + t))) is a pseudo-involution. Remark It can be seen that (d(t), h(t)) is a pseudo-involution if and only if there exist A-function and Z-function satisfying four relationships A(t) = t h(t) , A(t) = −t h(−t) , Z(t) = d(t) − 1 td(t) , and Z(t) = d(−t) − 1 h(−t) , which not only give the formulas shown in Corollary 2.3 but also the following useful formula in the construction of pseudo-involutions: h(t) = tZ(t) Z(−t)(1 − tZ(t)) and d(t) = h(t)Z(−t) tZ(t) . The proof is straightforward from the above four relationships and is omitted. From (1) and (3) of Theorem 2.1 we obtain the following result. Corollary 2.3 (1/(1−t), t/(1−t)) = n k n,k≥0 and (1/(1+t), t/(1+t)) = n k (−1) n−k n,k≥0 are pseudo-involutions. Hence, from (1) and (3) of Theorem 2.1, we generate the following four Riordan involutions, denoted by R 1 , R 2 , R 3 , and R 4 , respectively. Proof. Since (see, for example, Apostol [1] and Milton and Stegun [33]) R 1 = 1 1 − t , t 1 − t (1, −t) = (1, −t) 1 1 + t , t 1 + t = 1 1 − t , −t 1 − t = n k (−1) k n,k≥0 , R 2 = − 1 1 − t , t 1 − t (1, −t) = −(1, −t) 1 1 + t , t 1 + t = 1 t − 1 , t t − 1 = n k (−1) k+1 n,k≥0 , R 3 = (1, −t) 1 1 − t , t 1 − t = 1 1 + t , t 1 + t (1, −t) = 1 1 + t , −t 1 + t = n k (−1) n n,k≥0 , R 4 = −(1, −t) 1 1 − t , t 1 − t = − 1 1 + t , t 1 + t (1, −t) = − 1 1 + t , −t 1 + t = n k (−1) n+1 n,k≥0 .(22)n k=0 n k B k = B n (1) = (−1) n B n ,(23) {(−1) n B n } n≥0 and {B n } n≥0 are self-dual sequences with respect to D 1 and D 3 , respectively. Similarly, from (9) and (−1) n E n (−x) = −E n (x) + 2x n (see, for example [1,33]) there hold n k=0 n k (−1) k+1 E k 1 2 − 1 2 k = −(−1) n n k=0 n k − 1 2 − 1 2 n−k E k 1 2 + n k=0 n k (−1) k 1 2 k = −(−1) n E n − 1 2 + 1 2 n = E n 1 2 − 2 1 2 n + 1 2 n = E n 1 2 − 1 2 n Similarly, we have n k=0 n k (−1) n+1 (−1) k E k 1 2 − 1 2 k = − n k=0 n k (−1) n−k E k 1 2 + (−1) n n k=0 n k (−1) k 1 2 k = −E n − 1 2 + (−1) n 1 2 n = (−1) n E n 1 2 − 1 2 n , completing the proof of the theorem. We now consider the duals of Bernoulli and Euler numbers and the corresponding duals of Bernoulli and Euler polynomials with respect to different dual relationships shown in Theorem 2.4. Theorem 2.5 Let B * n be the duals of Bernoulli numbers B n with respect to R 1 , and let B * n (x) be the corresponding dual Bernoulli polynomials defined by (10). Then there hold B * n (x) = (−1) n B n (−x − 1)(24) for all n ≥ 0, and B * n = (−1) n B n + n.(25) Proof. (8) and (10) give B * n (x) = n k=0 n k (x) n−k B * k = n k=0 n k (x) n−k k j=0 k j (−1) j B j = n j=0 n j (−1) j B j n k=j n − j k − j (x) n−k = n j=0 n j (−1) j B j (x + 1) n−j = (−1) n B n (−x − 1). Hence, B * n = B * n (0) = (−1) n B n (−1) = B n (1) + n = (−1) n B n (0) + n, which implies (25). Corollary 2.6 Let B * n (x) be the duals of Bernoulli polynomials. Then their generating function is n≥0 B * n (x) t n n! = n≥0 (−1)B n (−x − 1) t n n! = −te (x+1)t e −t − 1 = e (x+1)t 1 + 1−t−e −t t ,(26) Other duals of Bernoulli and Euler numbers can be presented below. Theorem 2.7 With respect to the dual relationship D 3 , the duals of numbers (−1) n B n , denoted by ((−1) n B n ) * , may be written as ((−1) n B n ) * = n k=0 n k (−1) n (−1) k B k = B n + (−1) n n.(27) And the corresponding dual Bernoulli polynomials are n k=0 n k (−1) k B k * x n−k = B n (x) − n(x − 1) n−1 .(28) With respect to the dual relationship D 4 , the duals of numbers E n (1/2) − (1/2) n , denoted by (E n (1/2) − (1/2) n ) * , have the expressions E n 1 2 − 1 2 n * = n k=0 n k (−1) n+1 E k 1 2 − 1 2 k = (−1) n E n 1 2 + 3 n − 2 2 n .(29) The corresponding dual Euler polynomials are n k=0 n k x − 1 2 n−k E k 1 2 − 1 2 k * = (−1) n E n (−x+1)−2(x−1) n +(x−2) n . (30) With respect to the dual relationship D 2 , the duals of numbers (−1) n (E n (1/2) − (1/2) n ), denoted by ((−1) n (E n (1/2) − (1/2) n )) * , can be evaluated as (−1) n E n 1 2 − 1 2 n * = n k=0 n k (−1) k+1 (−1) k E k 1 2 − 1 2 k = E n 1 2 + 3 n − 2 2 n .(31) And the corresponding dual Bernoulli polynomials are n k=0 n k x − 1 2 n−k (−1) k E k 1 2 − 1 2 k * = E n (x) + (x + 1) n − 2x n .(32) Proof. The proofs are straightforward from the definitions and are omitted. Generating functions of self-dual number sequences In this section, we give some structures of self-dual number sequences, by using which the characterizations, relationships, and other properties of the self-dual number sequences can be obtained. The first half of the following theorem is presented in Prodinger [38]. Theorem 3.1 Let a(x) = n≥0 a n x n . Then its coefficient sequence is a self-dual sequence with respect to D 1 (D 2 ), i.e., it satisfies n k=0 n k (−1) k a k = a n (−a n ) for n ∈ N ∪ {0} if and only if a(x) satisfies the equation a x x − 1 = (1 − x)a(x) (−(1 − x)a(x)), respectively. The coefficient sequence of a(x) is a self-dual sequence with respect to D 3 (D 4 ), i.e., it satisfies n k=0 n k (−1) n a k = a n (−a n ) for n ∈ N ∪ {0} if and only if a(x) satisfies the equation a − x 1 + x = (1 + x)a(x) (−(1 + x)a(x)), respectively. Proof. Since the first half of the theorem is given in [38]. It is sufficient to prove the second half. ± 1 1 + x a − x 1 + x = ± ∞ k=0 a k (−1) k x k (1 + x) −k−1 = ± ∞ k=0 a k (−1) k x k ∞ j=0 −k − 1 j x j = ± ∞ k=0 ∞ j=0 a k (−1) k+j k + j j x k+j = ± ∞ n=0   (−1) n n j=0 n j a n−j   x n = ± ∞ n=0 a n x n = ±a(x). The proof is complete. Corollary 3.2 Let {a n } n≥0 be a given sequence with ordinary generating function a(x) = n≥0 a n x n . Then (1) [47] {a n } n≥0 is a self-dual sequence with respect to D 1 if and only if {2a n+1 − a n } n≥0 is a self-dual sequence with respect to D 2 . (2) {a n } n≥0 is a self-dual sequence with respect to D 3 if and only if {2a n+1 + a n } n≥0 is a self-dual sequence with respect to D 4 . Proof. The result shown in (1) is given in [47]. It is sufficient to prove (2). Let b n = 2a n+1 + a n with its ordinary generating function b(x) = n≥0 b n x n . Thus, b(x) = 2(a(x) − a 0 ) x + a(x) = x + 2 x a(x) − 2 x a 0 , which implies b − x 1 + x = − x + 2 x a − x 1 + x + 2 x (1 + x)a 0 = −(1 + x) x + 2 x 1 1 + x a − x 1 + x − 2 x a 0 . Therefore, a − x 1 + x = (1 + x)a(x) if and only if b − x 1 + x = −(1 + x) x + 2 x a(x) − 2 x a 0 = −(1 + x)b(x). Based on Theorem 3.1 we have finished the proof. Theorem 3.3 Let {a n } n≥0 be a given sequence with exponential generating function a * (x) = n≥0 a n x n /n!. Then (1) [47] {a n } n≥0 is a self-dual sequence with respect to D 1 if and only if a * (x)e −x/2 is an even function. (2) [47] {a n } n≥0 is a self-dual sequence with respect to D 2 if and only if a * (x)e −x/2 is an odd function. (3) {a n } n≥0 is a self-dual sequence with respect to D 3 if and only if a * (x)e x/2 is an even function. (4) {a n } n≥0 is a self-dual sequence with respect to D 4 if and only if a * (x)e x/2 is an odd function. Proof. The results shown in (1) and (2) are given in [47]. We only need to prove (3) and (4) here by using a similar argument: a * (−x)e −x = k≥0 (−1) k a k x k k! j≥0 (−1) j x j j! = k≥0 j≥0 (−1) k+j a k x k+j k!j! = n≥0 n k=0 (−1) n n k a k x n n! . Hence, if n k=0 (−1) n n k a k = a n and − a n , then a * (−x)e −x = a(x) and − a(x), respectively, which can be written briefly as a * (−x)e −x/2 = ±a(x)e x/2 . If the case of positive sign on the right-hand side holds, i.e., {a n } is a self-dual sequence with respect to D 3 , then a * (x)e x/2 is an even function; while the negative sign holds, or equivalently, {a n } is a self-dual sequence with respect to D 4 , then the function a * (x)e x/2 is odd. It is easy to see the sufficiencies of (3) and (4) are also true. This concludes the proof of the theorem. [47] uses Theorem 3.3 to derive a numerous identities. [30] uses umbral calculus to reprove and extend some of them. We now survey their results and extend them to other self-dual sequences. Theorem 3.4 For any function f , we have (1) [47] n k=0 n k f (k) − (−1) n−k k j=0 k j f (j) a n−k = 0 for n ∈ N ∪ {0} if {a n } n≥0 is a self-dual sequence with respect to D 1 . (2) [30] n k=0 n k f (k) + (−1) n−k k j=0 k j f (j) a n−k = 0 for n ∈ N ∪ {0} if {a n } n≥0 is a self-dual sequence with respect to D 2 . (3) n k=0 n k f (k) − k j=0 (−1) n−j k j f (j) a n−k = 0 for n ∈ N ∪ {0} if {a n } n≥0 is a self-dual sequence with respect to D 3 . (4) n k=0 n k f (k) + k j=0 (−1) n−j k j f (j) a n−k = 0 for n ∈ N ∪ {0} if {a n } n≥0 is a self-dual sequence with respect to D 4 . Proof. The proofs of (3) and (4) are similar as (1) and (2) by using either Theorem 3.3 or umbral calculus. Hence, we omit them. are self-dual sequences with respect to D 2 and D 4 , respectively. Therefore, we may use Theorem 3.4 to obtain the following identities. Theorem 3.5 Let B n and E n be Bernoulli numbers and Euler numbers, respectively. For any function f , we have (33) where the first identity is given in [47]. n k=0 n k   (−1) n−k f (k) − k j=0 k j f (j)   B n−k = 0, n k=0 n k   f (k) + (−1) n−k k j=0 k j f (j)   E n−k 1 2 − 1 2 n−k = 0, n k=0 n k   f (k) − k j=0 (−1) n−j k j f (j)   B n−k = 0, n k=0 n k   (−1) n−k f (k) + k j=0 (−1) k−j k j f (j)   E n−k 1 2 − 1 2 n−k = 0, Applications of dual sequences to Bernoulli and Euler polynomials Let α ∈ R. Denote (34) Then we have the following identities about C n,α (x) and C * n,α (x), which is an extension of the main results shown in Sun [49]. Theorem 4.1 Let C n,α (x) and C * n,α (x) be defined as (34), and let k, ℓ ∈ N ∪ {0} and x + y + z = 1 + 2α. Then there hold k j=0 (−1) j x k−j k j C ℓ+j+1,α (y) ℓ + j + 1 + ℓ j=0 (−1) j x ℓ−j ℓ j C * k+j+1,α (z) k + j + 1 = a 0 x k+ℓ+1 (k + ℓ + 1) k+ℓ k .(35) In addition, we have k j=0 (−1) j x k−j k j C ℓ+j,α (y) = ℓ j=0 (−1) j x ℓ−j ℓ j C * k+j,α (z)(36) and k j=0 (−1) j (ℓ + j + 1)x k−j+1 k + 1 j C ℓ+j,α (y) + ℓ j=0 (−1) j (k + j + 1)x ℓ−j+1 ℓ + 1 j C * k+j,α (z) = (k + ℓ + 2) (−1) k C k+ℓ+1,α (y) + (−1) ℓ C * k+ℓ+1,α (z) .(37) Proof. Substituting (34) into the left-hand side of (35) yields k j=0 k j (−1) j x k−j ℓ + j + 1 a 0 (−1) ℓ+j+1 (y − α) ℓ+j+1 + ℓ+j i=0 a i+1 ℓ + j + 1 i + 1 (−1) ℓ+j+1 (y − α) ℓ+j−i + ℓ j=0 ℓ j (−1) j x ℓ−j k + j + 1 k+j+1 r=0 (−1) k+j+1 k + j + 1 r r i=0 a i r i (−1) r (z − α) k+j+1−r = ca 0 + k+ℓ i=0 a i+1 k j=0 k j ℓ + j + 1 i + 1 (−1) j x k−j ℓ + j + 1 (−1) ℓ+j+1 (y − α) ℓ+j+1−(i+1) + k+ℓ+1 i=1 a i ℓ j=0 k+j+1 r=i ℓ j k + j + 1 r r i (−1) j x ℓ−j k + j + 1 (− (z − α)) k+j+1−r = ca 0 + k+ℓ i=0 c i i + 1 a i+1 , where c = k j=0 k j (−1) j x k−j ℓ + j + 1 (−1) ℓ+j+1 (y − α) ℓ+j+1 + ℓ j=0 k+j+1 r=0 ℓ j k + j + 1 r (−1) j x ℓ−j k + j + 1 (− (z − α)) k+j+1−r , which will be evaluated later, and (−1) ℓ c i = k j=0 k j ℓ + j + 1 i + 1 (i + 1) x k−j ℓ + j + 1 (y − α) ℓ+j−i + ℓ j=0 k+j+1 r=i+1 ℓ j k + j + 1 r r i + 1 (i + 1) (−1) ℓ−j+1 x ℓ−j k + j + 1 (− (z − α)) k+j+1−r = k j=0 k j ℓ + j i x k−j (y − α) ℓ+j−i + ℓ j=0 ℓ j k + j i (−1) ℓ−j+1 x ℓ−j k+j+1 r=i+1 k + j − i r − i − 1 (− (z − α)) k+j−i−(r−i−1) = x k+ℓ−i   k j=0 k j ℓ + j i y − α x ℓ+j−i − ℓ j=0 ℓ j k + j i (−1) ℓ−j 1 + y − α x k+j−i   = 0, where the last step is due to the expression in the last parenthesis is zero following from Lemma 3.1 of [49]. Finally, we evaluate c as follows. c = (−1) ℓ+1 k j=0 k j x k−j (y − α) ℓ+j+1 ℓ + j + 1 + ℓ j=0 ℓ j (−1) j x ℓ−j k + j + 1 k+j+1 r=0 k + j + 1 r (− (z − α)) k+j+1−r = (−1) ℓ+1 k j=0 k j x k−j y−α 0 t ℓ+j dt + ℓ j=0 ℓ j (−1) j x ℓ−j x+y−α 0 t k+j dt = (−1) ℓ+1 y−α 0 t ℓ (t + x) k dt + (−1) ℓ x+y−α 0 t k (t − x) ℓ dt = (−1) ℓ − x+y−α x (s − x) ℓ s k ds + x+y−α 0 (t − x) ℓ t k dt = (−1) ℓ x 0 (t − x) ℓ t k dt = (−1) ℓ 1 0 (tx − x) ℓ (tx) k xdt = x k+ℓ+1 1 0 (1 − t) ℓ t k dt = x k+ℓ+1 Γ(k + 1)Γ(ℓ + 1) Γ(k + ℓ + 2) = x k+ℓ+1 (k + ℓ + 1) k+ℓ k . By taking partial derivative with respect to y on the both sides of (35) and noting z = 1 + 2α − x − y, one may obtain (36). Replacing k and ℓ by k + 1 and ℓ + 1 in (36) taking the partial derivative with respect to y, one may obtain (37). The proof is complete. The following corollary of Theorem 4.1 is equivalent to the results shown in Theorems 1.1 and 1.2 of [49]. 49]) Let C n,0 (x) and C * n,0 (x) be defined as (34), and let k, ℓ ∈ N ∪ {0} and x + y + z = 1. Then there holds Corollary 4.2 ([k j=0 (−1) j x k−j k j C ℓ+j+1,0 (y) ℓ + j + 1 + ℓ j=0 (−1) j x ℓ−j ℓ j C * k+j+1,0 (z) k + j + 1 = a 0 x k+ℓ+1 (k + ℓ + 1) k+ℓ k . (38) In addition, we have k j=0 (−1) j x k−j k j C ℓ+j,0 (y) = ℓ j=0 (−1) j x ℓ−j ℓ j C * k+j,0 (z)(39) and k j=0 (−1) j (ℓ + j + 1)x k−j+1 k + 1 j C ℓ+j,0 (y) + ℓ j=0 (−1) j (k + j + 1)x ℓ−j+1 ℓ + 1 j C * k+j,0 (z) = (k + ℓ + 2) (−1) k C k+ℓ+1,0 (y) + (−1) ℓ C * k+ℓ+1,0 (z) . If a k = B k or (−1) k B k , then we have (−1) ℓ+1 k j=0 x k−j k j B ℓ+j+1 (y) ℓ + j + 1 + (−1) k+1 ℓ j=0 x ℓ−j ℓ j B k+j+1 (z) k + j + 1 = x k+ℓ+1 (k + ℓ + 1) k+ℓ k (40) In addition, we have (−1) ℓ k j=0 x k−j k j B ℓ+j (y) = (−1) k ℓ j=0 x ℓ−j ℓ j B k+j (z)(41) and (−1) ℓ k j=0 )ℓ + j + 1)x k−j k + 1 j B ℓ+j (y) + (−1) k ℓ j=0 (k + j + 1)x ℓ−j ℓ + 1 j B k+j (z) = (k + ℓ + 2) (−1) k B k+ℓ+1 (y) + (−1) ℓ B k+ℓ+1 (z) . The conjugate Bernoulli polynomialsB n (x) is introduced in [23] by their generating function as follows. e xt 1 + 1+t−e t t = ∞ n=0B n (x) t n n! ,(42) where the first few terms of the conjugate polynomial sequence {B n } n≥0 arẽ B 0 (x) = 1, B 1 (x) = x + 1 2 , B 2 (x) = x 2 + x + 5 6 , B 3 (x) = x 3 + 3 2 x 2 + 5 2 x + 2, B 4 (x) = x 4 + 2x 3 + 5x 2 + 8x + 191 30 , B 5 (x) = x 5 + 5 2 x 4 + 25 3 x 3 + 20x 2 + 191 6 x + 76 3 , etc. Let {B n (x)} n≥0 be the conjugate Bernoulli polynomial sequence defined by (42), and let the conjugate Bernoulli numbers {B n } n≥0 be defined byB n =B n (0). Then we may find thatB n (x) = n k=0 n k x n−kB k(43) for all n ≥ 0. We define dual sequence of the conjugate Bernoulli number sequence with respect to inverse matrix R 1 . Then the corresponding dual polynomial sequence is B * n (x) := n k=0 n k x n−kB * k(44) for all n ≥ 0. From Theorem 4.1, we obtain (−1) k k j=0 k j x k−j (−1) j+1B ℓ+j+1 (−y) ℓ + j + 1 + ℓ j=0 ℓ j x ℓ−jB * k+j+1 (−z) k + j + 1 = (−1) k+1 x k+ℓ+1 (k + ℓ + 1) k+ℓ k ,(45) whereB n (t) are defined by (42), andB * n (x) are defined by (44). Also (−1) k k j=0 k j (−x) k−jB ℓ+j (−y) = (−1) ℓ ℓ j=0 ℓ j (−x) ℓ−jB * k+j (−z)(46) and (−1) ℓ k j=0 k + 1 j (−x) k−j+1 (ℓ + j + 1)B ℓ+j (−y) +(−1) k ℓ j=0 ℓ + 1 j (−x) ℓ−j+1 (k + j + 1)B * k+j (−z) = (k + ℓ + 2)((−1) ℓ+1B k+ℓ+1 (−y) + (−1) k+1B * k+ℓ+1 (−z)).(47) In (34), substituting α = 1/2 and a k = E k 1 2 − 1 2 k and a * k = (−1) k E k 1 2 + 3 k − 2 2 k ,(48) where the duals a * k are derived by using (29) of Theorem 2.7, we obtain A n (x) = (−1) n E n (x) − (−1) n x n and A * n (x) = E n (−x + 1) + (2 − x) n − 2(1 − x) n . Hence, Theorem 4.1 implies Corollary 4.4 Let C n,α (x) and C * n,α (x) be defined as (34) with α = 1/2, and let k, ℓ ∈ N ∪ {0} and x + y + z = 2. For a k and a * k shown in (48), there hold k j=0 (−1) ℓ+1 x k−j k j E ℓ+j+1 (y) ℓ + j + 1 + ℓ j=0 (−1) j x ℓ−j ℓ j E k+j+1 (−z + 1) k + j + 1 = (−1) ℓ+1 y 0 t ℓ (t + x) k dt − x+y 0 t k (x − t) ℓ dt + 2 x+y−1 0 t k (x − t) ℓ dt.(49) In addition, we have (−1) ℓ k j=0 x k−j k j E ℓ+j (y) − (−1) ℓ y ℓ (x + y) k = ℓ j=0 (−1) j x ℓ−j ℓ j E k+j (1 − z) + (x + y) k (−y) ℓ − 2(x + y − 1) k (1 − y) ℓ .(50) The above identities of polynomial sequences can be used to establish identities of number sequences. For instance, if x = 1, y = 0, and z = 1, then (49) yields the number sequence identity k j=0 (−1) ℓ+1 x k−j k j E ℓ+j+1 (0) ℓ + j + 1 + ℓ j=0 (−1) j x ℓ−j ℓ j E k+j+1 (0) k + j + 1 = − 1 (k + ℓ + 1) k+ℓ k . If x = 1 and y = z = 1/2, from (49) there holds k j=0 (−1) ℓ+1 x k−j k j E ℓ+j+1 (1/2) ℓ + j + 1 + ℓ j=0 (−1) j x ℓ−j ℓ j E k+j+1 (1/2) k + j + 1 = − 1 (k + ℓ + 1) k+ℓ k + 2B 1 2 , k + 1, ℓ + 1 − 2B 3 2 , k + 1, ℓ + 1 , where B(α, a, b) = α 0 t a−1 (1 − t) b−1 dt is an incomplete beta function. From the last two identities, we have k j=0 (−1) ℓ+1 x k−j k j E ℓ+j+1 (1/2) − E ℓ+j+1 (0) ℓ + j + 1 + ℓ j=0 (−1) j x ℓ−j ℓ j E k+j+1 (1/2) − E k+j+1 (0) k + j + 1 = 2 B 1 2 , k + 1, ℓ + 1 − B 3 2 , k + 1, ℓ + 1 . Let A and B be any m × m and n × n square matrices, respectively. Cheon and Kim [6] use the notation ⊕ for the direct sum of matrices A and B: A ⊕ B = A 0 0 B .(51) Hence, we may obtain a matrix form of Lemma 3.2 of Pan and Sun [35] Theorem 4.5 Let n be any positive integer, and letB(t) and P [t] be defined as for all n ≥ 0 shown in [35]. Proof. Noting thatB 0 (y) = 1, the left-hand side of (52) can be written as x 2 + 2 1 xB 1 (y) + 2 2 B 2 (y) . . . x n + n 1 x n−1B 1 (y) + · · · + n n B n (y) . . .           = [X]                     1 x x 2 . . . x n . . .           +           0 1 1 B 1 (y)                    =           0 x (1 + 1 2 )x 2 . . . (1 + 1 2 + · · · + 1 n )x n . . . , {(−1) n B n }, {L n }, {nF n−1 },where {B n }, {L n }, and {F n } are the Bernoulli sequence, Lucas sequence, and Fibonacci sequence.Bernoulli polynomials B n (x) and Euler polynomials E n (x) for n = 0, 1, . . . Theorem 2. 4 − 1 ) 41Let R i (i = 1, 2, 3, 4) be the Riordan arrays shown in Corollary 2.3. Then there hold the following four dual relationships, denoted by D 1 , D 2 , D 3 , and D 4 , respectively. n+1 a k . 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[ "Exclusive vector meson photoproduction at the LHC and the FCC: A closer look on the final state", "Exclusive vector meson photoproduction at the LHC and the FCC: A closer look on the final state" ]	[ "G Gil Da Silveira \nInstituto de Física e Matemática\nUniversidade Federal de Pelotas Caixa Postal 354\n96010-090PelotasCEP, RSBrazil\n\nDepartamento de Física Nuclear e de Altas Energias\nUniversidade do Estado do Rio de Janeiro CEP\n20550-013Rio de JaneiroRJBrazil\n", "V P Gonçalves \nInstituto de Física e Matemática\nUniversidade Federal de Pelotas Caixa Postal 354\n96010-090PelotasCEP, RSBrazil\n\nDepartment of Astronomy and Theoretical Physics\nLund University\n223-62LundSweden\n", "M M Jaime \nInstituto de Física e Matemática\nUniversidade Federal de Pelotas Caixa Postal 354\n96010-090PelotasCEP, RSBrazil\n\nDepartamento de Física Nuclear e de Altas Energias\nUniversidade do Estado do Rio de Janeiro CEP\n20550-013Rio de JaneiroRJBrazil\n" ]	[ "Instituto de Física e Matemática\nUniversidade Federal de Pelotas Caixa Postal 354\n96010-090PelotasCEP, RSBrazil", "Departamento de Física Nuclear e de Altas Energias\nUniversidade do Estado do Rio de Janeiro CEP\n20550-013Rio de JaneiroRJBrazil", "Instituto de Física e Matemática\nUniversidade Federal de Pelotas Caixa Postal 354\n96010-090PelotasCEP, RSBrazil", "Department of Astronomy and Theoretical Physics\nLund University\n223-62LundSweden", "Instituto de Física e Matemática\nUniversidade Federal de Pelotas Caixa Postal 354\n96010-090PelotasCEP, RSBrazil", "Departamento de Física Nuclear e de Altas Energias\nUniversidade do Estado do Rio de Janeiro CEP\n20550-013Rio de JaneiroRJBrazil" ]	[]	Over the past years the LHC experiments have reported experimental evidences for processes associated to photon-photon and photon-hadron interactions, showing their potential to investigate the production of low-and high-mass systems in exclusive events. In the particular case of the photoproduction of vector mesons, the experimental study of this final state is expected to shed light on the description of the QCD dynamics at small values of the Bjorken-x variable. In this paper we extend previous studies for the exclusive J/Ψ and Υ photoproduction in pp collisions based on the nonlinear QCD dynamics by performing a detailed study of the final state distributions that can be measured experimentally at the LHC and at the Future Circular Collider. Predictions for the rapidity and transverse momentum distributions of the vector mesons and of final-state dimuons are presented for pp collisions at √ s = 7, 13, and 100 TeV.	null	[ "https://arxiv.org/pdf/1609.09854v1.pdf" ]	119,246,087	1609.09854	ac6adc245793a74fd1fcbfe63167b95900743744	Exclusive vector meson photoproduction at the LHC and the FCC: A closer look on the final state 30 Sep 2016 October 2016 G Gil Da Silveira Instituto de Física e Matemática Universidade Federal de Pelotas Caixa Postal 354 96010-090PelotasCEP, RSBrazil Departamento de Física Nuclear e de Altas Energias Universidade do Estado do Rio de Janeiro CEP 20550-013Rio de JaneiroRJBrazil V P Gonçalves Instituto de Física e Matemática Universidade Federal de Pelotas Caixa Postal 354 96010-090PelotasCEP, RSBrazil Department of Astronomy and Theoretical Physics Lund University 223-62LundSweden M M Jaime Instituto de Física e Matemática Universidade Federal de Pelotas Caixa Postal 354 96010-090PelotasCEP, RSBrazil Departamento de Física Nuclear e de Altas Energias Universidade do Estado do Rio de Janeiro CEP 20550-013Rio de JaneiroRJBrazil Exclusive vector meson photoproduction at the LHC and the FCC: A closer look on the final state 30 Sep 2016 October 2016numbers: 1238Bx1320Gd1360-r1360Le1385Dz1460Ef Keywords: vector mesonphotoproductionexclusive productionLHCFCCcolour dipole modelproton-proton collisions * gustavosilveira@cernch † Over the past years the LHC experiments have reported experimental evidences for processes associated to photon-photon and photon-hadron interactions, showing their potential to investigate the production of low-and high-mass systems in exclusive events. In the particular case of the photoproduction of vector mesons, the experimental study of this final state is expected to shed light on the description of the QCD dynamics at small values of the Bjorken-x variable. In this paper we extend previous studies for the exclusive J/Ψ and Υ photoproduction in pp collisions based on the nonlinear QCD dynamics by performing a detailed study of the final state distributions that can be measured experimentally at the LHC and at the Future Circular Collider. Predictions for the rapidity and transverse momentum distributions of the vector mesons and of final-state dimuons are presented for pp collisions at √ s = 7, 13, and 100 TeV. I. INTRODUCTION The exclusive production via photon interactions has attracted great interest given the experimental evidences reported by the Large Hadron Collider (LHC) experiments at CERN over the past years. The CMS experiment has reported results on the exclusive production of dileptons in pp collisions at 7 TeV [1,2], the two-photon production of W + W − pairs in pp collisions at 7 and 8 TeV [3,4], and the Υ photoproduction in p-Pb collisions at 5.02 TeV [5]. The other LHC experiments have reported similar results, e.g. the exclusive production of dileptons and W + W − pairs by the ATLAS experiment in pp collisions at 7 and 8 TeV [6,7], the Υ photoproduction in pp collisions at 7 and 8 TeV by the LHCb experiment [8][9][10], and the J/Ψ photoproduction in p-Pb collisions at 5.02 TeV and in Pb-Pb collisions at 2.76 TeV by the ALICE experiment [11][12][13]. These results demonstrate the capability of the LHC experiments in probing the kinematic region of photon interactions and exclusive photoproduction at high energies, extending the previous results obtained at the Tevatron [14], HERA [15][16][17][18][19][20][21], and RHIC [22] colliders. These set of measurements are the most comprehensive study of photon physics in hadronic collisions up to date, allowing more detailed studies of QED and the electroweak sector of the Standard Model and its extensions (for a recent review see, e.g. Ref. [23]). One of the main motivations for the study of the photoproduction of vector mesons in hadronic colliders is the possibility of probing the gluon distribution at small values of the Bjorken-x variable [24], as well as the QCD dynamics at high energies [25]. During the last decade, these ideas have been widely discussed in the literature considering different theoretical approaches that assume distinct underlying assumptions (see, e.g. Refs. [26][27][28]). Currently, we have that the experimental data are quite well described by models that consider the dipole approach and take into account nonlinear (saturation) effects in the QCD dynamics [29][30][31][32]. One important aspect of the colour dipole models is that the LHC predictions are parameter free, since their main elements -the dipole-proton scattering amplitude and vector meson wave function -have been constrained by the HERA data. Moreover, the energy dependence of the cross sections -accessible at higher energies in the LHC than at HERA -is determined by the QCD dynamics including nonlinear effects. The contribution of these effects increases with the photon-proton centre-of-mass (c.m.) energy (W γp ), being dependent on the mass of the vector meson. In particular, we expect a larger contribution for the J/Ψ than for the Υ production. On the other hand, the recent LHCb data have been used to extract the gluon distribution (xg) assuming the validity of the linear DGLAP evolution equation [33] in the kinematical range considered [34,35], with the resulting gluon distribution being used as input to predict the cross sections for larger energies. It is important to emphasise that the xg obtained through this procedure differs from that obtained in usual global analysis and also that the contribution of the next-to-leading order corrections for the exclusive vector meson photoproduction in collinear factorisation still are subjects of intense debate [36]. Very recently, the BFKL approach have been applied in Ref. [28] for the vector meson production in pp collisions. These authors have obtained that the linear BFKL evolution [37] is capable to describe the energy dependence of the cross sections in the current energy range probed by the LHCb data if a fit of the initial transverse momentum profile of the proton impact factor is performed. The studies performed in Refs. [28][29][30]34] demonstrated that a clear determination of the underlying QCD dynamics is still not feasible by the simple comparison of the predictions with the experimental data for the rapidity distributions of the pp cross sections and/or for the unfolded γp → V p cross section. Consequently, the complementary study of other final state distributions is important to shed light on this subject. In this paper we analyse in detail the photoproduction of vector mesons (J/Ψ and Υ) in pp collisions at LHC energies whereas for the kinematical range expected to be probed by the Future Circular Collider (FCC) [38]. We will focus in the final-state kinematics of the vector mesons decaying into muons and we will investigate the impact in the observables of two different assumptions for the energy dependence of the γp → V p cross section. In particular, we estimate the rapidity and transverse momentum distributions of the vector mesons, as well as the acoplanarity and transverse momentum balance distributions of the dimuons from the decay of the vector mesons. In our study we will use the SuperCHIC v2.0 Monte Carlo (MC) event generator [39], modified to include the predictions of the nonlinear QCD dynamics for the energy dependence of the vector meson photoproduction cross sections derived in Refs. [29,30]. Finally, it is important to emphasise that we will present the predictions for the photoproduction of vector mesons at the FCC for the first time in the literature. This paper is organised as follows. In the next Section we present a brief review of the photoproduction of vector mesons in proton-proton collisions and discuss the approaches used in our analysis to estimate the γp → V p (V = J/Ψ and Υ) cross sections. Moreover, we present some details regarding the Monte Carlo event generator used in our study. In Section III we present our predictions for the total cross sections and different final state distributions that can be analysed in the Run-II of the LHC as well as at FCC. Finally, in Section IV we summarise the main conclusions of this work. II. PHOTOPRODUCTION OF VECTOR MESONS IN PROTON -PROTON COLLI-SIONS The exclusive photoproduction is taken as the diffractive interaction between an emitted photon from one of the colliding protons and the second proton. The photon emission is described in the equivalent photon approximation [40][41][42], which provides a photon flux based on the energy spectrum of the emitted photons. This flux can be obtained in terms of the electric and magnetic form factor of the proton, given by [43] dn dω = α π dQ 2 Q 2 ω   1 − ω √ s 1 − Q 2 min Q 2 F E + ω 2 2s F M   ,(1) with √ s being the pp c.m. collision energy, ω and Q 2 the photon energy and virtuality, respectively, and F i are the proton electromagnetic form factors F E (Q 2 ) = 4m 2 p G 2 E (Q 2 ) + Q 2 G 2 M (Q 2 ) 4m 2 p + Q 2 ,(2)F M (Q 2 ) = G 2 M (Q 2 ),(3) where G i are the Sachs form factors [44,45] G 2 E = G 2 M (Q 2 ) 7.78 = 1 (1 + Q 2 /0.71 GeV 2 ) 4 ,(4) with the minimum photon virtuality being given by [43] Q 2 min = ω 2 √ s m 2 p √ s − ω ,(5) for √ s − ω ≫ m p . The total cross section is given by the convolution of the photon spectrum and the γp → V p cross section, which has to be evaluated with a minimum energy to produce the central system, as follows σ pp→p⊗V ⊗p = 2 dn dωσ γp→V p (ω, Q 2 ) dω ,(6) where ⊗ indicates the presence of a rapidity gap in the final state. In the case of two proton beams, the total cross section has to evaluated with a multiplicative factor of 2 to take into account the possibility of a photon emission from both protons. The rapidity distribution of the vector meson is given by dσ pp→p⊗V ⊗p dY = dn(ω + ) dω +σ γp→V p (+Y ) + dn(ω − ) dω −σ γp→V p (−Y ),(7) with the photon energies being given by ω ± = (M V /2)e ±Y . This formula accounts for the forward and backward emission for the production of a vector meson with rapidity Y . The main input in the calculations is the γp → V p cross section, which can be modelled in terms of different basic quantities, depending on the approach used to describe the process. In the collinear formalism the cross section is proportional to the square of the gluon distribution [46][47][48], which satisfies the linear DGLAP equation. In the k T -factorisation approach, it is given in terms of the square of the unintegrated gluon distribution, which satisfies the linear BFKL [37] or its nonlinear generalisations. On the other hand, in the dipole approach at x ≈ m V /W γpit is proportional to [N (x, r, b)] 2 , where N is the forward dipole -proton scattering amplitude, which describes the interaction of a qq dipole of size r with the proton at an impact parameter b and its evolution is given by the nonlinear Balitsky-Kovchegov (BK) equation [49] in the Colour Glass Condensate (CGC) formalism [50]. One have that in all approaches the energy dependence of the γp → V p cross section is strongly dependent on the description of the QCD dynamics at high energies. In particular, we expect that although their predictions can be similar in a limited energy range, their extrapolations for higher values become very distinct, such that the experimental analysis of the process can be used to discriminate between the approaches. Recent LHCb data have provided data in the range of W γp < 2 TeV, but the Run-II of the LHC is expected to probe values of order of 5 TeV, while the FCC (100 TeV) should probe W γp ≈ 15 TeV. As discussed before, the current data for the rapidity distributions and/or unfolded γp → V p cross section can be described by different approaches based on distinct assumptions for the QCD dynamics and it is not clear if the isolated analysis of this distribution could discriminate between the different models in the future. Therefore, it is important to investigate the behaviour of the cross sections in the kinematical range that will be probed in the Run-II of the LHC and in the FCC, with particular emphasis in complementary final state distributions that can be directly compared with the experimental data. In order to obtain some estimates of these distributions, in what follows we will consider a phenomenological approach for the process that describe the current experimental data and can be implemented in a MC event generator. Phenomenologically, the current experimental data for vector meson photoproduction can be described with a power-law function of the γp cross section with W γp , like σ ∝ W δ γp . In particular, in Ref. [39], the following power-law fit is assumed dσ γp→V p dt = N V W γp W 0 δ V β V e −β V \|t\| ,(8) where W 0 = 1 GeV and δ V gives the slope of the differential cross section in terms of W γp , which is obtained by [51] δ V = ∂ ln σ γp ∂ ln W γp Wγp=W 0 .(9) Additionally, the parameter β V in Eq. (8) is the slope of the proton-Pomeron vertex, which is parametrized by a function based on the Regge theory β V = β 0 + 4α ′ log W γp 90 GeV ,(10) where β 0 = 4.6 GeV −2 and α ′ = 0.2 GeV −2 is the slope of the Regge trajectory, which are assumed universal for the two mesons. In the case of the J/Ψ production, the parameters N Ψ and δ Ψ , which provide the normalisation and the slope of the differential cross section, respectively, were extracted in Ref. [39] from the power-law fit to the available data of the exclusive (or elastic) photoproduction The resulting values for N V and δ V discussed above are effective values constrained by the current data and are valid for a limited kinematical range. In particular, they are not associated to a specific model of the QCD dynamics. As a consequence, predictions for a higher energy range, where new dynamical effects are expected to modify the energy dependence of the cross sections, should be considered as educated guess. In order to investigate the impact of the nonlinear effects on the energy dependence and compare its predictions with the approaches that are available at SuperCHIC2, we have fitted the predictions to the results evaluated in Refs. [29,30] within the colour dipole framework [53] and using the bCGC model [54,55] for the dipole-proton scattering amplitude. Then, we obtain that the energy dependence can be described using Eq. kinematical distributions of the vector mesons decaying into µ + µ − , which have been not studied previously in the literature. III. RESULTS Our goal in this Section is to present our predictions for the exclusive vector meson photoproduction in pp collisions at the LHC ( √ s = 7 and 13 TeV) and FCC ( √ s = 100 TeV) considering two different assumptions for the energy dependence of the γp → V p cross section. However, before presenting the results, it is important to discuss the possible impact of soft interactions that can populate the rapidity gaps in the final state [56]. This subject have been intensively discussed in the last years [57,58], mainly motivated by the experimental data for dijet production in single diffraction events [59,60] that demonstrated that a gap survival factor should be taken into account in order to describe the data. The modelling and magnitude of this factor for the diffractive photon-hadron interaction still are themes of intense debate, with the possibility that it is equal to one being still valid. In particular, the four different models presented in Ref. [61] have been implemented in the SuperCHIC2 in a fully differential format. Thus, the kinematical distributions can be properly estimated by including such effect, since the survival factor depends on the particular subprocess under consideration. In Fig. 1 using the default parameters for N Ψ and δ Ψ . We have that for central rapidities the prediction obtained disregarding the extra soft reinteractions, represented by the solid line, is very similar to those calculated using the four different models present in Ref. [39]. However, the survival corrections have a strong impact at large-η, reducing the cross sections by ≈35%. We checked that similar suppressions are obtained using the bCGC model. As the treatment and magnitude of the survival probability remains a theme of intense debate, in what follows we will present our predictions assuming that it is equal to unity. In Fig. 2 and Table I we also present the predictions for Υ production obtained with the fitting parameters originally extracted from the HERA data. For the J/Ψ production, one have that the default and bCGC predictions are similar to LHC energies, which is expected since both models describe the current data. Moreover, they differ by ≈20% at the FCC energy regime, with the bCGC model predicting Process √ s (TeV) σ[pp → p + (γp) → p + V + p] HERA [64] Default [39] bCGC [29,30] J/ψ → µ + µ − with the bCGC one predicting larger values for the total cross section. As in the J/Ψ case, we would expect the opposite result. We believe that this difference is associated to the fact that the default values for (N Υ , δ Υ ) have been obtained by fitting the current set data, which is smaller in comparison to the J/Ψ one. As a consequence, these effective values have a large uncertainty. Surely, the Run-II data can improve this situation. Lets now analyse the final-state distributions. In Fig. 3 we present our predictions for the rapidity (upper panels) and transverse momentum (lower panels) distributions for different values of the c.m. energy. This is a useful result, since it can be directly obtained from the phenomenological models, as well as from the analysis of the experimental results, as seen in Refs. [9,10]. Considering initially the rapidity distribution for the J/Ψ production (upper left panel), one have that the default and bCGC predictions are very similar in the forward and backward regions at 7 and 13 TeV, but differ at mid-rapidities. In contrast, at FCC energies, they are similar at mid-rapidities and differ by ≈30% for \|Y \| = 5. In the case of the Υ production, we have that the difference between the predictions is smaller. In addition, we have analysed the transverse momentum distribution of Looking at the distributions of the muons coming from the decay of the vector mesons, applying specific kinematical cuts that may be useful in experimental analyses of exclusive events. We focus on muons in this work given that the LHC experiments are highly efficient in their detection in hadronic collisions [62,63]. In this case, one has to apply proper kinematical cuts to reduce the contamination from the inclusive background -i.e. Drell-Yan dimuon productionin order to enhance the signal region. Thus, for the current data-taking at the LHC, we focus this investigation on pp collisions at 13 TeV, selecting the events by applying a set of kinematic cuts on the single muons, such as η(µ ± ) < 2.5 and p ⊥ (µ ± ) > 0.5 GeV for the J/Ψ, whereas for the Υ(1S) we apply η(µ ± ) < 2.5 and p ⊥ (µ ± ) > 4.0 GeV. Considering that the continuum background from the exclusive two-photon γγ → µ + µ − production plays an important role in this kinematic range, we produce an event sample for this background and include it along with the predictions for the vector meson photoproduction to simulate a more realistic scenario. Together with the previous kinematic cuts, it is possible to improve the event selection by applying additional cuts on the dimuons to suppress the contribution from the continuum (non-resonant) background, like ∆p ⊥ (µ + µ − ) > 0.05 GeV, ∆φ(µ + µ − ) > 0.01, and p ⊥ (µ + µ − ) > 0.04 GeV for the J/Ψ, and ∆p ⊥ (µ + µ − ) > 0.05 GeV, ∆φ(µ + µ − ) > 0.01, and p ⊥ (µ + µ − ) > 0.12 GeV for the Υ(1S) photoproduction. In Fig. 4 we present our predictions for the transverse momentum distributions of the dimuons in Fig. 4, with a distribution centred around p ⊥ (µ + µ − ) ≃ 0.3 GeV. Given that the two-photon production has its largest contribution from back-to-back muons, it has a steep fall from low values of dimuon transverse momentum [1,2]. On the other hand, the vector meson photoproduction shows a distribution around larger values of p ⊥ (µ + µ − ) due to the Pomeron exchange in the t-channel, and then the continuum background can be easily suppressed with kinematic cuts. Figure 4 also shows the acoplanarity distributions of the dimuons at 13 TeV including the continuum background. It is clear that the difference between the default and bCGC predictions is more pronounced in the J/Ψ case than in the Υ(1S) one. Moreover, even the background from the two-photon production being more significant at low acoplanarity, the kinematical cuts applied can minimize its contribution in the J/Ψ case, favouring a kinematic region for the comparison of the two models. Nevertheless, this region is totally contaminated in the Υ(1S) case. Another kinematic variable of interest is ∆p ⊥ (µ + µ − ), which measures the transverse momentum balance in the event, as shown in bottom panel of Fig. 4. Once again, the J/Ψ photoproduction may provide the best scenario for a comparison of the phenomenological models, while they are similar for the Υ(1S) photoproduction. In both ranges, the contamination from inclusive background (such as Drell-Yan dimuon and resonant background) would be small or negligible, as shown in the reported analyses from the LHCb experiment [10]. Then, the signal-to-background (S/B) ratio obtained within these set of kinematic cuts results in around 21 for the J/Ψ photoproduction at the LHC at 13 TeV, showing the possibility of such study with the LHC data and comparison of the phenomenological models. Although the Υ(1S) photoproduction can be also studied, the S/B ratio goes down to 6. The same study have been performed for the FCC energy regime, where the predictions show a more favourable scenario for a comparison of the phenomenological models. Figure 5 shows the kinematics distributions of the muon pairs in both J/Ψ and Υ(1S) photoproduction at 100 TeV. \| π )/ - µ + µ ( φ ∆ 1-\| 0 0.1 0.2 0.3 ) [0.01] - µ + µ ( φ ∆ dN/d 0 1000 2000 at 13TeV ψ J/ \| π )/ - µ + µ ( φ ∆ 1-\| 0 0.1 0.2 0.3 ) [0.01] - µ + µ ( φ ∆ dN/d It is clear that the J/Ψ is the best observable that allows a comparison among the theoretical predictions with the Υ(1S) with nearly the same behaviour as in the LHC energy regime. Finally, the S/B ratio is around twice larger than the results at 13 TeV, showing that the future data coming from FCC will allow a better tuning of the phenomenological models in comparison with the experimental data. IV. SUMMARY The exclusive vector meson photoproduction in pp collisions is an important probe of the QCD dynamics at high energies. Recent studies at Tevatron, RHIC and LHC have demonstrated that the experimental analysis of this process is feasible and that data can be used to constrain several aspects associated to its theoretical description. Such situation should be improved in a near future with the installation of forward detectors. In this paper we have analysed the exclusive J/Ψ and Υ photoproduction in pp collisions at the energies of the Run-II of LHC and presented predictions for the FCC for the first time. Our main emphasis was in the final state distributions that can be measured by the experimentalists. In order to estimated these distributions we have used the SuperCHIC2 tuned by the LHCb data. Moreover, we have modified the energy dependence of the photon-hadron cross section in order to take into account the nonlinear effects in the QCD dynamics, as described by the bCGC model, and obtain more realistic predictions for higher energies that those probed in the Run-I. The impact of the modelling of the energy dependence of the γp → V p cross section have been investigated. Finally, several final state distributions for the dimuons generated from the decay of the vector mesons were estimated considering realistic kinematic cuts and compared with the background associated to the exclusive γγ → µ + µ − production. Our predictions can be directly compared with the Run-II data and shed light on the exclusive vector meson photoproduction at the FCC. of the J/ψ meson at HERA, resulting in N Ψ = (3.97 ± 0.05) nb and δ Ψ = 0.67 ± 0.03. The particular case of the Υ(1S) meson photoproduction is less accurate in the HERA energy regime, providing only a few data points with large uncertainties. In Ref.[39], the authors have included the recent LHCb data in the fitting and have obtained N Υ = 5.7 pb and δ Υ = 0.7. These values largely differ from previous values derived fitting only the HERA data, which were N Υ = 0.12 pb and δ Υ = 1.6. As indicated in[39], the resulting predictions for √ s = 13 TeV contain a 50% uncertainty due to the error in the extracted parameters of the power -law fit. Finally, it is important to emphasise that the SuperCHIC2 assumes that the photoproduction cross section is given by the Eq. (8) to produce event samples which can analysed using the ROOT data analysis framework[52], with the default values for N V and δ V being (N Ψ , δ Ψ ) = (3.97 nb, 0.67) and (N Υ , δ Υ ) = (5.7 pb, 0.7). FIG. 1 : 1set of parameters: (N Ψ , δ Ψ ) = (10.25 nb, 0.49) and (N Υ , δ Υ ) = (3.85 pb, 0.76). It allows to use SuperCHIC2 to extend the analyses performed in Refs. [29, 30] and obtain, e.g. the Rapidity distribution for the exclusive J/Ψ photoproduction in pp collisions at √ s = 13 (left panel) and 100 TeV (right panel) considering different models for the survival probability. The prediction derived disregarding the extra soft reinteractions between the incident protons is represented by the solid line. FIG. 2 : 2we estimate the impact of the extra soft reinteractions in the rapidity distributions for the J/Ψ production at √ s = 13 and 100 TeV Predictions for the energy dependence of the total cross sections for the photoproduction of J/ψ (left panel) and Υ (right panel) in pp collisions considering the default values of the SuperCHIC2 and thoseassociated to the bCGC model. For comparison, we also present the predictions for Υ production obtained with the fitting parameters originally extracted from the HERA data. we present our predictions for the energy dependence of the total cross sections for the photoproduction of J/Ψ (left panel) and Υ (right panel) in pp collisions considering the default values of the SuperCHIC2 and those associated to the bCGC model. For comparison, FIG. 3 : 3Rapidity (upper panels) and transverse momentum (lower panels) distributions at LHC and FCC energy regimes for the J/Ψ (left) and Υ (right) photoproduction. the vector mesons, as shown in the bottom panel of Fig. 3. The differences are more significant at 100 TeV, especially for the J/Ψ meson. Another observation in these results is the slightly shift of the transverse momentum distributions towards smaller values of p V ⊥ when the energy increases, which is expected from Eq. (10). FIG. 4 : 4Final-state distributions of the dimuons from the decay of the J/Ψ (left panels) and Υ(1S) (right panels) in pp collision at √ s = 13 TeV. FIG. 5 : 5Final state distributions of the dimuons from the decay of the J/Ψ (left panels) and Υ(1S) (right panels) in pp collision at √ s = 100 TeV. TABLE I : IExclusive vector meson photoproduction cross sections in the µ + µ − decay channels for pp collision energies of the LHC and the FCC.smaller values. Such aspect is also expected, since at larger energies the contribution of the nonlinear effects increases, modifying the energy dependence of the γp cross section. In the case of the Υ production, one have that if the old parameters, derived fitting the HERA data, are used in the calculations, the resulting predictions for the total cross section is one order of magnitude larger than those obtained with the parameters derived by the fitting of the HERA and LHCb data, simultaneously, that are the default parameters of the SuperCHIC2. Besides, we have that the default and bCGC predictions are similar at LHC energies and differ by ≈20% at the FCC,7. - 3.90 nb 4.58 nb 13. - 5.87 nb 6.43 nb 100. - 22.42 nb 18.65 nb Υ(1S) → µ + µ − 7. 23.91 pb 3.13 pb 3.02 pb 13. 64.34 pb 4.95 pb 4.92 pb 100. 1682.72 pb 21.50 pb 23.91 pb AcknowledgmentsThis research was supported by CNPq, CAPES and FAPERGS, Brazil. . S Chatrchyan, CMS CollaborationJHEP. 0152S. Chatrchyan et al. 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[ "Graphs that contain multiply transitive matchings", "Graphs that contain multiply transitive matchings" ]	[ "Alex Schaefer ", "Eric Swartz " ]	[]	[]	Let Γ be a finite, undirected, connected, simple graph. We say that a matching M is a permutable m-matching if M contains m edges and the subgroup of Aut(Γ) that fixes the matching M setwise allows the edges of M to be permuted in any fashion. A matching M is 2-transitive if the setwise stabilizer of M in Aut(Γ) can map any ordered pair of distinct edges of M to any other ordered pair of distinct edges of M. We provide constructions of graphs with a permutable matching; we show that, if Γ is an arc-transitive graph that contains a permutable m-matching for m 4, then the degree of Γ is at least m; and, when m is sufficiently large, we characterize the locally primitive, arc-transitive graphs of degree m that contain a permutable m-matching. Finally, we classify the graphs that have a 2-transitive perfect matching and also classify graphs that have a permutable perfect matching.2010 Mathematics Subject Classification. Primary 05C25, 20B25, 05C22.	10.1016/j.ejc.2020.103236	[ "https://arxiv.org/pdf/1706.08964v2.pdf" ]	119,255,912	1706.08964	541fd6b61e77264a86bee542a6f5f2d396d24654	Graphs that contain multiply transitive matchings 31 May 2018 Alex Schaefer Eric Swartz Graphs that contain multiply transitive matchings 31 May 2018arXiv:1706.08964v2 [math.CO] Let Γ be a finite, undirected, connected, simple graph. We say that a matching M is a permutable m-matching if M contains m edges and the subgroup of Aut(Γ) that fixes the matching M setwise allows the edges of M to be permuted in any fashion. A matching M is 2-transitive if the setwise stabilizer of M in Aut(Γ) can map any ordered pair of distinct edges of M to any other ordered pair of distinct edges of M. We provide constructions of graphs with a permutable matching; we show that, if Γ is an arc-transitive graph that contains a permutable m-matching for m 4, then the degree of Γ is at least m; and, when m is sufficiently large, we characterize the locally primitive, arc-transitive graphs of degree m that contain a permutable m-matching. Finally, we classify the graphs that have a 2-transitive perfect matching and also classify graphs that have a permutable perfect matching.2010 Mathematics Subject Classification. Primary 05C25, 20B25, 05C22. Introduction All graphs considered in this paper are finite, undirected, and simple, and are connected unless otherwise stated. A matching M is a set of edges of a graph Γ such that no two are incident with a common vertex. A matching M is a perfect matching of Γ if each vertex of Γ is incident with exactly one edge of M. In other words, a matching M is the edge set of a 1-regular subgraph of Γ, and M is perfect exactly when the 1-regular subgraph is spanning. Let Γ be a graph, let M be a matching in Γ with m edges, and let G be a subgroup of Aut(Γ). We will say that M is a G-permutable m-matching if the action of G on the edge set of M is that of the symmetric group S m , i.e., if G E(M) M ∼ = S m . If such a group G and matching M exist, we will say that the graph Γ contains a permutable m-matching. The concept of a permutable matching is due to Zaslavsky, motivated by a question involving signed graphs from [13]. A group G of permutations of a set Ω is 2-transitive on Ω if, given two ordered pairs of distinct elements (α, β), (γ, δ) ∈ Ω × Ω, there exists g ∈ G such that (α, β) g := (α g , β g ) = (γ, δ); in other words, G can map any ordered pair of distinct elements to any other ordered pair of distinct elements. We say that a matching M of a graph Γ is a 2-transitive matching if the setwise stabilizer of M in Aut(Γ) is 2-transitive on the edges of M. The purpose of this paper is to study graphs that contain a matching M such that the setwise stabilizer of M is multiply transitive on the edges of M. This paper is structured as follows. In Section 2, we provide background information necessary for the later sections. In Section 3, we provide various constructions for graphs with a permutable m-matching, showing that there is actually an abundance of such graphs for any m. Moreover, there are even numerous examples when the graph Γ is required to be G-arc-transitive for G Aut(Γ), that is, when G is transitive on the set A(Γ) of ordered pairs of adjacent vertices. In Section 4, we prove the following result, which shows that the degree of a vertex cannot be too small in a graph with a permutable matching, up to a single, known family of exceptions. THEOREM 1.1. If Γ is a connected G-arc-transitive graph with a G-permutable m-matching, then the degree of the graph Γ is at least m unless m = 3 and Γ is the cycle C 3k , where k 2. Many of the graphs with permutable matchings constructed in Section 3 contain a system of imprimitivity, i.e., the full automorphism group of the graph preserves a nontrivial partition of the vertex set (and, in some cases, the stabilizer of a vertex α even preserves a nontrivial partition of the neighbors of α). If a group G of permutations of a set Ω is transitive on Ω but G does not preserve any partition of Ω other than the trivial partitions of Ω into singleton sets and the single set Ω, then G is primitive on Ω. Given a graph Γ and G Aut(Γ), Γ is said to be G-locally primitive if, given any α ∈ V (Γ), the stabilizer of α in G is primitive on the neighbors of α. Given the constructions in Section 3 and Theorem 1.1, it makes sense to consider graphs with degree m that are locally primitive and arc-transitive containing a permutable m-matching. In Section 5, we provide a characterization of such graphs. The notation and terminology used in the following theorem are explained in depth in Section 2. THEOREM 1.2. Let Γ be a connected G-arc-transitive, G-locally primitive graph with degree m 6 that contains a G-permutable m-matching, and suppose G has a nontrivial normal subgroup N that has more than two orbits on vertices. If the normal quotient graph Γ N does not contain a permutable m-matching, then Γ N is a near-polygonal graph and (Γ N , G/N) is locally-S m . A group G is said to be quasiprimitive on a set Ω if every nontrivial normal subgroup of G is transitive on Ω, and a group G is said to be biquasiprimitive on a set Ω if if Ω has a G-invariant partition Ω = ∆ 1 ∪ ∆ 2 such that the setwise stabilizer G ∆ i is quasiprimitive on ∆ i for i = 1, 2. Using this terminology, Theorem 1.2 says that, if there exists a graph Γ that is G-arc-transitive and G-locally primitive with degree m 6 that contains a G-permutable m-matching, then one can keep taking normal quotients of this graph until reaching either (1) a vertex-quasiprimitive graph with a permutable m-matching, (2) a vertex-biquasiprimitive graph with a permutable m-matching, or (3) a near-polygonal graph such that the stabilizer of a vertex can permute the m neighbors in any way; see Section 2. Moreover, graphs in each case exist and are constructed in Section 3. We do not know if the theorem holds for m ≤ 5; the restriction on m is a result of the technique. Section 6 is devoted to the proof of the following theorem, which classifies the graphs with a 2-transitive perfect matching. Joins and matching joins are defined following the statement of the theorem. THEOREM 1.3. Let Γ be a connected graph on 2m vertices with a 2-transitive perfect matching M containing m edges. Then Γ is one of the following: (1) A join between two graphs that are either complete or edgeless: (a) K m ∨ K m ∼ = K 2m , (b) K m ∨ K m , (c) K m ∨ K m ∼ = K m,m . (2) A matching join between two graphs that are either complete or edgeless (but not both edgeless): (a) K m ⊻ K m , (b) K m ⊻ K m . (3) Let m = p f , where p is a prime and p f ≡ 3 (mod 4). Then either: (a) Γ is the incidence graph of the Paley symmetric 2-design over GF(p f ), i.e., V (Γ) = GF(p f ) × {0, 1}, and (x, i), (y, j) ∈ V (Γ) are adjacent if and only if i = 0, j = 1, and y − x is a square in GF(p f ); or (b) Γ is the graph obtained by taking the incidence graph of the Paley symmetric 2design over GF(p f ) and replacing the independent sets with copies of K p f ; that is V (Γ) = GF(p f ) × {0, 1}, and (x, i), (y, j) ∈ V (Γ) are adjacent if and only if either i = j and x = y or if i = 0, j = 1, and y − x is a square in GF(p f ). (4) Let m = 5. Then either (a) Γ is the Petersen graph; or (b) Γ = C 5 ∨ C 5 . Here, Γ 1 ∨ Γ 2 denotes the join of the graphs Γ 1 and Γ 2 , in which V (Γ 1 ∨ Γ 2 ) = V (Γ 1 ) ∪ V (Γ 2 ) and E(Γ 1 ∨ Γ 2 ) = E(Γ 1 ) ∪ E(Γ 2 ) ∪ {{α, β} : α ∈ V (Γ 1 ), β ∈ V (Γ 2 )}. The notation Γ 1 ⊻ φ Γ 2 denotes a matching join of Γ 1 and Γ 2 . In this case, both Γ 1 and Γ 2 must be graphs with \|V (Γ 1 )\| = \|V (Γ 2 )\| and φ : V (Γ 1 ) → V (Γ 2 ) is a bijection between the vertex sets. The graph Γ 1 ⊻ φ Γ 2 is defined to have vertex set V (Γ 1 ⊻ φ Γ 2 ) = V (Γ 1 ) ∪ V (Γ 2 ) and the edge set is E(Γ 1 ⊻ φ Γ 2 ) = E(Γ 1 ) ∪ E(Γ 2 ) ∪ {{α, α φ } : α ∈ V (Γ 1 )}. When Γ 1 or Γ 2 is a complete graph or an empty graph, then the resulting graph is unique up to isomorphism regardless of the choice of φ, and in this case we simply use the notation Γ 1 ⊻ Γ 2 . As an example of a matching join, consider two copies of C 5 : Γ 1 = {1, 2, 3, 4, 5} with x adjacent to y if and only if x − y ≡ ±1 (mod 5) and Γ 2 = {6, 7, 8, 9, 10} with, again, x adjacent to y if and only if x − y ≡ ±1 (mod 5). If we define φ to be x φ = x + 5, then the matching join Γ 1 ⊻ φ Γ 2 is isomorphic to the 5-prism, whereas if we define φ : Γ 1 → Γ 2 by 1 φ = 6, 2 φ = 9, 3 φ = 7, 4 φ = 10, and 5 φ = 8, then the matching join Γ 1 ⊻ φ Γ 2 is isomorphic to the Petersen graph. As a corollary of Theorem 1.3 we classify all connected graphs with a permutable perfect matching. COROLLARY 1.4. Let Γ be a connected graph on 2m vertices with a permutable perfect matching M. Then Γ is one of K 2m , K m ∨K m , K m,m , K m ⊻K m , K m ⊻K m , C 6 , or K 6 \{3·K 2 } ∼ = K 2,2,2 . In particular, Theorem 1.3 classifies the possible induced subgraphs on the vertex set of a 2transitive matching M of size m in an arbitrary graph: either the induced subgraph is disconnected and is m · K 2 (i.e., m vertex-disjoint edges) or it is connected and is one of the graphs listed in Theorem 1.3. Moreover, the induced subgraph on the vertex set of a permutable m-matching M in an arbitrary graph is either m · K 2 or one of the graphs listed in Corollary 1.4. Background In this section we review the terminology and theory that will be used in later sections. Let Γ be a graph. Given a subset X of the vertices of Γ, the induced subgraph of Γ on X is denoted by Γ[X]. We denote that the vertices α and β are adjacent by writing α ∼ β. We denote by Γ the complement of Γ. A walk W is defined to be a sequence of vertices (α 0 , α 1 , . . . , α n ) such that α i ∼ α i+1 for 1 i n − 1. For α ∈ V (Γ), we denote the set of neighbors of α in Γ by Γ(α). The degree of a vertex α is \|Γ(α)\|, and we say that the graph Γ is regular if every vertex has the same degree. A graph Γ is said to be (v, k, λ, µ)-strongly regular if Γ has v vertices; Γ is regular of degree k; if α, β ∈ V (Γ) and α ∼ β, then \|Γ(α) ∩ Γ(β)\| = λ; and if α, β ∈ V (Γ) and α ∼ β, then \|Γ(α) ∩ Γ(β)\| = µ. 2.1. Permutation groups and graph symmetry. Let Ω be a set and G a group of permutations of Ω, that is, let G Sym(Ω). For an element ω ∈ Ω, the orbit of ω under G is denoted by ω G . For a subset ∆ of Ω, we let G ∆ denote the setwise stabilizer of ∆ in G. When ∆ = {ω}, a single element of Ω, we write G ω := G {ω} . If ∆ = {ω 1 , ω 2 , . . . , ω k }, then G ω 1 ω 2 ...ω k := k i=1 G ω i , that is, G ω 1 ω 2 ...ω k fixes every ω i . For instance, G α∆ denotes G {α} ∩ G ∆ , i.e. the set of elements of G which stabilize both the element α pointwise and the set ∆ setwise. If H G ∆ , then we denote by H ∆ the induced action of H on ∆, i.e., H ∆ is the image of the natural homomorphism from H into Sym(∆). If G 1 is a group of permutations of Ω 1 and G 2 is a group of permutations of Ω 2 , then G 1 and G 2 are said to be permutation isomorphic if there are both a bijection ψ : Ω 1 → Ω 2 and a group isomorphism φ : G 1 → G 2 such that, for all g ∈ G 1 and ω ∈ Ω 1 , (ω g ) ψ = (ω ψ ) g φ . The group of permutations G is said to be transitive on Ω if, for every α, β ∈ Ω, there exists g ∈ G such that α g = β. A group G of permutations of a set Ω is said to be regular on Ω if G is transitive on Ω and G ω = 1 for all ω ∈ Ω. Additionally, G is said to be primitive on Ω is G is transitive on Ω and G preserves no nontrivial partition of Ω, that is, G preserves no partition of Ω other than the partition into singleton sets and the partition into the single set Ω. If Π is a nontrivial G-invariant partition of Ω, then Π is called a system of imprimitivity and the elements of Π are called blocks. Finally, a group G is said to be biprimitive on Ω if Ω has a G-invariant partition Ω = ∆ 1 ∪ ∆ 2 such that the setwise stabilizer G ∆ i is primitive on ∆ i for i = 1, 2. The group of permutations G is said to be quasiprimitive on the set Ω if every nontrivial normal subgroup of G is transitive on Ω. If G is primitive on Ω, then G is quasiprimitive on Ω; however, the converse is not true. A group G is said to be biquasiprimitive on Ω if Ω has a G-invariant partition Ω = ∆ 1 ∪ ∆ 2 such that the setwise stabilizer G ∆ i is quasiprimitive on ∆ i for i = 1, 2. Let Γ be a graph with vertex set V (Γ) and edge set E(Γ). An automorphism of a graph Γ is a permutation of the vertices that preserves adjacency. The set of automorphisms of Γ forms a group, which is denoted by Aut(Γ). Note that Aut(Γ) Sym(V (Γ)). Let G Aut(Γ). The graph Γ is G-vertex-transitive if G is transitive on the vertices of Γ, and Γ is G-edge-transitive if G is transitive on edges. Similarly, the graph Γ is G-vertex-quasiprimitive (respectively, G-vertex-biquasiprimitive) if G is quasiprimitive (respectively, biquasiprimitive) on the vertices of Γ. An arc is an ordered pair of vertices (α, β) such that {α, β} ∈ E(Γ), and Γ is G-arc-transitive if G is transitive on the set A(Γ) of arcs of Γ. In general, an s-arc of Γ is an ordered (s + 1)-tuple of vertices (α 0 , . . . , α s ) such that {α i , α i+1 } ∈ E(Γ) for 0 i s − 1 and α j−1 = α j+1 for 1 j s − 1. (Repeated vertices are allowed in the walk defined by the s-arc, but there are no returns in the walk.) The graph Γ is said to be (G, s)-arc-transitive if G is transitive on the set of s-arcs of Γ. Given vertices α, β of Γ, we define the distance between α and β to be the length of a shortest path between α and β (measured in edges), and we denote the distance between α and β by d(α, β). Since we are only considering connected graphs, there will always exist a path between any two vertices α and β, so distance is a well-defined, finite-valued function on pairs of vertices. Given a fixed vertex α, for every natural number i we let G [i] α := {g ∈ G α : β g = β for all β ∈ V (Γ) such that d(α, β) i}, that is, G [i] α is the group that fixes pointwise the set of all vertices at distance at most i from α. In particular, G [1] α = {g ∈ G α : β g = β for all β ∈ Γ(α)}, and G [1] α is often referred to as the kernel of the local action of G since, for the induced action G Γ(α) α of the vertex stabilizer G α on the neighbors of α, we have G Γ(α) α ∼ = G α /G [1] α . Finally, for vertices α 1 , α 2 , . . . , α k , we define G [1] α 1 ...α k := k i=1 G [1] α i , that is, G [1] α 1 ...α k is the pointwise stabilizer of the union of the Γ(α i ). Given a permutation group L, a graph Γ, α ∈ V (Γ), and G Aut( Γ) such that Γ is G-vertex- transitive, the pair (Γ, G) is said to be locally-L if G Γ(α) α is permutation isomorphic to L. The graph Γ is said to be G-locally primitive if G Γ(α) α is primitive on Γ(α). 2.2. Quotient graphs, voltage graphs, and regular covers. Let Γ be a graph with a group of automorphisms G, and let Π be a G-invariant partition of the vertices of Γ. We may define the quotient graph Γ Π with respect to the partition Π to be the graph that has as its vertex set the blocks of the partition Π with an edge between blocks B 1 and B 2 if and only if there exist vertices α ∈ B 1 and β ∈ B 2 such that α is adjacent to β in Γ. The original graph Γ is said to be a regular cover of Γ Π if each block B 2 adjacent to B 1 contains exactly one vertex adjacent to α in Γ for each α ∈ B 1 . Let Γ be a graph with group of automorphisms G, and let N be an intransitive normal subgroup of G. The N-orbits of vertices of Γ form a system of imprimitivity for G, and the normal quotient graph Γ N with respect to the normal subgroup N is the quotient graph with respect to this system of imprimitivity. The following lemma is a well-known result, and it shows that local primitivity is a sufficient condition for the original graph to be a regular cover of the normal quotient graph. LEMMA 2.1. [12,Theorem 10.4] Let Γ be a G-vertex-transitive and G-locally primitive graph, where G Aut(Γ), and let N be a normal subgroup of G with more than two orbits on V (Γ). Then Γ is a regular cover of the quotient graph Γ N , and the quotient graph Γ N is G/N-vertex-transitive and G/N-locally primitive. An equivalent definition of a regular cover is as follows. A covering projection p :Γ → Γ maps V (Γ) onto V (Γ), preserving adjacency, such that for any vertexα ∈ V (Γ), the set of neighbors of α is mapped bijectively onto the set of neighbors ofα p . For a vertex α of Γ, the set α p −1 of vertices that are mapped onto α by p is called the fiber over the vertex α. An automorphism g ∈ Aut(Γ) lifts tog ∈ Aut(Γ) if the following diagram commutes: ΓΓ Γ Γ g p p g The lift of the trivial group (identity) is known as the group of covering transformations and is denoted CT(p). The graphΓ is a regular cover of Γ if CT(p) acts regularly on the set α p −1 for all vertices α ∈ V (Γ). A voltage assignment is a map ξ : A(Γ) → H, where H is a group, such that (α, β) ξ = (β, α) ξ −1 . For ease of notation, the voltage of the arc (α, β) will be denoted ξ αβ , and ξ W will denote the total voltage of a walk W , that is, ξ W is the product (or sum, depending on the group operation) of the voltages of the edges in W . The derived covering graphΓ of a voltage graph has vertex set V (Γ) × H, where two vertices (α, h 1 ) and (β, h 2 ) are adjacent iff α is adjacent to β in Γ and h 2 = ξ αβ h 1 . The following theorem exhibits the deep connection between regular covers and derived covering graphs: . Every regular coverΓ of a graph Γ is a derived cover of a voltage graph (and conversely). In addition, suppose the voltage group is generated by the voltages on the edges of Γ. If the edges of a (fixed but arbitrary) spanning tree of Γ are assigned the identity voltage, thenΓ is connected. Fix a spanning tree T of a graph Γ. Choose α ∈ V (Γ), and assume that the edges of T have been assigned the identity voltage. This implies that the voltage assignment ξ induces a natural homomorphism of the fundamental group of Γ based at α (generated by all closed walks in Γ based at α) into the voltage group H. Let g ∈ Aut(Γ). For each closed walk W based at α, W g will be a closed walk based at α g . Moreover, the walk formed by the path in T from α to α g , followed by W g , followed by the path in T from α g back to α, is a closed walk based at α with the same voltage as W g . This induces a multivalued function g φα : H → H given by (ξ W ) g φα := ξ W g . This is not necessarily well-defined, as two walks W 1 and W 2 may have the same voltage while W g 1 and W g 2 may not. Furthermore, g φα may not be defined on all of H. With this in mind, the following lemma gives explicit criteria for an automorphism of a graph to lift. The following lemma also shows that it is quite possible to get the entire automorphism group of a graph to lift. ; H is a Z p -vector space. Let X be a basis for H, so \|X\| = \|E\| − \|T \|. Define Γ p to be the derived regular cover of the voltage graph defined by assigning a distinct element of X to each co-tree edge of Γ. Then Γ p is well-defined, unique up to graph isomorphism, and Aut(Γ) lifts. 2.3. Near-polygonal graphs. Following [11], we say that Γ is a near-polygonal graph if there exists a distinguished set of c-cycles C such that every 2-path of Γ is contained in a unique cycle in C. If c is the girth of Γ, then Γ is called a polygonal graph. Furthermore, if our collection C of c-cycles is in fact the set of all cycles of length girth(Γ), then Γ is called strict polygonal. Manley Perkel invented the notion of a polygonal graph in [10] and that of a near-polygonal graph in [11]. (Perkel's original definition of near-polygonal graphs required that the length c of the special cycles be greater than 3. In our definition, we allow c = 3.) Polygonal graphs are a natural generalization of the edge-and vertex-set of polygons and Platonic solids, and one immediately notes that these are themselves strict polygonal graphs, with the special set of cycles being the polygon itself or the faces of the solid, respectively. The complete graph on n points, K n , is a strict polygonal graph of girth 3, and the Petersen graph is a polygonal graph of girth 5 that is not a strict polygonal graph [14]. Very few examples of polygonal graphs are known; see [15,17,18]. Near-polygonal graphs have appeared in the past when studying quotient graphs of symmetric graphs [21,22]. We mention here the following result, which gives a sufficient condition for a graph Γ to be near-polygonal: LEMMA 2.5 ([23, Theorem 1]). Suppose that Γ is a connected (G, 2)-arc-transitive graph, where G Aut(Γ) . Let (α, β, γ) be a 2-arc of Γ and define H := G αβγ . Then the following are equivalent: (i) there exist both an integer c 3 and a G-orbit C on c-cycles of Γ such that Γ is a nearpolygonal graph with set of distinguished cycles C; (ii) H fixes at least one vertex in Γ(γ)\{β}; (iii) there exists g ∈ N G (H) such that (α, β) g = (β, γ). Constructions of graphs with a permutable matching In this section, we provide some constructions of graphs with permutable matchings. We begin with a construction that shows that, for any m 2, there are graphs that are neither edge-nor even vertex-transitive that contain a permutable m-matching. It is not difficult to see that QΓ contains a permutable m-matching. In particular, if Γ = K 1,m , then Aut(QΓ) ∼ = S m and that Aut(QΓ) has three orbits on vertices and two orbits on edges; the orbit of edges that do not all share a common endpoint is a permutable m-matching. Obviously, it is possible to construct other such examples; we mention another couple here. The graphs produced from these constructions may have less symmetry than the original graphs; for instance, these constructions may take vertex-, edge-, or arc-transitive graphs and produce graphs that are not vertex-, edge-, or arc-transitive. For this reason, we will henceforth restrict ourselves to graphs Γ containing a G-permutable matching that are also G-arc-transitive. PROOF. We take G = Aut(K m,m ) ∼ = S m wr S 2 . The group G preserves the partition of the vertices into two sets of size m, and any matching will be a permutable m-matching. Inspired by the example of complete bipartite graphs, the following construction demonstrates that it is quite easy to construct arc-transitive graphs with permutable matchings for any m: CONSTRUCTION 3.5. Let Γ be an arc-transitive graph with automorphism group H and let m be any fixed natural number. Define Γ(m) as the graph composition of Γ with K m : that is, V (Γ(m)) = {(η, i) : η ∈ V (Γ), 1 i m}, with (η, i) adjacent to (θ, j) if and only if η is adjacent to θ in Γ. If Γ(m) is constructed from an H-arc-transitive graph Γ as in Construction 3.5 with H = Aut(Γ), then Aut(Γ(m)) ∼ = S m wr H. For G := Aut(Γ(m)), Γ(m) is G-arc-transitive, and, for any edge {α, β} in Γ, the set M := {{(α, i), (β, i)} : 1 i m} is a G-permutable m-matching of Γ(m). Another construction which yields infinitely many such graphs from a G-arc-transitive graph Γ with a G-permutable m-matching is the following. ; H is a Z p -vector space. Let X be a basis for H, so \|X\| = \|E\| − \|T \|. Define Γ p to be the derived regular cover of the voltage graph defined by assigning a distinct element of X to each co-tree edge of Γ. CONSTRUCTION 3.6. Let Γ be a G-arc-transitive graph with a G-permutable m-matching M = {(α i , β i ) : 1 i m}. Let E denote By Lemma 2.4, if Γ is a G-arc-transitive graph with G-permutable m-matching M = {{α i , β i } : 1 i m}, then G lifts to a group G of automorphisms of Γ p , and it follows that M p := {{(α i , 1), (β i , 1)} : 1 i m} is itself a G-permutable m-matching of Γ p . What last these two constructions have in common is that the graphs that are produced are not quasiprimitive on vertices: in each case, the full automorphism group of the graph produced contains an intransitive normal subgroup. Moreover, the graphs produced by Construction 3.5 are always locally imprimitive. It makes sense, then, to study the G-arc-transitive graphs that have Gpermutable matchings that are G-vertex-quasiprimitive or G-vertex-biquasiprimitive. Indeed, such graphs exist. The odd graph O n has one vertex for each of the (n−1)-element subsets of a (2n−1)element set, and vertices are adjacent if and only if the corresponding subsets are disjoint. As the following result shows, there is at least one vertex-quasiprimitive (and, in fact, vertex-primitive) graph with a permutable m-matching for every m 3. (O m ) such that G ∼ = S 2m−1 , O m is G-vertex-primitive, and O m contains a G-permutable m- matching. PROOF. We identify the vertices of O m with subsets of size m − 1 of {1, 2, . . . , 2m − 1}. Then S 2m−1 is primitive on the sets of size m − 1: the stabilizer of each subset is isomorphic to S m−1 × S m , a maximal subgroup of S 2m−1 which is core-free (that is, the intersection of all conjugates of the subgroup is trivial; see [1]). Hence there is G Aut . . , m}) G. We note that H ∼ = S m and H stabilizes M setwise but allows the edges of M to be permuted as we please. Therefore, M is H-permutable, so M is G-permutable, as desired. (O m ) such that G ∼ = S 2m−1 and O m is G-vertex-primitive. One might expect that if Γ has a group of automorphisms G such that (i) Γ has a G-permutable m-matching, (ii) G has a nontrivial normal subgroup N that is intransitive on vertices, and (iii) Γ does not have an induced subgraph isomorphic to K m,m (i.e., if Γ does not arise from Construction 3.5), then the normal quotient graph Γ N should also have a permutable m-matching. However, as the following construction shows, more exotic examples can arise. where the action is induced on a generating set of all closed walks based at the vertex α. Since Γ is near-polygonal and (G, 2)-arc-transitive, each 2-arc (β i , α, β j ) is contained in a unique cycle C i,j , and G α is transitive on these cycles. Define h i to be the voltage of the walk W i , where W i is the concatenation of all cycles C i,j such that j = i. If g ∈ G α and β g i = β j , then the induced action of g on H sends h i to h j . If ξ i is the voltage of the arc (α, β i ), then the matching {{(α, h i ), (β i , ξ i + h i )} : 1 i m} is G-permutable.G such that (i) Γ is (G, 2)-arc-transitive, (ii) (Γ, G) is locally-S m ,(iii) Γ contains a G-permutable m-matching, and (iv) G has a nontrivial normal subgroup N that has more than two orbits on V (Γ), yet Γ N does not contain a G-permutable m-matching. PROOF. For each m 3, we can take Γ Π to be K m+1 , the m-dimensional hypercube Q m , or the folded m-dimensional hypercube, each of which satisfies the hypotheses of Construction 3.8. The result follows from Proposition 3.9. The local structure of graphs with a permutable matching In this section, we prove results about the local structure of a G-arc-transitive graph with a G-permutable m-matching, that is, we prove results about the stabilizer of a vertex and the size of the neighborhood of a vertex in such a graph. This first result, which has a similar proof to that of [16,Theorem 1.1], provides information about the edge stabilizer of an arc-transitive graph with a permutable m-matching when m is large enough. Now, consider the subgroup G αβ of G e . We have G αβ ⊳ G e since it has index at most two, and so G [1] αβ M ⊳ G αβM ⊳ G eM . Let P = (γ 0 = α, γ 1 = β, γ 2 , . . . , γ n ) be a path in Γ such that G [1] αβγ 2 ...γn = 1. Hence 1 = G [1] αβγ 2 ...γn M · · · G [1] αβγ 2 M G [1] αβ M G αβM . Since A m−1 is a composition factor of G eM and G αβ has index at most two in G e , A m−1 must be a composition factor of G αβM , so there are subgroups N, N 1 of G αβM such that N 1 ⊳N ⊳G αβM and N/N 1 ∼ = A m−1 . If N is not a subgroup of either G [1] α M or G [1] β M , then A m−1 is a composition factor of G Γ(α) αβM or G Γ(β) αβM , and we are done. Otherwise, N G [1] αβ M , and so there must exist a least l such that N G αβγ 2 ...γ l−1 M / G [1] αβγ 2 ...γ l M ∼ = G [1] αβγ 2 ...γ l−1 M Γ(γ l ) ⊳ ⊳ G Γ(γ l ) γ l−1 γ l M . Thus A m−1 is a composition factor of G Γ(γ l ) γ l−1 γ l M . Since Γ is G-arc-transitive, G γ l−1 γ l M ∼ = U G αβ , and hence A m−1 is a composition factor of U Γ(α) for some U G αβ , as desired. A consequence of this result is that the degree of a vertex in an arc-transitive graph with an permutable m-matching is at least m when m 6; in fact, we can classify the graphs with degree less than m and a permutable m-matching. PROOF OF THEOREM 1.1. By Proposition 4.1, when m ≥ 6, for an edge {α, β} of Γ there exists U G αβ such that U Γ(α) has a composition factor isomorphic to A m−1 . For m 5, the smallest faithful permutation representation of A m has degree m. Since U fixes β ∈ Γ(α), this implies that \|Γ(α)\| − 1 m − 1. When m = 1 and m = 2, the result is clear since Γ is connected. When m = 3, since the graph is connected and arc-transitive, the degree of the graph is at least two. If the degree of Γ is exactly two, then Γ is a cycle, and the result follows by noting that Γ must have at least six vertices and that Aut(Γ), which is a dihedral group, must have order divisible by three. We are left with the cases m = 4 and m = 5. In either case, if the degree of such a graph Γ is 2, then Γ is a cycle, and Aut(Γ) contains no subgroup isomorphic to S 4 . If the degree of such a graph Γ is 3, then, by a famous result of Tutte [19], the order of a vertex stabilizer divides 48, and hence the order of an edge stabilizer divides (48 * 2)/3 = 32. If m 4, G = Aut(Γ), and the permutable matching is M, then 3 divides the order of the stabilizer of an edge in G, since the stabilizer of an edge of M can permute three other edges of M in any way. Thus there is no graph of degree 3 with a permutable 4-matching. Finally, assume Γ is regular of degree 4 and that M is a G-permutable 5-matching for G = Aut(Γ). Let M = {e 1 , e 2 , e 3 , e 4 , e 5 }, where each e i = {α i , β i }. Consider a shortest path P 2 from a vertex of e 1 to a vertex of e 2 . Without loss of generality, the path is between α 1 and α 2 . Since M is permutable, there are elements g i in G M that fix e 1 and map e 2 to e i , 3 i 5, and so there exist shortest paths from e 1 to e i , where 2 i 5, that are all of the same length. Suppose first that there is no path from α 1 to α i with the same length as P 2 for some i. This means that α g i 1 = β 1 . Consider a fourth edge, e j . If α g j 1 = α 1 , then there is no element in G M fixing e 1 and e 2 and mapping e i to e j , a contradiction. If α g j 1 = β 1 , then there is no element in G M fixing e 1 and e i and mapping e 2 to e j , a contradiction. Thus we may assume that there are paths from α 1 to each other α i with the same length as P 2 , and we denote these paths by P i . For each i, let P i = (α 1 = γ i,0 , γ i,1 , . . . , γ i,n = α i ). Since \|Γ(α 1 )\{β 1 }\| = 3, at least two P i go through the same neighbor of α 1 , say γ = γ 2,1 = γ 3,1 . Consider h ∈ G M such that h acts on M as the permutation (3 4 5). Note that, since the induced action of h on M has order 3, if α h 1 = β 1 , then we could choose h 2 instead, so we may assume that α h 1 = α 1 . There are two cases: either γ h = γ or γ h = γ. Assume first that γ h = γ. This implies that Γ(α 1 ) = {β 1 , γ, γ h , γ h 2 } and that there is a shortest path from α 1 to α 2 through each of γ, γ h , and γ h 2 . However, this implies, by the permutability of M, that there is a shortest path from α 1 to each α i through each of γ, γ h , and γ h 2 . Thus we may choose each P i so that γ i,1 = γ. Moreover, if γ h = γ, then there is a path from α 1 to α i through γ for each i, namely, P ′ 4 := P h 3 goes from α 1 to α 4 and P ′ 5 := P h 2 3 goes from α 1 to α 5 . Hence, in any case we may assume that γ i,1 = γ for all i. However, γ has exactly three neighbors that are not α 1 . We apply a similar argument for the γ i,2 , γ i,3 , etc., and reach a contradiction: either Γ is disconnected or a vertex has degree greater than 4. Therefore, there is no connected graph of degree 4 with a permutable 5-matching, and the result holds. Locally primitive, arc-transitive graphs with degree m and a permutable m-matching Given that a graph with a permutable m-matching has degree at least m when m 4 and given the constructions from Section 3, it makes sense to study arc-transitive, locally primitive graphs of degree m that contain a permutable m-matching. The following results show that such graphs Γ with a group of automorphisms G do have a nice structure with respect to nontrivial normal subgroups N of G such that N is intransitive on vertices. Γ(α)\{β i } αβ i ∼ = G Γ(β i )\{α} αβ i ∼ = S m−1 . Because G αβ 1 has nontrivial layer (that is, the group generated by its subnormal quasisimple groups is nontrivial; see [1]), then, by [20, Theorem 2.12], G [1] αβ 1 = 1. Thus [G [1] α , G [1] β i ] G [1] α ∩ G [1] β i = G [1] αβ i = 1, i.e., for each i, the elements of G [1] α and G [1] β i commute. Since G [1] α ⊳ G αβ i , G [1] α ∼ = G [1] α /(G [1] α ∩ G [1] β i ) ∼ = G [1] α Γ(β i )\{α} ⊳ G Γ(β i )\{α} αβ i . Since the only normal subgroups of S m−1 when m − 1 5 are 1, A m−1 , and S m−1 , we conclude that G [1] α is isomorphic to one of 1, A m−1 , or S m−1 . 5.1.1: The case where G [1] α ∼ = S m−1 . Suppose first that G [1] α ∼ = S m−1 . Define L i := G [1] α G [1] β i ∼ = G [1] α × G [1] β i ∼ = S m−1 × S m−1 . We note that L i G αβ i . Since G [1] αβ i = 1, we have G αβ i /G [1] α ∼ = G Γ(α) αβ i ∼ = S m−1 , and so \|L i \| = \|G αβ i \| and hence G αβ i = L i . Moreover, when m − 1 5, Z(S m−1 ) = 1; hence G [1] β i Γ(β j )\{α} β j = 1. In other words, for any i = j we have G [1] β i β j = G [1] β j β i = G [1] β i ∩ G [1] β j . For each j 2, we have G [1] β 2 ...β j−1 β j+1 ...βm ∼ = S 2 , and so we let G [1] β 2 ...β j−1 β j+1 ...βm = g 1,j . If Γ(α) ∩ Γ(β 1 ) = ∅, since (Γ, G) is locally-S m , then Γ ∼ = K m+1 and G [1] α = 1, a contradiction. Thus we may pick γ 1 ∈ Γ(β 1 )\{α}, and we define γ i := γ α ∼ = A m−1 . Suppose next that G [1] α ∼ = A m−1 . We define L i := G [1] α G [1] β i ∼ = G [1] α × G [1] β i as above, only now L i ∼ = A m−1 × A m−1 . Since G Γ(α) αβ i ∼ = G αβ i /G [1] α ∼ = S m−1 , we have that \|G αβ i : L i \| = 2. As in the last case, G [1] β i β j = G [1] β j β i = G [1] β i β j ∼ = A m−2 . However, in this case, G [1] β 3 ...βm ∼ = A 2 = 1, and so G αβ 3 ...βm = G β 3 ...βm ∼ = G β 3 ...βm /G [1] β 3 ...βm and G {β 1 ,β 2 } αβ 3 ...βm ∼ = G αβ 3 ...βm /G [1] α ∼ = S 2 . Hence we choose g ∈ G αβ 3 ...βm such that β g 1 = β 2 and β g 2 = β 1 . We may also assume that γ g 3 = γ 3 for some γ 3 ∈ Γ(β 3 )\{α}; otherwise, we replace g by gx, where x ∈ G [1] α ; indeed, this in fact shows that we may assume that g acts as a transposition on Γ(β 3 )\{α}. We also remark that G αβ i = L i , g for i 3. Define H := G [1] β i : 1 i m . It is clear that H ⊳ G α , and, since G [1] α ∩ G [1] β i = 1 and G [1] β i β j = G [1] β j β i = G [1] β i ∩ G [1] β j for each i and j, we have H ∼ = H/(G [1] α ∩ H) ∼ = HG [1] α /G [1] α G Γ(α) α . Moreover, H ⊳ G α , so H is isomorphic to a normal subgroup of G Γ(α) α . Since H has a nontrivial action on Γ(α), either H ∼ = A m or H ∼ = S m . As in the previous case, Γ(α) ∩ Γ(β 1 ) = ∅. We claim now that H has m − 1 orbits of size m on D 2 (α) := m i=1 Γ(β i )\{α}. Suppose first that x i , y i ∈ G [1] β i and β x i k = β y i k . This means x i y −1 i ∈ G [1] β i β k G [1] β k . Hence, if γ ∈ Γ(β k ), then γ x i y −1 i = γ and γ x i = γ y i . Now suppose x i ∈ G [1] β i , x j ∈ G [1] β j , and β x i k = β x j k = β l . There exists some r ∈ {1, . . . , m}\{i, j, k, l} and there exist y i ∈ G [1] β i and y j ∈ G [1] β j such that β y i k = β x i k = β y j k = β x j k = β l and β y i r = β y j r = β r . Thus y i , y j ∈ G [1] βr , and, if γ ∈ Γ(β k ), then γ y i = γ y j from what we just proved above. Thus γ x i = γ y i = γ y j = γ x j , and so H has exactly m − 1 orbits of size m on D 2 (α). Now, define X := H, g . Since X is a 2-transitive group on Γ(α) that contains a transposition, X Γ(α) ∼ = S m . Now, X G α and H ⊳ G α , so H is a normal subgroup of X. This implies that the orbits of H on D 2 (α) are an X-invariant partition, which we use to find our matching: indeed, suppose γ H is such an orbit. Then (γ H ) g = (γ g ) H . Select the orbit γ H 3 , where γ 3 ∈ Γ(β 3 ) and γ g 3 = γ 3 as above. Define γ i := γ H 3 ∩ Γ(β i ). This implies that γ X 3 = γ H 3 , and hence X stabilizes M = {{β i , γ i } : 1 i m} setwise and M is an X-permutable m-matching, as desired. 5.1.3: The case where G [1] α ∼ = 1. The final case is when G [1] α = 1. This implies that G α ∼ = S m and G αβ 1 ∼ = S m−1 with a faithful action on each of Γ(α)\{β 1 } and Γ(β 1 )\{α}. Hence G αβ 1 β 2 fixes a vertex in Γ(β 1 )\{α}. Moreover, since Γ has degree m and (Γ, G) is locally-S m , Γ is a (G, 2)-arc-transitive graph. By Lemma 2.5, Γ is near-polygonal, as desired. We can now prove Theorem 1.2, which essentially characterizes arc-transitive, G-locally primitive graphs of degree m with a permutable m-matching. PROOF OF THEOREM 1.2. Suppose that Γ is a G-arc-transitive, G-locally primitive graph with degree m 6 that contains a G-permutable m-matching M such that G contains an intransitive normal subgroup N that has more than two orbits of vertices. Let {α, β} be an edge of M, let A be the N-orbit containing α, and let B 1 be the N-orbit containing β. Since G is edge-transitive and the N-orbits of vertices are G-invariant, all edges of Γ are between N-orbits; that is, if {γ, δ} ∈ E(Γ), then γ, δ are in different N-orbits. Thus A = B 1 . Up to relabeling, there are three possibilities for {γ, δ}, where {γ, δ} is another edge of M: (i) Neither γ nor δ is in either A or B 1 . (ii) γ ∈ A, δ ∈ B 1 . (iii) γ ∈ A, δ ∈ B 1 . 5.2.1: Neither γ nor δ is in either A or B 1 . Since {α, β}, {γ, δ} ∈ M, the four vertices α, β, γ, δ are in distinct N-orbits, and, since M is a G-permutable m-matching, no two vertices of V (M) are in the same N-orbit. We may thus view the action of G M on V (M) as an action on the N-orbits containing the vertices. This means there will be a G/N-permutable m-matching in the quotient graph Γ N , and we are done. 5.2.2: γ ∈ A, δ ∈ B 1 . Here, M is of the form {{α 1 , β 1 } = {α, β}, {α 2 , β 2 }, . . . , {α m , β m }} ,{{α 1 , β 1 } = {α, β}, {α 2 , β 2 }, . . . , {α m , β m }} , where α i ∈ A and β i ∈ B 1 for all i. Moreover, since Γ is G-locally primitive, the induced subgraph Γ[V (A) ∪ V (B 1 )] is a matching. Since Γ is connected, we may select a shortest path P 2 from α 1 to α 2 , say P 2 = (α 1 , γ 1,1 , γ 1,2 , . . . , γ 1,k , α 2 ). For each i, let the vertex γ 1,i lie in the N-orbit C 1,i . For each i, consider the orbit C G M 1,i , and let C 1,0 := A. If, for any i, \|C G M 1,i \| > 1, then, if l is the least such i, \|C G M 1,l \| = m (since M is a G-permutable m-matching) and \|C G M 1,l−1 \| = 1. This implies further that (Γ N , G/N) is locally-S m , and the result follows by Lemma 5.1. Finally, if \|C G M 1,i \| = 1 for all i, then Γ cannot be connected without some vertex having more than one neighbor in an N-orbit, a contradiction to the G-local primitivity of Γ. Therefore, in any case either Γ N contains a G/N-permutable m-matching or Γ N is a nearpolygonal graph with (G/N) A ∼ = S m . As discussed after the statement of Theorem 1.2, this provides a characterization of graphs with degree m containing a permutable m-matching, in the sense that under these conditions, one can keep taking normal quotients of this graph until reaching either a graph with a permutable mmatching or a near-polygonal graph where the stabilizer of a vertex acts on its m neighbors like S m . Moreover, Theorem 1.2 is a best-possible characterization in the sense that graphs in each case do exist. When combined with Construction 3.6, Theorem 3.7 shows that for any m 6 there exists a connected G-arc-transitive, G-locally-primitive graph with a G-permutable m-matching such that G has an intransitive normal subgroup N with more than two orbits of vertices such that Γ N is G/N-vertex-quasiprimitive and contains a G/N-permutable m-matching. When combined with Construction 3.6, Proposition 3.4 shows that for any m 6 there exists a connected G-arctransitive, G-locally-primitive graph with a G-permutable m-matching such that G has an intransitive normal subgroup N that has more than two orbits on vertices such that Γ N is G/N-vertexbiquasiprimitive and contains a G/N-permutable m-matching. Finally, Corollary 3.10 shows that for any m 6 there exists a connected G-arc-transitive, G-locally-primitive graph with a Gpermutable m-matching such that G contains an intransitive normal subgroup N that has more than two orbits on vertices, where Γ N is near polygonal, (Γ N , G/N) is locally-S m , but Γ N does not contain a permutable m-matching. A classification of graphs with a 2-transitive perfect matching This section is devoted to the proof of Theorem 1.3, which classifies the connected graphs that contain a 2-transitive perfect matching of size m. Throughout this section, we will use the following notation. We define Γ to be a graph with a perfect matching M of m edges such that Aut(Γ) is 2-transitive on M. This implies that \|V (Γ)\| = 2m and V (Γ) = V (M). We write M as follows: M = {e i = {α i , β i }} m i=1 . We also define M Aut(Γ) to be the subgroup of Aut(Γ) preserving M setwise, i.e., M := Aut(Γ) M . We begin with the following observation, which allows us to subdivide the problem into cases. LEMMA 6.1. For any i, j such that 1 i < j m, Γ[α i , β i , α j , β j ] ∼ = Γ[α 1 , β 1 , α 2 , β 2 ]. PROOF. This follows immediately from the 2-transitivity of Aut(Γ) on M. PROOF. Assume that M contains more than one edge, and let e i = {α i , β i } ∈ M. Since Γ is connected, either α i or β i has another neighbor, say γ. Since M is a perfect matching, γ is α j or β j for some j. Since there is at least one edge from an endpoint of e i to an endpoint of e j , the result follows by Lemma 6.1. We now subdivide the problem based on the induced subgraph Γ[α 1 , β 1 , α 2 , β 2 ]. LEMMA 6.3. If Γ has a matching M such that Aut(Γ) is 2-transitive on the edges of M, then the induced subgraph Γ[α 1 , β 1 , α 2 , β 2 ] will be isomorphic to one of K 4 , C 4 , K 4 \{e}, P 4 , or a triangle with a pendant edge. PROOF. This follows from Lemmas 6.1 and 6.2 and exhausting the graphs on four vertices. See Figure 1 for these induced subgraphs. We consider these cases one by one. LEMMA 6.4. If Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 , then Γ ∼ = K 2m . PROOF. Consider any two vertices γ, δ ∈ V (Γ). In M, they are either matched or not. If they are, they are the endpoints of some e i . If not, one is an endpoint of some e i and the other is an endpoint of some e j . But by by Lemma 6.1, they are adjacent in this case as well. Then every pair of vertices is adjacent and Γ ∼ = K 2m . LEMMA 6.5. There is no graph Γ such that Γ[α 1 , β 1 , α 2 , β 2 ] is a triangle with a pendant edge. PROOF. Without loss of generality, we let α 1 be the vertex with degree 3 and β 1 be the vertex with degree 1. By the 2-transitivity of Aut(Γ) on M, there is a g ∈ Aut(Γ) such that {α 1 , β 1 } g = {α 2 , β 2 } and {α 2 , β 2 } g = {α 1 , β 1 }. Without loss of generality, β g 2 = β 1 and α g 2 = α 1 . But because α 1 ∼ β 2 , α g 1 ∼ β g 2 , we have α g 1 ∼ β 1 . But α g 1 ∈ {α 2 , β 2 }, so we have a contradiction. 6.1. The case Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 . In order to characterize the graphs when the induced subgraph Γ[α 1 , β 1 , α 2 , β 2 ] is isomorphic to C 4 , we first need some preliminary results. A vertex γ ∈ V (Γ) is contained in a unique edge e i of M, so we define γ c := V (e i )\{γ}, i.e., γ c is the unique vertex adjacent to γ in the matching M. PROOF. We first need to show that x ∈ M; that is, we need to show that x ∈ Aut(Γ) and x preserves the matching M setwise. Suppose γ, δ ∈ V (Γ) and γ ∼ δ. If δ = γ c , then γ x = δ and δ x = γ, and so γ x ∼ δ x . If δ = γ c , then, since Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 and γ ∼ δ, we have that γ c ∼ δ c , and hence γ x ∼ δ x . On the other hand, if γ ∼ δ, then, since Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 and γ ∼ δ, γ ∼ δ c and δ ∼ γ c . Hence γ x ∼ δ, and so γ x ∼ δ x . Thus x is a permutation of the vertex set preserving both adjacency and nonadjacency, and so x ∈ Aut(Γ). Since x fixes each edge e i , x ∈ M. We will now show that x ∈ Z(M). Let g ∈ M. For any γ ∈ V (Γ), we have: γ gxg −1 = ((γ g ) x ) g −1 = ((γ g ) c ) g −1 = (γ c ) gg −1 = γ c = γ x . Therefore, gxg −1 = x for all g ∈ M, and so x ∈ Z(M). Let e i , e j ∈ M\{e}. Then there exists g ∈ M e such that e g i = e j . If g ∈ M γ , then gx ∈ M γ and e gx i = e j , where x is as in Lemma 6.6. The result follows. LEMMA 6.8. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 . Define A i := {α i } ∪ {γ ∈ V (Γ) : i = j, γ ≁ α i } = {γ ∈ V (Γ) : γ ∼ β i } and B i := {γ ∈ V (Γ)\|γ ∼ α i }. If g ∈ Aut(Γ) and e g i = e i , then g preserves the partition of V (Γ) into A i ∪ B i . Moreover, if α g i = α i , then A g i = A i and B g i = B i ; if α g i = β i , then A g i = B i and B g i = A i . PROOF. If α g i = α i , then it is clear that A g i = A i and B g i = B i . If α g i = β i , then suppose γ ∈ B i . This means γ ∼ α i , and so γ g ∼ α g i = β i . Then γ g ∈ A i , and the result follows as A i and B i are an equicardinal partition of V (Γ). g i 2 = α i . Note that A = A 1 = (A 1 ∩ A 2 ) ∪ (A 1 ∩ B 2 ), B = B 1 = (B 1 ∩ A 2 ) ∪ (B 1 ∩ B 2 ), where A i and B i are defined as in the statement of Lemma 6.8. Since α h 2 = α 2 , A h 2 = A 2 and B h 2 = B 2 , and so either (i) A h = A and B h = B or (ii) h swaps (A 1 ∩ A 2 ) and (B 1 ∩ A 2 ) and h swaps (A 1 ∩ B 2 ) and (B 1 ∩ B 2 ). However, α 2 ∈ A 1 ∩ A 2 , so we have A h = A and B h = B. Let α i , α j ∈ A, i = j. Since A is invariant under M α 1 and h, there is α k ∈ A such that α hg −1 3 g i k = α j . Since α 1 = γ ∼ α k , we have α j = α hg −1 3 g i k ∼ α hg −1 3 g i 1 = α i . Since i, j were arbitrary, A is a coclique. Since Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 , it immediately follows that B is a coclique as well. Suppose now that γ h = β 3 . Let γ = β 1 , and let A := {α i : 1 i m} and B := {β i : 1 i m}. The proof now proceeds as above. By Lemma 6.7, for each i 2 there exists g i ∈ M β 1 such that β g i 1 = β 1 and β g i 2 = β i . Note that B = B 1 = (B 1 ∩ A 2 ) ∪ (B 1 ∩ B 2 ), A = A 1 = (A 1 ∩ A 2 ) ∪ (A 1 ∩ B 2 ), where A i and B i are defined as in the statement of Lemma 6.8. Since β h 2 = β 2 , B h 2 = B 2 and A h 2 = A 2 , and so either (i) B h = B and A h = A or (ii) h swaps (B 1 ∩ A 2 ) and (A 1 ∩ A 2 ) and h swaps (B 1 ∩ B 2 ) and (A 1 ∩ B 2 ). However, β 2 ∈ B 1 ∩ B 2 , so we have B h = B and A h = A. Let β i , β j ∈ B, i = j. Since B is invariant under M β 1 and h, there is β k ∈ B such that β hg −1 3 g i k = β j . Since β 1 = γ ∼ β k , we have β j = β hg −1 3 g i k ∼ β hg −1 3 g i 1 = β i . Since i, j were arbitrary, B is a clique. Since Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 , it immediately follows that A is a clique as well. LEMMA 6.10. If Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 , then either Γ = K m ⊻ K m or Γ = K m,m . PROOF. This follows immediately from Lemma 6.9 and a consideration of the degree of each vertex in the induced subgraph Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 . 6.2. The cases Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}. The two remaining cases are actually very closely related. We begin with a helpful lemma. LEMMA 6.11. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 or Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}. Either (1) Γ is regular, or (2) Γ has two orbits of vertices: one orbit is a clique, the other a coclique. PROOF. We know that G is transitive on M, so Aut(Γ) has at most 2 orbits of vertices. If Aut(Γ) is also transitive on V (Γ), then (1) holds. If not, Γ has exactly 2 orbits of vertices, and Aut(Γ) will be 2-transitive on each of these orbits. Thus each orbit is either a clique or a coclique. Both cannot be cliques, because otherwise Γ would be regular. Both cannot be cocliques, because otherwise Γ is not connected. So we are in case (2). This allows us immediately to classify these graphs in the event that they are not regular. LEMMA 6.12. If Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and Γ is not regular, then Γ ∼ = K m ⊻ K m . If we have Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e} and Γ is not regular, then Γ ∼ = K m ∨ K m . PROOF. This follows immediately from Lemma 6.11. The remaining cases are when Γ is regular. This implies that m is odd. PROOF. Assume that Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 . For each i 2, the vertices of e i contribute 0 to the degree of one endpoint of e 1 and 1 to the other, i.e., each e i for i 2 contributes 1 to the sum of the degree of α 1 and the degree of β 1 . Since Γ is regular, 2 · \|Γ(α 1 )\| = \|Γ(α 1 )\| + \|Γ(β 1 )\| = 1 + 1 + (m − 1). The result follows for Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 . The proof is analogous in the case when we have Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}. In fact, in these remaining cases when Γ is regular, Γ must be vertex transitive. LEMMA 6.14. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 or Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}. If Γ is regular, then M is transitive on V (Γ). PROOF. We know that M is transitive on the edges of M, so it suffices to show that there is g i ∈ Aut(Γ) such that α g i i = β i for each i. Assuming Γ contains more than a single edge, it must contain at least three edges since m is odd. In each case we may choose three edges as follows: α j β j α i β i α k β k e j e i e k α j β j α i β i α k β k e j e i e k By the 2-transitivity of M on M, there is g ∈ M such that e g i = e i and e g j = e k . The g i that we seek is this g, and the result follows. We now show that there is a bijection between regular graphs in these two cases, i.e. that the two cases correspond. LEMMA 6.15. There exists a regular graph Γ 0 on 2m vertices with Γ 0 [α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 if and only if there exists a regular graph Γ 1 on 2m vertices with Γ 1 [α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}, and there is a natural bijection between such graphs. PROOF. Suppose we have such a graph Γ 1 . Note that M is the setwise stabilizer of M in Aut(Γ 1 ), which is transitive on V (Γ 1 ) but preserves the matching M. However, M has (at least) two orbits on the edges of Γ 1 : the edges of M and the edges not in M. The complement Γ 1 also has M as a group of automorphisms. We define Γ 0 to be the graph with vertex set V (Γ 1 ) and edge set E(Γ 1 ) ∪ M. The group M is still 2-transitive on a perfect matching in this case, but Γ 0 [α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 . The proof in the other direction is analogous. After considering Lemma 6.15, there are really only three cases left. We may assume that Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 , and one of the following holds: (i) Aut(Γ) is primitive on V (Γ), (ii) Γ is bipartite, or (iii) M itself is a system of imprimitivity. (Any other system of imprimitivity is ruled out by the 2-transitivity of M on M.) 6.3. The case where Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and G = Aut(Γ) is primitive on vertices. LEMMA 6.16. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and G = Aut(Γ) is primitive on vertices. Then Γ is a (G, 2)-arc-transitive graph. PROOF. Since G is primitive on V (Γ), G α 1 is a maximal subgroup of G. On the other hand, since there is g ∈ M such that α g 1 = β 1 , β g 1 = α 1 (see Lemma 6.14), we have M α 1 < M e 1 < M G, and so M α 1 is not a maximal subgroup of M. Thus M < G. By the 2-transitivity of M on M, M has two orbits on E(Γ): M and E(Γ)\M. Since M < G, there is h ∈ G\M, i.e., there is an automorphism that does not preserve M. This implies that h takes an edge in M to an edge in E(Γ)\M, and so G is transitive on E(Γ). Finally, we note that (i) Γ is G-vertex-transitive, (ii) Γ is G-edge-transitive, (iii) there is an element sending the arc (α 1 , β 1 ) to the arc (β 1 , α 1 ), and (iv) G α 1 β 1 is transitive on Γ(α 1 )\{β 1 }, which implies that Γ is a (G, 2)-arc-transitive graph. Consider the labeling of the vertices as in Figure 2. If we define D i (γ) := {δ ∈ V (Γ) : d(γ, δ) = i}, i.e., if D i (γ) is the set of vertices at distance i from the vertex γ, we can guarantee the distance of all vertices in the graph from α 1 except for the set X; all we know is that X ⊆ D 2 (α 1 ) ∪ D 3 (α 1 ). · · · α 1 β 1 · · · e 1 D 1 (α 1 ) Y 2 ⊆ D 2 (α 1 ) Y 1 ⊆ D 2 (α 1 ) X ⊆ D 2 (α 1 ) ∪ D 3 (α 1 ) FIGURE 2. Labeling of Γ when Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and G is primitive on vertices. LEMMA 6.17. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 , G = Aut(Γ) is primitive on vertices, and the subset X is as defined above. Then X ∩ D 2 (α 1 ) = ∅. PROOF. Suppose that X ∩ D 2 (α 1 ) = ∅, that is, X = D 3 (α 1 ). By Lemma 6.16 and the fact that M is a 2-transitive perfect matching, Γ is distance-transitive with diameter 3. We will show that, if γ, δ ∈ X, then γ ∼ δ. Indeed, suppose γ ∈ D 3 (α 1 ) = X. This means that d(β 1 , γ) = 2. Since X ∩ D 2 (α 1 ) = ∅, there are no edges from D 1 (α 1 ) to X. Similarly, since Γ is vertex-transitive, there are no edges from Figure 2). Hence, if δ ∈ D 1 (β 1 ), δ has no neighbors in Y 2 . Since Γ is distance-transitive, this means that no vertex in D 2 (α 1 ) has any neighbors in D 2 (α 1 ), i.e., all edges in Γ are from D 1 (β 1 ) = Y 1 ∪ {α 1 } to D 3 (β 1 ) = Y 2 (seeA = {α 1 } ∪ Y 1 ∪ Y 2 to X ∪ D 1 (α 1 ). However, this means that Γ is bipartite, in contradiction to Γ being vertex-primitive. Therefore, X ∩ D 2 (α 1 ) = ∅. PROOF. We again assume that vertices are labeled as in Figure 2. By the 2-transitivity of M on M, M α 1 is transitive on X, and so X ⊆ D 2 (α 1 ). Hence Γ has diameter 2, and, by Lemma 6.16, Γ is a distance-transitive, diameter 2, triangle-free strongly regular graph. (These are known as rank 3 graphs since, for any vertex α ∈ V (Γ), the stabilizer of α is a primitive group of rank 3 on vertices.) We note that Γ is a (4k + 2, k + 1, 0, µ)-strongly regular graph. By the classic equation relating the parameters (see [2]), (k + 1)k = [(4k + 2) − (k + 1) − 1]µ, and so µ = (k + 1)/3. The eigenvalues of the adjacency matrix for this graph and their multiplicities are known (again, see [2]). There are three eigenvalues: k+1, with multiplicity one, and two others. The multiplicities of these other two eigenvalues are 1 2   (4k + 1) ± (4k + 1) k+1 3 − (2k + 2) k+1 3 2 + 8 k+1 3   = 1 2 (4k + 1) ± 4k 2 − k − 5 (k + 1)(k + 25) ∈ Z. This implies that (4k 2 − k − 5) 2 (k + 1)(k + 25) = 16k 2 − 424k + 10585 − 264000 k + 25 is a perfect square. The last term allows us, via factoring, to come up with a list of values of k to check, which yields k = 2 or k = 24. But, together with µ = k+1 3 ∈ Z, we rule out k = 24, so the only graph in this case is strongly regular with parameters (10, 3, 0, 1) (corresponding to k = 2), which is the Petersen graph. It can be verified that the Petersen graph has a 2-transitive perfect matching by direct inspection. PROOF. Suppose Π = {{α i , β i } : 1 i m} is a system of imprimitivity on V (Γ). This implies that G = M. We remove the edge orbit M from Γ to create a new graph Γ ′ ; since G = M, M is an orbit of the edges of Γ under G, and G still acts 2-transitively on the system of imprimitivity Π. However, each block in Π is now an independent set. The quotient graph Γ ′ Π will be the complete graph K m , and there is exactly one edge between any two blocks in Γ ′ . By the 2-transitivity of M on Π, Γ ′ is M-arc-transitive. Hence Γ ′ is a symmetric spread of the complete graph K m (see [5]). By inspection of [5,Tables 1,2], the only possibility for Γ is the incidence graph of the Paley symmetric 2-design over GF(p f ). Moreover, if Γ is such a graph, then V (Γ) = GF(p f )×{0, 1}, and Aut(Γ) acts 2-transitively on each copy of GF(p f ) (simultaneously). Hence the matching {{(x, 0), (x, 1)} : x ∈ GF(p f )} is a 2-transitive perfect matching. Our final case is when Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and Γ is bipartite. · · · α 1 β 1 · · · e 1 D 1 (α 1 ) Y 2 ⊆ D 2 (α 1 ) Y 1 ⊆ D 2 (α 1 ) X = D 3 (α 1 ) FIGURE 3. Labeling of Γ when Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 , M < G, Γ bipartite LEMMA 6.20. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and that Γ is bipartite. Then m = p f , where p is a prime and p f ≡ 3 (mod 4), and Γ is isomorphic to the incidence graph of the Paley symmetric 2-design over GF(p f ). PROOF. Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and that Γ is bipartite. If G := Aut(Γ) = M, then this case has been resolved by Lemma 6.19. Hence we may assume that M < G. By the 2-transitivity of M on M, M has two orbits on E(Γ): M and E(Γ)\M. Since M < G, there is h ∈ G\M, i.e., there is an automorphism that does not preserve M. This implies that h takes an edge in M to an edge in E(Γ)\M, and so G is transitive on E(Γ). Since (i) G is transitive on the edges of Γ, (ii) G is transitive on the vertices of Γ, (iii) there is an element sending the arc (α 1 , β 1 ) to the arc (β 1 , α 1 ) by Lemma 6.14, and (iv) G α 1 β 1 is transitive on Γ(α)\{β 1 }, we have that Γ is a (G, 2)-arc-transitive graph. Since Γ is bipartite, using the labeling of Figure 3, we have D 2 (α 1 ) = Y 1 ∪ Y 2 and D 3 (α 1 ) = X. Since Γ is (G, 2)-arc-transitive and M α 1 is transitive on X, Γ is a distance-transitive graph of diameter 3. By [6, Theorem 5.10.3], Γ is the incidence graph of a symmetric 2-design. The points of the design are represented by one of the biparts of Γ. The stabilizer of a point (i.e., of α 1 , say) has at most three orbits on points: (i) {α 1 }, (ii) the set of all points incident with "block" β 1 , and (iii) set of all points not incident with "block" β 1 . This means that Γ is the incidence graph of a rank 2 or 3 symmetric 2-design. Such symmetric 2-designs have been classified [3,4,8]. The only possibilities, other than the Paley symmetric 2designs, are: the Hadamard design with 11 points where each point is incident with 5 blocks, which gives the same incidence graph as the Paley symmetric 2-design on 11 points; the design with 35 points where each point is incident with exactly 17 blocks, which is ruled out since the only 2transitive groups on 35 points are A 35 and S 35 , which are not involved in the automorphism group of this design (the unique minimal normal subgroup of the automorphism group of this design is isomorphic to A 8 ); and the design with 15 points where each point is contained in exactly 7 blocks. In this last case, the unique minimal normal subgroup of the automorphism group of the design is isomorphic to A 6 . While A 6 has a rank 3 action on 15 points, the stabilizer of a point in this action has orbits of size 1, 6, and 8. However, if the incidence graph of this design had a 2-transitive perfect matching, then the stabilizer of a point would have orbits of size 1, 7, and 7. Therefore, the only such graphs Γ with Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and Γ bipartite are isomorphic to incidence graphs of Paley symmetric 2-designs. We are now ready to complete the proof of Theorem 1.3. PROOF OF THEOREM 1.3. The result follows from Lemmas 6.3, 6.4, 6.5, 6.10, 6.12, 6.15, 6.18, 6.19, and 6.20. Finally, we prove Corollary 1.4. PROOF OF COROLLARY 1.4. The result follows from Theorem 1.3 and noting which graphs in cases (3) and (4) have an induced symmetric group on the matching. Since the group acting on the matching in each of (3) and (4) has a minimal normal subgroup that is elementary abelian and acts regularly on an odd number of edges, we conclude that the only option in cases (3) and (4) is when m = 3. The result follows. ACKNOWLEDGEMENTS. The authors wish to thank Thomas Zaslavsky for his numerous editorial suggestions and comments on earlier versions of this paper. LEMMA 2. 3 ([ 9 , 39Propositions 3.1, 5.1]). Fix a spanning tree T of a graph Γ and α ∈ V (Γ). Assume the edges of T are assigned the identity voltage and that the voltage group H is generated by the edge voltages of Γ. An automorphism g of Γ lifts to an automorphismg ofΓ if and only if g φα is a group automorphism. Moreover, if H is abelian, the automorphism g φα does not depend on the choice of base vertex α. LEMMA 2. 4 ([ 9 , 49Proposition 6.4]). Let Γ be a graph with edge set E, and let T denote the set of edges of a spanning tree of Γ. Let Z p denote the cyclic group of order p, where p is a prime. Let H := Z \|E\|−\|T \| p CONSTRUCTION 3 . 1 . 31Let Γ be a graph with a vertex α of degree m and a group of automorphisms G such that G Γ(α) α ∼ = S m . Define a new graph QΓ to be the graph obtained by subdividing every edge of Γ into a path of length 2. CONSTRUCTION 3. 2 . 2Let Γ be a graph with a permutable matching M. Let Q M Γ be the graph obtained by subdividing every edge not in M into a path of length 2. CONSTRUCTION 3. 3 . 3Let Γ be a graph with a permutable matching M. Let Q M Γ be the graph obtained by subdividing every edge in M into a path of length 3. Perhaps the most obvious examples of graphs with permutable m-matchings are also examples of vertexbiprimitive graphs with permutable m-matchings. PROPOSITION 3 . 4 . 34For every m 2, the complete bipartite graph K m,m has degree m and there exists G Aut(K m,m ) such that K m,m is G-vertex-biprimitive and K m,m contains a Gpermutable m-matching. the edge set of Γ and let T denote the set of edges of a spanning tree of Γ that contains each of the edges of M. Let Z p denote the cyclic group of order p, where p is a prime. Let H := Z \|E\|−\|T \| p THEOREM 3 . 7 . 37For every m 3, the odd graph O m has degree m and there exists G Aut define the sets S i := {1, . . . m}\{i} and T i := {i} ∪ {m + 1, . . . , 2m − 2}. Since vertices of O m are identified with subsets of {1, . . . , 2m − 1} of size m − 1, each S i and each T i is a vertex of O m , and, furthermore, M := {{S i , T i } : 1 i m} is a matching of size m. Let H := Sym({1, . CONSTRUCTION 3 . 8 . 38Let Γ be a (G, 2)-arc-transitive, near-polygonal graph of degree m such that Γ does not contain a G-permutable m-matching and (Γ, G) is locally-S m . Let E denote the edge set of Γ and let T denote the set of edges of a spanning tree of Γ. Let Z p denote the cyclic group of order p, where p is a prime. Let H := Z \|E\|−\|T \| p; H is a Z p -vector space. Let X be a basis for H, so \|X\| = \|E\| − \|T \|. Define Γ p to be the derived regular cover of the voltage graph defined by assigning a distinct element of X to each co-tree edge of Γ. PROPOSITION 3 . 9 . 39The graph Γ p created from Construction 3.8 contains a permutable mmatching. PROOF. Let α be a vertex of Γ with Γ(α) = {β 1 , . . . , β m }. By Lemmas 2.3 and 2.4, G lifts to a group of automorphisms G of Γ p and there is a group homomorphism φ : G → Aut(H), COROLLARY 3 . 10 . 310For each m 3, there exist infinitely many graphs Γ with a group of automorphisms PROPOSITION 4. 1 . 1Let Γ be a G-arc-transitive graph with a G-permutable m-matching, where m 6. If {α, β} is an edge of Γ, then there is a subgroup U G αβ such that U Γ(α) has a composition factor isomorphic to A m−1 . PROOF. Let M be a G-permutable m-matching containing {α, β}, where M = {e = e 1 = {α, β}, e 2 , . . . , e m } . Note that G M M ∼ = S m and G M\e eM ∼ = S m−1 . Let K := {g ∈ G eM : e g i = e i , 1 i m}, the kernel of the action of G eM on M. We have K ⊳ G eM , G eM /K ∼ = S m−1 , and hence A m−1 is a composition factor of G eM (since m 6, A m−1 is simple). LEMMA 5 . 1 . 51Let Γ be a G-arc-transitive graph with degree m 6 and let (Γ, G) be locally-S m . Either Γ contains a G-permutable m-matching or Γ is near-polygonal.PROOF. Let Γ be such a graph, and let α be a vertex with Γ(α) = {β 1 , . . . , β m }. Since (Γ, G) is locally-S m , Γ is G-locally primitive; in fact, G Γ(α) α ∼ = S m and, for each 1 i m, G . If H = g 1,i : 2 i m , H stabilizes M = {{β i , γ i } : 1 i m} setwise, and H M ∼ = S m , so M is an H-permutable m-matching, as desired. where α i ∈ A and β i ∈ B i for each i. Since M is permutable, this implies thatB 1 , . . . , B m are distinct N-orbits. Moreover, if B = {β = β 1 , β 2 , . . . , β m }, since each edge contains a vertex in the N-orbit A and M is permutable, the stabilizer of A in G acts as the full symmetric group on B, that is, G B AB ∼ = S m . Moreover, since Γ is G-locally primitive of degree m, (Γ N , G/N) is locally-S m , the neighbors of the vertex A of Γ N are precisely B 1 , . . . , B m , and the vertex α has a unique neighbor in each of B 1 , . . . , B m . By Lemma 5.1, the result follows. 5.2.3: γ ∈ A, δ ∈ B 1 .Now, M is entirely contained within the N-orbits A and B 1 , i.e. M is of the form LEMMA 6. 2 . 2Every edge of M has at least one endpoint adjacent to at least one endpoint of every other edge of M. FIGURE 1 . 1The possibilities for Γ[α 1 , β 1 , α 2 , β 2 ] LEMMA 6 . 6 . 66Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 . Let x ∈ Sym(V (Γ)) be the permutation of the vertices of Γ defined by γ x = γ c for all γ ∈ V (M) = V (Γ), that is, α x i = β i and β x i = α i for all i. Then x ∈ Z(M). LEMMA 6. 7 . 7Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 . For γ ∈ V (Γ), if e is the edge of M containing γ, then M γ is transitive on M\{e}.PROOF. Let e ∈ M and e = {γ, δ}. Since M is 2-transitive on M, M e is transitive on M\{e}. LEMMA 6 . 9 . 69Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = C 4 . The vertices of Γ can be partitioned into two sets, A and B, such that \|A\| = \|B\| = m, each of A and B contains exactly one endpoint from each edge of M, and either Γ[A] ∼ = Γ[B] ∼ = K m or Γ[A] ∼ = Γ[B] ∼ = K m . PROOF. By Lemmas 6.7 and 6.8, for any vertex γ ∈ V (Γ), M γ has four orbits on vertices: {γ}, {γ c }, Γ(γ)\{γ c }, and Γ(γ c )\{γ}. Without loss of generality, we may let {γ, γ c } = e 1 , Γ(γ c )\{γ} = {α i : i 2}, and Γ(γ)\{γ c } = {β i : i 2}. Moreover, by Lemma 6.7, there exists h ∈ M α 2 such that α h 2 = α 2 and e h 1 = e 3 . Suppose first that γ h = α 3 . Let γ = α 1 , and let A := {α i : 1 i m} and B := {β i : 1 i m}. By Lemma 6.7, for each i 2 there exists g i ∈ M α 1 such that α g i 1 = α 1 and α LEMMA 6 . 13 . 613Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 or Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}. If Γ is regular, then m is odd. Moreover, if m = 2k +1, then the degree of each vertex is k +1 if Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and the degree of each vertex is 3k + 1 if Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = K 4 \{e}. LEMMA 6 . 18 . 618Assume Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and G = Aut(Γ) is primitive on vertices. Then Γ is isomorphic to the Petersen graph. For instance, if the vertices of the Petersen graph P are represented as subsets of size two of {1, 2, 3, 4, 5}, then Aut(P) = S 5 is 2-transitive on the matchingM = {{{1, 2}, {3, 4}}, {{3, 5}, {2, 4}}, {{1, 4}, {2, 5}}, {{2, 3}, {1, 5}}, {{4, 5}, {1, 3}}} .6.4. The case where Γ[α 1 , β 1 , α 2 , β 2 ] ∼ = P 4 and G = Aut(Γ) is imprimitive on vertices.LEMMA 6.19. 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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS, 1460 JAYHAWK BLVD, LAWRENCE, KS 66045 E-mail address: [email protected] DEPARTMENT OF MATHEMATICS, COLLEGE OF WILLIAM AND MARY, P.O. BOX 8795, WILLIAMSBURG, VA 23187-8795 E-mail address: [email protected]	[]
[ "Graded Prime Ideals Over Graded Near Ring", "Graded Prime Ideals Over Graded Near Ring", "Graded Prime Ideals Over Graded Near Ring", "Graded Prime Ideals Over Graded Near Ring" ]	[ "Malik Bataineh ", "Tamem Al-Shorman ", "Eman Al-Kilany ", "Malik Bataineh ", "Tamem Al-Shorman ", "Eman Al-Kilany " ]	[]	[]	In this paper, we consider graded near-rings over a monoid G as a generalizations of graded rings over groups. We introduce certain innovative graded prime ideals and study some of its basic properties over graded near-rings.	null	[ "https://arxiv.org/pdf/2204.05638v2.pdf" ]	248,118,564	2204.05638	6994a104bf854cd82b88839f538fbd3b4edb20bf	Graded Prime Ideals Over Graded Near Ring 14 Apr 2022 Malik Bataineh Tamem Al-Shorman Eman Al-Kilany Graded Prime Ideals Over Graded Near Ring 14 Apr 2022arXiv:2204.05638v2 [math.GM] In this paper, we consider graded near-rings over a monoid G as a generalizations of graded rings over groups. We introduce certain innovative graded prime ideals and study some of its basic properties over graded near-rings. INTRODUCTION Near-rings are generalizations of rings: addition is not necessarily abelian and only one distributive law holds. They arise in a natural way in the study of mappings on groups: the set M(G) of all maps of a group (G, +) into itself endowed with pointwise addition and composition of functions is a near-ring. For general background on the theory of near-rings we refer the reader to the monographs written by Pilz [5] and Meldrum [3]. In fact Pilz defines it as : A near-ring (N, +, * ) is a set N with two binary operations + and * that satisfy the following axioms: (1) (N, +) is a group. (2) (N, * ) is semi group. (semi group: a set together with an associative binary operation); (3) * is right distributive over + (i.e. (a + b) * y = ay + by). The graded rings were introduced by Nastasescu and Van Oystaeyen [4]. Also graded near-rings were introduced and studied by Dumitru, Nastasescu and Toader [1] in which they defined it as: Let G be a multiplicatively monoid (an algebraic structure with a single associative binary operation) with identity, a near-ring N is called a G-graded if there exists a family of additive normal subgroups {N σ } of N satisfying that: 1. N = ⊕ σ∈G N σ ; 2. N σ N τ ⊆ N στ , ∀σ, τ ∈ G. . Graded prime ideals over graded rings have been introduced and defined as: Let R be a commutative G-graded ring where G is an abelian group, a graded proper ideal I of R is called graded prime ideal if a g b h ∈ I gh , then a g ∈ I g or b h ∈ I h . (see [2]). GRADED PRIME IDEALS In this Section, we present the concept of graded prime ideals, study some of its properties in graded near rings and in special graded near-rings such as graded near-rings with zero ideal is graded prime and graded near-rings in which every graded ideal is graded prime. Also, determine the shape of graded prime ideal in the graded quotient near-ring and in the product of two G-graded near-rings. Definition 2.1. Let G be a multiplicatively monoid with identity element and N be a G-graded near-ring. A proper graded ideal P is called graded prime ideal if whenever A g B h ⊆ P gh , then either A g ⊆ P g or B h ⊆ P h , for any ideals A and B in N. Example 2.2. Let N be the near-ring (Z 0 [x], +, o) the set of all polynomial with integer coefficients and zero constant term, where + is the usual addition and o is the composition of functions. Let G = (N * , * ) the set of natural number without zero and * is usual multiplication. N is G-graded near-ring where (Z 0 [x]) n = ZX n . (See [4]). Consider the ideal P , the set of all polynomials in Z 0 [x] with even coefficient. Let A and B be two ideals in Z 0 [x] such that A g B h ⊆ P gh where g, h ∈ N * . Suppose that A g P g and B h P h . Then there exists a ∈ A g and b ∈ B h such that neither a nor b has even coefficient. So ab does not have even coefficient, contradiction. So A g ⊆ P g or B h ⊆ P h and hence P is graded prime More general examples of graded prime ideals are the maximal ideal in near-ring with multiplicative identity and the intersection of totally ordered graded prime ideals by inclusion. Proposition 2.3. Let N be a graded near-ring with multiplicative identity. If P is a proper graded ideal does not properly contained in any proper ideal, then P is graded prime. Proof. Let A and B be ideals such that A g B h ⊆ P gh . Suppose that neither A g ⊆ P g nor B h ⊆ P h , then there exists a ∈ A g , b ∈ B h such that a / ∈ P g , b / ∈ P h . P is subset of both P + < a > and P + < b >. So, by assumption, N = P + < a > = P + < b > . Thus, 1 = p + an and 1 = q + bm where n, m ∈ N and p, q ∈ P. But P is an ideal, so a(q + bm) − abm ∈ P and since 1 = (q + bm), then (q + bm)a − abm ∈ P. Thus, q + bma − abm ∈ P. Therefore , bma ∈ P since qa, abm ∈ P. But, 1 = 1 * 1 = (p + an)(q + bm) = p(q + bm) + an(q + bm) = p(q + bm) + (q + bm)an, and since q + bm = 1. Then, 1 = p(q + bm) + qan + bman ∈ P. Hence, N = P, contradiction. So, A g ⊆ P g or B h ⊆ P h . Hence, P is graded prime ideal. Proposition 2.4. Let N be a G-graded near-ring. Let A be a totally ordered set. Let (P a ) a∈A be a family of graded Prime ideals with P a ⊆ P b for any a, b ∈ A with a b. Then P = a∈A P a is a graded prime ideal. Proof. Let I and J be ideals of N with I g J h ⊆ ( a∈A P a ) gh . Then ∀a ∈ A, we have I g J h ⊆ (P a ) gh . If ∃a ∈ A such that I g (P a ) g , then J h ⊆ (P a ) h . Hence, ∀ b a, we have J h ⊆ (P b ) h . If ∃ c < a such that J h (P c ) h , then I g ⊆ (P c ) g . So, I g ⊆ (P a ) g . A contradiction. Hence, ∀a ∈ A we have J h ⊆ (P a ), therefore, J h ⊆ a∈A (P a ) h . Proposition 2.4, states that if (P a ) a∈A is a family of graded prime ideals with P a ⊆ P b for a, b ∈ A with a b, where A is totally ordered set. Then P = a∈A P a is graded prime. In general, it is not necessary that the intersection is graded prime if we drop the condition (A is totally ordered set and P a ⊆ P b for a, b ∈ A with a b). For example, consider the near-ring (Z 6 , +, * ) with (G = 0, 1, +) where + defined as 0+0 = 0, 0+1 = 1, 1+0 = 1 and 1 + 1 = 1. Then N is G-graded near-ring defined by N 0 = Z 6 , N 1 = {0}. Note that the intersection of P 1 = {0, 2, 4} and P 2 = {0, 3} is not graded prime ideal, although P 1 and P 2 are. However, the next two theorems give some properties of any intersection of graded prime ideals. We denoted to the I g I g ... I g n-times by (I g ) n . Theorem 2.5. Let N be a graded near-ring. Let P be an intersection of graded prime ideals. For any ideal J with (J g ) n ⊆ P g n , for some n ∈ N we have J g ⊆ P g . Proof. Let N be a graded near-ring and let P α ′ s be a set of graded prime ideals in N. Let P be the intersection of P α ′ s . Let I be an ideal such that (I g ) n ⊆ P g n . Then (I g ) n is subset of each (P α ) g n . Since each P α is graded prime, we have I g is subset of each (P α ) g or I g n−1 is subset of each (P α ) g n−1 . If I g n−1 is subset of each (P α ) g n−1 , then we have I g is subset of each (P α ) g or I g n−2 is subset of each (P α ) g n−2 . Consequently, we guarantee that I g is subset of each (P α ) g . Hence I g is subset of their intersection which means I g is subset of P g . Theorem 2.6. Let N be a graded near-ring. Let P be an intersection of graded prime ideals. For any ideal J with J n ⊆ P for some n ∈ N, we have J ⊆ P . Proof. Let P be the intersection of graded prime ideals and let J be an ideal with J n ⊆ P . We have (J g ) n ⊆ [J n ∩N g n ] ⊆ P g n . By Theorem 2.5, we have for each g ∈ G, J g ⊆ P g . Therefore, J ⊆ P. Note that, if P is a prime ideal and P is graded ideal, then it is graded prime ideal. (To see that, take two ideals A and B such that A g B h ⊆ P gh , then A g B h ⊆ P . Let C be the ideal generated by A g , D be the ideal generated by B h , and E be the ideal generated by A g B h . By the definition of the ideal generated by set, we have CD ⊆ E ⊆ P . Since P is prime ideal, then C ⊆ P or D ⊆ P . Therefore, A g ⊆ P g or B g ⊆ P g ). Next Example shows that it is not necessary for graded prime ideal to be prime ideal. Example 2.7. Let N be Z[i] with usual addition and multiplication. N is Z 2 -graded with N 0 = Z and N 1 = Zi. Since < 1 + i >< 1 − i >⊆ 2N with neither (1 + i) nor (1 − i) belongs to 2N, P = 2N is not prime. However, P is graded prime. Next Theorem gives an equivalent conditions for a graded ideal to be graded prime ideal. Theorem 2.8. Let N be a near-ring and P be an ideal of N. Then the following are equivalent: 1. For any two homogeneous elements i and j with i / ∈ P g and j / ∈ P h then < i > g < j > h P gh ; 2. For all ideals I, J with P g ⊂ I g and P h ⊂ J h then I g J h P gh ; 3. For all ideals I, J with I g P g and J h P h then I g J h P gh ; 4. P is graded prime ideal. (4) : Follows directly from the definition of graded prime ideal. Proof. (1) ⇒ (2) : If P g ⊂ I g and P h ⊂ J h . Take i ∈ I g and j ∈ J h with i / ∈ P g and j / ∈ P h . Then < i > g < j > h P gh . Hence I g J h P gh . (2) ⇒ (3) : If I g P g and J h P h . Take i ∈ I g and j ∈ J h with i / ∈ P g and j / ∈ P h . Then (< i > g + P g ) (< j > h + P h ) P gh . Therefore, ∃i 1 ∈ < i > g , j 1 ∈ < j > h , p 1 ∈ P g and p 2 ∈ P h such that: (i 1 + p 1 )(j 1 + p 2 ) / ∈ P gh . So, i 1 (j 1 + p 2 ) − i 1 j 1 + i 1 j 1 + p 1 (j 1 + p 2 ) P gh . So, since i 1 (j 1 + P 2 ) − i 1 j 1 ∈ P gh and i 1 j 1 / ∈ P g we get I g J h P gh . (3) ⇒(4) ⇒ (1) : If < i > g < j > h ⊆ P gh , then < i > g ⊆ P g or < j > h ⊆ P h . Therefore, i ∈ P g or j ∈ P h . Next proposition gives another equivalent condition that guarantees a graded ideal to be graded prime. Proposition 2.9. Let P be a graded prime ideal of a graded near-ring N. Then the following are equivalent: 1. For a ∈ N h and b, c ∈ N g , with a(< b > g + < c > g ) ⊆ P hg , we have a ∈ P h , or b and c ∈ P g . 2. For x ∈ N but x g / ∈ P g , we have (P hg :< x > g + < y > g ) h = P h for any y ∈ N g . 3. P is graded prime. Proof. (1) ⇒ (2) : Let t ∈ N h and t ∈ (P hg :< x > g + < y > g ) for any y and x ∈ N g but x / ∈ P g . Therefore, t(< x > g + < y > g ) ⊆ P hg . Thus, by hypothesis, t ∈ P h . (2) ⇒ (3) : Let A and B be graded ideals of N such that A h B g ⊆ P hg . Suppose that A h P h and B g P g , then there exists b ∈ B g with b / ∈ P g . Now we claim that A h B g = 0. Let b 1 ∈ B g , then A h (< b > g + < b 1 > g ) ⊆ P hg , and then A h ⊆ (P hg :< b > h + < b 1 > h ) = P h , which is a contradiction. Hence, P is graded prime. (3) ⇒ (1) : If a(< b > g + < c > g ) ⊆ P hg , then < a > h (< b > g + < c > g ) ⊆ P hg . Hence, < a > h ⊆ P h or (< b > g + < c > g ) ⊆ P h since P is graded prime. Therefore, a ∈ P h and b, c ∈ P g , since a ∈ N h and b, c ∈ N g . Next, we use the properties of graded prime ideals to study some properties of some special graded near-rings such as graded quotient near-rings, graded near-rings with zero ideal is graded prime and the product of two graded near-rings. Theorem 2.10. Let G be a multiplicatively monoid with identity element and let N and M be two G-graded near-rings. Let Φ be a surjective homomorphism such that Φ(I g ) = (Φ(I)) g for any ideal I ∈ N and any g ∈ G. Then: (i). The pre-image of graded prime ideal is graded prime. (ii). The image of graded prime ideal which contains the kernal of Φ is graded prime. Proof. (i) Suppose that A g B h ⊆ (Φ −1 (P )) gh , where A and B are ideals of N, and P is graded prime ideal of M. Then we have (Φ(A)) g (Φ(B)) h ⊆ P gh . Since P is graded prime ideal, (Φ(A)) g ⊆ P g or (Φ(B)) h ⊆ P h . Then, A g ⊆ (Φ −1 (P )) g or B h ⊆ (Φ −1 (P )) h . Hence, Φ −1 (P ) is graded prime ideal. (ii) Suppose that A g B h ⊆ (Φ(P )) gh , where A and B are ideals of M and P is graded prime ideal of N. Then we have (Φ −1 (A)) g (Φ −1 (B)) h ⊆ (P gh + KerΦ) ⊆ P . However, (Φ −1 (A)) g (Φ −1 (B)) h ⊆ N gh hence (Φ −1 (A)) g (Φ −1 (B)) h ⊆ P gh . Since P is graded prime ideal, (Φ −1 (A)) g ⊆ P g or (Φ −1 (B)) h ⊆ P h . Then, A g ⊆ (Φ(P )) g or B h ⊆ (Φ(P )) h . Hence, Φ −1 (P ) is graded prime ideal. Lemma 2.11. Let N be a graded near-ring. If h : N →N := N/I is a homomorphism with h(J g ) = h(J) g , then h −1 (J g ) = h −1 (J) g . Proof. h −1 (J) g = h −1 (J) ∩ N g = h −1 (h(h −1 (J) ∩ N g )) = h −1 (h(h −1 (J)) ∩ (N/I) g ) = h −1 (J ∩ (N/I) g ) = h −1 (J g ). Theorem 2.12. Let N be a graded near-ring, P be a graded ideal, and Q be a graded ideal of N with Q ⊆ P . Let π : N →N := N/Q be the canonical epimorphism. Then P is graded prime if and only if π(P ) is graded prime. Proof. Let P be a graded prime ideal of N. Let J and I be two ideals ofN with J g I h ⊆ (π(P )) gh . Let j g := (π −1 (J)) g and i h := (π −1 (I)) h . Then we have, by Lemma 2.11, j g := (π −1 (J g )) and i h := (π −1 (I h )). So, j g i h = (π −1 (J g ))(π −1 (I h )) ⊆ (π −1 (J g I h )) ⊆ π −1 ((π(P )) gh ) = (π −1 (π(P ))) gh = (P + Q) gh = P gh . and hence j g ⊆ P g or i h ⊆ P h . Therefore, J g = (π(π −1 (J))) g = π(π −1 (J) g ) = π(j g ) ⊆ π(P g ) = (π(P ) g or I h ⊆ (π(P )) h . Thus, π(P ) is graded prime. For the other direction, let π(P ) be prime ideal ofN . If j g i h ⊆ P gh , then (π(j)) g (π(i)) h = π(j g i h ) ⊆ (π(P gh )) = (π(P )) gh . And hence (π(j)) g ⊆ (π(P )) g or (π(i)) h ⊆ (π(P )) h . Therefore, J g ⊆ (j+Q) g = (π −1 (π(j))) g = π −1 ((π(j)) g ) ⊆ π −1 ((π(P )) g ) = (π −1 (π(P ))) g = (P + Q) g = P g or I h ⊆ P h . Next proposition gives a property that could be useful to determine weather some ideals are graded prime or not. Furthermore, the next proposition, gives characteristics of the quotient graded near-ring N/P , whenever P is graded prime ideal in the graded near-ring N. Proposition 2.13. Let N be a graded near-ring and A, B be ideals. P is graded prime ideal if and only ifĀ gBh = 0 if bothĀ gBh = 0 in N/P . Proof. Let P be a graded prime ideal and A, B be any ideals such thatĀ g = 0 and B h = 0 in N/P for some g, h ∈ G. Hence, neither A g ⊆ P g nor B h ⊆ P h . To show that A gBh = 0. Suppose thatĀ gBh = 0. Hence, A g B h ⊆ P gh , contradicts Theorem 2.8. Conversely, Let A g B h ⊆ P gh , then,Ā gBh = 0 in N/P . By assumption,Ā g = 0 or B h = 0 in N/P . Therefore, A g ⊆ P g or B h ⊆ P h . Hence, P is graded prime. We know that in some near-rings, for instance integral near-rings, the zero ideal is prime ideal and we know that these near-rings are called prime near-rings. However, we can note that in some graded near-rings, the Zero ideal is graded prime. For example, I = {0} is graded prime in the graded near-ring which defined in Example 2.2. In Next, we study few interesting properties of such near-rings. If a graded near-ring N is simple, then there is no proper ideal unless {0}. Hence, by definition of graded prime ideal, {0} is graded prime ideal or N is a zero near-ring. More general results is discussed in the Proposition 2.15. Before that we need the following lemma. Lemma 2.14. Let N be a graded near-ring and let I be a graded ideal of N. Zero is graded prime ideal in the graded near-ring N/I if and only if the ideal I is a graded prime ideal of N. Proof. Follows by taking I = P in Theorem 2.12. Proposition 2.15. Let N be a graded near-ring and I be a maximal ideal. If I is graded ideal, then either I is graded prime or N 2 ⊆ I. Proof. Let N be a graded near-ring and I be a maximal ideal which can be graded. N/I is simple near-ring. Hence, {0} is graded prime in N/I or N/I is a zero near-ring, which implies that either I is graded prime ideal by Lemma 2.14, or N 2 ⊆ I. Corollary 2.16. Let N be a graded near-ring with unity and let I be a graded ideal. If I is a maximal ideal in N, then I is a graded prime ideal. Proof. Let I be a maximal ideal. If 1 is a unity of N, then for any n ∈ N, n = n.1 ∈ N 2 . Hence, N 2 = N I. By previous theorem, I is graded prime. Proposition 2.17. Let G be a multiplicatively monoid with identity element and let N and M be two G-graded near rings. Let Φ be a surjective homomorphism from N into M, such that Φ(I g ) = (Φ(I)) g for any ideal I ∈ N and any g ∈ G. If {0} is graded prime ideal in M, then the kernel is graded prime in N. with N 0 = Z , N 1 = {0} and M 0 = Z 8 , M 1 = {0}. Then we have {0} is graded prime ideal in Z, but it is not graded prime in Z 8 . Although, Φ(x) = x, is surjective homomor- phism from Z into Z 8 . Let G be a multiplicatively monoid with identity. Let (N 1 , + 1 , * 1 ) and (N 2 , + 2 , * 2 ) both be G-graded near-rings. Define N = N 1 × N 2 = {(n 1 , n 2 ) : n 1 ∈ N 1 , n 2 ∈ N 2 }. The Direct Product is the set N paired with the operations addition + and multiplication * defined as (n ′ 1 , n ′ 2 ) + (n ′ 1 ′ , n ′ 2 ′ ) = (n ′ 1 + 1 n ′ 1 ′ , n ′ 2 + 2 n ′ 2 ′ ) and (n ′ 1 , n ′ 2 ) * (n ′ 1 ′ , n ′ 2 ′ ) = (n ′ 1 * 1 n ′ 1 ′ , n ′ 2 * 2 n ′ 2 ′ ), for each (n ′ 1 , n ′ 2 ), (n ′ 1 ′ , n ′ 2 ′ ) ∈ N. N is graded nearring by N g = (N 1 ) g × (N 2 ) g . Since it is known that if (N 1 ) g and (N 2 ) g are normal subgroups of N 1 and N 2 , respectively, then N g = (N 1 ) g × (N 2 ) g is normal subgroup of N. Also for any n = (n 1 , n 2 ) ∈ N is written uniquely as sum of finite elements of N g = (N 1 ) g × (N 2 ) g , g ∈ G since n 1 and n 2 are written uniquely as sum of finite elements of (N 1 ) g and (N 2 ) g , g ∈ G respectively. Furthermore, N g N h = [(N 1 ) g × (N 2 ) g ][(N 1 )h × (N 2 ) h ] = ((N 1 ) g (N 1 ) h ) × ((N 2 ) g (N 2 ) h ) ⊆ (N 1 ) gh × (N 2 ) gh = N gh Note that If P 1 and P 2 are graded prime ideals in N 1 and N 2 , respectively, then P 1 ×P 2 need not be graded prime ideal of N 1 × N 2 . For example, {0} is graded prime in N = Z 2 where G = Z 2 with N 0 = {0} and N 1 = Z 2 , while {0} × {0} is not graded prime in Z 2 × Z 2 since (Z 2 × {0}) 1 ({0} × Z 2 ) 1 ⊆ ({0} × {0}) 1 although neither (Z 2 × {0}) 1 ⊆ ({0} × {0}) 1 nor ({0} × Z 2 ) 1 ⊆ ({0} × {0}) 1 .A g C h × B g D h ) ⊆ (P gh × M gh ). So, A g C h ⊆ P gh . Which implies A g ⊆ P g or C h ⊆ P h . Hence, (A × B) g ⊆ (P × M) g or (C × D) h ⊆ (P × M) h . Hence P × M is graded prime ideal. Conversely, suppose that (P × M) is a graded prime ideal of N × M and let I and J be ideals of N such that I g J h ⊆ P gh . Then (I × M) g (J × M) h ⊆ (P × M) gh . By assumption, we have (I × M) g ⊆ (P × M) g or (J × M) h ⊆ (P × M) h . So, I g ⊆ P g or J g ⊆ P g . Hence, P is graded prime ideal. If P is of the form (I × M) or of the form (N × J), then, by Theorem 2.18, P is graded prime ideal if and only if both I and J are graded prime. Let P = (I × J) be a graded prime ideal with I g = N g and J h = M h , for some g and h belongs to G. Suppose a ∈ I gh . Then (< a > gh × {0}) ⊆ P gh This implies that either (< a > × M) g ⊆ P g or (N × {0}) h ⊆ P h . If (< a > × M) g ⊆ P g , then M g = J g and if (N × {0}) h ⊆ P h , then N h = I h . Contradiction. Hence I × J can not be graded prime ideal if both I and J are proper ideals. Note that, from previous theorem, we can not find a graded near-ring in which every proper graded ideal is graded prime if the G-graded near-ring N with unity and of the form N 1 × N 2 , where (N 1 × N 2 ) g = (N 1 ) g × (N 2 ) g unless one of N 1 or N 2 is the Zero near-ring. More general, we can not find graded near-ring with every proper graded ideal is graded prime regardless it has unity or not. Since the ideal {0}×{0} can not be graded prime ideal if both N 1 and N 2 are non trivial near-rings. Corollary 2.22. If every graded ideal is graded prime in some G-graded near-ring N ×M with (N × M) g = N g × M g , ∀g ∈ G, then every graded ideal is graded prime of one of N and M while the other is the trivial near-ring. Proof. Suppose that every graded ideal is graded prime in some N × M with neither N nor M is Zero near-ring. Hence, {0} × {0} is graded prime which contradicts previous proposition. in the previous proposition, it is not necessary that {0} is graded prime ideal in N if {0} is graded prime in M. For example, if (G = {0, 1}, +) where + then M = Z 2 and N = Z 8 both are G-graded near rings, where, M 0 = Z 2 , M 1 = {0} and N 0 = Z 8 , N 1 = {0}. Φ : Z 8 → Z 2 , Φ(x) = x satisfies the mentioned conditions in the previous proposition. However, {0} is not graded prime in Z 8 although it is graded prime ideal in Z 2 . We can also note that it is not necessary that {0} is graded prime ideal in M if {0} is graded prime in N. For example, Let G be defined as Example 2.2, and let N = Z and M = Z 8 Theorem 2 . 18 . 218Let N and M be any two G-graded near-rings with identity and P be a graded proper ideal of N. Then P is graded prime if and only if P × M is a graded prime ideal of N × M Proof. Let P be a graded prime ideal of N and let (A × B) and (C × D) be ideals of N × M such that (A × B) g (C × D) h ⊆ (P × M) gh . Then ( Corollary 2 . 19 . 219Let N and M be two G-graded near-rings with identity. If every proper ideal of N and M is a product of graded prime ideals, then every proper ideal of N × M is a product of graded prime ideals. Proof. Let I be a proper ideals of N and J be a proper ideals of M, Such that I = A 1 ... A n and J = B 1 ... B m where each A i and B j is graded prime. If the proper ideal is of the form I×M, then I×M = A 1 ...A n ×M can be written as (A 1 × N 2 ) ... (A n × M) which is by Theorem 2.18 a product of graded prime ideals. Similarly, If the proper ideal is of the form N × J, then it is a product of graded prime ideals. If the proper ideals is of the form I × J, then it can be written as A 1 ... A n × B 1 ... B m = (A 1 ... A n × M) (N × B 1 ... B m ) = (A 1 × M) ... (A n × M) (N × B 1 ) ... (N × B m ). Which is a product of graded prime ideals. Theorem 2.20. Let N and M be two G-graded near-rings with unity. Then a graded ideal P of N × M with (N × M) g = N g × M g , ∀g ∈ G is graded prime if and only if it has one of the following two forms: 1. (I × M), where I is a graded prime ideal of N. 2. (N × J), where J is a graded prime ideal of M. Proof. Let P be a proper graded ideal of N × M. Then P has one of the following three forms: (I × M) where I is proper ideal of N, (N × J), where J is proper ideal of M, or I × J, where I g = N g and J h = M h for some g and h belongs to G. Proposition 2 . 21 . 221Let N and M be two non trivial G-graded near-rings. Then {0}×{0} can not be graded prime ideal of N × M with (N × M) g = N g × M g , ∀g ∈ G. Proof. Suppose that {0} × {0} is graded prime ideal for some G-graded near-ring N × M with (N × M) g = N g × M g , ∀g ∈ G. Take g and h belongs to G such that N g = {0} and M h = {0}. Since ({0} × N) g (M × {0}) h ⊆ ({0} × {0}) gh for any g and h belongs to G, then ({0} × N) g ⊆ ({0} × {0}) g or (M × {0}) h ⊆ ({0} × {0}) h . Therefore, N g = {0} or M h = {0}. Contradiction and hence neither ({0} × N) g ⊆ ({0} × {0}) g nor (M × {0}) h ⊆ ({0} × {0}) h , which implies {0} × {0} can not be graded prime ideal. Graded Near Rings. M Dumitru, L Nastasescu, B Toader, 24Analele Stiintifice ale Universitatii Ovidius ConstantaM. Dumitru, L. Nastasescu and B. Toader, Graded Near Rings, Analele Stiintifice ale Universitatii Ovidius Constanta, 24(1), pp.201-216, 2016. Almost Graded Prime Ideals. A Jaber, M Bataineh, H Khashan, Journal of Mathematics and Statistics. 44231A. Jaber, M. Bataineh and H. Khashan, Almost Graded Prime Ideals, Journal of Mathematics and Statistics, 4(4), p.231, 2008. Near-Rings and Their Links with Groups. J D P Meldrum, Pitman Pub-lishing CoLondonJ.D.P. Meldrum, Near-Rings and Their Links with Groups , Pitman Pub-lishing Co., London, 1985. . C Nastasescu, F Van Oystaeyen, Methods of Graded Rings. Springer-VerlagC. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Springer-Verlag, Berlin- Heidelberg, 2004. G Pilz, Near-Rings: The Theory and Its Applications. AmsterdamNorth Holland Publishing CoRevised EditionG. Pilz, Near-Rings: The Theory and Its Applications, Revised Edition, North Hol- land Publishing Co., Amsterdam, 1983.	[]
[ "System Identification of a Multi-timescale Adap- tive Threshold Neuronal Model 1", "System Identification of a Multi-timescale Adap- tive Threshold Neuronal Model 1" ]	[ "Amirhossein Jabalameli ", "Aman Behal ", "\nDepartment of Electrical Engineering and Computer Science\n‡ NanoScience Technology Center\nUniversity of Central Florida\n32816OrlandoFL\n", "\nUniversity of Central Florida\n32826OrlandoFL\n" ]	[ "Department of Electrical Engineering and Computer Science\n‡ NanoScience Technology Center\nUniversity of Central Florida\n32816OrlandoFL", "University of Central Florida\n32826OrlandoFL" ]	[ "Interna-tional Conference on Computational Advances in Bio and Medical" ]	1 An abridged version(Jabalameli and Behal, 2015)of this paper was presented at the In this paper, the parameter estimation problem for a multi-timescale adaptive threshold (MAT) neuronal model is investigated. By manipulating the system dynamics, which comprise of a non-resetting leaky integrator coupled with an adaptive threshold, the threshold voltage can be obtained as a realizable model that is linear in the unknown parameters. This linearly parametrized realizable model is then utilized inside a prediction error based framework to identify the threshold parameters with the purpose of predicting single neuron precise firing times. The iterative linear least squares estimation scheme is evaluated using both synthetic data obtained from an exact model as well as experimental data obtained from in vitro rat somatosensory cortical neurons. Resultsshow the ability of this approach to fit the MAT model to different types of fluctuating reference data. The performance of the proposed approach is seen to be superior when comparing with existing identification approaches used by the neuronal community.	null	[ "https://arxiv.org/pdf/1803.04236v1.pdf" ]	3,883,662	1803.04236	d30111f71d2d0965e24b3579ed34bcd2193def6f	System Identification of a Multi-timescale Adap- tive Threshold Neuronal Model 1 2015 Amirhossein Jabalameli Aman Behal Department of Electrical Engineering and Computer Science ‡ NanoScience Technology Center University of Central Florida 32816OrlandoFL University of Central Florida 32826OrlandoFL System Identification of a Multi-timescale Adap- tive Threshold Neuronal Model 1 Interna-tional Conference on Computational Advances in Bio and Medical 20151Spiking Neuron ModelPredicting Spike TimesMAT modelAdaptive ThresholdParameter EstimationSystem IdentificationLinear Parameterization 1 An abridged version(Jabalameli and Behal, 2015)of this paper was presented at the In this paper, the parameter estimation problem for a multi-timescale adaptive threshold (MAT) neuronal model is investigated. By manipulating the system dynamics, which comprise of a non-resetting leaky integrator coupled with an adaptive threshold, the threshold voltage can be obtained as a realizable model that is linear in the unknown parameters. This linearly parametrized realizable model is then utilized inside a prediction error based framework to identify the threshold parameters with the purpose of predicting single neuron precise firing times. The iterative linear least squares estimation scheme is evaluated using both synthetic data obtained from an exact model as well as experimental data obtained from in vitro rat somatosensory cortical neurons. Resultsshow the ability of this approach to fit the MAT model to different types of fluctuating reference data. The performance of the proposed approach is seen to be superior when comparing with existing identification approaches used by the neuronal community. Introduction As large-scale detailed network modeling projects are appearing in the computational neuroscience area, it becomes essential to construct easily identifiable lower dimensional models of single neurons. While lower dimensional models allow for construction of large-scale computational neuronal networks that can be simulated with ease, identifiability of the underlying neuronal models is necessary for the neuronal network to be able to mimic the computational properties of the biological structure being modeled in silico. Fortunately, a wide variety of single neuron models are available in literature. These models can broadly be categorized into two main groups: detailed biophysical models and simple phenomenological models, e.g., (Izhikevich, 2004;Jolivet et al., 2008b). Detailed biophysical Hodgkin-Huxley (Hodgkin and Huxley, 1952) type neuron models can accurately reproduce most behaviors of neurons, however, their complex dynamics and a high-dimensional parameter space make them an impractical choice as building blocks for large scale neuronal networks (Zhi et al., 2012). In spiking neuron models, since isolated spikes of a given neuron are similar, the shape of the action potential does not represent any information. Instead, it is the spike train, as a sequence of spikes, which is important (Gerstner et al., 2014). Due to this reason and high computational costs, simple phenomenological models such as leaky integrate-and-fire models have been proposed (Brunel and van Rossum, 2007;Stein, 1965) and developed to study the dynamics of neural networks (Gerstner and Kistler, 2002;Izhikevich, 2004). Recently, substantial efforts have been put in the expansion of leaky integrate and fire models for fitting of such models to data in order to reproduce quantitative features (Prinz et al., 2003;Huys et al., 2006). Several methods have been proposed that can accurately predict the timing of spikes, e.g., (Jolivet and Gerstner, 2004;Kobayashi et al., 2009;Kobayashi and Shinomoto, 2007;Clopath et al., 2007). Identification of model parameters can be performed by several methods. Although hand tuning of parameters may yield reasonable results, this process is labor intensive and impractical. In our previous work (Zhi et al., 2012), the versatile quadratic model proposed by Izhikevich (Izhikevich, 2003) was utilized to automatically identify experimentally obtained neuronal firing data. However, as noted in (Chen et al., 2011(Chen et al., , 2016, that model cannot be utilized for identification because it is unable to quantitatively represent the upstroke of the spike unless it is assumed that the model parameters are voltage-dependent (Izhikevich, 2007). In Jabalameli (2015), a two-stage linear parameter estimation strategy was explored based on a five parameter subthreshold model coupled with a voltage dependent threshold model but the estimation results were seen to be unsatisfactory. Another proposed model, viz., the Multi-timescale Adaptive Threshold model (MAT) (Kobayashi et al., 2009) shows great performance for both stationary and non-stationary fluctuating currents to replicate spike trains of experimental data (Yamauchi et al., 2011). While the threshold is dependent on five independent parameters, the authors a priori fix two time constant related model parameters and identify the other three parameters by maximizing a non-convex performance metric encoding for the coincidence of spikes between the model predictions and the experimental data. Thus, a systematic technique is still needed to automatically identify all five parameters so that the threshold model firing pattern is consistent with that of the experimental data. Furthermore, a convex cost function is needed in order to guarantee parameter convergence. The advantage of such a technique is that it can be utilized in a fully automated system that can identify the underlying neuronal models that are needed to design and implement a realistic biological computational structure. This work focuses on the development of the aforementioned automated identification technique. Specifically, we manipulate the original threshold equation (which is nonlinearly dependent on its parameters) into a linear-in-the-parameters model and then proceed to estimate the parameters in order to minimize in a least squares sense the error between the subthreshold voltage and the threshold estimate at the experimentally observed spiking times. To ensure meaningfulness of the values of the obtained parameter estimates, convex constraints are generated and imposed on the optimization. Results show that the proposed scheme outperforms existing strategies in terms of reproducing spike locations. Another novelty of the proposed method is that unlike many other methods that require the input and the reference membrane voltage of the neuron to tune their models, this method only needs the input current and the spike locations to fit the model to the reference data. The remainder of this paper is organized as follows. The basics of the multitimescale adaptive threshold (MAT) model are introduced in Section 2. Next, in Section 3, we pursue the algebraic manipulations needed to arrive at the proposed scheme to identify the MAT model parameters. Section 4 describes the steps of implementation. In Section 5, results are provided on model identification from different types of reference data followed by comparisons with results from existing approaches. Pertinent conclusions are drawn in Section 6. The MAT Model The Multi-timescale Adaptive Threshold (MAT) model (Kobayashi et al., 2009) was proposed by Kobayashi et al. for the purpose of predicting the timing of output spikes of neurons. The MAT model can be described by a subthreshold voltage V and a multitimescale adaptive threshold f . While it is possible in general to have an arbitrary number of timescales, the analysis in this paper will be limited to two timescales as similarly done in (Kobayashi et al., 2009). The subthreshold voltage can be obtained by the leaky integrator which is given by a first order differential equation (1) τ m dV dt = −V (t) + RI(t)(1) where V (t) denotes a membrane potential, I(t) is the injected input current, while τ m and R are parameters that describe leaky time constant and input resistance, respectively. While Equation (1) is the foundation of Generalized Linear Models and Spike Response Model (Gerstner et al., 2014), however, in the MAT model, the variable V (t) is not reset after reaching a constant or time/state dependent threshold. Instead, at the spiking instants defined by the intersection of V and f , the threshold variable f resets to a different value in the manner shown below f (t) = α 1 t k exp(−k 1 (t − t k )) + α 2 t k exp(−k 2 (t − t k )) + ω(2) where t k is the k th spike time, k 1 and k 2 denote inverses of the two threshold time constants, α 1 and α 2 denote the increments of the threshold at the spike instants, while ω is the threshold resting value. Furthermore, an absolute refractory period τ R is defined to prevent consecutive firing; consequently, within τ R period after a spike, the model cannot fire more spikes even if the subthreshold voltage is above the threshold. As seen in Figure (1), spikes (represented by arrows) are assumed to be generated whenever the subthreshold voltage reaches the threshold voltage from below. The moment of crossing is the so-called firing time at which instant the threshold voltage increases a certain amount and then starts decaying exponentially to its resting value. The Proposed Estimation Technique Pursuant to the development in the previous section, the MAT model can be completely characterized by 7 free parameters; where {τ m , R} are the leaky integrator parameters and {α 1 , α 2 , k 1 , k 2 , ω} are the spike threshold parameters. The resistance R does not affect the spike time prediction and only scales the subthreshold voltage (Kobayashi et al., 2009); furthermore, a common membrane time constant (τ m ) is extracted from the data and preselected for all simulations. Therefore, we focus our work to estimate the threshold parameters and we assert that θ 0 [α 1, α 2 , k 1 , k 2 , ω] T completely describes the model. In what follows, we develop an automatic method for estimating the MAT model parameters. Linear Parametrization While the static threshold representation of (2) is nonlinear with respect to the parameters k 1 and k 2 , a dynamic representation can be developed to acquire a linearly parameterized model. By taking Laplace transform of (2) and rearranging the terms, one can obtain (s 2 )F (s) = −[(k 1 + k 2 ) s + k 1 k 2 ]F (s) + [(α 1 + α 2 ) s + (α 1 k 2 + α 2 k 1 )] t k exp ( − t k s) +[ws 2 + w (k 1 + k 2 ) s + wk 1 k 2 ] 1 s (3) where s is the Laplace variable and F (s) represents the Laplace transform of f (t). A second order low pass filter 1 A = 1 s 2 + β 1 s + β 0(4) is employed in order to eliminate the model dependency on derivatives of the measurable signals (Zhi et al., 2012). The following can be obtained by applying the filter to both sides of (3): F (s) = ( β 1 s+β 0 A )F (s) − (k 1 + k 2 ) s A F (s) − k 1 k 2 1 A F (s) + (α 1 + α 2 ) s A t k exp(−t k s) + (α 1 k 2 + α 2 k 1 ) 1 A t k exp(−t k s) + wk 1 k 2 1 A 1 s + [w s 2 A 1 2 + w (k 1 + k 2 ) s A 1 s ] .(5) Since the last row of (5) vanishes beyond an initial transient, we can neglect it in the subsequent calculations to reduce the dimension of the parameter vector. Ultimately, the linearly parameterized (LP) model is described as follows f (t) = Ψ T (f, t, t k )θ + Φ(f, t) (6) where Φ(f, t) is a signal that is independent with respect to the model parameters and Ψ(f, t, t k ) ∈ R 5 is a regression vector. These signals are defined as follows Φ(f, t) L −1 ( β 1 s + β 0 A )F (s) (7) Ψ(f, t, t k ) L −1 [ s A F (s), 1 A F (s), s A t k exp(−t k s), 1 A t k exp(−t k s), 1 sA ] T(8) while θ ∈ R 5 is an unknown auxiliary parameter vector that is a nonlinear function of θ 0 and defined as follows θ = [−(k 1 + k 2 ), −k 1 k 2 , α 1 + α 2 , α 1 k 2 + α 2 k 1 , wk 1 k 2 ] T .(9) Algorithm Development According to the MAT model process, firing happens whenever the subthreshold voltage reaches the threshold voltage. In other words, f (t) and V (t) are equal at the spike instants. Since the main objective in this work is to fit the MAT model to the reference data in order to obtain a predictive model for the location of spike times, we define a prediction error variable as follows for each spike moment e t k f (t k ) − V (t k )(10) wheref (t) denotes an estimate of f (t). A natural cost function based on the prediction error for all spike instants can be defined as J t k e 2 t k and developed as follows J = t k (f (t) − V (t)) 2 = t k (Ψ(f, t, t k )θ + Φ(f, t) f (t) − V (t)) 2(11) based on the LP model derived in (6). Here,θ denotes an estimate for θ. Therefore, the objective is to find the MAT model parameters that minimize the prediction error θ = arg min J = t k (f (t) − V (t)) 2(12) However, certain constraints need to be introduced so that the parameter estimatesθ converge to physically meaningful values when mapped back to the actual parameter space θ 0 . The actual model parameters can be obtained in closed form as follows k 1 = max( −θ 1 ± θ 2 1 + 4θ 2 2 ) (13) k 2 = min( −θ 1 ± θ 2 1 + 4θ 2 2 ) (14) α 1 = θ 4 − k 1 θ 3 k 2 − k 1 , α 2 = θ 3 − α 1 , w = −θ 5 θ 2(15) where θ i denotes the i th component of θ. While there are no restrictions on α 1 , α 2 , and w other than that they are real (which is trivially enforced), k 1 and k 2 , being time constants, are needed by the model to be positive and real. Thus, the set of equations above suggest the inequalities θ 1 , θ 2 < 0 and θ 2 1 + 4θ 2 ≥ 0 to ensure the real positiveness of k 1 and k 2 . As seen in Figure 2, these inequalities define a non-convex region. Since k 1 and k 2 denote different timescales, a separation based on available data and studies suggests the choice 20 ≤ k 1 ≤ 500 , 2 ≤ k 2 ≤ 40 (the unit is 1/ sec). Based on this feasible range, the following set of convex constraints can be obtained −540 ≤ θ 1 ≤ −22 , − 2 × 10 4 ≤ θ 2 ≤ −40 38.5θ 1 − θ 2 ≤ −1482 −1.7θ 1 + θ 2 ≤ 0(17) which is a triangular area demarcated in Figure However, it is impractical to check for and enforce this condition at all times. Instead, we endeavor to enforce this constraint practically by observing the maximum value of V (t) \| t=tm between each pair of spikes and enforcing the following convex constraint between each pair of spikes: f (t m ) − V (t m ) = Ψ T (t m )θ + Φ(t m ) − V (t m ) > 0.(18) This constraint essentially states that the subthreshold voltage at its peak between any pair of experimentally observed spikes is not allowed to cross over the threshold. Of course, this set of constraints does not avoid intersection altogether since the threshold is time varying but it is an easily implementable set of constraints that serves well to improve the model efficacy by reducing false positives as will be seen in the sequel. Implementation Procedure Although no iterative procedure is apparent at first glance when solving the constrained least squares problem defined by (12), (17) and (18) f on the RHS of (11) forces us instead to replace f withf θ k−1 at the k th iteration and apply the algorithm iteratively until the parameters converge. The implementation proceeds according to the following steps: Step 1: Generate subthreshold voltage V : Since the objective function (12) requires sub-threshold voltage V , we solve (1) by assuming R = 50M Ω and τ m = 5 ms. Following (Jolivet et al., 2008b), the excitation is performed with current generated from an Ornstein-Uhlenbeck process as follows I(t + dt) = I(t) − I(t) τ I dt + m I dt + s I ζ(t) √ dt(19) where m I and s I are parameters and ζ(t) is a zero-mean, unit variance Gaussian random variable. Similar to (Jolivet et al., 2008b), the process is generated and injected at a rate of 5 kHz and the correlation length τ I is 1 ms. The resulting current I(t) has a stationary Gaussian distribution with mean µ I = m I τ I and variance σ 2 I = s 2 I τ I /2. Step 2: Build signals Ψ(·) and Φ(·) by using (7) and ( 8): Since both are dependent on f which is not available for measurement, we begin with an initial guessθ 0 for θ 0 and buildf f (θ 0 , t, t k ) based on which we buildΨ(·) Ψ(f , t, t k ) andΦ(·) Φ(f , t). During estimation computations, f (t) does not reset at the times where the threshold crosses the subthreshold, instead, the generated f (t) fires at the reference (experimentally obtained) spike times, t k . Step 3: Minimize the objective function (11) to obtainθ subjects to the set of constraints (17) and (18). The loop error is defined as follows: e(f ) t k (Ψ(f , t, t k )θ + Φ(f , t) − V (t)) 2(20) where e(f ) denotes the error off that is generated byθ at the end of the loop. Since the error function is a quadratic function of variables which is subject to linear constraints on those variables, the optimization problem (12) at each step is formulated as a Quadratic Programming problem. Step 4: Solveθ to obtainθ 0 according to (13)-(15). Step 5: Finally,θ 0 is updated at Step 2 with new parameters and the procedure is repeated until all parameters converge to constant values. Evaluation of Prediction The error function defined in (11) is useful for estimating parameters but it cannot be an evaluation criterion since the main aim of this work is to predict the spike train produced by the neurons. While the firing rate of a spike train provides helpful information, yet, the evaluation of similarity between two spike trains is needed to capture local artifacts. Several measures exist for comparing the spike train predicted by the model and the spike train generated by the reference data. A popular index known as the coincidence factor has been proposed in (Jolivet et al., 2008b). This coincidence factor Γ measures both the similarity and dissimilarity of two spike train by considering the spiking rate and coincident spikes. Γ is calculated as follows (Jolivet et al., 2008b) Γ = N Coinc − < N Coinc > N Data + N M odel × 2 1 − 2υ∆(21) where N Data is the number of spikes in the reference spike train, N M odel is the number of spikes in the predicted spike train, N Coinc is the number of coincidences with precision ∆ between the two spike trains, and < N coinc > is the expected number of coincidences generated by a homogeneous Poisson process with the same rate υ as the spike train of the model. The factor 2/(1 − 2υ∆) normalizes Γ to a maximum value of 1 which is reached if and only if the spike train of the model reproduces exactly the reference spike train. Hence, after identifying the model parameters, we calculate the value of the Coincidence Factor to evaluate the predicted spike times. Results and Discussion In this section, the parameters of the MAT model are identified by the proposed method to match three types of reference data: (a) data from the MAT model, (b) noisy version of data from the MAT model, and (c) experimental data. The synthetic datasets are used to test validity and robustness of our approach. For our experimental data, we utilized a standard dataset from a Quantitative Neuron Modeling competition (Jolivet et al., 2008b) which includes the excitation input and single-electrode data recorded from a cortical pyramidal neuron in slices of rat barrel cortex. The details of the experimental protocol are available in (Rauch et al., 2003) and (La Camera et al., 2006). The injected input current is generated based on (19) and stimulation is done with currents of differ-ent means and fluctuation amplitudes. Additionally, for computing Coincidence Factor, the value of ∆ is set to 2 ms, since it is in the same range as the accuracy of measuring synaptic rise times in the soma of cortical pyramidal neurons (Jolivet et al., 2008b). Results from MAT Reference Data First, to show the ability of the proposed method to identify the model parameters, a reference train of spikes is produced by the exact MAT model using a known set of parameters shown in Table (1). The subthreshold voltage V is generated via the method described in Section 4 and is assumed to be noiseless. An initial guess for θ 0 (reasonably far away from the actual parameter values) is chosen as follows α 1 = 10 α 2 = 5 k 1 = 50 k 2 = 8 ω = 13 Following the procedure in Section 4, the parameters are seen to converge to final values as shown in Table (1). It is seen that the estimated values are very close to the actual model parameters. Figures (4) and (5) Results from Noisy MAT Reference Data In a neural recording experiment in vivo, the input current received by the neuron is divided into two components, a deterministic one and a stochastic one (Gerstner et al., 2014). The deterministic part does not vary during the trials with the same stimulus and the stochastic part represents all the remaining inputs which change during the trials. Therefore, to consider noise in the biological system, the stochastic component of the input is treated as noise which is added to the right hand side of the subthreshold voltage dynamics. Specifically, we modified the subthreshold voltage by adding Gaussian White Noise (GWN) to it. Results from a simulation with SN R = 35dB can be seen in (9) show the convergence of, respectively, the objective function value, the estimated parameters, and the threshold function along the estimation loops. We performed simulations for different SNR values to measure robustness of the proposed method against noise. Our simulations show that the randomness of the generated noise causes the parameters to not converge during some trials. Table (3) describes the obtained results for 10 trials of performed simulations. It is seen that the rate of Results from Experimental Data After confirming the ability of the proposed method to identify the exact model reference data, the estimation procedure is applied to in vitro experimental data from the Single Neuron Competition (Challenge A, 2007). Table ( After the identification process, the estimated parameters were utilized in a MAT model to predict the spike train. Obtained results for comparing the reference spike train and predicted spike train are shown in Table (5) for the three data samples as shown above. Figures (10) and (11) display the experimental traces and predicted MAT model voltages while the similarity of the spike trains is pointed out by the spike times. A similar figure for data#1 is not provided because the neuron membrane potential data corresponding to that current has not been made available in the published datasetonly firing times are available. As discussed earlier in the paper, a set of intersection constraints was applied to the objective function per (18). To clarify the contribution of these intersection constraints, comparisons were made between spike trains predictions from models estimated with and without the use of constraints. Table (6) shows the comparative results using an experimental data sample. It can be seen that the use of constraints increases the accuracy of predicting spike times by more than 20%. In fact, the intersection constraints drive the estimator to avoid undesirable spikes and as a result the firing rate is decreased. Although unconstrained estimator predicts higher percentage of target spikes correctly, its performance, represented by coincidence factor, is worse than the constrained one. Comparison with Existing Methods In (Jolivet et al., 2008a), a benchmark test was established to facilitate a systematic comparison of methods and models in predicting the activity of rat cortical pyramidal neurons. The provided data set includes four different input currents which were generated based on (19). For each injected current, four trials were recorded to observe if the neuron fires with high reliability. To evaluate the quantitative predictive feature of our approach, we compare our proposed method performance with the benchmark test reported results. Figure (12) indicates the average performance of our method along with the benchmark test results on the whole data set. The rawΓ is computed by averaging the values of Γ over the whole test set. Since for a certain input, the pyramidal neuron is more reliable than for the others (Mainen and Sejnowski, 1995), the normalized Γ A is also introduced. Γ A scales the raw Γ according to reliability of the neuron which is evaluated with trial-to-trial variation of the neuron recordings (Kobayashi et al., 2009). Of the four submissions for the challenge, the auto regressive method (AR) demonstrated the best performance. The AR method uses a mathematical model to estimate the membrane potential of the neuron and then the spike times were predicted by ad-justing a dynamic threshold to the estimated membrane potential. The parameters of this model were determined so that the coincidence factor Γ would be maximized. The carbon-copy method (CC) also yielded a good performance; however it does not benefit from any mathematical model. The CC method utilizes the mean and variance of fluctuating current and evokes a sequence of spike by considering the training data set. There were also two other anonymous submissions in the challenge whose performances are presented here. It is seen that the proposed method is superior to the challenge submissions by both theΓ and the Γ A metrics. Discussion In this study, we took advantage of the MAT model which comprises two dynamics. Although the simplified subthreshold leaky integrator dynamics fail to consider many aspects of neuronal dynamics (Gerstner et al., 2014), the leaky integrator free parameters provide adequate strength to track the neuron membrane potential trace. Furthermore, the threshold dynamic of MAT model makes effective use of its multi-timescale feature. Biologically speaking, the multiple timescales can be regarded as surrogates for ionic currents such that different timescale values represent fast transient current, non-inactivating current, etc. (Yamauchi et al., 2011). The proposed linear representation of the MAT model along with the novel objective function and constraints provides a framework for fitting the model to a single neuron recording in order to predict the quantitative features. The results for prediction of reference data generated from the exact MAT model confirm the validity of the manipulated equations and demonstrate the ability of the approach to find the best parameters even in the presence of noise. While the obtained results from experimental data demonstrate a high performance in predicting reference spike times, a detailed discussion is merited to analyze our approach more clearly. Our utilization of the objective function of (12) clarifies that the defined error function does not have an inherent mechanism to consider the non-spike moments. To overcome this issue, we added innovative constraints in our estimation procedure to control the threshold voltage from potential intersections with the subthreshold voltage during the period between two spikes. As shown above, the application of the constraints allows the identified model to generate the spike train with more confidence such that, by avoiding potential false positives, the firing rate is reduced which in turn leads to a higher predictive performance. To evaluate of the proposed approach, we also compared the predictive performance of our identified model to the submissions for Quantitative Neuron Modeling. Our results show the superiority of the proposed linear method to predict reference spike trains. In addition to improvements in prediction, our method benefits from a convex cost function while at least some of the other submissions in the challenge utilize the non-convex Γ Coincidence Factor as the cost function to maximize their prediction per-formance. Thus, our proposed approach also has a lower computational cost and a guaranteed global minimum which further underscores its superiority among the other methods. Finally, an obvious deficit of the MAT model is its inability to respond appropriately to rectangular and ramp currents. Therefore, future work will focus on model modification so it can response to not only fluctuating currents but also different input types. Conclusions In this paper, we proposed a constrained linear least squares algorithm to identify MAT model parameters, for predicting single neuron spike times. Our results show that the proposed identification method is robust to system noise and has the ability to find the best parameters to replicate the spike train. Moreover, the obtained experimental results indicates that our method has excellent performance in comparison to reported results from the Quantitative Neuron Modeling competition (Jolivet et al., 2008b). Convexity of the cost function is another advantage when compared with similar fitting approaches utilized currently by the neuronal community. While the MAT model succeeds in reproducing quantitative features of single neurons, it still lacks the capability to replicate different firing patterns; hence, there is room to investigate possible modifications of the model in the future. Figure 1 : 1Dynamics of the MAT model. When the threshold voltage (blue) intersects the subthreshold voltage (green), a spike is generated and the threshold jumps. 2 by the dashed blue and green lines and the vertical solid black line. The above set of constraints does not preclude intersection Figure 2 : 2Feasible Region for Solution of V and f between the experimentally observed spiking instants as shown inFigure 3. Figure 3 : 3Per the objective function, the method only minimizes the error at the spike times without considering possible intersections of subthreshold (green) and threshold voltage (red) between spike times. represent the convergence of the objective function value and the estimated parameters along the estimation loops. To illustrate the evolution of the threshold function, Figure (6) shows the generated f (t) in a few sample process loops. Figure 4 :Figure 5 :Figure 6 : 456Objective function error. The vertical axis is log scale and indicates the value of the error function along the estimation loops Evolution of identified parameters α 1 , α 2 , k 1 , k 2 and ω using noise-free synConvergence of estimated threshold during the loops Figure 8 : 8Evolution of Parameter Errors with Noisy Synthetic Data non-convergence and the average error per spike over the converged trials (ē) decreases as SNR becomes better. Figure 9 : 9Convergence of Threshold Trace during Iterative Process Figure 10 :Figure 11 : 1011Model prediction for fluctuating current according to experimental data#2. The top row is the subthreshold (green) and threshold (blue) voltages trace of MAT model while predicted spike times are specified by blue triangles. The bottom row, indicates the experimental membrane potential (magenta) from a single neuron, while the actual spike train is marked by magenta triangles. Model prediction for fluctuating current according to experimental data#11. The top row is the subthreshold (green) and threshold (blue) voltages trace of MAT model while predicted spike times are specified by blue triangles. The bottom row indicates the experimental membrane potential (magenta) from a single neuron, while the actual spike train is marked by magenta triangles. Figure 12 : 12Comparison of the proposed method to results of the challenge. Table 1 : 1Identified and Actual Parameters for the MAT reference data.θ 0 α 1 α 2 k 1 k 2 ω Identified Val. 3.93 0.48 98.39 4.71 15.13 Actual Val. Table ( 2 () and Figures (7)-(9).Table (2) shows that the estimated values are very close to the actual model parameters. A comparison with Table (1) shows that the estimates are quite robust with respect to noise at this level of SNR. Similar to Section 5.1, Figures(7), (8), and Table 2 : 2Identified and Actual Parameters for noisy (SN R = 35) MAT reference data.θ 0 α 1 α 2 k 1 k 2 ω Identified Val. 4.05 0.48 100.30 5.49 15.34 Actual Val. 4 0.5 100 5 15 Table 3 : 3Estimation results for noisy MAT model reference data SN R = 40 SN R = 35 SN R = 30Convergence 8/10 7/10 5/10 e 0.0273 0.0502 0.1669 Table 4 : 4Identified Parameters for the experimental reference data.θ 0 I (nA) α 1 α 2 k 1 k 2 ω data#1 0.62±0.34 15.4 -1.49 76 11.6 33.5 data#2 0.16±0.32 7.9 1.24 190 38 14.6 data#11 0.15±0.33 15.3 -5.7 62 38 16.3 Table 5 : 5Comparison of the predicted spike train similarity and reference spike train.N Data N M odel N Coinc Γ data#1 875 558 496 0.60 data#2 268 246 230 0.88 data#11 212 162 144 0.76 Table 6 : 6Comparison of predicted spike train for unconstrained/constrained estimationUnconstrained Constrained N Coinc /N M odel 69% 92% N Coinc /N Data 88% 51% Spikes / sec 176 77 Γ Coinc. 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[ "Two localization lengths in the Anderson transition on random graphs", "Two localization lengths in the Anderson transition on random graphs" ]	[ "I García-Mata \nInstituto de Investigaciones Físicas de Mar del Plata (IFIMAR)\nCONICET-UNMdP\nB7602AYL Mar del Plata3350FunesArgentina\n\nConsejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET)\nArgentina\n", "J Martin \nInstitut de Physique Nucléaire\nAtomique et de Spectroscopie\nCESAM\nUniversité de Liège\nBât. B15B -4000LiègeBelgium\n", "R Dubertrand \nInstitut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany\n", "O Giraud \nLPTMS\nCNRS\nUniv. Paris-Sud\nUniversité Paris-Saclay\n91405OrsayFrance\n", "B Georgeot \nLaboratoire de Physique Théorique\nIRSAMC\nUniversité de Toulouse\nCNRS\nUPS\nFrance\n", "G Lemarié \nLaboratoire de Physique Théorique\nIRSAMC\nUniversité de Toulouse\nCNRS\nUPS\nFrance\n" ]	[ "Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)\nCONICET-UNMdP\nB7602AYL Mar del Plata3350FunesArgentina", "Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET)\nArgentina", "Institut de Physique Nucléaire\nAtomique et de Spectroscopie\nCESAM\nUniversité de Liège\nBât. B15B -4000LiègeBelgium", "Institut für Theoretische Physik\nUniversität Regensburg\n93040RegensburgGermany", "LPTMS\nCNRS\nUniv. Paris-Sud\nUniversité Paris-Saclay\n91405OrsayFrance", "Laboratoire de Physique Théorique\nIRSAMC\nUniversité de Toulouse\nCNRS\nUPS\nFrance", "Laboratoire de Physique Théorique\nIRSAMC\nUniversité de Toulouse\nCNRS\nUPS\nFrance" ]	[]	We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two localization lengths: the largest one describes localization along rare branches and diverges at the transition, while the second one describes localization along typical branches and remains finite at criticality. We show numerically that both quantities can be extracted from several different physical quantities: wavefunction moments, correlation functions and spectral statistics. These different localization lengths are associated with two different critical exponents, which control the finite-size scaling properties of the system close to the transition. Our approach could be directly applied to the many-body localization transition and more generally to nonergodic properties of states in Hilbert space. arXiv:1904.08869v1 [cond-mat.dis-nn]	10.1103/physrevresearch.2.012020	[ "https://arxiv.org/pdf/1904.08869v2.pdf" ]	121,129,414	1904.08869	41e9b9ee3e632a7ad5c4d4b4e2674464f52e4f2f	Two localization lengths in the Anderson transition on random graphs (Dated: April 19, 2019) I García-Mata Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR) CONICET-UNMdP B7602AYL Mar del Plata3350FunesArgentina Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET) Argentina J Martin Institut de Physique Nucléaire Atomique et de Spectroscopie CESAM Université de Liège Bât. B15B -4000LiègeBelgium R Dubertrand Institut für Theoretische Physik Universität Regensburg 93040RegensburgGermany O Giraud LPTMS CNRS Univ. Paris-Sud Université Paris-Saclay 91405OrsayFrance B Georgeot Laboratoire de Physique Théorique IRSAMC Université de Toulouse CNRS UPS France G Lemarié Laboratoire de Physique Théorique IRSAMC Université de Toulouse CNRS UPS France Two localization lengths in the Anderson transition on random graphs (Dated: April 19, 2019) We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two localization lengths: the largest one describes localization along rare branches and diverges at the transition, while the second one describes localization along typical branches and remains finite at criticality. We show numerically that both quantities can be extracted from several different physical quantities: wavefunction moments, correlation functions and spectral statistics. These different localization lengths are associated with two different critical exponents, which control the finite-size scaling properties of the system close to the transition. Our approach could be directly applied to the many-body localization transition and more generally to nonergodic properties of states in Hilbert space. arXiv:1904.08869v1 [cond-mat.dis-nn] Introduction. There has been a huge interest recently in the nonergodic properties of many-body states [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], in particular related to many-body localization [17,18] (see [19][20][21][22] for recent reviews). In such problems, the structure of Hilbert space is tree-like, and it was found that in many cases the states do not explore ergodically all the branches. The problem of Anderson localization on random graphs is a simple one-particle model which is believed to capture this physics, and has attracted recently a strong interest . On the finite Bethe lattice (tree with boundary) there is now a consensus that with decreasing disorder there is a transition from a localized phase to a nonergodic delocalized multifractal phase [27,33,35,38,39,41,43,49]. For generic random graphs (with loops and without boundary) the situation is still debated but several numerical and analytical studies point toward a transition from a localized to an ergodic delocalized phase with however nonergodic properties below a certain scale which diverges exponentially at the transition [32, 35-37, 43, 48]. However, a precise description of such a nonergodic behavior is still lacking. In this article, we clarify the nonergodic properties of the Anderson transition on random graphs showing, by a combination of extensive numerical simulations and scaling theory arguments, that the localized and critical regimes are characterized by two localization lengths associated with two different critical exponents. This is in striking contrast with the standard finite-dimensional case, where only a single length is known to control the transition. We show that each localization length controls the behavior of specific observables: statistics of wavefunction amplitudes, correlation functions, and spectral statistics, through which they can be extracted numerically. In the context of the many-body localization transition, some theoretical arguments have pointed towards the existence of two localization lengths in the localized regime [51,52]. Our results may provide a way to identify such length scales in this related problem, and also more generally to study nonergodic properties of states in Hilbert space Physical picture. At strong disorder, there are theoretical arguments (forward scattering approximation) [23,[53][54][55] that relate Anderson localization to the problem of directed polymers. Directed polymers on trees display a glassy phase with strong non-ergodic properties, exploring only few branches instead of the exponentially many available [56][57][58]. The analogy thus suggests that Anderson localized states on random graphs are located on rare branches, along which they are exponentially localized with a localization length ξ , which is the usual localization length diverging at the transition. To describe the behavior on other branches, we postulate the existence of a much smaller length ξ ⊥ characterizing the exponential decay away from the rare branches. Each of these two localization lengths can be characterized (numerically) using specific observables. First, the wavefunction amplitude moments P q = i \|ψ i \| 2q for large values of q (with q > 1) focus on large amplitude values and therefore reflect the localization properties of the rare populated branches governed by ξ . In contrast, values of q < 0.5 focus on small wavefunction amplitudes, and reflect the bulk localization properties controlled by ξ ⊥ . Second, correlation functions of different types reflect these two different lengths. The standard average correlation function [37,59] is dominated by the rare branches and thus by ξ whereas a suitably defined typical correlation function is controlled by ξ ⊥ . Lastly, the behavior < l a t e x i t s h a 1 _ b a s e 6 4 = " v u z R 6 p 0 2 g E Y r W E y F q l w w D G 4 V Y k c = " > A A A C 2 n i c d V H L b h M x F H W G V w m v F C R Y s L E Y I b G K Z t I F L C v a B c t W J W 2 l e B Q 8 n j u p F d s z s j 1 t I 9 c b d o g t f 8 A S + C A 2 f A u e C Z F o 0 1 7 J 8 t E 9 5 + i + 8 l p w Y 5 P k d y + 6 d f v O 3 X s b 9 / s P H j 5 6 / G S w + f T Q V I 1 m M G a V q P R x T g 0 I r m B s u R V w X G u g M h d w l M 9 3 W v 7 o F L T h l f p o F z V k k s 4 U L z m j N q S m g x d E U n v C q H A 7 f u q I h X P r 6 K n 3 0 0 G c D L e S N v A 6 S I f d n 8 T b z / f / f E I I 7 U 0 3 e z 9 I U b F G g r J M U G M m a V L b z F F t O R P g + 6 Q x U F M 2 p z O Y B K i o B J O 5 b g K P X 4 d M g c t K h 6 c s 7 r L / O x y V x i x k H p R t v + Y q 1 y a v 4 y a N L d 9 l j q u 6 s a D Y s l D Z C G w r 3 K 4 D F 1 w D s 2 I R A G W a h 1 4 x O 6 G a M h u W 1 i c K z l g l J V W F I 3 O w f p J m 7 g L H K d F U z c J U + P q 4 b M w 1 7 Y x E d K Z g 9 2 s C d k V x E S Q F l I Q r B T p O 4 5 F b k Y G L R 6 v 6 N z W w 3 s T B r n c H 4 c J a 4 t 3 u v K s b 4 p v B 4 W i Y B r w f 7 v w e L W M D v U S v 0 B u U o r d o G 3 1 A e 2 i M G P L o O / q J f k U k + h x 9 i b 4 u p V H v n + c Z u h T R t 7 + Y h 9 z Z < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + F E 8 E T o U 1 A K Y I K v m I S y j p L d s p u w = " > A A A C 2 n i c d V H L b h M x F H W G V w m v F C R Y s L E Y I b G K Z t I F L K u 2 C 5 a t S t p K 8 S j y e O 6 k V m z P y P Y U I t c b d o g t f 8 A S + h f 8 B B u + B c + E S L R p r 2 T 5 6 J 5 z d F 9 5 L b i x S f K 7 F 9 2 6 f e f u v Y 3 7 / Q c P H z 1 + M t h 8 e m S q R j M Y s 0 p U + i S n B g R X M L b c C j i p N V C Z C z j O 5 7 s t f 3 w G 2 v B K f b C L G j J J Z 4 q X n F E b U t P B C y K p P W V U u F 0 / d c T C J + v o m f f T Q Z w M t 5 I 2 8 D p I h 9 2 f x N v P D / 7 w i 5 1 f + 9 P N 3 g 9 S V K y R o C w T 1 J h J m t Q 2 c 1 R b z g T 4 P m k M 1 J T N 6 Q w m A S o q w W S u m 8 D j 1 y F T 4 L L S 4 S m L u + z / D k e l M Q u Z B 2 X b r 7 n K t c n r u E l j y 3 e Z 4 6 p u L C i 2 L F Q 2 A t s K t + v A B d f A r F g E Q J n m o V f M T q m m z I a l 9 Y m C j 6 y S k q r C k T l Y P 0 k z d 4 7 j l G i q Z m E q f H 1 c N u a a d k Y i O l O w + z U B u 6 I 4 D 5 I C S s K V A h 2 n 8 c i t y M D F o 1 X 9 m x p Y b + J w z 7 v D c G E t 8 V 5 3 3 t U N 8 c 3 g a D R M A z 4 I d 9 5 B y 9 h A L 9 E r 9 A a l 6 C 3 a R u / R P h o j h j z 6 j n 6 i i 4 h E n 6 M v 0 d e l N O r 9 8 z x D l y L 6 9 h d j / t 6 V < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + F E 8 E T o U 1 A K Y I K v m I S y j p L d s p u w = " > A A A C 2 n i c d V H L b h M x F H W G V w m v F C R Y s L E Y I b G K Z t I F L K u 2 C 5 a t S t p K 8 S j y e O 6 k V m z P y P Y U I t c b d o g t f 8 A S + h f 8 B B u + B c + E S L R p r 2 T 5 6 J 5 z d F 9 5 L b i x S f K 7 F 9 2 6 f e f u v Y 3 7 / Q c P H z 1 + M t h 8 e m S q R j M Y s 0 p U + i S n B g R X M L b c C j i p N V C Z C z j O 5 7 s t f 3 w G 2 v B K f b C L G j J J Z 4 q X n F E b U t P B C y K p P W V U u F 0 / d c T C J + v o m f f T Q Z w M t 5 I 2 8 D p I h 9 2 f x N v P D / 7 w i 5 1 f + 9 P N 3 g 9 S V K y R o C w T 1 J h J m t Q 2 c 1 R b z g T 4 P m k M 1 J T N 6 Q w m A S o q w W S u m 8 D j 1 y F T 4 L L S 4 S m L u + z / D k e l M Q u Z B 2 X b r 7 n K t c n r u E l j y 3 e Z 4 6 p u L C i 2 L F Q 2 A t s K t + v A B d f A r F g E Q J n m o V f M T q m m z I a l 9 Y m C j 6 y S k q r C k T l Y P 0 k z d 4 7 j l G i q Z m E q f H 1 c N u a a d k Y i O l O w + z U B u 6 I 4 D 5 I C S s K V A h 2 n 8 c i t y M D F o 1 X 9 m x p Y b + J w z 7 v D c G E t 8 V 5 3 3 t U N 8 c 3 g a D R M A z 4 I d 9 5 B y 9 h A L 9 E r 9 A a l 6 C 3 a R u / R P h o j h j z 6 j n 6 i i 4 h E n 6 M v 0 d e l N O r 9 8 z x D l y L 6 9 h d j / t 6 V < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p o 9 l C c d J V W C g V C W Q w j F B 2 A s O 6 E k = " > A A A C 2 n i c d V H L b h M x F H W G V w m v F B Y s 2 F i M k F h F M 2 E B y 4 p 2 w b K o T V s p H k V 3 P D e p F d s z s j 2 F y J 0 N O 8 S W P 2 A J f B B / U 8 8 0 k W j T X s n y 0 T 3 n 6 L 7 y S g r r k u R f L 7 p z 9 9 7 9 B 1 s P + 4 8 e P 3 n 6 b L D 9 / M i W t e E 4 5 q U s z U k O F q X Q O H b C S T y p D I L K J R 7 n i 9 2 W P z 5 D Y 0 W p D 9 2 y w k z B X I u Z 4 O B C a j p 4 y R S 4 U w 7 S 7 z Z T z x x + d R 7 O m m Y 6 i J P h u 6 Q N u g n S Y f c n M V n F / n S 7 9 5 s V J a 8 V a s c l W D t J k 8 p l H o w T X G L T Z 7 X F C v g C 5 j g J U I N C m / l u g o a + C Z m C z k o T n n a 0 y / 7 v 8 K C s X a o 8 K N t + 7 X W u T d 7 E T W o 3 + 5 B 5 o a v a o e a X h W a 1 p K 6 k 7 T p o I Q x y J 5 c B A D c i 9 E r 5 K R j g L i y t z z R + 4 a V S o A v P F u i a S Z r 5 c x q n z I C e h 6 n o z X H V m B v o j E x 2 p m B v N g T 8 m u I 8 S A q c M a E 1 m j i N R 3 5 N B i 4 e r e v f 1 s B m E w d 7 j T 8 I F z a K 7 n X n X d + Q 3 g 6 O R s M 0 4 M 9 J v P N x d e g t 8 o q 8 J m 9 J S t 6 T H f K J 7 J M x 4 a Q h v 8 g f 8 j d i 0 b f o e / T j U h r 1 V p 4 X 5 E p E P y 8 A b M / a 9 A = = < / l a t e x i t > C typ < l a t e x i t s h a 1 _ b a s e 6 4 = " v Y b x I P / C z W k x c M e x l j Z s f S k Y U 2 4 = " > A A A C 2 3 i c d V H L b t Q w F L 0 T X m V 4 T W G D x M Y i I L E a J d M F L C v a B c u i M m 2 l S T R y n J u p N b Y T 2 U 7 L y A 0 b d o g t f 8 A O w Z I 1 v 8 G H s M f J M B L t t F e y f H T P O b q v r B L c 2 C j 6 3 Q u u X b 9 x 8 9 b G 7 f 6 d u / f u P x h s P j w w Z a 0 Z j l k p S n 2 U U Y O C K x x b b g U e V R q p z A Q e Z v O d l j 8 8 Q W 1 4 q d 7 Z R Y W p p D P F C 8 6 o 9 a n p 4 H E i q T 1 m V L i d Z u o S i + + t 8 7 q m m Q 7 C a L g V t U H W Q T z s / i j c f v b n 5 y 8 A 2 J t u 9 r 4 l e c l q i c o y Q Y 2 Z x F F l U 0 e 1 5 U x g 0 0 9 q g x V l c z r D i Y e K S j S p 6 0 Z o y H O f y U l R a v + U J V 3 2 f 4 e j 0 p i F z L y y b d h c 5 N r k Z d y k t s W r 1 H F V 1 R Y V W x Y q a k F s S d p 9 k J x r Z F Y s P K B M c 9 8 r Y c d U U 2 b 9 1 v q J w l N W S k l V 7 p I 5 2 m Y S p + 6 M h H G i q Z r 5 q c j l c d 6 Y a d o Z E 9 G Z v L 1 Z E 7 A L i j M v y b F I u F K o w z g c u R X p u X C 0 q n 9 V A + t N 7 O 8 2 b t + f W E u y 2 5 1 3 d U N y N T g Y D W O P 3 / o 7 v 4 Z l b M A T e A o v I I a X s A 1 v Y A / G w O A D f I X v 8 C N I g 4 / B p + D z U h r 0 / n k e w b k I v v w F C P H e U w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " v Z D p H Y V h z u z m q 4 h F J 5 6 0 q R M y q w c = " > A A A C 2 3 i c d V H L b t Q w F P W E V 5 n y m M I G i Y 1 F Q G I 1 S o Y F L C v a B c u i M m 2 l S T R y n J u p N b Y T 2 T e F k R s 2 7 B B b / o A d g i U / w F f w I b D G y T A S 7 b R X s n x 0 z z m 6 r 6 y S w m I U / e o F V 6 5 e u 3 5 j 4 2 Z / 8 9 b t O 3 c H W / c O b F k b D m N e y t I c Z c y C F B r G K F D C U W W A q U z C Y T b f a f n D E z B W l P o N L i p I F Z t p U Q j O 0 K e m g w e J Y n j M m X Q 7 z d Q l C O / Q e V 3 T T A d h N H w W t U H X Q T z s / i j c f v z 7 x 8 + T z T 9 7 0 6 3 e 1 y Q v e a 1 A I 5 f M 2 k k c V Z g 6 Z l B w C U 0 / q S 1 U j M / Z D C Y e a q b A p q 4 b o a F P f C a n R W n 8 0 0 i 7 7 P 8 O x 5 S 1 C 5 V 5 Z d u w P c + 1 y Y u 4 S Y 3 F i 9 Q J X d U I m i 8 L F b W k W N J 2 H z Q X B j j K h Q e M G + F 7 p f y Y G c b R b 6 2 f a H j L S 6 W Y z l 0 y B 2 w m c e p O a R g n h u m Z n 4 p e H G e N m W G d M Z G d y d u b N Q E / p z j 1 k h y K R G g N J o z D k V u R n g t H q / q X N b D e x P 5 u 4 / b 9 i Y 2 i u 9 1 5 V z e k l 4 O D 0 T D 2 + L W / 8 0 u y j A 3 y k D w i T 0 l M n p N t 8 o r s k T H h 5 D 3 5 Q r 6 R 7 0 E a f A g + B p + W 0 q D 3 z 3 O f n I n g 8 1 9 D h N / N < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " v Z D p H Y V h z u z m q 4 h F J 5 6 0 q R M y q w c = " > A A A C 2 3 i c d V H L b t Q w F P W E V 5 n y m M I G i Y 1 F Q G I 1 S o Y F L C v a B c u i M m 2 l S T R y n J u p N b Y T 2 T e F k R s 2 7 B B b / o A d g i U / w F f w I b D G y T A S 7 b R X s n x 0 z z m 6 r 6 y S w m I U / e o F V 6 5 e u 3 5 j 4 2 Z / 8 9 b t O 3 c H W / c O b F k b D m N e y t I c Z c y C F B r G K F D C U W W A q U z C Y T b f a f n D E z B W l P o N L i p I F Z t p U Q j O 0 K e m g w e J Y n j M m X Q 7 z d Q l C O / Q e V 3 T T A d h N H w W t U H X Q T z s / i j c f v z 7 x 8 + T z T 9 7 0 6 3 e 1 y Q v e a 1 A I 5 f M 2 k k c V Z g 6 Z l B w C U 0 / q S 1 U j M / Z D C Y e a q b A p q 4 b o a F P f C a n R W n 8 0 0 i 7 7 P 8 O x 5 S 1 C 5 V 5 Z d u w P c + 1 y Y u 4 S Y 3 F i 9 Q J X d U I m i 8 L F b W k W N J 2 H z Q X B j j K h Q e M G + F 7 p f y Y G c b R b 6 2 f a H j L S 6 W Y z l 0 y B 2 w m c e p O a R g n h u m Z n 4 p e H G e N m W G d M Z G d y d u b N Q E / p z j 1 k h y K R G g N J o z D k V u R n g t H q / q X N b D e x P 5 u 4 / b 9 i Y 2 i u 9 1 5 V z e k l 4 O D 0 T D 2 + L W / 8 0 u y j A 3 y k D w i T 0 l M n p N t 8 o r s k T H h 5 D 3 5 Q r 6 R 7 0 E a f A g + B p + W 0 q D 3 z 3 O f n I n g 8 1 9 D h N / N < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " m A v Y p 6 Q G 6 0 x e U D R v y d 8 4 z b 6 t N i g = " > A A A C 2 3 i c d V H L b t Q w F P W k P M r w m p Y N E h u L C I n V K B k W s K x o F 1 0 W l W k r T a L R j X M z t c Z 2 I t u h H b l h 0 x 1 i y x + w Q / A / / A 1 O O i P R T n s l y 0 f 3 n K P 7 y i r B j Y 2 i v 7 1 g 4 9 7 9 B w 8 3 H / U f P 3 n 6 7 P l g a / v I l L V m O G a l K P V J B g Y F V z i 2 3 A o 8 q T S C z A Q e Z / P d l j / + g t r w U n 2 2 i w p T C T P F C 8 7 A + t R 0 8 D K R Y E 8 Z C L f b T F 1 i 8 d w 6 r 2 u a 6 S C M h u + i N u g 6 i I f d H 4 V k G Q f T r d 6 v J C 9 Z L V F Z J s C Y S R x V N n W g L W c C m 3 5 S G 6 y A z W G G E w 8 V S D S p 6 0 Z o 6 B u f y W l R a v + U p V 3 2 f 4 c D a c x C Z l 7 Z N m x u c m 3 y N m 5 S 2 + J D 6 r i q a o u K X R U q a k F t S d t 9 0 J x r Z F Y s P A C m u e + V s l P Q w K z f W j 9 R e M Z K K U H l L p m j b S Z x 6 i 5 o G C c a 1 M x P R W + P 6 8 Z M Q 2 d M R G f y 9 m Z N w G 4 o L r w k x y L h S q E O 4 3 D k V q T n w t G q / l 0 N r D d x u N e 4 Q 3 9 i L e l e d 9 7 V D e n d 4 G g 0 j D 3 + F I U 7 H 5 e H 3 i S v y G v y l s T k P d k h + + S A j A k j X 8 l P 8 p v 8 C d L g M v g W f L + S B r 2 l 5 w W 5 F s G P f 9 o O 2 4 Q = < / l a t e x i t > 19 ; the straight lines for large \|ψ\| 2 correspond to a fit P(log 10 \|ψ\| 2 ) ∼ (\|ψ\| 2 ) −β . Last panels: average and typical correlation functions Cav and Ctyp for N = 2 19 . The fitting lines correspond to Cav ∼ K −r e −r/ξ /r α and Ctyp ∼ e −r/ξ ⊥ . of spectral statistics at small energy distance is dominated by bulk localization and thus by ξ ⊥ . Importantly, we find that ξ and ξ ⊥ are associated to two different critical exponents. Model. We consider a generic class of random graphs [60][61][62] built by taking a one-dimensional Anderson model of N sites with periodic boundary conditions, and adding pN shortcut links between random pairs of sites ( pN is the integer part of pN ). The Hamiltonian reads H = N i=1 ε i \|i i\| + \|i i + 1\| + pN k=1 \|i k j k \| + h.c.(1) The on-site disorder is described by random variables ε i of zero mean with a Gaussian distribution of standard deviation W . The second term describes nearest-neighbor hopping and the third term the long-range links between (i k , j k ) with \|i k − j k \| > 1. Such a graph has locally a tree-like structure with an average branching number K ≈ 1 + 2p and an average branching distance ≈ 1/(2p). This type of graph ("small world networks" [63]) is similar to random regular graphs when p → 1/2. For all p, it displays loops of typical size ∼ log N , hence has no boundary, and the diameter (maximal distance between sites) d N increases as ∼ log N , making the system effectively infinite-dimensional. For our numerical investigations we use exact diagonalization of very large sparse matrices of sizes up to N = 2 21 with the Jacobi-Davidson method [64,65], using in general, for each disorder and graph realization, 16 eigenvalues and eigenfunctions closest to E = 0. We average over 3000 to 15000 disordered graphs realizations, depending on N . It has been established [62] that the model (1) presents an Anderson transition at a certain value of disorder W c (p). Recently, the critical properties of the transition through the finite-size scaling properties of the wavefunction moments P q for large q > 1 were investigated [36]. On the localized side these high q moments are controlled by ξ which were found to diverge at the transition as ξ ∼ (W − W c ) −ν with ν ≈ 1. On the delocalized side, an ergodic behavior at a scale larger than a nonergodic volume Λ diverging exponentially at the transition as log Λ ∼ (W c − W ) −κ with κ ≈ 0.5 was found. These observations agree with the analytical predictions of [66][67][68]. A simple model for ξ ⊥ . For small values of q, at the root of a finite Bethe lattice, a strong multifractal behavior [24,59] occurs in the localized phase [29,35], that is, for large system sizes, moments scale as P q ∼ N −τ * q with τ * q = q q * − 1 for q < q * ; τ * q = 0 for q > q * . (2) One has q * = q * c = 0.5 at the transition [59] and q * decreases with increasing W away from W c [35]. This behavior can be interpreted as a manifestation of ξ ⊥ . Indeed, let us consider a wavefunction exponentially localized at the root of a tree with connectivity K and depth d, with the same localization length ξ ⊥ along all the branches. A simple calculation shows that for ξ ⊥ < 1/ ln K, the moment P q for such a wavefunction is: P q = d−1 r=0 K r [e −r/ξ ⊥ ] q d−1 r=0 K r e −r/ξ ⊥ q ∼ N −τ * q(3) where N = K d and τ * q is given by (2) with q * = ξ ⊥ ln K < 1 . In this picture, the strong multifractal behavior (2) is due to the exponential proliferation of sites at distance r from the root which compensates, for q < q * , the exponential decrease of the localized wavefunction. As q * = q * c = 0.5 at the transition, this simple model also suggests that the critical behavior is localized with ξ c ⊥ = q * c / ln K for the Bethe lattice. Wavefunction moments. For our model (1), we show in Fig. 1 2 20 ), β (squares) and ξ ⊥ ln K (circles) vs W for p = 0.06, extracted from the three panels of Fig. 1: left,τq; middle, P(\|ψ\| 2 ); and right, Ctyp, respectively. In the localized and critical regimes, q * ≈ β ≈ ξ ⊥ ln K. Red dotted line is a fit with the critical behavior we predict: localized, critical (W = W c ) and delocalized. In the localized regime,τ q clearly tends when N → ∞ towards τ * q in (2) with a q * < 0.5. q * can be determined by a linear fit ofτ q at small q where finite-size effects are negligible. q * = q * c − C(W − Wc) ν ⊥ , with q * c = 0.5, ν ⊥ = In the critical regime the same behavior is observed, with q * ≈ 0.5 = q * c . In the delocalized regime, the behavior clearly tends to the ergodic limit τ q = q − 1 for large N . Indeed, our determination of q * at small q gives 1 in the delocalized phase (see Fig. 2). Wavefunction amplitude distribution. In the Bethe lattice it was predicted [35] (see also [28,29]) that in the localized phase q * = β where β describes the right tail exponent of the distribution of wavefunction amplitudes at the root of the tree, P(\|ψ\| 2 ) ∼ (\|ψ\| 2 ) −1−β for large \|ψ\| 2 . Fig. 1 (middle) shows that for our system in the localized phase the distribution P(\|ψ\| 2 ) also follows an algebraic law. We obtain β ≤ 0.5 for all W ≤ W c . We note that in the directed polymer problem on random graphs, an exponent β < 1 is a marker of replica symmetry breaking [26,57,58], a characteristic property of many spin glasses [69]. In the delocalized phase P(\|ψ\| 2 ) is clearly not algebraic for large \|ψ\| 2 . Typical correlation function. In the last panels of Fig. 1 we show the average and typical correlation functions. We calculated them along the 1D lattice which can be seen as a typical branch. Without loss of generality, doing so also avoids the computation of correlations on the whole graph and allows us to reach very large system sizes. We define C av (r) = N i=1 \|ψ i \| 2 \|ψ i+r \| 2 and C typ (r) = exp ln( i \|ψ i \| 2 \|ψ i+r \| 2 ) . In the localized phase, we find C av ∼ K −r exp(−r/ξ )/r α , in accordance with [37]. The physical picture implies, and numerics confirm, that the typical correlation function C typ behaves in a different way C typ ∼ exp(−r/ξ ⊥ ). Indeed, C typ gives the typical exponential decay with ξ ⊥ along an arbitrarily chosen branch, namely the 1D lattice, whereas C av is dominated by the configurations where the rare populated branches coincide with the 1D lattice, and is thus controlled by ξ . At criticality ξ ⊥ remains finite while ξ diverges. This reflects in the fact that in the localized and critical regimes C typ C av . In the delocalized regime, C av and C typ have a very similar behavior, in accordance with the ergodic nature of this phase. In Fig. 2 we show the variation of q * , β and ξ ⊥ as a function of W for different system sizes N . In the delocalized regime W < W c , q * tends towards a plateau at q * = 1 (dashed line in Fig. 2), confirming ergodicity, with strong finite-size effects close to the transition. In the localized regime W > W c the curves of q * for different N collapse onto a single curve, indicating negligible finitesize effects. In this regime, the data for q * , β and ξ ⊥ ln K agree very well with each other, and thus, as in the simple model (3), we have q * = ξ ⊥ ln K [70]. These results confirm that the wavefunction moments, their amplitude distribution and the typical correlation function are controlled by ξ ⊥ . Moreover, q * , β and ξ ⊥ ln K are well-fitted by q * c − C(W − W c ) ν ⊥ (red dotted line in Fig. 2), with q * c = 0.5, ν ⊥ = 0.5 and only two fitting parameters C and W c ≈ 1.74, a critical behavior that we will derive below. Finite size scaling. The description up to now has not addressed the strong finite-size effects clearly present close to the transition. A controlled procedure to address them uses finite-size scaling analysis [71]. This is well understood in finite dimension [72][73][74][75], but requires some care in random graphs, due to the exponential growth of the volume with linear system size (diameter). In [36], it was shown that surprisingly the moments P q for large values of q follow a different scaling depending on the side of the transition. On the localized side, P q = P c q F lin (d N /ξ ) with d N = log N , indicating a linear scaling, whereas on the delocalized side P q = P c q F vol (N/Λ) indicating an unusual volumic scaling. This volumic scaling, together with the observed ergodic behavior at small W , implies an ergodic behavior for N Λ in the entire delocalized phase W < W c . Here P c q ≡ P q (W c ) is the critical moment at W = W c . In infinite dimension the two types of scaling are distinct. We note that similar behaviors have been observed recently for the many-body localization transition [10]. To describe the finite-size scaling of the P q for small q < 0.5, we generalize the scaling assumptions of [36] and assume a two-parameter scaling function: with ξ and Λ two scaling parameters. We can then recover the observed large size behavior given by (2) in the localized phase W > W c if we further assume that F in (4) has the asymptotic behavior P q P c q = F (X, Y ) , X = d N ξ , Y = N Λ ,(4)F (X, Y ) ∼ V (X) −Aq + Y τ c q ; W > W c ; X, Y 1,(5) with A a positive constant and V (X) ∼ e X the volume associated with the length X. Since P c q ∼ N −τ c q , with τ c q given by (2) for q * = q * c , then P c q V (X) −Aq ∼ N 1−q/q * with q * ≈ 1 q * c + A ξ −1 or 1 ξ ⊥ ≈ 1 ξ c ⊥ + A ln K ξ .(6) Together, (4) and (5) give P q ∼ N 1−q/q * + Λ −τ c q .(7) Thus P q ∼ N −τ * q , as observed in Fig. 1, which justifies the asymptotic form (5). In the delocalized regime, a volumic scaling (F (X, Y ) ∼ function of Y ) was observed at large q [36], together with an ergodic behavior P q ∼ N −(q−1) for N Λ, implying ergodicity at all q. However we found an exponentially large nonergodic volume Λ close to the transition, which leaves room to a nonergodic multifractal behavior at intermediate scales and thus a linear scaling (F (X, Y ) ∼ function of X) associated with it. The finite-size scaling of the P q for q = 0.25 and p = 0.06 displayed in Fig. 3 shows a very good collapse with a linear scaling on both sides of the transition. The asymptotic behavior of the scaling function is well-fitted by F (X, Y ) ∼ V (X) −Aq (see (5)) with A ≈ 1.9. The scaling parameter ξ is shown in the inset to diverge at the transition as ξ ∼ \|W − W c \| −ν ⊥ , with ν ⊥ ≈ 1 2 on both sides of the transition. This is in striking contrast with the result for large q > 1 where the localization length ξ diverges with an exponent ν ≈ 1 at the transition W h x i Q p q f I W W Z a g y o t m H 5 c G q 1 g d k G H y 1 9 v 1 O 2 a F 1 7 r r 3 U d j S g 9 b R 9 7 b 1 2 r J 1 o b u 1 7 7 X f t T + 3 v 2 s 9 6 v b 5 R 3 y y g q y s l 5 6 m 2 s O r P / g G J r t s L < / l a t e x i t > W < l a t e x i t s h a 1 _ b a s e 6 4 = " A o n I l r o b N h 1 + J D k g a S U 7 z G G K s j A = " > A A A F V n i c b V T b b t M w G M 7 W l Y 1 y 6 u C S m 0 K K N K S o i r N 2 b e 8 G T B o X C I 3 D D t I y J s d x M 2 u O E 8 X u I V j Z q / E a 8 A B w C W + A s J N 0 t N k s W X L + / / u + / 5 D f 9 m J K u L D t H y u r t b X 6 n f W N u 4 1 7 9 x 8 8 f N T c f H z E o 3 G C 8 C G K a J S c e J B j S h g + F E R Q f B I n G I Y e x c f e 5 R v t P 5 7 g h J O I f R Z p j M 9 C G D A y I g g K Z T r f r J 2 4 C W Z 4 i q I w h M y X 7 g S j 7 B S c S T e E 4 s I b S R N k W c N d R B A F c B R A 4 J n I E 9 A Y a T p V 3 F 4 m X Z / w m M K U i 5 T i i t t P / Z l C j B K I p J 9 m U n 3 d g m B F M g X q i 4 6 U Q / O T x v t 4 p C o v 0 g j h e I I z + X H / d S Y H w L I t 0 N u u Q I I E p i X C 2 b a t c l d A v k f H c x 3 Q 1 z p O V Y c u Q B w w t J x u 1 3 J 6 3 a p Q k G D M S t j O j j X Y t v q D q t I i x n E G W s Z y 7 J 0 q L M W U R t M 5 r t e z 9 A a D o c J x r H 4 7 C 8 S F O / K i W T K m W N q d f i 8 W l X 6 O k F c E K 1 q a C y v C d Z r z V G 7 + c s X M C 6 4 S i y 6 U z b i V V q Z d I X o U o s v s f 1 H X 1 K I O 6 T J a 1 O 9 y B C k 2 g e l I X a a I p s Q v / X r i 1 O x y 8 h V r S R P k x m f z S d T k M I 2 h j m e C W 9 g g c w v v q T h T / R o O 5 + a s 0 X j h s o g w H z P R d k V C I A s o T k h w I d p X V 4 0 F G g A o z G S j v T T n r e N 2 Q 2 m c N 0 2 7 Y + e r d f M A y o N p l O v g v P n L 9 S M 0 D l V U R C H n p 8 C O V Q y Y C I L y 2 z P m O F Z 9 g w G W M O Q 8 D b 0 b R n 1 p l 4 3 a w m N 1 q Z e s U 8 h T J b B s n I 0 Z Q Z F f C T W j Y i Y S W M x Z C A k b R U z I V x P M S N J 6 r 9 6 A T L U r d 6 l A 2 r e 1 R w I i u P V O P T H M 2 l c z d f l S 7 h N K W 5 8 g 4 5 l u D K i 2 4 e b h y O k A u w M + O O Z u t 2 z R h v H U e G 5 s G c D o G 7 v G W + P A O D R Q 7 V v t Z + 1 3 7 c / a 9 7 W / 9 X p 9 v Y C u r p S c J 8 b S q j f / A R J A 2 X 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A o n I l r o b N h 1 + J D k g a S U 7 z G G K s j A = " > A A A F V n i c b V T b b t M w G M 7 W l Y 1 y 6 u C S m 0 K K N K S o i r N 2 b e 8 G T B o X C I 3 D D t I y J s d x M 2 u O E 8 X u I V j Z q / E a 8 A B w C W + A s J N 0 t N k s W X L + / / u + / 5 D f 9 m J K u L D t H y u r t b X 6 n f W N u 4 1 7 9 x 8 8 f N T c f H z E o 3 G C 8 C G K a J S c e J B j S h g + F E R Q f B I n G I Y e x c f e 5 R v t P 5 7 g h J O I f R Z p j M 9 C G D A y I g g K Z T r f r J 2 4 C W Z 4 i q I w h M y X 7 g S j 7 B S c S T e E 4 s I b S R N k W c N d R B A F c B R A 4 J n I E 9 A Y a T p V 3 F 4 m X Z / w m M K U i 5 T i i t t P / Z l C j B K I p J 9 m U n 3 d g m B F M g X q i 4 6 U Q / O T x v t 4 p C o v 0 g j h e I I z + X H / d S Y H w L I t 0 N u u Q I I E p i X C 2 b a t c l d A v k f H c x 3 Q 1 z p O V Y c u Q B w w t J x u 1 3 J 6 3 a p Q k G D M S t j O j j X Y t v q D q t I i x n E G W s Z y 7 J 0 q L M W U R t M 5 r t e z 9 A a D o c J x r H 4 7 C 8 S F O / K i W T K m W N q d f i 8 W l X 6 O k F c E K 1 q a C y v C d Z r z V G 7 + c s X M C 6 4 S i y 6 U z b i V V q Z d I X o U o s v s f 1 H X 1 K I O 6 T J a 1 O 9 y B C k 2 g e l I X a a I p s Q v / X r i 1 O x y 8 h V r S R P k x m f z S d T k M I 2 h j m e C W 9 g g c w v v q T h T / R o O 5 + a s 0 X j h s o g w H z P R d k V C I A s o T k h w I d p X V 4 0 F G g A o z G S j v T T n r e N 2 Q 2 m c N 0 2 7 Y + e r d f M A y o N p l O v g v P n L 9 S M 0 D l V U R C H n p 8 C O V Q y Y C I L y 2 z P m O F Z 9 g w G W M O Q 8 D b 0 b R n 1 p l 4 3 a w m N 1 q Z e s U 8 h T J b B s n I 0 Z Q Z F f C T W j Y i Y S W M x Z C A k b R U z I V x P M S N J 6 r 9 6 A T L U r d 6 l A 2 r e 1 R w I i u P V O P T H M 2 l c z d f l S 7 h N K W 5 8 g 4 5 l u D K i 2 4 e b h y O k A u w M + O O Z u t 2 z R h v H U e G 5 s G c D o G 7 v G W + P A O D R Q 7 V v t Z + 1 3 7 c / a 9 7 W / 9 X p 9 v Y C u r p S c J 8 b S q j f / A R J A 2 X 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A o n I l r o b N h 1 + J D k g a S U 7 z G G K s j A = " > A A A F V n i c b V T b b t M w G M 7 W l Y 1 y 6 u C S m 0 K K N K S o i r N 2 b e 8 G T B o X C I 3 D D t I y J s d x M 2 u O E 8 X u I V j Z q / E a 8 A B w C W + A s J N 0 t N k s W X L + / / u + / 5 D f 9 m J K u L D t H y u r t b X 6 n f W N u 4 1 7 9 x 8 8 f N T c f H z E o 3 G C 8 C G K a J S c e J B j S h g + F E R Q f B I n G I Y e x c f e 5 R v t P 5 7 g h J O I f R Z p j M 9 C G D A y I g g K Z T r f r J 2 4 C W Z 4 i q I w h M y X 7 g S j 7 B S c S T e E 4 s I b S R N k W c N d R B A F c B R A 4 J n I E 9 A Y a T p V 3 F 4 m X Z / w m M K U i 5 T i i t t P / Z l C j B K I p J 9 m U n 3 d g m B F M g X q i 4 6 U Q / O T x v t 4 p C o v 0 g j h e I I z + X H / d S Y H w L I t 0 N u u Q I I E p i X C 2 b a t c l d A v k f H c x 3 Q 1 z p O V Y c u Q B w w t J x u 1 3 J 6 3 a p Q k G D M S t j O j j X Y t v q D q t I i x n E G W s Z y 7 J 0 q L M W U R t M 5 r t e z 9 A a D o c J x r H 4 7 C 8 S F O / K i W T K m W N q d f i 8 W l X 6 O k F c E K 1 q a C y v C d Z r z V G 7 + c s X M C 6 4 S i y 6 U z b i V V q Z d I X o U o s v s f 1 H X 1 K I O 6 T J a 1 O 9 y B C k 2 g e l I X a a I p s Q v / X r i 1 O x y 8 h V r S R P k x m f z S d T k M I 2 h j m e C W 9 g g c w v v q T h T / R o O 5 + a s 0 X j h s o g w H z P R d k V C I A s o T k h w I d p X V 4 0 F G g A o z G S j v T T n r e N 2 Q 2 m c N 0 2 7 Y + e r d f M A y o N p l O v g v P n L 9 S M 0 D l V U R C H n p 8 C O V Q y Y C I L y 2 z P m O F Z 9 g w G W M O Q 8 D b 0 b R n 1 p l 4 3 a w m N 1 q Z e s U 8 h T J b B s n I 0 Z Q Z F f C T W j Y i Y S W M x Z C A k b R U z I V x P M S N J 6 r 9 6 A T L U r d 6 l A 2 r e 1 R w I i u P V O P T H M 2 l c z d f l S 7 h N K W 5 8 g 4 5 l u D K i 2 4 e b h y O k A u w M + O O Z u t 2 z R h v H U e G 5 s G c D o G 7 v G W + P A O D R Q 7 V v t Z + 1 3 7 c / a 9 7 W / 9 X p 9 v Y C u r p S c J 8 b S q j f / A R J A 2 X 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A o n I l r o b N h 1 + J D k g a S U 7 z G G K s j A = " > A A A F V n i c b V T b b t M w G M 7 W l Y 1 y 6 u C S m 0 K K N K S o i r N 2 b e 8 G T B o X C I 3 D D t I y J s d x M 2 u O E 8 X u I V j Z q / E a 8 A B w C W + A s J N 0 t N k s W X L + / / u + / 5 D f 9 m J K u L D t H y u r t b X 6 n f W N u 4 1 7 9 x 8 8 f N T c f H z E o 3 G C 8 C G K a J S c e J B j S h g + F E R Q f B I n G I Y e x c f e 5 R v t P 5 7 g h J O I f R Z p j M 9 C G D A y I g g K Z T r f r J 2 4 C W Z 4 i q I w h M y X 7 g S j 7 B S c S T e E 4 s I b S R N k W c N d R B A F c B R A 4 J n I E 9 A Y a T p V 3 F 4 m X Z / w m M K U i 5 T i i t t P / Z l C j B K I p J 9 m U n 3 d g m B F M g X q i 4 6 U Q / O T x v t 4 p C o v 0 g j h e I I z + X H / d S Y H w L I t 0 N u u Q I I E p i X C 2 b a t c l d A v k f H c x 3 Q 1 z p O V Y c u Q B w w t J x u 1 3 J 6 3 a p Q k G D M S t j O j j X Y t v q D q t I i x n E G W s Z y 7 J 0 q L M W U R t M 5 r t e z 9 A a D o c J x r H 4 7 C 8 S F O / K i W T K m W N q d f i 8 W l X 6 O k F c E K 1 q a C y v C d Z r z V G 7 + c s X M C 6 4 S i y 6 U z b i V V q Z d I X o U o s v s f 1 H X 1 K I O 6 T J a 1 O 9 y B C k 2 g e l I X a a I p s Q v / X r i 1 O x y 8 h V r S R P k x m f z S d T k M I 2 h j m e C W 9 g g c w v v q T h T / R o O 5 + a s 0 X j h s o g w H z P R d k V C I A s o T k h w I d p X V 4 0 F G g A o z G S j v T T n r e N 2 Q 2 m c N 0 2 7 Y + e r d f M A y o N p l O v g v P n L 9 S M 0 D l V U R C H n p 8 C O V Q y Y C I L y 2 z P m O F Z 9 g w G W M O Q 8 D b 0 b R n 1 p l 4 3 a w m N 1 q Z e s U 8 h T J b B s n I 0 Z Q Z F f C T W j Y i Y S W M x Z C A k b R U z I V x P M S N J 6 r 9 6 A T L U r d 6 l A 2 r e 1 R w I i u P V O P T H M 2 l c z d f l S 7 h N K W 5 8 g 4 5 l u D K i 2 4 e b h y O k A u w M + O O Z u t 2 z R h v H U e G 5 s G c D o G 7 v G W + P A O D R Q 7 V v t Z + 1 3 7 c / a 9 7 W / 9 X p 9 v Y C u r p S c J 8 b S q j f / A R J A 2 X 4 = < / l a t e x i t > g Y I h z + X 7 v R S 5 7 w L I t 0 N m s Q c I U Z h X C 2 b S t 6 q m B f I 8 O J j q g q 3 W c u g 6 d g j i g b z l b W 5 b T 2 a o L h S n G r I J t b 1 u 9 T a v b q y t N Y x y n p 2 U s x 9 6 u w z J M a T y a 4 D o d S z + g 1 1 c 4 j t V n Z 6 E 4 d w M v H q c D i q W 9 0 e 0 k o j b P A H l l s n K k h b A i X J U 5 K W X + k y t m 0 X C d W E 6 h G s a 1 t K r s G t G j E F 3 k / 5 u 6 o p Z 9 S J f R s n + X I 0 i x C U x H 6 j Z F P C J + F d c b p 3 a X k 8 9 Y S 5 q g c D 6 e b K I m R 1 k C d T 4 T X M M G u V t G T 8 S p m l e / P 3 H n r d Z T l 8 W E + Z i J t i t S A l l I c U r C c 9 G + v G x N 0 Q B g Y I h z + X 7 v R S 5 7 w L I t 0 N m s Q c I U Z h X C 2 b S t 6 q m B f I 8 O J j q g q 3 W c u g 6 d g j i g b z l b W 5 b T 2 a o L h S n G r I J t b 1 u 9 T a v b q y t N Y x y n p 2 U s x 9 6 u w z J M a T y a 4 D o d S z + g 1 1 c 4 j t V n Z 6 E 4 d w M v H q c D i q W 9 0 e 0 k o j b P A H l l s n K k h b A i X J U 5 K W X + k y t m 0 X C d W E 6 h G s a 1 t K r s G t G j E F 3 k / 5 u 6 o p Z 9 S J f R s n + X I 0 i x C U x H 6 j Z F P C J + F d c b p 3 a X k 8 9 Y S 5 q g c D 6 e b K I m R 1 k C d T 4 T X M M G u V t G T 8 S p m l e / P 3 H n r d Z T l 8 W E + Z i J t i t S A l l I c U r C c 9 G + v G x N 0 Q B g Y I h z + X 7 v R S 5 7 w L I t 0 N m s Q c I U Z h X C 2 b S t 6 q m B f I 8 O J j q g q 3 W c u g 6 d g j i g b z l b W 5 b T 2 a o L h S n G r I J t b 1 u 9 T a v b q y t N Y x y n p 2 U s x 9 6 u w z J M a T y a 4 D o d S z + g 1 1 c 4 j t V n Z 6 E 4 d w M v H q c D i q W 9 0 e 0 k o j b P A H l l s n K k h b A i X J U 5 K W X + k y t m 0 X C d W E 6 h G s a 1 t K r s G t G j E F 3 k / 5 u 6 o p Z 9 S J f R s n + X I 0 i x C U x H 6 j Z F P C J + F d c b p 3 a X k 8 9 Y S 5 q g c D 6 e b K I m R 1 k C d T 4 T X M M G u V t G T 8 S p m l e / P 3 H n r d Z T l 8 W E + Z i J t i t S A l l I c U r C c 9 G + v G x N 0 Q B A U S 5 b 7 a N 2 S 7 H O V k x 7 w y 7 O 2 r w B K s M 0 q r N / t v L L 9 W M 0 i F Q e R C H n J 8 B O l C p M B U H F f R l w n K h J w R B L G H G e R d 6 c U 1 / T W a f 2 8 E R d 4 x n v C P J M C c w 6 x w N G U O z X U o 2 p G I s U l p s V Q c K C m A n 5 f I g Z S d f e q l u f q w E V I Z V I x 9 Z 3 S U g E t 9 6 o n w q z 9 t Q W X T y T e 4 T S t Q + Q 8 V w P B t Spectrum: finite-size scaling analysis of ηr (see text) for p = 0.06. N = 2 10 to N = 2 18 . Each symbol is a different size. Left: Raw data for ηr; inset: example of P (r) (see text) for localized, critical and delocalized phases. Right: Collapse of the data after a rescaling of the form ηr(W )/ηr(Wc) = F lin (log 2 N/ξ) with ξ the scaling parameter; inset: ξ vs W across the transition at Wc = 1.65. Solid lines are ξ ∼ \|W − 1.65\| −0.49 (delocalized branch) and ξ ∼ \|W − 1.65\| −0.51 (localized branch). [36]. From (6) we infer that q * = q * c − C(W − W c ) ν ⊥ with C a constant, in perfect agreement with numerical data (see Fig. 2). These results describe explicitly how ξ ⊥ approaches the finite ξ c ⊥ close to the transition. Spectral statistics. The new length scale ξ ⊥ can also be probed through spectral statistics. We study the distribution of the ratios r of spacings [76][77][78] between consecutive energy levels. The transition manifests itself through a change from Poisson statistics in the localized phase to a random matrix distribution in the delocalized phase, as seen in Fig. 4 left. We define the parameter η r as η r = min(r,1/r)−IP IWD−IP where the overline denotes ensemble average and I P ≈ 0.386 (resp. I WD ≈ 0.536) is min (r, 1/r) for Poisson statistics (resp. random matrix statistics). At the transition spectral statistics is expected [37] to converge to Poisson logarithmically slowly, which is compatible with our numerical data. Figure 4 right shows the result of a finite-size scaling analysis of η r for different system sizes N and disorder strengths W for p = 0.06. The raw data shown in the left panel are found to collapse after a rescaling of the form η r (W )/η r (W c ) = F lin (log 2 N/ξ) with ξ = A \|W − W c \| −ν ⊥ and ν ⊥ ≈ 0.5 for both sides of the transition. We note that similar scaling laws where reported in [79] for the Bethe lattice and scale-free networks. These results indicate that the behavior at small energy distance (level repulsion) is dominated by ξ ⊥ , the localization length associated with ν ⊥ . Indeed, wavefunctions at different but closeby energies are located on different branches and their overlap is controlled by ξ ⊥ . Universality. We have checked (see forthcoming article [80]) that our results are valid for p up to p = 0.49, which corresponds to random graphs with K = 1.98, and other types of disorder distributions. We have also checked that the new critical exponent ν ⊥ does not vary significantly as a function of q < 0.5 and 0 < p < 0.5. Conclusion. Our results clearly show that there exist two different localization lengths in the Anderson transition on random graphs, ξ describing rare branches and ξ ⊥ describing the bulk, which control the critical behavior of different physical observables and are associated with distinct critical exponents ν = 1 and ν ⊥ = 0.5. This clarifies the nature of the Anderson transition in the limit of infinite dimensionality, which remains, for the bulk properties, a continuous, second-order phase transition, while rare events, characteristic of random graphs, are responsible for the discontinuous properties described up to now. We further note that in finite dimension, the localization length critical exponent tends to 1/2 in the limit of large dimensionality [81], which suggests that the rare branch mechanism may not be present in large but finite dimension. We believe that our approach could be used to describe the nonergodic properties of the many-body localization transition [10] (in particular to extract the putative typical localization length [51,52]), and more generally many-body states in Hilbert space [11,15,16]. FIG. 1 . 1First panel:τq (calculated as log 2 Pq(N/2) − log 2 Pq(N ) ) vs q for W = 1.05 (delocalized), W = 1.725 (critical), W = 2.3 (localized) for p = 0.06, N = 2 12 , 2 15 , 2 20 ; the straight lines for small q are fits by q/q * − 1. To avoid huge fluctuations of Pq for small values of q, we have averaged as is usual over boxes of size 4 along the 1D chain (1). Second panel: probability distribution P(log 10 \|ψ\| 2 ), same W values, N = 2 of Pq for q = 0.25, p = 0.06, N between 2 9 and 2 20 ; inset is ξ vs W with the fit ξ ∼ \|W − 1.64\| −ν ⊥ (solid line), ν ⊥ ≈ 0.43. FIG. 4. Spectrum: finite-size scaling analysis of ηr (see text) for p = 0.06. N = 2 10 to N = 2 18 . Each symbol is a different size. Left: Raw data for ηr; inset: example of P (r) (see text) for localized, critical and delocalized phases. Right: Collapse of the data after a rescaling of the form ηr(W )/ηr(Wc) = F lin (log 2 N/ξ) with ξ the scaling parameter; inset: ξ vs W across the transition at Wc = 1.65. Solid lines are ξ ∼ \|W − 1.65\| −0.49 (delocalized branch) and ξ ∼ \|W − 1.65\| −0.51 (localized branch). (left) the localτ q = − d log Pq d log N in different regimes, FIG. 2. q * (continuous lines, for N between 2 15 andN W q * , β, ξ ⊥ ln K 2.4 2.2 2 1.8 1.6 1.4 1.2 1 1 0.75 0.5 0.25 0.5 and only two fitting parameters C and Wc. Wc ≈ 1.74 in good agreement with the other determinations (see[36],Figs. 3 and 4). Dashed line shows the asymptotic behavior, when N → ∞, of q * for W < Wc: q * = 1, indicating an ergodic delocalized phase. This study has been supported through the EUR grant NanoX No. ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir", by the ANR grant MANYLOK (Grant No. ANR-18-CE30-0017), by the ANR grant COCOA (Grant No. C We Thank, N Castellani, T Laflorencie, ANR-17-CE30-0024-01We thank CalMiP for access to its supercomputer and the Consortium des Équipements de Calcul Intensif (CÉCI). by the CONICET (Grant No. PIP 11220150100493CO), by ANCyPT (Grant NoWe thank C. Castellani, N. Laflorencie and T. Thiery for fruitful discussions. We thank CalMiP for access to its supercomputer and the Consortium des Équipements de Calcul Intensif (CÉCI). 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[ "The Luminosity Distribution of Local Group Galaxies", "The Luminosity Distribution of Local Group Galaxies" ]	[ "Christopher J Pritchet \nDepartment of Physics & Astronomy\nDominion Astrophysical Observatory\nUniversity of Victoria\nP. O. Box 3055V8W 3P6VictoriaBritish ColumbiaCanada\n", "Sidney Van Den Bergh \nHerzberg Institute of Astrophysics\nNational Research Council\nVictoria, British ColumbiaV8X 4M6Canada\n" ]	[ "Department of Physics & Astronomy\nDominion Astrophysical Observatory\nUniversity of Victoria\nP. O. Box 3055V8W 3P6VictoriaBritish ColumbiaCanada", "Herzberg Institute of Astrophysics\nNational Research Council\nVictoria, British ColumbiaV8X 4M6Canada" ]	[]	From a rediscussion of Local Group membership, and of distances to individual galaxies, we obtain M V values for 35 probable and possible Local Group members. The luminosity function of these objects is well fitted by a Schechter function with faint end slope α = −1.1 ± 0.1. The probability that the luminosity distribution of the Local Group is a single Schechter function with α steeper than −1.3 is less than 1 per cent. However, more complicated luminosity functions, such as multi-component Schechter functions with steep faint-end slopes, cannot be ruled out. There is some evidence that the luminosity distribution of dwarf spheroidal galaxies in the Local Group is steeper than that of dwarf irregular galaxies.	10.1086/300977	[ "https://arxiv.org/pdf/astro-ph/9904250v2.pdf" ]	7,321,657	astro-ph/9904250	e6ae0894eaab4b896f9e0bfc30959a56d811340f	The Luminosity Distribution of Local Group Galaxies 26 Apr 1999 Christopher J Pritchet Department of Physics & Astronomy Dominion Astrophysical Observatory University of Victoria P. O. Box 3055V8W 3P6VictoriaBritish ColumbiaCanada Sidney Van Den Bergh Herzberg Institute of Astrophysics National Research Council Victoria, British ColumbiaV8X 4M6Canada The Luminosity Distribution of Local Group Galaxies 26 Apr 1999arXiv:astro-ph/9904250v2Subject headings: galaxies: Local Group -galaxies: luminosity function From a rediscussion of Local Group membership, and of distances to individual galaxies, we obtain M V values for 35 probable and possible Local Group members. The luminosity function of these objects is well fitted by a Schechter function with faint end slope α = −1.1 ± 0.1. The probability that the luminosity distribution of the Local Group is a single Schechter function with α steeper than −1.3 is less than 1 per cent. However, more complicated luminosity functions, such as multi-component Schechter functions with steep faint-end slopes, cannot be ruled out. There is some evidence that the luminosity distribution of dwarf spheroidal galaxies in the Local Group is steeper than that of dwarf irregular galaxies. Introduction The galaxy luminosity distribution, or luminosity function 1 , φ(L) plays an important role in our understanding of the properties of galaxies, galaxy evolution, and galaxy formation. The connection between φ(L) and galaxy formation is through the primordial density fluctuation spectrum, δρ/ρ ∝ k n , where k is wavenumber. If the Universe consisted only of weakly interacting particles (e.g. "cold dark matter"), then the mass function of "halos" would be determined solely by n, and could be computed using a simple physical recipe for the gravitational clustering and merging of halos (e.g. Press & Schechter 1974). However, it is the baryons, rather than the dark matter, that we observe directly; hence the luminosity function φ(L) depends on gas physics and radiation processes (e.g. cooling, radiative transfer, star formation, energy input from supernovae, to name just a few). It follows that the luminosity function is sensitive not only to the fluctuation spectrum δρ/ρ, but also to the detailed history of galaxy formation and evolution in different environments. Generally the luminosity function of galaxies is parametrized by a Schechter (1976) function, φ(L) = φ * e −(L/L * ) (L/L * ) α ,(1) where L * is a characteristic luminosity defining the transition between a power-law at faint magnitudes and an exponential cutoff at bright magnitudes. [Further information concerning the Schechter function can be found in Felten (1985).] L * corresponds roughly to the brightness of the Milky Way. For a CDM fluctuation spectrum with n = 1, the power-law exponent α is theoretically predicted to be ≈ −2 on the scale of galaxies (Bardeen et al. 1986), though this result is very sensitive to the detailed physical processes involved in the calculation (cf. Babul and Ferguson 1996, Frenk et al. 1996, Kauffmann et al. 1998. What is known empirically about the shape of the luminosity distribution in the nearby Universe? Values of α in the range −0.7 to −1.0 have been obtained for bright galaxies within 2-3 magnitudes of L * (e.g. Loveday et al. 1992;Marzke et al. 1994 a, b;Lin et al. 1996), with a hint of a turnup in the LF at magnitudes fainter than about M R = −17 in the work of Lin et al.. The field luminosity distribution derived from the SSRS2 redshift survey (Marzke et al. 1998) indicates a relatively flat value of α ≃ −1 for E/S0's (including dwarf spheroidals) and spirals, and a steep α = −1.8 value for dwarf irregulars and peculiars; this steepening of the LF for late-type star forming systems also appears in the work of Bromley et al. (1998), who subdivided the Las Campanas Redshift Survey according to emission-line strength. Côté et al. (1998) have found a very steep (α = −2.1) luminosity distribution for nearby H I-rich low surface brightness galaxies, and Schneider, Spitzak, & Rosenberg (1998) find a steep upturn in the H I mass function for low mass objects (M H I ∼ < 10 8 M ⊙ ). Turning to other environments, Loveday (1997) finds that the luminosity distribution of dwarf galaxies surrounding luminous (∼ L * ) galaxies is steep: this can be interpreted either as a turnup in a supposedly universal field luminosity distribution, or alternately as an enhanced probability that dwarfs form in the vicinity of luminous, massive galaxies. Galaxies in groups exhibit a slope α ≃ −1 (Muriel, Valotto & Lambas 1998). There is evidence that compact group luminosity distributions cannot be fitted by a single Schechter function (Hunsberger, Charlton & Zaritsky 1998), but instead show α = −0.5 at the bright end and α = −1.2 below M R ≃ −16. There is also some indication that cluster LF's must be fitted with multiple Schechter functions (e.g. Trentham 1998a, Lopez-Cruz et al. 1997, with a steep upturn at faint magnitudes (e.g. Trentham 1998b;Phillipps et al. 1998a). Phillipps et al. (1998b) suggest the existence of an environmental dependence of dwarf-to-giant ratio (i.e. α) in clusters. The faintest galaxies in the steep luminosity distribution population in clusters are, based on their colors (Trentham 1998a, b), dwarf spheroidals, whereas in the field they appear to be gas-rich dwarf irregulars (Marzke et al. 1998). (It should, however, be noted that the Marzke et al. dIr/peculiar sample is drawn from a small local volume and comprises only 4% of the total SSRS2 sample.) On the basis of the discussion given above, it appears that the simple paradigm of a universal Schechter function (that fits the LF of galaxies in all environments) is now untenable. There is growing evidence that the LF is not a simple Schechter function, that it depends on environment, and that it also depends on galaxy morphological class and/or gas content within a given environment. No clear physical picture has emerged that would allow one to understand current observational evidence on the shape and environmental dependence of the LF. The Local Group represents a unique opportunity for the study of the luminosity distribution in relatively low density environments. The manner in which faint Local Group galaxies are detected is completely different from that for other groups (because most Local Group galaxies are easily resolved into stars); hence surface brightness selection effects operate differently in the Local Group than they do in more distant groups. Subtraction of foreground and background contaminating objects is irrelevant for Local Group galaxies, something that is of course not the case for more distant clusters. The numbers of galaxies in poor groups are so low that it has usually been possible to study only the composite LF of poor groups (rather than the LF of any group individually) -here the Local Group again is an exception. Furthermore, the census of Local Group members extends to considerably fainter absolute magnitudes (M V ≃ −8.5) than do any other samples for which the luminosity distribution has been measured (though incompleteness must be severe at the faint end). Thus a study of the Local Group luminosity distribution is of considerable importance. What is known about the Local Group luminosity distribution? Tully (1988) derived a composite luminosity function for six nearby groups (including the Local Group), and found α = −1 ± 0.2. Van den Bergh (1992) demonstrated that the integral luminosity function of the Local Group was consistent with α = −1.1, but did not rule out the possibility that other values of α fitted the data equally well. More recently, Mateo (1998) has shown that the LF of galaxies in the vicinity of the Local Group (but extending out beyond the usually accepted LG boundary of R = 1 Mpc) is consistent with that derived for poor groups (Ferguson and Sandage 1991). Again, this statement does not preclude the possibility that other luminosity distributions fit equally well. Over the past few years substantial additional data have been accumulated on Local Group membership and absolute magnitudes, and so the time seems ripe for a fresh, and more detailed, attack on the problem of the Local Group luminosity distribution. The Local Group Catalog The Local Group of galaxies was first decribed by Hubble (1936) in his book The Realm of the Nebulae. He listed M 31, M 32, M 33, the Magellanic Clouds, NGC 205, NGC 6822, and IC 1613 as probable members of the small group of galaxies associated with our Milky Way system. Inspection of the prints of the Palomar Sky Survey (van den Bergh 1962) shows that a large fraction of all of galaxies occur in small groups and clusters that resemble the Local Group. This shows that our Galaxy is located in a rather typical region of space. Since Hubble's pioneering work the number of galaxies that are known to belong to the Local Group has increased by 4 or 5 per decade to over thirty. A listing of data on presently known Local Group members (van den Bergh 2000) is given in Table 1. Selection of Local Group members proceeded in three steps. First galaxies with distances from the Local Group centroid (Courteau and van den Bergh 1999) less than or about 1.5 Mpc were regarded as suspected Group members. Secondly it was required that Local Group members should lie close to the relation between between radial velocity V r and cos θ for well-established Local Group members, where θ is the distance from the solar apex (Courteau and van den Bergh 1999). Finally, Local Group members should not appear to be associated with groups of galaxies that are centered well beyond the limits of the Local Group. On the basis of these criteria van den Bergh (1994Bergh ( , 2000 concluded that the following objects should be excluded from Local Group membership: (1) UKS 2323-326, (2) Maffei 1 and its companions, (3) UGC A86, (4) NGC 1560, (5) NGC 1569, (6) NGC 5237, (7) DDO 187, (8) Cassiopeia 1, and (9) NGC 55. A particularly strong concentration of these Local Group suspects, which includes (2), (3), (4), (5) and (8) listed above, occurs in the direction of the IC 342/Maffei group (Krismer, Tully & Gioia 1995). (1) and (9) appear in the direction of the Sculptor (=South Polar) group; in the case of (9), Jergen, Freeman & Binggeli (1998) find D=1.66 ± 0.2 kpc, which gives a distance 1.65 Mpc from the Local Group centroid. Finally, the discovery of a Cepheid (Tolstoy et al. 1995) in DDO 155 (=GR 8) suggests that this object is located at a distance of 2.2 Mpc, which places it well beyond the usually accepted limits of the Local Group. Dohm-Palmer et al. (1998) obtain a similar distance to DDO 155 from the tip of the red giant branch. Also excluded from Local Group membership are the galaxies NGC 3109, Antlia, Sextans A and Sextans B. These objects, which are located relatively close together on the sky, all have distances of 1.3 -1.5 Mpc from the Milky Way, and, of more relevance, distances of ∼ 1.7 Mpc from the Local Group centroid. Furthermore, these objects possess a mean radial velocity of +114 ± 12 km s −1 relative to the relation between V r and cos θ for well-established Local Group members (van den Bergh 1999). This suggests that these galaxies form a small group just beyond the zero velocity surface of the Local Group. (This surface is at a distance R(LG) = 1.18 ± 0.15 Mpc from the Local Group centroid [Courteau and van den Bergh 1999].) How does the Local Group membership defined above compare with that of Mateo (1998)? The principal differences are that the Mateo catalog does not contain several recently-discovered satellites of M 31, but does include nine objects beyond 1 Mpc (NGC 55, EGB 0427+63, Sextans A, Sextans B, NGC 3109, Antlia, GR 8, IC 5152, UKS 2323-326). Most of these were discussed above. EGB 0427+63 has a distance of 2.2 Mpc (Karachentsev, Tikhonov & Sazonova 1994) and thus lies well outside the Local Group. From a color-magnitude diagram, Zijlstra & Minniti (1999) find that IC 5152 has a distance from the Milky Way of 1.70 ± 0.16 Mpc, a result that agrees with the Cepheid distance of 1.6 Mpc (Caldwell & Schommer 1988); the distance of this galaxy from the Local Group centroid is therefore 1.8 Mpc, again beyond the Local Group. A more detailed discussion of Local Group membership and of the outer boundary of the Local Group can be found in van den Bergh (2000). Luminosity Distribution of the Local Group Because Local Group galaxies are situated so nearby it is possible to study their luminosity distribution down to very faint absolute magnitudes. Nevertheless these data are, no doubt, still quite incomplete for M V > −10. This is shown most clearly by the fact that only one galaxy fainter than this limit has so far been discovered in the Andromeda subgroup of the Local Group, whereas five such faint objects are presently known in the Milky Way subgroup of the Local Group. On the other hand, a survey of a twenty thousand square degree area at high Galactic latitudes by Irwin (1994) resulted in the discovery of only a single new Local Group member. Furthermore, no new optically visible Local Group galaxies have turned up in the survey of compact high latitude high velocity clouds (Braun and Burton 1999). Taken at face value these results might be taken to suggest that the luminosity distribution of the Local Group no longer increases below M V ≃ −8. It is noted in passing that very large low surface brightness galaxies in the Local Group, like those that have been discovered in the Virgo cluster (Impey, Bothun & Malin 1988), in the Fornax cluster (Bothun, Impey & Malin 1991) and in the M 81 group (Caldwell et al. 1998), may have also eluded us. The data in Table 1 can be used to study the luminosity distribution of Local Group galaxies. Histograms plotting this distribution are shown in Figure 1. The upper histogram clearly shows an increased number of objects at faint absolute magnitudes. The separation by morphological type (lower two panels) shows that most of this increase is due to galaxies that are dwarf spheroidals. A somewhat smoother visual impression of the Local Group luminosity distribution may be obtained by plotting the cumulative luminosity distribution, which is compared in Figure 2 to several different cumulative Schechter functions. In Fig. 2, the cumulative numbers are normalized at M V =−10 because it is unlikely that the data are complete at fainter magnitudes. This figure shows that a Schechter function with α ≃ −1.1 and M * V = −20 is an acceptable fit to the data (cf. van den Bergh 1992, Mateo 1998), and that there is some evidence for a steepening to α < −1.3 at faint magnitudes. (Because of incompleteness effects, this is an upper limit on the faint-end slope). To parametrize the luminosity distribution of the Local Group, we fit the data from Table 1 to a Schechter (1976) function [Eqn. (1)]. As discussed in §1, this function possesses a power-law luminosity dependence with exponent α, and an exponential cut-off at L>L * . In a plot of log φ(M) vs. absolute magnitude M, the faint end of the Schechter function is linear, with slope a = −0.4(α + 1). In detail, we fit the unbinned Local Group absolute magnitude data to a Schechter function using maximum likelihood techniques. The small number of objects involved dictates that we use a Poisson, rather than Gaussian, estimator of likelihood. The very small population of luminous galaxies (L ∼ > L * ) also means that it is not possible to obtain a robust estimation of M * V ; hence we fit only α (and of course a normalization constant proportional to φ * ), with a few different trial values of M * V (which make virtually no difference to the fitted value of α). The sparsenesss of the dataset furthermore prevents us from considering luminosity distributions that are a combination of two or more fitting functions (as found for Coma and other rich clusters by Trentham 1998b; see also Ferguson and Sandage 1991;Binggeli, Sandage and Tammann 1987). However, as will be seen, it is nevertheless possible to constrain such functions by isolating different magnitude ranges. The maximum likelihood program was tested with artificial data sets drawn from a distribution of absolute magnitudes that followed a Schechter function. From thousands of simulations, the maximum likelihood program was found to return almost precisely the input value of α in the mean, even for very small numbers of objects (N < 10). Furthermore, the error estimates (see below) were also found to be accurate. Table 2 gives the maximum likelihood value of α for various fits to the Local Group data (different magnitude ranges, and selections of morphological types), together with several different error estimators for this quantity. The first error estimate is simply a 1σ error: this is derived by finding that region of the maximum likelihood probability distribution that is centred on the fitted value of α, and that includes 68% of the probability. Also given in Table 2 are 95% and 99% confidence limits for the upper bound on α (i.e. the values of α for which the probability is 95% and 99% that α is steeper than this). Finally, Table 2 also shows the percentage probability that α is steeper than (less than) -1.3. Considering the entire data set (faint limit M V = −8), it is apparent that a value α = −1.07 ± 0.05 is a best fit, with a 95% probability that α < −0.98. Since the Local Group luminosity distribution is known to be incomplete at such faint magnitudes, we instead consider limiting the choice of objects at the faint end. However, regardless of the choice of parameters, the value α ≃ −1.1 ± 0.1 emerges. The probability that α is steeper than -1.3 is < 1%. The only exception to this is for galaxies fainter than M V = −15; for such objects slopes as steep as α = −1.5 are derived. However: (1) these slopes have large errors (±0.2 or even greater) because they are based both on small numbers of objects, and also on a limited range of M V ; (2) the 95% probability upper limit for α (lower limit on steepness) continues to hover around α = −1, and the probability that α < −1.3 is only 85%; and (3) the effect goes away if one instead considers objects brighter than M V = −16. Thus we consider this apparent steepening of the luminosity distribution at faint absolute magnitudes to be tantalizing, but not proven. Note that the derived slope is not very sensitive to the precise value of the faint cutoff for the fit. This is probably because of incompleteness at the faint end, but also because, even with a faint end upturn in the luminosity distribution, the majority of objects contributing to the fit are at brighter magnitudes. The derived value of the slope α is not sensitive to the assumed outer boundary of the Local Group. Relative to the Local Group centroid, the shell between 1.18 Mpc (the zero velocity surface according to Courteau and van den Bergh 1999) and 1.6 Mpc contains only a single galaxy, SagDIG. Removing this galaxy from our sample does not alter any of the results above. Mateo (1998) includes nine galaxies in his Local Group catalog that, because of their distance, do not appear in our catalog (see Section 2). Including these nine objects in our fits makes α less steep by < 0.1. It should be noted that even this small effect can be entirely explained by incompleteness at the low luminosity end of the sample of galaxies beyond 1.18 Mpc. We also stress that the available evidence does not support the inclusion of these nine galaxies in our Local Group catalog (see discussion in Section 2). Most of the apparent steepening in α for faint objects is due to the dwarf spheroidals in the Local Group, as can be seen from Fig. 1. Fitting a power-law slope to these objects alone shows a steeper value of α than for the entire dataset, but again the effect is only marginally significant. From a Kolmogorov-Smirnov two-sample test, the difference between the luminosity distributions of dIr's and dSph's is significant only at the 90% level. Unfortunately, the observations of M81 group dwarfs (e.g. van Driel et al. 1998) do not enable us to throw additional light on this problem. This is because these authors were not able to determine morphological classes for the two faintest magnitude bins in their survey. Trentham (1998c) has derived a composite luminosity function for galaxies in clusters, and has shown that it can be applied to galaxies in the field as well. The cumulative form of this empirical luminosity function (for which α steepens towards fainter M V ) is plotted as the dashed line in Fig. 2. A Kolmogorov-Smirnov test excludes the possibility that Local Group galaxies are drawn from this parent population at > 99% probability. Finally, we have compared the distribution of B magnitudes of galaxies in the M 81 group (van Driel et al. 1998) with the luminosity distribution of M V of Local Group galaxies, under the assumptions that (m-M) B = 28.8 and < B − V >= 0.5 for the M 81 galaxies. From a comparison between the (presumably more-or-less complete) data on galaxies with M V brighter than -10 and B brighter than 17.5, a Kolmogorov-Smirnov test shows no significant difference between the M 81 (N=38) and Local Group luminosity distributions (N=27). This suggests that the Local Group and M 81 LFs are broadly similar, and are drawn from similar parent populations. Discussion and Conclusions A Schechter function with α ≃ −1.1 ± 0.1 provides a good fit to new data for the luminosity distribution of the Local Group. This result is in agreement with the luminosity distribution found for poor groups (e.g. Sandage 1991, Muriel et al. 1998), and is probably consistent with the work of Hunsberger et al. (1998), who found α ≃ −0.5 for M R < −18 and −1.2 for M R > −18. Our result is comparable to various determinations of α in the field (e.g. Loveday et al. 1992;Marzke et al. 1994 a, b;Lin et al. 1996;Marzke et al. 1998), and is insensitive to the manner in which the Local Group is defined. There is evidence for a steepening in α below M V =−15; as discussed in §1, this effect has been observed in other environments. However, the steepening of the field luminosity distribution observed by Marzke et al. (1998) is in the dIr population, in contrast to the situation in the Local Group, for which the dIr population possesses a flat α ≈ −1, and for which the dSph population appears to be possess steeper α (though this difference is significant only at the ∼ 90% level in our work). The steepening in α that we observe at faint magnitudes is limited in significance by small number statistics, and almost certainly α is steeper than our fits would indicate, because of magnitude dependent incompleteness. Clearly much further observational work is needed to improve the completeness of the census of Local Group members at faint absolute magnitudes. Note. -M V (br) and M V (ft) give the range of absolute magnitude over which the fit was done. The error in α corresponds to a 1σ error. α(95%) and α(99%) are the values for which the probability is 95% (99%) that the true value of α lies below the tabulated values. P(α < −1.3) is the probability that that α < −1.3. Fig. 1.-Histogram of the luminosity distribution of Local Group members, with absolute magnitude data taken from Table 1. The upper panel gives the luminosity distribution for all Local Group members, and the lower panels show the luminosity distributions for Ir/dIr and Sph/dSph morphological types. Four galaxies appear in the top panel but not in the lower panels: the spirals M31, the Milky Way, and M33; and the elliptical M32. The galaxies Pisces and Phoenix (dIr/dSph) are counted with weight 0.5 in each of the lower two panels. Trentham (1998c). FIGURE CAPTIONS C.J.P. acknowledges the hospitality of the South African Astronomical Observatory, where part of this work was completed, and the financial support of the Natural Sciences and Engineering Research Council of Canada. The authors are grateful to George Jacoby and Taft Armandroff for pointing out an error in an earlier version ofFig. 2. Fig. 2 . 2-Cumulative distribution of absolute magnitudes of Local Group members (solid line), compared with cumulative Schechter functions (dotted lines). The Schechter functions are computed with M * V =−20 and five different α values (-0.9 to -1.3 in steps of 0.1). The cumulative functions are all normalized at M V = −10 because it seems likely that the data are incomplete below this level. The dashed line shows the empirical luminosity function (in cumulative form) of Table 1 . 1Derived Properties of Probable Local Group Galaxies Note. -Colons denote uncertain values. * Membership in Local Group not yet firmly established.Name Alias DDO Type (m-M) 0 M V ℓ b D[kpc] cosθ M 31 N 224 Sb I-II 24.4 -21.2 121.17 -21.57 760 0.88 Milky Way Galaxy S(B)bc I-II: 14.5 -20.9: 000.00 00.00 8 -0.15 M 33 N 598 Sc II-III 24.5 -18.9 133.61 -31.33 795 0.73 LMC ... Ir III-IV 18.5 -18.5 280.19 -33.29 50 -0.80 SMC ... Ir IV/IV-V 18.85 -17.1 302.81 -44.33 59 -0.61 M 32 NGC 221 E2 24.4 -16.5 121.15 -21.98 760 0.88 NGC 205 ... Sph 24.4 -16.4 120.72 -21.14 760 0.88 IC 10 ... Ir IV: 24.1 -16.3 118.97 -03.34 660 0.94 NGC 6822 ... Ir IV-V 23.5 -16.0 025.34 -18.39 500 0.29 NGC 185 ... Sph 24.1 -15.6 120.79 -14.48 660 0.91 IC 1613 ... Ir V 23.3 -15.3 129.73 -60.56 725 0.47 NGC 147 ... Sph 24.1 -15.1 119.82 -14.25 660 0.92 WLM DDO 221 Ir IV-V 24.85 -14.4 075.85 -73.63 925 0.32 Sagittarius ... dSph(t) 17.0 -13.8:: 005.61 -14.09 24 -0.04 Fornax ... dSph 20.7 -13.1 237.24 -65.66 138 -0.25 Pegasus DDO 216 Ir V 24.4 -12.3 094.77 -43.55 760 0.76 Leo I Regulus dSph 22.0 -11.9 225.98 +49.11 250 -0.44 And I ... dSph 24.55 -11.8 121.69 -24.85 810 0.86 And II ... dSph 24.2 -11.8 128.91 -29.15 700 0.78 Leo A DDO 69 Ir V 24.2 -11.5 196.90 +52.41 690 -0.14 Aquarius* DDO 210 Ir V 25.05 -11.3 034.04 -31.35 1025 0.40 SagDIG* ... Ir V 25.7: -10.7: 021.13 -16.23 1300: 0.22 Pegasus II And VI dSph 24.45 -10.6 106.01 -36.30 830 0.83 Pisces LGS 3 dIr/dSph 24.55 -10.4 126.77 -40.88 810 0.71 And III ... dSph 24.4 -10.2 119.31 -26.25 760 0.86 And V ... dSph 24.55 -10.2 126.22 -15.12 810 0.87 Leo II ... dSph 21.6 -10.1 220.14 +67.23 210 -0.26 Phoenix ... dIr/dSph 23.0 -9.8 272.19 -68.95 395 -0.30 Sculptor ... dSph 19.7 -9.8 287.69 -83.16 87 -0.06 Tucana ... dSph 24.7 -9.6 322.91 -47.37 870 -0.44 Cassiopeia And VII dSph 24.2 -9.5 109.46 -09.95 690 0.98 Sextans ... dSph 19.7 -9.5 243.50 +42.27 86 -0.65 Carina ... dSph 20.0 -9.4 260.11 -22.22 100 -0.85 Draco ... dSph 19.5 -8.6 086.37 +34.71 79 0.77 Ursa Minor ... dSph 19.0 -8.5 104.88 +44.90 63 0.66 Table 2 . 2Maximum Likelihood Fits to Local Group DataM V (br) M V (ft) α α(95%) α(99%) P(α < −1.3) (a) All morphological types -22 -8 -1.07 ± 0.05 -0.98 -0.94 0.00% -22 -10 -1.09 ± 0.07 -0.96 -0.91 0.50% -22 -11 -1.02 ± 0.08 -0.88 -0.83 0.18% -22 -12 -0.91 ± 0.10 -0.74 -0.66 0.02% -18 -8 -1.10 ± 0.07 -0.99 -0.93 0.40% -18 -10 -1.17 ± 0.10 -0.99 -0.94 11.31% -18 -11 -1.09 ± 0.13 -0.87 -0.77 6.52% -18 -12 -0.92 ± 0.18 -0.61 -0.45 2.13% -16 -8 -1.09 ± 0.09 -0.94 -0.88 1.82% -16 -10 -1.21 ± 0.14 -0.95 -0.87 30.11% -16 -11 -1.06 ± 0.20 -0.72 -0.57 15.15% -16 -12 -0.62 ± 0.35 0.05 0.32 2.38% -15 -8 -1.18 ± 0.11 -0.98 -0.92 17.93% -15 -10 -1.50 ± 0.21 -1.14 -1.01 85.87% -15 -11 -1.48 ± 0.33 -0.94 -0.71 73.27% (b) Sph/dSph data only -17 -9 -1.33 ± 0.12 -1.13 -1.05 62.53% -17 -10 -1.31 ± 0.16 -1.04 -0.94 58.55% -17 -11 -1.11 ± 0.22 -0.73 -0.57 23.66% -15 -9 -1.52 ± 0.18 -1.24 -1.12 90.72% -15 -10 -1.66 ± 0.29 -1.23 -1.05 92.01% -15 -11 -1.44 ± 0.46 -0.72 -0.42 66.92% (c) Ir/dIr data only -16 -9 -1.02 ± 0.17 -0.70 -0.58 7.90% -16 -10 -1.09 ± 0.22 -0.72 -0.56 20.69% -16 -11 -1.01 ± 0.31 -0.48 -0.26 20.49% -15 -10 -1.40 ± 0.33 -0.90 -0.66 66.20% The luminosity function is expressed in units of number density (e.g. number per Mpc 3 ), whereas the luminosity distribution simply gives the shape of the luminosity function without density normalization. . 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[ "NEW RESTRICTIONS ON THE TOPOLOGY OF EXTREME BLACK HOLES", "NEW RESTRICTIONS ON THE TOPOLOGY OF EXTREME BLACK HOLES" ]	[ "Marcus Khuri ", "Eric Woolgar ", "William Wylie " ]	[]	[]	We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross-sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries and the mathematical notion of m-quasi Einstein metrics, in addition to generalizations of the classical splitting theorem from Riemannian geometry. Consequences are analyzed and refined classifications are given for the possible topologies of these black holes.	10.1007/s11005-018-1121-9	[ "https://arxiv.org/pdf/1804.01220v3.pdf" ]	119,276,161	1804.01220	703fe80ea5d098508e3b8b72a5858dd810cf25c4	NEW RESTRICTIONS ON THE TOPOLOGY OF EXTREME BLACK HOLES 17 Apr 2018 Marcus Khuri Eric Woolgar William Wylie NEW RESTRICTIONS ON THE TOPOLOGY OF EXTREME BLACK HOLES 17 Apr 2018 We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross-sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries and the mathematical notion of m-quasi Einstein metrics, in addition to generalizations of the classical splitting theorem from Riemannian geometry. Consequences are analyzed and refined classifications are given for the possible topologies of these black holes. Introduction Stephen Hawking revolutionized the theory of black holes. One part of that legacy is his horizon topology theorem [17,18] which states that, in spacetime dimension D = 4, cross-sections of the event horizon of asymptotically flat stationary vacuum black holes that obey the dominant energy condition must have spherical topology S 2 . In higher dimensions such a simple characterization does not hold, as is illustrated by the D = 5 ring solutions of Emparan-Reall [5] and Pomeransky-Sen'kov [41] which have cross-sectional horizon topology S 1 × S 2 . A natural question then arises: what are the possible horizon topologies for asymptotically flat stationary vacuum black holes in dimensions D > 4? A primary observation that underpins much of the study of this problem is the fact that such horizons, or more generally stable marginally outer trapped surfaces, are of positive Yamabe type [10,12,13]. This means that these manifolds admit Riemannian metrics of positive scalar curvature. In D = 5 this provides strong restrictions on the possible topologies of the 3-dimensional horizons. More precisely, by the prime decomposition theorem [20] along with a result of Gromov-Lawson [16] and resolution of the Poincaré conjecture, a compact orientable 3-manifold having positive Yamabe invariant is diffeomorphic to a spherical space, S 1 × S 2 , or a finite connected sum thereof. Here a spherical space refers to a quotient of the 3-sphere S 3 /Γ where Γ ⊂ O(4) is a discrete subgroup, an example is the lens space L(p, q) in which Γ = Z p . Another tool used to study the topology of black holes is the topological censorship theorem [9] which states that any curve beginning and ending in the asymptotically flat end can be deformed continuously to lie entirely within the asymptotic region. An example of a result relying on topological censorship is a refinement of the classification given above when D = 5. In [21] it is assumed that in addition to stationarity a U (1) symmetry is present, which is guaranteed for analytic solutions [22,24,38], and it is proven that the horizon is one of several possible quotients of S 3 by isometries, or a connected sum between copies S 1 × S 2 and lens spaces. If multiple axial symmetries are present, namely the isometry group contains U (1) D−3 , then Hollands-Yazadjiev [25] asymptotic flatness in dimensions D = 4, 5. Furthermore, it should be pointed out that it is not known whether all of these possible topologies are realized by stationary vacuum solutions, even in dimension five. More precisely, while S 3 and S 1 × S 2 have been realized by the Myers-Perry [39] and ring solutions listed above respectively, the only explicit examples of vacuum lenses [3] are known to have conical singularities or possess naked singularities. On the other hand, vacuum lenses and other configurations have been produced in multitude through abstract existence results for singular harmonic maps [26], although it is not known at this time whether any of these solutions are void of conical singularities; in addition, there has been some progress in the asymptotically Kaluza-Klein and locally Euclidean cases [27] as well. In other theories, such as D = 5 minimal supergravity, geometrically regular asymptotically flat lens black holes have been produced [1,35,42]. The latter examples are supersymmetric and hence extremal. Cobordism theory has also aided in classifying the topologies of black holes. Recall that two compact manifolds of the same dimension are called cobordant if their disjoint union is the boundary of a compact manifold of one higher dimension. The idea is that, in the asymptotically flat case, horizon cross-sections will be cobordant through a simply connected spacelike hypersurface to a sphere S D−2 sitting in the asymptotic end, and this should yield conditions on the possible topologies of the horizon. The lowest dimension for which this line of investigation can yield topological restrictions is D = 6. In [19] Helfgott-Oz-Yanay combined the methods of cobordism theory with Freedman's classification of 4-manifolds [8] and results of Donaldson [4] to show that if the horizon cross-section is simply connected then it must be homeomorphic to S 4 , to a finite connected sum of S 2 × S 2 's or CP 2 #CP 2 's-the 1-point blow-up of CP 2 where CP 2 denotes the complex projective plane with opposite orientation. If in addition the horizon is a spin manifold, then the connected sum of complex projective planes may be removed from this list. In [33] explicit examples of vacuum near-horizon geometries having horizon cross-sections realizing the topologies S 2 × S 2 and CP 2 #CP 2 have been constructed. Surveys concerning the various mathematical techniques used to study black hole topology may be found in [11,23]. Although much progress has been made in understanding black hole topology, the methods utilized so far have substantial limitations. Indeed, while the primary observation that horizon cross-sections are of positive Yamabe type provides strong constraints in dimensions D = 4, 5, this condition is considerably more flexible in higher dimensions. For instance any manifold of the form S n × M , where n ≥ 2 and M is compact, admits a metric of positive scalar curvature by scaling the round metric of S n properly. The purpose of this article is to introduce a new technique into the study of horizon topology in the context of extreme black holes, and to analyze its consequences. The approach here is based on a relation between the associated near-horizon geometries of these black holes and the mathematical notion of m-quasi Einstein metrics first exploited in [28,29], as well as results emanating from the classical splitting theorem of Riemannian geometry [40]. (iii) If Λ > 0 then π 1 (H) is finite. Item (iii) has previously been established in [28] by different methods. As an illustration of the additional topological restrictions Theorem 1 places on H, in Section 3 we use topological arguments stemming from (i) to refine classification results when D = 5 to show that H must be diffeomorphic to a spherical space, S 1 × S 2 , or RP 3 #RP 3 , see Corollary 7. The connected sum of projective spaces can be removed from this list if an additional U (1) symmetry is present. Furthermore it is also shown that in all dimensions, horizon topologies arising from a 'nontrivial' connected sum can be ruled out. It should be pointed out that the topological restrictions of this theorem also hold for near-horizon geometries which may be studied separately from extreme black holes, and are thus of independent interest. In addition, Theorem 1 is valid when matter is present as long as an appropriate energy condition is satisfied. The appropriate energy condition is described in [28,Inequality (8)], and in the case of perfect fluids it reduces to the dominant energy condition. Also, the inequality holds for pure electric fields normal to a static degenerate horizon. Main results Preliminaries. Consider a stationary black hole spacetime of dimension D satisfying the vacuum Einstein equations (1) R µν = Λg µν . Our normalization is such that Λ is 2 (n−2) times the usual cosmological constant. Stationarity gives an asymptotically timelike Killing field, and generically the rigidity theorem [22,24,38] yields one or more additional rotational symmetries which altogether produces a Killing field V that is normal to the event horizon. The event horizon is then a Killing horizon and on this surface (2) ∇ V V = κV, where the constant κ denotes the surface gravity. Near each horizon component Gaussian null coordinates (r, v, x i ) may be imposed such that V = ∂ v , r = 0 represents the horizon, and x i are coordinates on the D − 2-dimensional compact horizon cross-section H. In the degenerate case when κ = 0, the spacetime metric then has the form [34] (3) g = 2dv dr + 1 2 r 2 F (r, x)dv + rh i (r, x)dx i + γ ij (r, x)dx i dx j . This allows for a near-horizon limit φ * ǫ g → g N H as ǫ → 0, where the diffeomorphisms φ ǫ are defined by v → v ǫ , r → ǫr. The resulting near-horizon geometry may then be expressed as (4) g N H = 2dv dr + 1 2 r 2 F (x)dv + rh i (x)dx i + γ ij (x)dx i dx j , where γ ij is the induced metric on H. As a consequence of the Einstein equations, the near-horizon data (F, h i , γ ij ) satisfy the near-horizon geometry equations on H (5) R ij = 1 2 h i h j − ∇ (i h j) + Λγ ij , F = 1 2 \|h\| 2 − 1 2 ∇ i h i + Λ. Near-horizon geometries are closely related to the notion of m-quasi-Einstein metrics studied in the mathematical literature. These are solutions to the equation (6) Ric m X = λγ, where the generalized m-Bakry-Émery-Ricci tensor on H is given by (7) Ric m X = Ric + 1 2 £ X γ − 1 m X ⊗ X, in which X is a 1-form/vector field and £ X is Lie differentiation along X. It should be noted that some authors reserve this terminology for the special case when the vector field is a gradient X = ∇f . Clearly a vacuum near-horizon geometry defines an m-quasi-Einstein metric for m = 2, λ = Λ, and X = h. Splitting theorem. A classical result of Riemannian geometry known as the splitting theorem [2] asserts that a complete Riemannian manifold with nonnegative Ricci curvature and containing a line (an inextendible curve which minimizes the distance between any two of its points), must isometrically split off a Euclidean factor. From this several topological consequences follow. In order to take advantage of this, however, a version of the splitting theorem under the hypothesis of nonnegative m-Bakry-Émery-Ricci curvature is needed. Indeed, a suitable splitting theorem is known to hold in the case that X is a gradient vector field [ Theorem 2. Let (M, g) be a complete connected Riemannian manifold of dimension n admitting a complete C 1 vector field X. If Ric m X ≥ 0 for some m > 0, then M splits isometrically as R k × N where N is a complete Riemannian manifold without a line. Moreover the projection of X onto the R k -factor vanishes, and N has nonnegative m-Bakry-Émery-Ricci curvature. One can prove this by repeating the arguments of [7] with df replaced by X, once two underlying lemmata are suitably modified. The first lemma is a modification of an inequality on the Laplacian [2, Lemma 2] to our setting, and is derived from the second variation formula for arclength of curves. The second lemma is a straightforward identity. Using these results, we follow a standard approach. In the presence of a line we construct Busemann functions and show that they are linear. Level sets of these functions then manifest the desired splitting. To set notation let (8) Lu = ∆u − ∇ X u, where ∆u = tr Hess u is the Laplace-Beltrami operator acting on functions. Moreover let p ∈ M be fixed and set ρ(q) = dist(p, q). Lemma 3. Let Ric m X ≥ λg for some λ ≥ 0 and m > 0. If x ∈ M is not in the cut locus of p then (9) Lρ(x) ≤ n + m − 1 ρ(x) . Proof. Let γ : [0, ρ] → M be a unit speed minimal geodesic connecting p = γ(0) to x = γ(ρ). Following [7, derivation of equation (2.1)], we recall [2, Lemma 2] that the second variation of arclength along a geodesic γ away from the cut locus of p = γ(0) yields (10) ∆ρ(x) ≤ ρ 0 (n − 1) ρ 2 − t 2 ρ 2 Ric(γ,γ) dt . Using Ric m X ≥ λg(γ,γ) = λ and performing some trivial integrals, we obtain ∆ρ(x) ≤ (n − 1) ρ − λρ 3 + ρ 0 t 2 ρ 2 1 2 £ X g(γ,γ) − 1 m g(X,γ) 2 dt = (n − 1) ρ − λρ 3 + g(X,γ) − 1 ρ 2 ρ 0 2tg(X,γ) + t 2 m g(X,γ) 2 dt = (n − 1) ρ − λρ 3 + g(X,γ) − 1 ρ 2 ρ 0 √ m + tg(X,γ) √ m 2 dt + m ρ 2 ρ 0 dt = (n + m − 1) ρ − λρ 3 + g(X,γ) − 1 ρ 2 ρ 0 √ m + tg(X,γ) √ m 2 dt ≤ (n + m − 1) ρ + g(X,γ) ,(11) when λ ≥ 0. Then clearly Lρ(x) = ∆ρ(x) − ∇ X ρ(x) ≤ (n + m − 1) ρ(x) + g(X,γ)(x) − ∇ X ρ(x) ≤ n + m − 1 ρ(x) ,(12) since along the minimizing curve γ we have g(X,γ) = ∇ X ρ. When a line γ is present, we apply this result to the Busemann function (13) b γ (q) := lim t→∞ [t − dist(q, γ(t))] . More precisely, we apply it to Busemann support functions b γ r (q) as defined in [7, proof of Lemma 1]. By similar reasoning to that proof but using our estimate (12), we obtain in our case (14) − Lb γ r (x) ≤ − (n + m − 1) dist(δ(r), x) , where, as in [7, proof of Lemma 1], δ(r) is the ray obtained by the usual limiting procedure for Busemann support functions. From this, we may conclude that Lb γ ≥ 0 in the barrier sense when Ric m X ≥ 0 for some m > 0. Furthermore, Lemmata 2.4 and 2.5 of [7] can then be invoked to yield Lb ± = 0 for b ± (x) := lim t→∞ [t − dist(x, γ(±t))], t ≥ 0. The next identity is the second component required to modify the proof of [7]. We now have all the tools necessary to establish the splitting theorem. Proof of Theorem 2. If (M, g) contains no line, we are done, so assume otherwise. Then we may use the line to construct the associated Busemann functions as above. We may apply equation (15) with u = b ± , together with the earlier result that Lb ± = 0 and the condition Ric m X ≥ 0, to obtain L \|∇b ± \| 2 ≥ 2\| Hess b ± \| 2 ≥ 0. But then the strong maximum principle forces \|∇b ± \| 2 = const, so we may normalize ∇b ± to have unit length and moreover now we have (17) 0 = L \|∇b ± \| 2 ≥ 2\| Hess b ± \| 2 ≥ 0. Thus, ∇b ± are parallel and the functions b ± are linear, as desired. Level sets of b ± are totally geodesic and, by the completeness of (M, g) are complete in the induced metric. Let N be the zero level surface of b + (x) for some x (we could work equally with b − ; in fact the level sets coincide). It now follows that (18) F : N × R : (p, t) → e t∇b + (p) =: φ t (p) is an isometry; see [7] or [6]. We may now identify (N × R, g N ⊗ dt 2 ) with (M, g) where g N is the induced metric, and we have that the Ricci curvature splits as Ric(g) = 0 · dt 2 ⊕ Ric(g N ). Applying (15) to u = b + and using that ∇b + is parallel as well as Lb + = 0 produces (19) 0 = L \|∇b + \| 2 = Ric m X (∇b + , ∇b + ) + 1 m X(b + ) 2 . Since Ric m X ≥ 0 and m > 0, we have Ric m X (∇b + , ∇b + ) = 0 = X(b + ). Then X is tangent to N and (20) 1 2 £ X (∇b + , ∇b + ) = ∇ ∇b + g(X, ∇b + ) = 0. Furthermore, for any Y ⊥ ∇b ± we have £ X (Y, ∇b ± ) = g(Y, ∇ ∇b ± X) = 0 by an easy calculation using the splitting. So the condition Ric m X (g) ≥ 0 descends to Ric m X (g N ) ≥ 0. If (N, g N ) does not contain a line, then we have obtained the desired splitting. If (N, g N ) does contain a line then we can apply the splitting to N and split off a Euclidean factor of N . Applying the splitting iteratively, after finitely many steps we obtain the isometric splitting of M as R k × N where N does not contain any lines. Now that Theorem 2 has been proven we make some remarks about Bakry-Émery Ricci curvature and m-Quasi Einstein metrics. First note that without further assumptions Theorem 2 is not true, even in the gradient case, when the parameter m is negative or infinite, where we interpret Ric ∞ X = Ric + 1 2 £ X g. A splitting theorem can be proven in cases when m = ∞ or m < 0 if an additional energy condition is placed on the vector field X, see [45,Theorem 6.3]. We also point out that while the splitting theorem holds for non-gradient vector fields, there are a number of results for m-quasi Einstein metrics that require X = ∇f . One that is relevant for the considerations in this paper is a result of Kim-Kim [30] which implies that if a compact manifold admits Ric m X = 0, for some m > 0 and X = ∇f , then X ≡ 0 and (M, g) is Ricci flat. Vacuum near-horizon geometries with zero cosmological constant on spherical spaces of dimensions 2 and 3 (arising from the extreme Kerr and Myers-Perry solutions) show that the assumption that X is gradient is necessary in this result, since S 2 and S 3 do not admit Ricci flat metrics. This may be interpreted as illustrating how the non-gradient case is more flexible, as expected. The fact that the splitting theorem still holds in this setting is then somewhat surprising. In addition, to our knowledge it is not known if there is a compact manifold which admits Ric m X ≥ 0 but does not support a metric of nonnegative Ricci curvature. This indicates that it could be true that vacuum near-horizon geometries always admit metrics of non-negative Ricci curvature. Proofs of main results. Proof of Theorem 1 (i). When the cosmological constant Λ ≥ 0, Theorem 2 may be applied to the universal cover H of the horizon cross-section to show that H = R k × N , where N is compact [40,Theorem 69]. As a consequence, the subgroup G ≤ π 1 (H) of isometries of N is finite. Hence the kernel of the homomorphism π 1 (H) → G is a subgroup of finite index, and acts discretely as well as cocompactly on R k so that it is a crystallographic group. By a well-known theorem of Bieberbach [44], any such group must contain an Abelian subgroup isomorphic to Z k of finite index. A priori k ≤ dim H = D − 2, however this may be refined further. Suppose that k = D − 2, then H = R D−2 and H = H/π 1 (H) is flat. Standard arguments [40] then show that the inclusion Z D−2 ֒→ π 1 (H) is an isomorphism, and thus H is a torus. However tori do not admit metrics of positive scalar curvature [16], and therefore cannot be of positive Yamabe type, yielding a contradiction. Now suppose that k = D − 3. In this case N is a 1-dimensional compact manifold, and hence must be S 1 . This, however, contradicts the fact that H is simply connected. It follows that k ≤ D − 4. Proof of Theorem 1 (ii). By part (i) there is a subgroup Z k ≤ π 1 (H) of finite index, with k ≤ D − 4. Since the first homology group H 1 (H, Z) is isomorphic to the abelianization of the fundamental group, this subgroup is also of finite index H 1 (H, Z). Therefore the rank of the torsion free part (first Betti number) satisfies b 1 (H) = k ≤ D − 4. Proof of Theorem 1 (iii). If Λ > 0, then the 2-Bakry-Émery-Ricci curvature is strictly positive Ric 2 h > 0. It is then impossible for the the universal cover to split as H = R k × N with k ≥ 1. Thus the universal cover is compact, which again implies that π 1 (H) is finite. 2.4. A remark on curvature-dimension conditions. It is not central to our applications, but for completeness we now show that our 'energy condition' form of the curvature-dimension condition is equivalent to a more usual form of the curvature-dimension condition. Lemma 5. Let m > 0 and assume that Ric m X ≥ λg for some λ ∈ R, then (21) L \|∇u\| 2 ≥ 2∇ ∇u (Lu) + 2 (n + m) (Lu) 2 + λ\|∇u\| 2 . Conversely, if (21) holds for all u ∈ C 3 (M ), then Ric m X ≥ λg. Proof. A direct computation yields (Lu) 2 (n + m) = (∆u) 2 (n + m) + [X(u)] 2 (n + m) − 2 (n + m) X(u)∆u = (∆u) 2 n + [X(u)] 2 m − 1 (n + m) m n ∆u + n m X(u) 2 ,(22) so that (23) 1 m [X(u)] 2 = (Lu) 2 (n + m) − (∆u) 2 n + 1 (n + m) m n ∆u + n m X(u) 2 . Substituting this into (15) produces L \|∇u\| 2 = 2 Hess u − 1 n (∆u) g 2 + 2∇ ∇u (Lu) + 2 Ric m X (∇u, ∇u) + 2 (n + m) (Lu) 2 + 2 (n + m) m n ∆u + n m X(u) 2 .(24) Equation (21) now follows from the assumption on Ric m X . To prove the converse, simply follow the argument of [43, pp 387-388], noting that no use whatsoever is made in that argument of the assumption X = ∇V . Applications In this section we will use Theorem 1 to refine the topological classification of horizons described in the Introduction. This will be accomplished with the aid of results from geometric group theory. Recall that in a finitely generated group, the smallest number of generators needed to express an element is referred to as the length of the element. Furthermore, such a group is said to have polynomial growth if the number of elements of length at most α is bounded above by a polynomial function p(α), and the minimum degree of any polynomial having this property is the order of growth. An important characterization for groups of polynomial growth was given by Gromov [15,31]. It asserts that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup of finite index. In particular, a finitely generated group of exponential growth cannot have an Abelian (more generally nilpotent) subgroup of finite index. Therefore in light of Theorem 1 (i), horizon crosssectional topology cannot take the form of a manifold whose fundamental group is of exponential growth. We note that in the case of a horizon with nonnegative Ricci curvature, a result of Milnor [37] implies directly that the fundamental group is of polynomial growth of order no larger than D − 2. The above arguments state that, heuristically, horizons must have limited topology. More precisely, we can rule out some basic constructions that appear in many classifications. Consider the connected sum M #N of two manifolds M and N . The fundamental group of the connected sum is the free product of the individual fundamental groups π 1 (M #N ) = π 1 (M ) * π 1 (N ), when the dimension of M , N is greater than 2. In general the free product of two nontrivial groups is quite large, and thus should not be able to serve as the fundamental group of a horizon. Indeed, it can be shown [36,Theorem 4] that the free product of two nontrivial groups is always of exponential growth or better, whenever at least one of the two groups making up the free product has order greater than 2. This may be established by showing that such groups contain a non-Abelian free subgroup on two generators. Altogether this establishes the following result. In dimension D = 5 this theorem gives a strong refinement of previous horizon topology classifications. Namely, it rules out all possible connected sums of prime 3-manifolds. Corollary 7. Let H be a degenerate horizon (cross-section) component of a 5-dimensional stationary vacuum spacetime with nonnegative cosmological constant Λ ≥ 0. Then H is diffeomorphic to either a spherical space, S 1 × S 2 , or RP 3 #RP 3 . Proof. According to Theorem 6 and the previous classification, the only possible topology which is not immediately ruled out is H = RP 3 #RP 3 . In this case π 1 (H) = Z 2 * Z 2 is the infinite dihedral group, which has an index 2 infinite cyclic subgroup, and thus the statement of Theorem 1 (i) cannot be used to exclude this possibility. Remark 8. In the asymptotically flat or asymptotically Kaluza-Klein case, if there is a U (1) symmetry then RP 3 #RP 3 may be removed from the statement of Corollary 7. 1 This follows from [21, Result 1], which implies that in such a situation the connected sum of two projective spaces can only appear through a connected sum with at least one S 1 × S 2 . The presence of a U (1) symmetry is generic in the sense that it is assured by the rigidity theorem when the solution is analytic [22,24,38]. The results above suggest that if a degenerate horizon cross-section is decomposed as a nontrivial connected sum, then at least one member of the sum should be simply connected. Manifolds which cannot be written as a nontrivial connected sum are referred to as prime. Therefore, to a certain extent perhaps horizons of stationary black holes may be described as almost prime. Conjecture 9. Each connected component of a degenerate horizon cross-section in an asymptotically flat stationary vacuum spacetime is 'almost prime', 2 in the sense that if it is expressed as a connected sum of two manifolds then at least one member of the sum must be simply connected. This may be delicate. For instance, we claim that there exists a solution of the vacuum nearhorizon geometry equations with Λ = 0 on H = RP 3 #RP 3 , although it is not clear whether this solution arises from the near-horizon limit of an asymptotically flat stationary vacuum spacetime. To verify this claim, observe that the universal cover of the connected sum of two projective spaces is an infinite connected sum of S 3 's, which is diffeomorphic to R × S 2 . The group Z 2 * Z 2 acts naturally on this space as follows. The generator for the first Z 2 consists of a reflection in the R component across 1 together with an application of the antipodal map in the S 2 component. The generator for the second Z 2 in the free product is defined similarly, but with a reflection across −1. We then have H = R × S 2 /Z 2 * Z 2 . Consider now the product metric γ = dx 2 + γ kerr on R × S 2 in which γ kerr is the horizon metric from the near-horizon geometry of the extreme Kerr black hole. Let h denote the natural extension to R × S 2 of the 1-form near-horizon data associated with extreme Kerr. By defining F according to (5), a solution (F, h, γ) of the near-horizon geometry equations is produced on the universal cover. Since this set of near-horizon data is invariant under the Z 2 * Z 2 action described above, by passing to the quotient a solution is obtained on RP 3 #RP 3 . It should be pointed out that this construction could be achieved by performing a similar quotient of an extreme Kerr string black hole, and taking the near-horizon limit. Thus, while it is not known whether this solution arises from an asymptotically flat parent black hole, it does arise from an asymptotically Kaluza-Klein black hole. What of nondegenerate horizons? Should Conjecture 9 also extend to them? In fact, it may be more interesting if the question were answered in the negative, so that degenerate horizons had a topological rigidity not seen in nondegenerate ones. One can then contemplate a quasi-stationary system having a horizon consisting of a nontrivial connected sum, being 'spun up' adiabatically so that the mass is fixed and the system is always stationary to a good approximation. On approach to extremality, i.e. as the horizon nears a degenerate state, the horizon would have to undergo a dynamical instability, perhaps forming neckpinches to break apart the connected sum. The adiabatic approximation would likely fail at some point but, much worse, so could cosmic censorship. This picture is consistent with the 3rd law of black hole thermodynamics, which asserts that it is not possible to produce a black hole with vanishing surface gravity (temperature) through a physical process. Any neckpinch region might well resemble a black string undergoing the Gregory-Laflamme instability [14]. Of course, one possible resolution is that this scenario will not occur because such instabilities prevent (stationary vacuum) connected sum horizons from forming in the first place. This view might be seen as supporting the extension of Conjecture 9, except perhaps for unstable horizons. It's a question worth pursuing. have shown that the only possible horizon topologies are S 3 × T D−5 , S 2 × T D−4 , or L(p, q) × T D−5 , where T D−5 denotes the (D − 5)-dimensional torus. Note that this amount of axisymmetry is only compatible with M. Khuri acknowledges the support of NSF Grant DMS-1708798. E. Woolgar was supported by a Discovery Grant RGPIN 203614 from the Natural Sciences and Engineering Research Council. W. Wylie acknowledges the support of Simons Foundation Grant #355608 and NSF Grant DMS-1654034. Theorem 1 . 1Let H be a degenerate horizon (cross-section) component of a stationary vacuum spacetime with nonnegative cosmological constant Λ ≥ 0. ( i ) iThe fundamental group π 1 (H) contains an Abelian subgroup of finite index which is isomorphic to Z k with k ≤ D − 4.(ii) The first Betti number satisfies b 1 (H) ≤ D − 4. Lemma 4 . 4For any function u ∈ C 3 (M ) it holds that(15) L \|∇u\| 2 = 2\| Hess u\| 2 + 2∇ ∇u (Lu) + 2 Ric m X (∇u, ∇u) + 2 m [X(u)] 2 .Proof. By straightforward manipulations we have L \|∇u\| 2 = ∆ \|∇u\| 2 − ∇ X \|∇u\| 2 = 2\| Hess u\| 2 + 2g (∇u, ∆∇u − ∇ X ∇u) = 2\| Hess u\| 2 + 2g (∇u, ∆∇u + Ric(∇u, ·) − ∇∇ X u + ∇ ∇u X) = 2\| Hess u\| 2 + 2∇ ∇u (Lu) + 2 Ric(∇u, ∇u) + £ X g(∇u, ∇u) = 2\| Hess u\| 2 + 2∇ ∇u (Lu) + 2 Ric m X (∇u, ∇u) + 2 m [X(u)] 2 . Theorem 6 . 6Let H be a degenerate horizon (cross-section) component of a stationary vacuum spacetime with nonnegative cosmological constant Λ ≥ 0. Then H cannot be expressed as a connected sum M #N for any compact manifolds M and N both having nontrivial fundamental group, except possibly in the case that π 1 (M ) = π 1 (N ) = Z 2 . 7, Theorem 1.3]. Here we show that the arguments of [7, Theorem 1.3] extend to the non-gradient case. This observation is due independently to Stefan Hollands and James Lucietti.2 The related concept of stably prime manifold appears in[32]. Acknowledgements. The authors would like to thank Hari Kunduri for comments, as well as Stefan Hollands and James Lucietti for independently pointing out that RP 3 #RP 3 can be ruled out in Corollary 7 when an extra U (1) symmetry is present. The first author would also like to thank Luca Di Cerbo for useful discussions. Moduli space of supersymmetric solitons and black holes in five dimensions. 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[ "The Dark Z and Sterile Neutrinos Behind Current Anomalies", "The Dark Z and Sterile Neutrinos Behind Current Anomalies" ]	[ "A Hammad \nInstitute of Convergence Fundamental Studies\nSeoul National University of Science and Technology\n01811SeoulKorea\n\nCentre for Theoretical Physics\nBritish University in Egypt\nP.O. Box 4311837CairoEgypt\n", "Ahmed Rashed \nDepartment of Physics\nFranklin Science Center\nShippensburg University of Pennsylvania\n1871 Old Main Drive17257PennsylvaniaUSA\n", "S Moretti \nSchool of Physics and Astronomy\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUK\n\nDepartment of Physics and Astronomy\nUppsala University\nBox 516SE-751 20UppsalaSweden\n" ]	[ "Institute of Convergence Fundamental Studies\nSeoul National University of Science and Technology\n01811SeoulKorea", "Centre for Theoretical Physics\nBritish University in Egypt\nP.O. Box 4311837CairoEgypt", "Department of Physics\nFranklin Science Center\nShippensburg University of Pennsylvania\n1871 Old Main Drive17257PennsylvaniaUSA", "School of Physics and Astronomy\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUK", "Department of Physics and Astronomy\nUppsala University\nBox 516SE-751 20UppsalaSweden" ]	[]	We show how, in the B − L extension of the SM (BLSM) with an Inverse Seesaw (IS) mechanism for neutrino mass generation, a light Z state with moderate couplings to SM objects, hence 'dark' in its nature, can be associated, in conjunction with light sterile neutrinos, to some present day data anomalies, such as the anomalous magnetic moment of the muon as well as a possible signal indicating the existence of sterile neutrinos in neutrino beam experiments.	10.1016/j.physletb.2022.136945	[ "https://arxiv.org/pdf/2110.08651v2.pdf" ]	239,016,750	2110.08651	e4f33d568ac331fd8187f7c98e34eae247967870	The Dark Z and Sterile Neutrinos Behind Current Anomalies February 8, 2022 A Hammad Institute of Convergence Fundamental Studies Seoul National University of Science and Technology 01811SeoulKorea Centre for Theoretical Physics British University in Egypt P.O. Box 4311837CairoEgypt Ahmed Rashed Department of Physics Franklin Science Center Shippensburg University of Pennsylvania 1871 Old Main Drive17257PennsylvaniaUSA S Moretti School of Physics and Astronomy University of Southampton HighfieldSO17 1BJSouthamptonUK Department of Physics and Astronomy Uppsala University Box 516SE-751 20UppsalaSweden The Dark Z and Sterile Neutrinos Behind Current Anomalies February 8, 2022 We show how, in the B − L extension of the SM (BLSM) with an Inverse Seesaw (IS) mechanism for neutrino mass generation, a light Z state with moderate couplings to SM objects, hence 'dark' in its nature, can be associated, in conjunction with light sterile neutrinos, to some present day data anomalies, such as the anomalous magnetic moment of the muon as well as a possible signal indicating the existence of sterile neutrinos in neutrino beam experiments. Introduction Despite its huge successes, the Standard Model (SM) of particle physics has several drawbacks which require one to conceive some Beyond the SM (SM) physics. Its Achilles' heel is probably the leptonic sector, though, as neutrino masses are forbidden in the SM, yet, experiments have verified that neutrino flavours oscillate which in turn implies that neutrinos have finite masses. Neutrinos are strictly massless in the SM essentially due to two reasons: (i) the absence of their right-handed eigenstates; (ii) an exact global Baryon minus Lepton (B − L) number conservation. However, a modification of the SM, based on the gauge group SU (3) C ×SU (2) L ×U (1) Y ×U (1) B−L , nicknamed the B − L extension of the SM (BLSM), wherein the additional Abelian group is elevated to be a local symmetry, can account for light neutrino masses through an Inverse Seesaw (IS) mechanism [1,2]. In such a construct, the aforementioned right-handed neutrinos would acquire Majorana masses at the B − L symmetry breaking scale, but they are not allowed to do so by the discussed B − L gauge symmetry and another pair of SM gauge singlet fermions with tiny masses, of O(1 keV), must be introduced. Therefore, such a small scale can be considered as a slight breaking of the underlying gauge symmetry, hence, according to 't Hooft criteria, its dynamics becomes natural. One of these two singlet fermions couples to right-handed neutrinos and is involved in generating the light neutrino masses. The other singlet (usually called inert or sterile neutrino) is completely decoupled and interacts only through the B − L gauge boson, a Z , ensuing from the spontaneous breaking of the additional U (1) B−L group [3], so that it may account for warm Dark Matter (DM) [4] (see also Ref. [5]), the lack of a viable candidate for it being another significant flaw of the SM. This construct, BLSM-IS for short, predicts several testable signals at the Large Hadron Colider (LHC) through some of the new particles that it embeds: the Z (neutral gauge boson) associated with U (1) B−L , an extra Higgs boson (h , in fact, an additional (pseudo)scalar singlet state is introduced to break the gauge group U (1) B−L spontaneously) and heavy neutrinos (ν h , which are required to cancel the associated new gauge anomalies and are thus necessary for the consistency of the whole model). Particle l L l R Q L u R d R ν R φ χ S 1 S 2 B − L charge -1 -1 1 3 1 3 1 3 -1 0 -1 -2 +2 Ref. [6] reviewed the LHC potential to access the BLSM-IS, including its Supersymmetric extension [7,8,9], when the Z mass is of order TeV and such a state is relatively strongly coupled to SM states. In this paper, we aim instead at considering the case of a very light Z , of MeV scale, very mildly coupled to SM objects, specifically, whether it can be responsible, together with the aforementioned sterile neutrinos, of data anomalies that have emerged from the E821 experiment at BNL and the Muon g − 2 one at FNAL as well as the MiniBooNE (MB) collaboration also at FNAL. In fact, the former two hinted at statistically significant deviations from the SM predictions of the anomalous magnetic moment of the muon, (g − 2) µ for short, which could be explained by a very light Z state, while the latter one was taken as a sign of the possible existence of sterile neutrinos. The plan of the paper is as follows. In the next section, we describe the BLSM-IS. In Sect. III, we discuss both direct and indirect experimental constraints on light Z and sterile neutrino states. We then move on to present our results for (g − 2) µ . After this, we discuss our explanation for the MB excess. Finally, in the last section, we present our summary. The model To start with, in the BLSM-IS that we consider here, we assume that the SM singlet scalar χ, which spontaneously breaks U (1) B−L , has B − L charge = −1. Also, the three pairs of SM singlet fermions, S 1,2 with B − L charge = ∓2, respectively, are introduced (see tab. I, wherein l L,R refer to leptons, Q L , u R , d R identify quarks and φ is the Higgs state of the SM). Limited to the leptonic sector, the BLSM-IS Lagrangian is given by L (B−L) = − 1 4 F µν F µν + il L D µ γ µ l L + il R D µ γ µ l R + iS 1R D µ γ µ S 1 + iS 2R D µ γ µ S 2 + iν R D µ γ µ ν R + (D µ φ) † (D µ φ) + (D µ χ) † (D µ χ) − V (φ, χ) − λ llL φl R + λ νlLφ ν R + λ sνR χS 2 + h.c.,(1) withφ = iσ 2 φ * . Using the unitary gauge parameterisation, the kinetic terms become (D µ φ) † (D µ φ) = 1 2 ∂ µ h∂ µ h + 1 8 (φ + υ) 2 g 2 \|W µ 1 − iW µ 2 \| 2 + (gW µ 3 − g 1 B µ −gB µ ) 2(2) and (D µ χ) † (D µ χ) = 1 2 ∂ µ h ∂ µ h + 1 2 (h + υ ) 2 2g (B−L) B µ ,(3) where g (  B µ W µ 3 B µ   =   cos θ W − sin θ W cos θ sin θ W sin θ sin θ W cos θ W cos θ − cos θ W sin θ 0 sin θ cos θ     A µ Z µ Z µ   ,(4) with θ W the weak mixing angle while −π 4 ≤ θ ≤ −π 4 such that tan 2θ = 2g g 2 + g 2 1 g + 16 υ υ 2 g 2 (B−L) − g 2 − g 2 1 .(5) The neutral gauge boson masses are determined by fixing the values of the new parameters as M Z,Z = υ g 2 + g 2 1 2    1 2   g 2 + 16 υ υ 2 g 2 (B−L) g 2 + g 2 1 + 1    ∓g sin 2θ g 2 + g 2 1    1 2 .(6) In the BLSM-IS model, the Majorana neutrino Yukawa interaction induces the masses onto the SM neutrinos after U (1) B−L symmetry breaking via the Lagrangian terms L ν = m DνL ν R + m NνR S 2 + h.c ,(7) with a Dirac mass m D = 1 √ 2 λ ν υ and a Majorana mass m N = 1 √ 2 λ S υ . The 9 × 9 neutrino mass matrix can be written as M ν =   0 m D 0 m T D 0 m N 0 m T N µ s   .(8) In order to avoid a possible large mass term mS 1 S 2 in the Lagrangian, that would spoil the IS structure, one assumes a Z 2 symmetry under which ν R , χ, S 2 and the SM particles are even while S 1 is an odd particle. The neutrino mass matrix M ν can be diagonalised by the matrix V as 1 V T M ν V = M diag ν ,(9) with V a 9 × 9 matrix defined as [10] V = V 3×3 V 3×6 V 6×3 V 6×6 .(10) The upper 3 × 3 block are the parameters for the effective Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix with its elements given by V 3×3 = 1 − 1 2 θ T θ U PMNS ,(11) in terms of the actual one. The off-diagonal blocks of the V matrix are defined via V 3×6 = (0 3×3 , θ) V 6×6 ,(12) with θ ∼ m D m −1 N . The matrix V 6×6 diagonalises the right-handed Majorana neutrinos and S 2 . The diagonalisation of the entire neutrino mass matrix leads to the following light and heavy neutrino masses: m light ∼ m D m −1 N µ s (m T N ) −1 m T D , m heavy ∼ m 2 N + m 2 D 1 2 ,(13) 1 Assuming there are no complex Majorana phases and the Lagrangian parameters are real. with the latter being pair degenerate. With this structure, the light neutrinos can be of order eV, as required by flavour oscillation experiments, and, with a small µ s value, the ensuing Yukawa coupling is no longer restricted to be very small, indeed, it can be of order one. Moreover, the mixing between light and heavy neutrinos is constrained from lepton flavour violation measurements to be of order O(0.01) as discussed in [11,12,13] and references therein. The tree level coupling of the Z with charged and neutral fermions is expressed as g (Z l ilj ) = iδ ij 2 2 g (B−L) cos θ +g sin θ W sin θ + g 1 sin θ W sin θ − g 2 cos θ W sin θ γ µ 1 − γ 5 2 + iδ ij g (B−L) cos θ + (g 1 +g) sin θ W sin θ γ µ 1 + γ 5 2 ,(14) while the ones with active and sterile (light and heavy) neutrinos are given by g (Z ,νi,νj ) = i 2 2g (B−L) cos θ + (2g + g 1 ) sin θ W sin θ + g 2 cos θ W sin θ 3 a=1 V * ja V ia − 2g (B−L) cos θ + 2g sin θ W sin θ 3 a=1 V * j3+a V i3+a γ µ 1 − γ 5 2 + + (− i 2 ) 2g (B−L) cos θ + (2g + g 1 ) sin θ W sin θ + g 2 cos θ W sin θ 3 a=1 V * ia V ja − 2g (B−L) cos θ + 2g sin θ W sin θ 3 a=1 V * i3+a V j3+a γ µ 1 + γ 5 2 ,(15) with θ constrained from LEP experiment to be ∼ 3 × 10 −3 [14]. Direct and Indirect constraints on light Z and sterile neutrinos In this section we discuss the direct and the indirect constraints for low mass Z and sterile neutrinos. In fig. 1 we show the most severe constraints on the light Z mass as a function of the Z gauge coupling g (B−L) and the gauge kinetic mixing parameterg from existing low energy experiments. To recast these bounds on those applicable to our model we used the method of Ref. [15] and produced fig. 1 by using the code advertised in the same paper. The key here is that low energy experiments setting bounds on the low mass photon (a dark photon, A ) considered therein also set limits on a light dark Z , so long that one accounts for the gauge kinetic mixing and axial coupling. Recasting a dark photon search that used the final state F in constraints onto our model can be done, for each Z mass, by equating the upper limit total cross section of dark photon models to the Z one in our model as follows: σ Z × BR(Z → F ) × Z = σ A × BR(A → F ) × A ,(16) with σ Z /A being the production cross section and BR(Z /A → F ) the Branching Ratio (BR) of the light gauge boson into the final state F while Z /A is the detector efficiency. Therefore, one can see that, in order to recast the aforementioned experimental limits in terms of our model parameters, we only need the ratios σ Z /σ A , BR(Z → F )/BR(A → F ) and Z / A . In the following, we are going to discuss how these ratios can be obtained for each experiment. • The BaBar detector at the PEP-II B-factory [16] has collected 53 fb −1 of e − e + collisions looking for events with a single high-energy photon and large missing (transverse) momentum or energy which is consistent with the process e − e + → γX and X → invisible, with X being a light gauge boson with spin equal to 1. Further, in [17], the BaBar experiment searched for a single high energy photon plus a dilepton final state, e − e + → γX and X →ll, with l = e, µ. In both searches no statistically significant deviations from the SM predictions have been observed and a 90% Confidence Level (CL) upper limit on the light gauge boson coupling to leptons in the mass range of 0.02 − 10.2 GeV has been set. Recasting this limit onto our model we obtain g 2 Z × BR(Z → ll) g 2 X × BR(X → ll) = 1 ,(17) with g Z being the Z coupling to charged and neutral leptons, eqs. (1) and (15), and g X being the measured gauge boson coupling to charged and neutral leptons. • The A1 Collaboration at the Mainz Microtron (MAMI) [18] searched for the signal of a new light U (1) gauge boson in electron-positron pair production. Since no deviation from the SM value for the corresponding cross section has been observed, A1 set a limit on the light gauge boson coupling over the mass range 40 − 300 MeV. To recast this limit on our model parameters, we have again made use of eq. (17). • Electron beam dump experiments (like E141, E774 and those at KEK and Orsay) also have sensitivity to a new light gauge boson. An overview of the different electron beam dump experiments and their properties is given in [19]. For the SLAC E141 experiment [20], an upper limit is set for neutral particles with masses in the range 1 − 15 MeV following the nonobservation of any excess above the SM bremsstrahlung rate for events of the type e + N → e + N + X. From the Fermilab E774 experiment [21], an upper limit for neutral particles which decay into electron-positron pairs was set. In the electron beam dump experiment at KEK [22], no signal was observed in their search for axion-like particles. The electron beam dump experiment in Orsay [23] also found no positive signal when looking for light Higgs bosons decaying into electron-positron pairs. Combining and reinterpreting these last three experiments, one is able to exclude a light boson over the mass range 1.2 − 52 MeV. • Proton beam dump experiments, like NOMAD [24] and CHARM [25], also found no positive signal while looking for axion like-particles decaying to leptonic pairs, following which the 0.1 − 20 MeV mass range is also precluded to a dark photon or Z . • The NA64 experiment at the CERN SPS [26] found no deviation from the SM expectation while looking for dark photons in the process e − N → e − N A . Hence, a new limit has been set on the A (dark photon) mixing and the absence of invisible A decays excluded the mass range M A ≤ 100 MeV. • The DELPHI experiment at LEP2 [27] analysed single photon events in looking for extra dimension gravitons. As in [28], since the measured single-photon cross sections are in agreement with the expectations from the SM, an upper limit on the coupling and mass of the dark candidate was set, the latter being above 10 GeV. Before moving on to study the relevant experimental observables, we should mention that we have used SPheno [29,30] to generate the model spectrum as well as HiggsBounds and HiggsSignals [31,32,33,34,35] to check the constraints on the Higgs sector of it. Also, we have used FlavourKit [36] to check lepton flavour violation constraints. The muon anomalous magnetic moment The Lande g factor for muons, and its deviation from the tree level value of 2, represents one of the most precisely measured quantities in the SM. Therefore, it is also an excellent probe for new physics. Currently, there exists a long standing and statistically significant discrepancy between its measurement and the theoretically predicted value [37,38,39,40] 2 : ∆a µ = ∆a ex µ − ∆a th µ = (2.51 ± 0.59) × 10 −9 .(18) In this section, we focus on a light Z as a means to solve the current muon anomalous magnetic moment anomaly. Following the general formula in [42], the interaction Lagrangian of a Z with muons can be rewritten as L int =μγ µ C V + C A γ 5 µZ ,(19) where C V and C A are the vector and axial couplings introduced in eq. (14). The Z modifies the muon magnetic moment via the one loop diagram in fig. 2. The Z contribution can be obtained as [43] ∆a µ = m 2 µ 4π 2 m 2 Z   C 2 V 1 0 x 2 (1 − x) 1 − x + x 2 m 2 µ m 2 Z dx − C 2 A 1 0 x(1 − x)(4 − x) + 2x 3 m 2 µ m 2 Z 1 − x + x 2 m 2 µ m 2 Z dx    ,(20) with x being the Feynman parameter. For a low mass Z , its contribution to the muon anomalous magnetic moment is This viable region of model parameters is also compliant with the constraints given in [43], which included the following ones. ∆a µ (m 2 Z C 2 V − 2m 2 µ C 2 A ) 8π 2 m 2 Z .(21) 1. Cosmological and astrophysical bounds: Big Bang Nucleosynthesis (BBN) [44] as well as Cosmic Microwave Background (CMB) from "Planck 2018" [44] in addition to the astrophysical experiments (e.g SN1987A) studied by [45]. 2. Neutrino scattering bounds: several neutrino scattering experiments results on couplings to muons and muon neutrinos, the most stringent ones of these being from Borexino [46] and CHARM-II [47]. Observation of energy loss in supernovae due to Z − interactions set constraints on the B − L model parameters [45,48,49,50]. A Z mass up to 100 MeV is constrained in the (M Z , g (B−L) ) plane [45]. Both cosmological (BBN) and astrophysical (SN1987A) limits are model dependent. For instance, the chameleon effect due to the environmental matter density and late reheating can weaken the SN1987A [51] and BBN [52] limits, respectively. In the study of the astrophysical limit, a pure Z model has been considered [45], but in our model we have an extended scalar sector. In the presence of new scalar states, the limits on Z change dramatically and can be avoided when a neutral state couples to a dark matter particle [53], as is the case in our model. A model-independent fit to all such experimental data (thus including (g − 2) µ ) reveals the following parameter values as viable. 2. An axial coupling of the Z to electrons larger than the vector one: \|C Ae \| ∼ [1 − 3.2] × 10 −4 > \|C V e \| 7.7 × 10 −5 . 3. A large vector coupling to muons, 5 × 10 −4 < \|C V µ \| 0.05, and an axial coupling C Aµ that is smaller by at least a factor of a few. 4. Tiny Z couplings to neutrinos: \|C νe, νµ \| 10 −5 . We now move on to study the MB anomaly and its theoretical implications. MiniBoone In this section, we will study the anomaly registered by the MB experiment, wherein the beam primarily consists of ν µ 's produced via pion decay. The relevant process leading to an electron excess is ν µ N (k) → N (k )ν 4 e + e − −, as shown in fig. 4, where ν 4,5 are sterile neutrinos, with m ν5 > m ν4 . (Previous work explaining the results in Ref. [54] using sterile neutrinos can be found in [55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71].) The process is mediated by the new Z producing a collimated e + e − pair (producing the visible light that makes up the signal) through the decay of ν 5 into ν 4 , which then makes the cross section proportional to \|U ν5ν4 \| 2 . By calculating the liftime of ν 4 , it is found to be 1.76 second. The corresponding decay length is 5.3 × 10 8 m which is way greater than the MB dimention of 12.2 m. Thus, ν 4 will decay away outside the detector. So that there is no additional EM deposition in the detector due to the ν 4 decay. The form factor of the coupling of the Z with nucleons N is N (k )\|J µ Z \|N (k) = g Bū (k )Γ µ Z (k − k)u(k), where k and k are the initial and final nucleon momenta whereas Γ µ Z (q) = γ µ F 1 V (q 2 ) + i 2 m N σ µν q ν F 2 V (q 2 ).(22) The isoscalar form factors F 1 V (q 2 ) and F 2 V (q 2 ) for the nucleon are given by [72] F 1 V (q 2 ) F D (q 2 ) = 1 − q 2 (a p + a n ) 4m 2 N − q 2 , F 2 V (q 2 ) F D (q 2 ) = 4m 2 N (a p + a n ) 4m 2 N − q 2 ,(23) where m N = 0.938 GeV, F D (q 2 ) = (1 − q 2 /0.71 GeV 2 ) −2 with a p ≈ 1.79 and a n ≈ −1.91 being coefficients related to the magnetic moments of the proton and neutron, respectively. The total differential cross section has two components, an incoherent and a coherent one, which we will both consider. The total differential cross section, for the target in MB, i.e., CH 2 , is given by dσ dE h CH2 = 14 × dσ dE h incoherent + 144 × exp(2b(k − k) 2 ) dσ dE h coherent .(24) The incoherent contribution from the single nucleon cross section is multiplied by the total number of the nucleons present in CH 2 , i.e., 14. However, the entire carbon nucleus contributes to the coherent process weighted by the exponential factor exp(2b(k − k) 2 ) [73], where b is a numerical parameter, which for C 12 has been found to be 25 GeV −2 [74,73]. The coherent process decreases as q 2 = (k − k) 2 increases, where q 2 is negative. The number of events is given by [75] N events = η dE ν dE ν5 dΦ ν dE ν dσ dE ν5 ×BR(ν 5 → ν 4 Z )×BR(Z → e − e + ) × n,(25) with E h ∈ [E h , E h + ∆E h ] and where Φ ν is the incoming muon neutrino flux. Here, n is the number of nuclei in the fiducial volume of the detector. In the case of MB, the target is 818 tons of mineral oil (CH 2 ) with atomic mass 14 [54], as mentioned, so that n = 3.5174 × 10 31 . Furthermore, η = 0.2 contains all the detector related information like efficiencies, Protons-on-Target (POT), etc. The latest data set for the neutrino mode, corresponding to 18.75 × 10 20 POT, as detailed in [54,75], has been used in our fit. Finally, for these values, the calculated lifetimes of the ν 5 and Z states in their rest frame are 10 −17 s and 1.8 × 10 −12 s, respectively. The value of E ν5 is related to the visible energy, E vis = E e + + E e − , as follows E ν5 = E Z −M 2 ν4 + M 2 ν5 + M 2 Z − (E 2 Z − M 2 Z ) M 4 ν4 − 2M 2 ν4 M 2 ν5 + M 2 Z + M 2 ν5 − M 2 Z 2 2M 2 Z . (26) Furthermore, the Mandelstam variables in terms of the neutrino(lepton) energy E ν (E l ) are s = M 2 + 2M E ν , t = 2M (E l − E ν ), s − u = 4M E ν + t − m 2 l .(27) Then, t and E l lie in the intervals m 2 l − 2E cm ν (E cm l + p cm l ) ≤ t ≤ m 2 l − 2E cm ν (E cm l − p cm l ) ,(28)E ν + m 2 l − 2E cm ν (E cm l + p cm l ) 2M ≤ E l ≤ E ν + m 2 l − 2E cm ν (E cm l − p cm l ) 2M ,(29) where the energy and momentum of the neutrino and lepton in the center of mass (cm) system are E cm ν = (s − M 2 ) 2 √ s , p cm l = (E cm l ) 2 − m 2 l , E cm l = (s − M 2 + m 2 l ) 2 √ s .(30) The threshold neutrino energy to create the charged lepton partner is given by E th ν l = (m l + M p ) 2 − M 2 n 2M n ,(31) where m l , M p and M n are the masses of the charged lepton, proton and neutron, respectively. The differential cross section in the laboratory frame is given by dσ tot (ν l ) dt = \|M\| 2 32πE 2 ν M 2 f (t),(32) where f (t) = M 2(E ν + M ) − Eν 5 p 2 ν 5 (t − m 2 ν5 + 2E ν E ν5 ) .(33) We have then verified our analytic calculations with MadGraph [76], where the nucleon form factor in eq. (22) is implemented effectively in the Universal FeynRules Output (UFO) files [77], by fixing q 2 = M 2 Z . To measure the goodness of the fit between the BLSM-IS and the measured data, we constructed a χ 2 test function as χ 2 = bins (Events th − Events ex ) 2 δ 2 ij ,(34) with δ ij the covariance matrix that contains the uncorrelated intrinsic experimental statistic and systematic uncertainties in its diagonal entries. Fig. 5 we show the prediction for two BLSM-IS signals obtained by adopting two benchmark points with M Z = 20, 30 MeV and fixed g (B−L) = −10 −4 ,g = 0.2, θ = 3 × 10 −3 , m ν4 = 60 MeV and m ν5 = 110 MeV, together with the background and against the data collected by MB which appear anomalous. We find a good agreement between predictions and data up to a 5σ CL. Finally, in fig. 6, shows the result of the above fit to the measured MB data extracted from [54] over the Z mass range 2 − 130 MeV with fixed m ν4 , m ν5 and θ values while g (B−L) andg have been chosen at their maximal allowed values for the given M Z (as in fig.1). The fit shows that we can reach the 5σ CL for Z masses in the range of 15 − 25 MeV. Conclusions In summary, in this letter, we have argued that two anomalies presently stemming from noncollider experiments, specifically, in the measurement of the anomalous magnetic moment of the muon at the E821 experiment at BNL and the Muon g − 2 one at FNAL as well as in the study of appearance data in the MB short-baseline neutrino experiment at FNAL, hint at a common explanation relying on some BSM physics that might involve both a light Z and light neutrinos, all being extremely weakly coupled to the visible sector (so as to being dubbed dark and sterile, respectively). There is a BSM scenario that can incorporate these new force and matter states in a minimal formulation, thereby being notionally able to explain the aforementioned data sets without invoking an excessing number of new parameters. This is the so-called BLSM-IS, wherein the SM gauge group is supplemented by an additional, spontaneously broken U (1) B−L invariance, obtained by localising the accidental global B − L conservation of quantum numbers that appears in the SM, in combination with an IS mechanism for neutrino mass generation. The requirement of theoretical self-consistency of this BSM scenario in fact imposes the simultaneous presence of a Z state following the B − L breaking, which can be made light rather naturally, and of multiple sterile neutrinos, which are per se rather light. Herein, we have put the BLSM-IS explanations to the aforementioned data anomalies on firm quantitative grounds. In fact, solutions have been found to both anomalies simultaneously for the following ranges of BLSM-IS parameters: M Z = 15 − 25 MeV, g (B−L) ∼ −10 −4 ,g ≈ 0.2, θ = 3 × 10 −3 , m ν4 = 60 MeV and m ν5 = 110 MeV. B−L) is the coupling strength of the new Z boson,g is the gauge kinetic mixing parameter and υ, υ are the SM and U (1) B−L vacuum expectation values. The mass eigenstates of the gauge boson fields are linear combinations of B µ , W µ 3 and B µ . The explicit expression for the mass mixing matrix is  Figure 1 : 1Bounds on the plane (g, g (B−L) ) for different Z mass values in the BLSM-IS for fixed θ = 3 × 10 −3 . The allowed region is the non-shaded area with allowed values of M Z less than each corresponding value (in GeV) on the red dots. Figure 2 :Figure 3 : 23Feynman diagram for the Z contribution to the muon anomalous magnetic moment. The ∆a Z µ dependence on the two couplingg and g (B−L) . Fig. 3 3shows the ∆a Z µ dependence on the two couplingg and g (B−L) . The density plot is confined between the upper and lower experimental values ((2.51 ± 0.59) × 10 −9 ), respectively, of ∆a Z µ within 1σ CL. The plot represents the allowed region of the model parameters that satisfies the experimental data on ∆a Z µ . Here, \|g\| is taken from 10 −4 to 10 −1 with the allowed range of \|g (B−L) \| being from(9.93 − 10.79) × 10 −4 to (9.87 − 10.86) × 10 −4 , respectively. The corresponding range of vector and axial couplings for M Z = 30 MeV and \|g\| = 10 −4 are \|C V \| = (10.14 − 10.59) × 10 −4 and \|C A \| ≈ 1.88 × 10 −4 whereas for \|g\| = 10 −1 they are \|C V \| = (9.67 − 11.06) × 10 −4 and \|C A \| = 1.88 × 10 −4 . Figure 4 : 4Feynman diagram of the scattering process in the BLSM-IS which leads to the excess in MB. 1. A lightZ in the mass range 16 MeV M Z 38 MeV. Figure 5 :Figure 6 : 56fit mZ = 20 M eV Our fit mZ = 30 M eV Data Left Panel: The MB electron-like anomalous data and total background events [54] versus the visible energy. 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[ "arXiv:1307.6148v2 [physics.plasm-ph] Electromagnetic and gravitational radiation from the coherent oscillation of electron-positron pairs and fields", "arXiv:1307.6148v2 [physics.plasm-ph] Electromagnetic and gravitational radiation from the coherent oscillation of electron-positron pairs and fields" ]	[ "Wen-Biao Han \nShanghai Astronomical Observatory\n80 Nandan Road200030ShanghaiChina\n", "She-Sheng Xue \nPhysics Department\nICRANet\nPiazza della Repubblica 10I-65122PescaraItaly\n\nICRA\nUniversity of Rome\nLa Sapienza, Piazzale Aldo Moro 5I-00185RomeItaly\n" ]	[ "Shanghai Astronomical Observatory\n80 Nandan Road200030ShanghaiChina", "Physics Department\nICRANet\nPiazza della Repubblica 10I-65122PescaraItaly", "ICRA\nUniversity of Rome\nLa Sapienza, Piazzale Aldo Moro 5I-00185RomeItaly" ]	[]	Integrating equations of particle-number and energy-momentum conservation and Maxwell field equations, we study the oscillation and drift of electron and positron pairs coherently with fields after these pairs are produced in external electromagnetic fields. From the electric current of oscillating pairs, we obtain the energy spectrum of electromagnetic dipole radiation. This narrow spectrum is so peculiar that the detection of such radiation can identify pair production and oscillation in strong laser fields. We also obtain the energy spectrum of gravitational quadrapole radiation from the energy-momentum tensor of oscillating pairs and fields. Thus, we discuss the generation of gravitational waves on the basis of rapid development of strong laser fields.	10.1103/physrevd.89.024008	[ "https://arxiv.org/pdf/1307.6148v2.pdf" ]	118,530,941	1307.6148	d8c21e3bed1b5dff82f3d2ae062fb35701ca1316	arXiv:1307.6148v2 [physics.plasm-ph] Electromagnetic and gravitational radiation from the coherent oscillation of electron-positron pairs and fields 17 Mar 2014 Wen-Biao Han Shanghai Astronomical Observatory 80 Nandan Road200030ShanghaiChina She-Sheng Xue Physics Department ICRANet Piazza della Repubblica 10I-65122PescaraItaly ICRA University of Rome La Sapienza, Piazzale Aldo Moro 5I-00185RomeItaly arXiv:1307.6148v2 [physics.plasm-ph] Electromagnetic and gravitational radiation from the coherent oscillation of electron-positron pairs and fields 17 Mar 2014(Dated: Received version May 11, 2014)numbers: 5227Ep5240Db0430-w Integrating equations of particle-number and energy-momentum conservation and Maxwell field equations, we study the oscillation and drift of electron and positron pairs coherently with fields after these pairs are produced in external electromagnetic fields. From the electric current of oscillating pairs, we obtain the energy spectrum of electromagnetic dipole radiation. This narrow spectrum is so peculiar that the detection of such radiation can identify pair production and oscillation in strong laser fields. We also obtain the energy spectrum of gravitational quadrapole radiation from the energy-momentum tensor of oscillating pairs and fields. Thus, we discuss the generation of gravitational waves on the basis of rapid development of strong laser fields. Integrating equations of particle-number and energy-momentum conservation and Maxwell field equations, we study the oscillation and drift of electron and positron pairs coherently with fields after these pairs are produced in external electromagnetic fields. From the electric current of oscillating pairs, we obtain the energy spectrum of electromagnetic dipole radiation. This narrow spectrum is so peculiar that the detection of such radiation can identify pair production and oscillation in strong laser fields. We also obtain the energy spectrum of gravitational quadrapole radiation from the energy-momentum tensor of oscillating pairs and fields. Thus, we discuss the generation of gravitational waves on the basis of rapid development of strong laser fields. Introduction. Positron and electron pairs are produced from the vacuum in a constant electromagnetic field and the production rate is sizable when the field strength reaches the critical value (E c = 1.3×10 16 V/cm); see Refs. [1,2]. To reach this critical value in laboratory, based on recent advanced laser technologies, there are many ongoing experiments: x-ray free-electron laser (XFEL) facilities [3], optical high-intensity laser facilities such as Vulcan or ELI [4], and SLAC E144 using nonlinear Compton scattering [5] for details, see Refs. [6][7][8]. This leads to the physics of ultrahigh intensity lasermatter interactions in the critical field [10]. We focus on the backreaction and screening effects of electron and positron pairs on external electric fields that lead to the phenomenon of plasma oscillation: electrons and positrons moving back and forth coherently with alternating electric fields. In a constant electric field E ext , the phenomenon of plasma oscillations is studied in two frameworks: (1) the semiclassical QED with a quantized Dirac field and classical electric field [11,12]; and (2) the kinetic description using the Boltzmann-Vlasov equation (or equations of particle-number and energy-momentum conservation) and the Maxwell equations [13,14]. In Ref. [11], two frameworks are discussed. The first framework is semiclassical, where quantized fermion fields ψ satisfy the Dirac equation in an external classical potential A µ , which satisfies the Maxwell equation coupling to the mean value of charged fermion current. These equations are numerically integrated in (1+1)-dimensional case. The second framework is classical -the description of particle distribution or density is adopted, and the kinetic equation of the Boltzmann-Vlasov for particle density and the Maxwell equation for fields are numerically integrated. The results obtained in two frameworks are in good quantitative agreement [11], for details, see Refs. [2]. In this paper, we adopt the second framework to investigate the plasma oscillations of electron and positron pairs in the (2+1) space-time with the presence of both electric and magnetic fields. We obtain not only the frequencies of plasma oscillations, but also the oscillating pattern of electron and positron pairs in the (2+1) space-time. In addition, we obtain the energy spectra of electromagnetic and gravitational radiation from plasma oscillations. PLASMA OSCILLATION. In 1931 Sauter [15] and four years later Heisenberg and Euler [16] provided a first description of the vacuum properties in constant electromagnetic fields. They identified a characteristic scale of strong field E c = m 2 e c 3 /e , at which the field energy is sufficient to create electron positron pairs from the vacuum. In 1951, Schwinger [17] gave an elegant quantum-field theoretic reformulation of their result in the spinor and scalar QED framework (see also [18]). The special attention was given for the presence of magnetic fields [19]. In the configuration of constant electromagnetic fields, the pair-production rate per unit volume is given by Γ V = αε 2 π 2 n=1 1 n 2 nπβ/ε tanh nπβ/ε exp − nπE c ε ,(1) where the two Lorentz invariants ε and β are ε ≡ (S 2 + P 2 ) 1/2 + S, β ≡ (S 2 + P 2 ) 1/2 − S. (2) In terms of the two Lorentz invariants, the scalar S ≡ (E 2 − B 2 )/2 = (ε 2 − β 2 )/2, and the pseudoscalar P = E · B = εβ. In order to focus on studying the phenomenon of plasma oscillations, as a model for quantitative calculations, we postulate an initial configuration of constant electromagnetic fields: (i) The electric and magnetic fields are perpendicular to each other (E ⊥ B). (ii) Their amplitudes are different (\|E\| > \|B\| = 0) in the laboratory frame, i.e., the rest frame of electron-positron pair production. For this electromagnetic configuration P = 0 and leading term (n = 1), Eq. (1) yields S = m 4 e 4π 3 2S E 2 c exp − πE c (2S) 1/2 ,(3) where the critical field E c ≡ m 2 e /e and m e (−e) is the electron mass (charge). Note that Eq. (3) is valid only for S > 0, i.e., \|E\| > \|B\| and E ⊥ B. In this case β = 0 and ε 2 = 2S, Eq. (3) is equivalent to the case for a purely electric field E = 2S. Equation (3) approaches zero as \|B\| approaches \|E\| + 0 − . As shown below, we have chosen an electric field strength E that is significantly larger than the magnetic one B; otherwise, Eq. (3) would approximately vanish for S ≈ 0 and P = 0, analogously to the field configuration of a monochromatic laser beam (plane wave S = P = 0). We will also discuss the situation in which electromagnetic fields are parallel. It is an important issue for future investigations how initial configurations are dynamically generated from the outset. We adopt = c = 1 and Compton units of length λ C = /m e c, time τ C = /m e c 2 , energy scale m e c 2 and critical field strength E c . In the kinetic description for plasma fluids of positrons (+) and electrons (−), with single-particle spectra p 0 ± = (p 2 ± + m 2 e ) 1/2 , we define the number densities n ± (t, x) and "averaged" velocities v ± (t, x) of the fluids: n ± ≡ d 3 p ± (2π) 3 f ± , v ± ≡ 1 n ± d 3 p ± (2π) 3 p ± p 0 ± f ± ,(4) where f ± = f ± (t, p ± , x) is the distribution function in phase space. The four-velocities of the electron and positron fluids are U µ ± = γ ± (1, v ± ), the Lorentz factor γ ± = (1 − \|v ± \| 2 ) −1/2 , and the comoving number densitiesn ± = n ± γ −1 ± . The collisionless plasma fluid of electrons and positrons coupling to electromagnetic fields is governed by the equations of particle-number and energymomentum conservation and the Maxwell equations: ∂ n ± U µ ± ∂x µ = S; ∂T µν ± ∂x ν = −F µ σ (J σ ± + J σ ±pola ), (5) ∂F µν ∂x ν = −4π(J µ cond + J µ pola + J µ ext ),(6) where we have an external electric current J µ ext = (ρ ext , J ext ), electron and positron fluid currents J µ ± = ±en ± U µ ± , and energy-momentum tensors T µν ± =p ± g µν + (p ± +ǭ ± )U µ ± U ν ± , T µν m = ± T µν ± . (7) Here the pressurep ± and energy densityǭ ± are related by the equation of statep ± =p ± (ǭ ± ) in the fluid comoving frame. In the laboratory frame, the electron and positron energy density p 0 ± ≡ T 00 ± and momentum density p i ± ≡ T i0 ± are given by p 0 ± = (ǭ ± +p ± v 2 ± )γ 2 ± and p ± = (ǭ ± + p ± )γ 2 ± v ± . The conducting four-current density is J µ cond ≡ e(n + U µ + −n − U µ − ), ∂ µ J µ cond = 0,(8) and the polarized four-current density J µ pola = ± J µ ±pola with J µ ±pola = ρ ± pola , J ± pola defined by [14] F ν µ J µ ±pola = Σ ν ± , Σ ν ± = d 3 p ± (2π) 3 p 0 ± p ν ± A,(9) where A is related to Eq. (3) by S = d 3 p ± /[(2π) 3 p 0 ± ]A. F µν and T µν em are the field strength and the energymomentum tensor of electromagnetic fields. We now assume external electromagnetic fields E ext = E extẑ and B ext = B extx , where E ext and B ext are constant fields in space and time. As will be shown below, in this system, the electron-positron fluid velocities [Eq. (4)] haveẑ andŷ components v ± = (v y ±ŷ + v z ±ẑ ) in the y − z plane, and the total electromagnetic fields are E = E yŷ + E zẑ and B = B xx , which are the superposition of two contributions: E z = E ext +Ẽ z (t, y, z), E y =Ẽ y (t, y, z); B x = B ext +B x (t, y, z), B y = 0,(10) where the space-and time-dependentẼ z,y (t, y, z) and B z (t, y, z) are the electromagnetic fields created by the motion of electron and positron pairs. We adopt the approximationsp ± ≈ 0,ǭ ± ≈ m en± , and ǫ ± =ǭ ± γ 2 ± when the pair number density is not very large for E ≃ E c . Using Eqs. (4) and (9), we obtain J z,y pola ≈ E z,y E 2 (m e γ ± S) , J 0 ±pola ≈ v z ± E z + v y ± E y E 2 m e γ ± S, where E 2 = E 2 z + E 2 y . The total electric current and charge densities of the electron-positron fluid are composed by Eqs. (8) and (9) as J z = e + n + v z + + e − n − v z − + J z +pola + J z −pola ,(11) J y = J z (z → y) and ρ = ± (e ± n ± + J 0 ±pola ), where the positron and electron charge e ± ≡ ±e. It turns out to be a (1 + 2)-dimensional problem in space-time coordinates (t, y, z). Equations (5) and (6) are reduced to (i) the particle-number and energy conservation, ∂n ± ∂t + ∂n ± v z ± ∂z + ∂n ± v y ± ∂y = S,(12)∂ǫ ± ∂t + ∂p z ± ∂z + ∂p y ± ∂y = e ± n ± v z ± E z +e ± n ± v y ± E y +m e γ ± S; (ii) the momentum conservation, We are in the position of numerically integrating the basic equations (12) and (13). The initial conditions (t = 0) are given by the constant electromagnetic fields E z = E ext and B x = B ext . To simplify numerical integrations, we assume the (z − y) homogeneity that the electron-positron fluid quantities and electromagnetic fields are independent of y and z. As a result, Eqs. (12) and (13) are reduced to ordinary differential equations, and Eq. (13) leads toB x = 0, i.e., the magnetic field B x of Eq. (10) is a constant in space and time. The initial condition E y = 0 leads to the solutionẼ y = 0 and J y = 0 for t = 0, because v z − = −v z + and v y − = v y + > 0. This is verified in the following numerical calculations. ∂p z ± ∂t + ∂p z ± v z ± ∂z + ∂p z ± v z ± ∂y = e ± n ± E z −e ± n ± v y ± B x +E z J 0 ±pola with (z ↔ y, B x → −B x ); (iii) Maxwell equations ∇·E = 4πρ, ∇ · B = 0, ∂Ẽ z ∂t + ∂B x ∂y = −4πJ z , ∂Ẽ z ∂y − ∂Ẽ y ∂z = − ∂B x ∂t ,(13) To illustrate the plasma oscillations of pairs and fields, we consider two cases: (i) E ext = E c and B ext = 0.1 E c ; (ii) E ext = E c and B ext = 0.3 E c . Due to the presence of the magnetic field B x , pairs are not only oscillating up and down in theẑ direction, as first shown in Ref. [11], but they also move in theŷ direction. In Fig. 1, we show the trajectory and velocity of pairs produced at z = y = 0 and t = 0. When B x = 0 and dv y ∼ ev z B x dt, v y increases for v z > 0 and decreases for v z < 0. In the case of B x being small enough compared with E z , v y does not change its sign (see Fig. 1, B x = 0.1E c ) in the period of one circle oscillation in theẑ direction; therefore pairs move forward in theŷ direction. When B x = 0.3 E c , v y changes its sign (see Fig. 1, B x = 0.3E c ); therefore pairs also oscillate back and forth, while they are moving in thê y direction. In contrast to the case B x = 0, the negative E z amplitude is smaller than the positive E z amplitude (see Fig. 1). The reason is that v y increases in the phase of positive decreasing v z when E z < 0; i.e., the electric energy goes to the kinetic energy of the motion in theŷ direction. In Fig. 3, we plot (i) the electric current density of pairs J z as a function of the time, which is the source of electromagnetic radiation, and (ii)the total energy-momentum tensor of pairs and fields T µν = T µν m +T µν em as functions of the time, which are the sources of gravitational radiation. Before ending this section, we would like to present some discussions on the role of magnetic fields. In the particular initial configuration of fields E ⊥ B and \|E\| > \|B\| considered in this paper, by integrating Eqs. (3), (5) and (6), we show the oscillating electric field strength (see Fig. 1), and the number and current densities of pairs (see Fig. 2) are suppressed by magnetic fields, compared with their counterparts in the absence of magnetic fields. However, we cannot conclude that such magnetic suppression is generally true. For example, when electromagnetic fields are parallel (E × B = 0 and \|E\| > \|B\|), Eq. (1) yields (see Ref. [19]) Γ V ≃ α\|B\|\|E\| π coth π\|B\| \|E\| exp − πE c \|E\| ,(14) indicating that the pair-production rate receives an enhancement (π\|B\|/\|E\|) coth(π\|B\|/\|E\|) to the prefactor, compared with the rate in the absence of magnetic fields [19,20] (see also [1,2]). It is worthwhile to study the phenomenon of plasma oscillations by numerically integrating Eqs. (5), (6) and (14) consistently with the initial configuration of parallel electromagnetic fields [21]. Electromagnetic and gravitational radiation. We attempt to study electromagnetic and gravitational radiation generated, respectively, by the electric current and energy-momentum tensor of pairs and fields. Suppose that we observe this radiation in the wave zone; that is, at distances much larger than the dimension R of the plasma oscillations, and also much larger than ωR 2 and 1/ω, where ω is the typical frequency of radiation. For definiteness we think of the electric current and energy-momentum tensor of the plasma oscillations occurring in the volume V and for a finite interval of time T , so that the total energy radiated is finite. Thus, the electromagnetic energy radiated per unit solid angle per frequency interval is given by [23] d 2 E em dωdΩ = 2 V d 3 x ′ T dt ′ e iωt ′ −ikx ′ ∂J z (x ′ , t ′ ) ∂t ′ 2 .(15) The gravitational energy radiated per unit solid angle per frequency interval is then given by [24] d 2 E grav dωdΩ = 2Gω 2 T µν * (k, ω)T µν (k, ω) − 1 2 \|T ν ν (k, ω)\| 2 , T µν (k, ω) = V d 3 x ′ T dt ′ T µν (x ′ , t ′ )e iωt ′ −ikx ′(16) where \|k\| = ω and T µν ( x ′ , t ′ ) = T µν m (x ′ , t ′ ) + T µν em (x ′ , t ′ ) . We consider ωR ≪ 1 and e −ikx ′ ≈ 1 for dipole electromagnetic radiation in Eq. (15), and for quadrapole gravitational radiation in Eq. (16). In the calculations of Eq. (16), we set B ext = 0 and E ext = E c , and then the nonvanishing components are T 00 = T 00 m + T 00 em and T zz = T zz m + T zz em . Using the approximation of spatial homogeneity in Eqs. (15) and (16), we can factorize out the volume V = V d 3 x ′ , in which the total energy density T 00 = T 00 m + T 00 em = E 2 ext /(8π) is conserved (see Fig. 3). Let T and V also be the time and volume of strong fields E ext E c created by coherent laser beams. Se- lecting different T values, in Fig. 4 we plot the electromagnetic and gravitational radiation spectra (15) and (16) with V 2 factored out. These two energy spectra are narrow, and the locations (ω peak ) of their peaks are related to the coherent oscillation frequency (ω p ) of pairs and fields, which depend on T and E ext (see Ref. [25]). The peculiar energy spectrum of electromagnetic radiation is clearly distinguishable from the energy spectra of the bremsstrahlung radiation, electron-positron annihilation and other possible background events. Therefore, it is sensible and distinctive to detect such peculiar radiative signatures to identify the production and oscillation of electron-positron pairs in strong laser fields. As shown in Fig. 4, gravitational radiation is much smaller than the electromagnetic one for the reason that the gravitational coupling Gm 2 e = 2.5 × 10 −45 is much smaller than the electromagnetic coupling e 2 = 1/137. In order to achieve a sizable radiation intensity from the plasma oscillation, the volume V of oscillating pairs and strong electric fields should be large enough and/or the strength of strong fields should be enhanced (E ext E c ) to increase the pair density. It is worthwhile to point out that Fig. 4 shows the numerical results of Eqs. (15) and (16) being consistent with the approximate relation between Eqs. (15) and (16) in the ultrarelativistic limit of charged particles moving in external electromagnetic fields [26]. Gravitational waves from an inflationary cosmos [29] are in the high frequency band (10 8 − 10 11 Hz). Gravitational waves originating from some sources in ground laboratories are also in this frequency band, and several proposals have been made to detect high-frequency gravitational waves up to 5 GHz [30]. Gravitational waves generated from the high-energy particle beam [26,31] in the ground experiments of the Stanford Linear Collider and LHC have much higher frequencies of O(10 23 ) Hz. The frequency of gravitational wave discussed here is O(10 19−20 ) Hz, i.e., O(10 0−1 )KeV, or sub nanometer O(10 −(9−10) ) cm. It is not clear whether such gravitational waves could ever be detected, or have observable effects. One would have to build an atom-sized gravitational wave detector to response incoming gravitational wave with such high frequencies (for some more details, see Ref. [32]) To end this paper, we remark again that the intensity of electromagnetic radiation emitted by the plasma oscillations is tens of orders of magnitude larger than their gravitational radiation; therefore any detectable signal is enormously more likely to result from the electromagnetic interaction. The prospect of detecting gravitational radiation of such ultrahigh frequencies looks dim. Nevertheless, our theoretical investigation of the gravitational radiation from the electron-positron plasma oscillation would be useful for the study of gravitational radiation emitted from particles and antiparticles in the very early Universe. PACS numbers: 52.27.Ep; 52.40.Db; 04.30.-w with (z ↔ y, B x → −B x ). The pair-production rate [Eq. (3)] can be approximately used for varying electromagnetic fields [Eq. (10)], providedẼ(t, y, z) and B(t, y, z) created by electron-positron pair oscillations vary very slowly compared with the rate of electronpositron pair productions O(m e c 2 / ). This is justified if the inverse adiabaticity parameter[22] η = me ωp E Ec ≫ 1,where ω p is the frequency of plasma oscillations. FIG. 1 : 1With initial conditions Ez = Ec, Bx = 0.1Ec (left) and Bx = 0.3Ec (right), we plot the electron and positron trajectories and velocities vy vs vz and the electric field Ez vs the time t from t = 0 to t = 10 4 . FIG. 2 :FIG. 3 : 23The pair number density n± and current density Jz vs the time t for Ez = Ec and different Bx field values. Ez = Ec and Bx = 0.0. The charged current density Jz vs time t (left). The total energy-momentum tensor T 00 and T zz vs time t (right). FIG. 4 : 4Ez = Ec and Bx = 0.0. By factoring V 2 out, the electromagnetic radiation (left) of Eq. (15) and the gravitational radiation (right) of Eq. (16) are plotted as functions of the frequency ω, for different T values . Acknowledgements: Wen-Biao Han is supported by NSFC Grant No.11273045. 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[ "Chiral symmetry breaking and vacuum polarization in a bag", "Chiral symmetry breaking and vacuum polarization in a bag" ]	[ "S Yasui \nPhysics Department\nTokyo Institute of Technology\nOokayama 2-12-1152-8551MeguroTokyoJapan\n" ]	[ "Physics Department\nTokyo Institute of Technology\nOokayama 2-12-1152-8551MeguroTokyoJapan" ]	[]	We study the effects of a finite quark mass in the hedgehog configuration in the two phase chiral bag model. We discuss the chiral properties, such as the fractional baryon number and the chiral Casimir energy, by using the Debye expansion for the analytical calculation and the Strutinsky's smearing method for the numerical computation. It is shown that the fractional baryon number carried by massive quarks in the vacuum is canceled by that in the meson sector. A finite term of the chiral Casimir energy is obtained with subtraction of the logarithmic divergence term. * [email protected]	10.1103/physrevd.74.114003	[ "https://arxiv.org/pdf/hep-ph/0606001v1.pdf" ]	11,988,092	hep-ph/0606001	aa36ebac912b6d23196527eafd060265d032574e	Chiral symmetry breaking and vacuum polarization in a bag May 2006 April 20, 2018 S Yasui Physics Department Tokyo Institute of Technology Ookayama 2-12-1152-8551MeguroTokyoJapan Chiral symmetry breaking and vacuum polarization in a bag May 2006 April 20, 2018arXiv:hep-ph/0606001v1 31 We study the effects of a finite quark mass in the hedgehog configuration in the two phase chiral bag model. We discuss the chiral properties, such as the fractional baryon number and the chiral Casimir energy, by using the Debye expansion for the analytical calculation and the Strutinsky's smearing method for the numerical computation. It is shown that the fractional baryon number carried by massive quarks in the vacuum is canceled by that in the meson sector. A finite term of the chiral Casimir energy is obtained with subtraction of the logarithmic divergence term. * [email protected] Introduction Strangelets of finite volume quark matter with strangeness has been one of the most interesting subjects of exotic particles in hadron and quark physics [1,2,3,4]. In recent studies of strangelets, a model has been presented for a discussion of chiral symmetry breaking inside a quark droplet of finite volume quark matter [5,6,7,8]. There, in addition to quark confinement for a finite volume system, it was allowed that the quarks acquire a finite mass by dynamical chiral symmetry breaking. In this picture, it was assumed that the quark interaction was provided by the Nambu-Jona-Lasinio (NJL) type interaction inside a quark droplet. In earlier works, the quark wave function was given by the MIT bag in Refs. [5,6,7]. Later, in order to overcome the chiral symmetry breaking at the bag surface, the chiral bag model was introduced in Ref. [8]. The latter model was called as the NJL chiral bag model. Such an idea was first presented by T. Kunihiro in 1983 [9, 10]. In the discussion of strangelets by the NJL chiral bag model, the hedgehog ansatz was used for the pion and ud quarks to include non-linear interaction of pions and quarks [8]. In the history of the study of the chiral bag model since [11,12], effects of vacuum polarization, such as the anomalous baryon number and the chiral Casimir effects, have been discussed extensively for massless quarks [13,14,15,16,17,18]. It was shown that the correct behavior of the baryon number was provided by the contribution from the sea quarks [17,19,20]. In the development of the chiral bag model, several techniques for calculation of the sea quark contribution was also developed. In Ref. [17], the fractional baryon number was derived by using the Multiple Reflection Expansion method. After that, the Debye expansion was used [19,20]. The Casimir energy is also affected by the pion cloud. It was shown that the chiral Casimir energy had a term of logarithmic divergence [22,26]. However, these properties for massive quarks have not been investigated so far. In the framework of the NJL chiral bag model, the dynamical quark mass is an order parameter for the chiral symmetry breaking in the chiral bag. In order to search the real stable state in the energy variation with respect to the dynamical quark mass, we need to clarify the effect of vacuum polarization for the finite quark mass in the hedgehog configuration in the chiral bag. This is the motivation in the present paper. We organize this paper as follows. In Section 2, we propose a model lagrangian of the chiral bag with massive quarks. In this paper, we describe the quarks with a finite Dirac mass, which is generated through the NJL type interaction in the chiral bag. In Section 3, we discuss a fractional baryon number for massive quarks by using both of numerical computation and analytical formulation. In Section 4, we discuss the chiral Casimir energy. Deriving the terms of logarithmic divergence the finite term of the chiral Casimir energy is obtained numerically. We summarize our discussion in Section 5. The chiral bag with massive quarks In the NJL chiral bag model [8,9,10], the quark mass in a finite volume system is induced by the NJL point-like interaction [23]. In the NJL chiral bag model, we consider the ud quark sector with a finite quark mass induced by the mean field approximation in the scalar channel. The lagrangian in the ud quark sector is written as L =ψ(i∂ / −m)ψθ(R − r) − 1 2ψ U γ 5 ψδ(r − R),(1) which includes a boundary condition explicitly. Here, ψ = (u, d) t is the ud quark fields, and the ud quark mass matrix is given bym = diag(m u , m d ) with flavor symmetry m = m u = m d . The ud quark mass is the sum of the current mass and the constituent mass which is generated in the non-zero expectation value of the quark scalar condensatē qq. In general, the constituent quark mass can be position dependent. Here, in order to simplify the essential discussion of the finite quark mass, we treat the quark mass as a constant. For a bag configuration, we assume a static spherical bag with radius R. The step function is multiplied in the first term in order to confine quarks inside the bag. Here, r is a distance from the center of the bag. The second term with the delta function realizes the chiral invariant boundary condition at the bag surface, where we define U γ 5 = e i τ · πγ 5 ,(2) with the π meson field π and the Pauli matrix τ . In this paper, we do not explicitly show the lagrangian in the pion sector, since our current interest is to study the chiral vacuum polarization of the quarks in the bag. In order to consider the non-linear effect of the pion, we assume the hedgehog ansatz in the π meson sector, where the pion field conserves the grand spin K = J + I with total angular momentum J and isospin I. The hedgehog pion field for r > R is written as π = F (r) n,(3) with a chiral angle F (r) and a unit radial vector n in the real space [28,29]. According to the π meson sector, we introduce the hedgehog basis set in the ud quark sector [18,27]. We construct the quark wave function ψ (κ) for natural (κ = +) and unnatural (κ = −) assignment for grand spin K, respectively; ψ (+) = a 0 j K (pr)\|0 + a 1 j K (pr)\|1 a 2 j K+1 (pr)\|2 + a 3 j K−1 (pr)\|3 ,(4) and ψ (−) = b 2 j K+1 (pr)\|2 + b 3 j K−1 (pr)\|3 b 0 j K (pr)\|0 + b 1 j K (pr)\|1 .(5) The coefficients a i and b i (i = 0, · · · , 3) are determined by satisfying the equation of motion of the ud quark for κ = ±, respectively. j K (pr) is the spherical Bessel function and p is the ud quark momentum. Here, the two-component spinors \|0 , · · · , \|3 are given by \|0 = Y KM (θ, φ)χ 0 0 , \|1 = µ=−1,0,1 (KM − µ1µ\|KM)Y KM −µ (θ, φ)χ 1 µ , \|2 = µ=−1,0,1 (K + 1M − µ1µ\|KM)Y K+1M −µ (θ, φ)χ 1 µ , \|3 = µ=−1,0,1 (K − 1M − µ1µ\|KM)Y K−1M −µ (θ, φ)χ 1 µ ,(6) where Y L M (θ, φ) is the spherical harmonics with spherical coordinate (θ, φ). χ G µ are eigenstates of G = S + I = σ/2 + τ /2, χ 0 0 = 1 √ 2 (\| ↑ \|d − \| ↓ \|u ),χ 1 1 = \| ↑ \|u , χ 1 0 = 1 √ 2 (\| ↑ \|d + \| ↓ \|u ),χ 1 −1 = \| ↓ \|d .(7) The sign of naturalness κ corresponds to the parity P = (−) K+κ . In order to satisfy the equation of motion, the coefficients a i for naturalness assignment κ = + are subjected to      E − m 0 −iP p −iQp 0 E − m −iQp iP p iP p iQp E + m 0 iQp −iP p 0 E + m           a 0 a 1 a 2 a 3      = 0,(8) where we define P = K + 1 2K + 1 , Q = K 2K + 1 ,(9)E = p 2 + m 2 . Then, we obtain two independent solutions a ′ =      a ′ 0 a ′ 1 a ′ 2 a ′ 3      = N ′       p E−m iP p E−m iQ 1 0       , a ′′ =      a ′′ 0 a ′′ 1 a ′′ 2 a ′′ 3      = N ′′       p E−m iQ − p E−m iP 0 1       .(10) Here, the normalization constants N ′ and N ′′ are determined by the condition d 3 xψ (+) † ψ (+) = 1, N ′−2 = R 3 E E − m j K (pR) 2 + j K+1 (pR) 2 − 1 pR 2(K + 1) − m E j K (pR)j K+1 (pR) , N ′′−2 = R 3 E E − m j K−1 (pR) 2 + j K (pR) 2 − 1 pR 2K + m E j K−1 (pR)j K (pR) . (11) The final solution is expressed as a linear combination of a ′ and a ′′ , a = c ′ a ′ + c ′′ a ′′ = c ′ N ′       p E−m iP p E−m iQ 1 0       + c ′′ N ′′       p E−m iQ − p E−m iP 0 1       ,(12) with constants c ′ and c ′′ . We obtain a solution for the unnaturalness assignment κ = − in the same way. The eigenenergy of the hedgehog ud quark is given by the boundary condition at the bag surface r = R, i n · γψ (κ) = −e i τ · nF (R)γ 5 ψ (κ) .(13) By substituting the solution of Eq. (12), we obtain an equation for the eigenvalue, which is given by cos F (R)    j K (pR) 2 − E − κm p 2 j K+1 (pR)j K−1 (pR)    −κ E − κm p j K (pR) {j K+1 (pR) − j K−1 (pR)} + E − κm p sin F (R) 2K + 1 j K (pR) {j K+1 (pR) + j K−1 (pR)} = 0.(14) The equation for K = 0 is obtained by setting j −1 (pR) = 0. Eq. (14) determines the energy spectrum of a quark in the bag. We mention that there is a symmetry of the energy level in the positive and negative energy E K P (F ) = −E K −P (−F ).(15) This relation has its origin in the invariance of the lagrangian (1) under the transformation U → U * or F → −F . Another symmetry under F → F + π and κ → −κ is conserved in the massless case. However, this symmetry is not conserved for the finite mass. This is due to an asymmetry of the quark energy levels, which is shown in Fig. 1 in [8]. Baryon number conservation In the limit of the zero bag radius, the meson field in the Skyrme lagrangian gives a mapping from R 3 ∼ S 3 to SU(2) ∼ S 3 by imposing a boundary condition U = 1 at r → ∞. The winding number in the mapping is identified with the baryon number [28,29]. The fractional baryon number due to the pion cloud is given as [28,29] B π = − 1 π F − 1 2 sin 2F ,(16) where F is the chiral angle at the bag surface. In the pure Skyrmion, the chiral angle is given as nπ with integer n. On the other hand, in the chiral bag with finite bag radius, the chiral angle is not generally equal to nπ. Therefore, the meson carries only a fractional baryon number. There, it has been known that the total baryon number is composed of the fractional baryon numbers of the pion, of the vacuum and of the valence quarks in a bag [18,27]. The baryon number carried by the vacuum quarks is defined as B q (m, F ) = 1 2 O\|[ψ † , ψ]\|O = − 1 2 lim t→0 n sgn(E n )e −t\|En\| .(17) Here, \|O is a vacuum state of an empty bag filled with negative energy quarks, E n = E n (m, F ) an n-th state energy of a quark with mass m, and at the chiral angle F at the bag surface. The quantum number n labels the grand spin and parity. The sum is taken over for all the quark states n with positive and negative energies in the bag. In order to obtain the convergence in the sum, an exponential type regularization is multiplied. In the studies of the chiral bag model with massless quarks, it was shown that the baryon number of the vacuum quarks was canceled by that of the meson cloud [13,14,15,17,18,26,20]. Accordingly, the total baryon number is a conserved quantity. This result should also be the case for the chiral bag with massive quarks. It is expected that the baryon number conservation is not affected by the finite quark mass, because U(1) B symmetry is conserved in our lagrangian (1). In the following, we show explicitly the cancellation of the fractional baryon number between the vacuum massive quarks and the pion cloud. We use both numerical computation and analytical procedure. First, we show the numerical computation. In the studies of chiral bag model with massless quarks, the regularization, such as the Gaussian type [22], heat kernel type [26], have been used with success. In this paper, we use the Strutinsky's smearing method [24,25]. This method has an advantage that the states necessary for computation are limited up to the grand spin K max ∼ 40 in order to obtain a good convergence, while we need K max ∼ 100 at least for the other regulators. We rewrite the baryon number (17) by introducing a delta function for a density of states to pick up the discrete levels B q (m, F ) = − 1 2 ∞ −∞ dx n sgn(E n )δ(x − E n R)(18) Then, we replace the delta function by a gaussian function, B q (m, F ) = − 1 2 ∞ −∞ dxρ x (m, F ),(19) where we define a function for a density of states ρ x (m, F ) = n sign(E n ) e −(x−EnR) 2 /γ 2 γ √ π ,(20) where γ ≃ 2 − 3 is a smearing parameter. We obtain B q = 0 at F = 0, since the energy spectrum is symmetric for positive and negative energies. In oder to obtain a rapid convergence in the sum, it is convenient to define the difference of ρ x (m, F ) between finite F and F = 0,ρ x (m, F ) = ρ x (m, F ) − ρ x (m, 0).(21) Therefore, the baryon number defined by Eq. (17) is given by an alternative formulation B q (m, F ) = ∞ −∞ dxρ x (m, F ).(22) For a numerical computation, we restrict the range of the x-integral in a finite interval x ∈ [−x max , x max ]. B q (m, F ) = xmax −xmax dxρ x (m, F ).(23) We use x max ≃ 20 to obtain sufficient convergence. Carrying out numerical computation, we obtain the result which is given by B q (m, F ) = 1 π F − 1 2 sin 2F(24) for F ∈ [F 0 , F 0 + π]. The value of F 0 depends on the quark mass; F 0 /π = −0.5, −0.695, −0.816 and −0.878 for mR = 0, 1, 2 and 3, respectively. The equation (24) is independent of quark mass m, and coincides with the opposite sign of Eq. (16) in the meson sector. Therefore, the fractional baryon number by meson and vacuum quark cancel each other for any quark mass. The total baryon number is equal to A for a quark droplet with 3A valence quarks for any chiral angle. Let us investigate the case of a baryon. The baryon number carried by valence and vacuum quarks for chiral angle F 0 < F < 0 is given as a sum of the valence quark and the fractional baryon number B(m, F ) val+sea = 1 + 1 π F − 1 2 sin 2F .(25) For −π < F < F 0 , the valence quark is absent since the 0 + state is absorbed in the vacuum. Then, the baryon number is obtained by substituting F + π instead of F from a periodicity, B(m, F ) val+sea = 1 π (F + π) − 1 2 sin 2(F + π) = 1 + 1 π F − 1 2 sin 2F .(26) Therefore, the total baryon number is one for any chiral angle. In the same way, the conservation of baryon number holds for any quark droplets. Second, we present an analytical calculation of the baryon number (17) by using the Debye expansion. This method was originally developed in the chiral bag with massless quark [19,20]. We apply this formalism to the massive case. We rewrite Eq. (17) in terms of the quark propagator as in [19,20]. Then, we obtain B q (m, F ) = 1 2 lim τ →0 + ∞ −∞ dx 2iπ e iτ x ∞ K=0 (2K + 1) d dx ln S K (m, F ; ix) S * K (m, F ; ix)(27) Here, S K (m, F ; p) is given as a product of the left hand side of the eigenvalue equation Eq. (14) for natural (κ = +1) and unnatural (κ = −1) states. For the sake of using the Debye expansion, we substitute an imaginary number ix for pR in the spherical Bessel function. We explicitly show S 0 (F ; ix) = cos F j 1 (ix) 2 − j 0 (ix) 2 − 2 E sin F +m p j 0 (ix)j 1 (ix)(28) for K = 0, and S K (m, F ; ix) = Ē p cos F j K (ix) 2 − j K+1 (ix)j K−1 (ix) + sin F 2K + 1 j K (ix) (j K+1 (ix) + j K−1 (ix)) 2 + m p j K (ix) 2 + j K+1 (ix)j K−1 (ix) − j K (j K+1 (ix) − j K−1 (ix)) 2 .(29) for K = 0. For a short notation, we writem = mR,p = x andĒ = √m 2 + x 2 . The x-integral picks up an eigenvalue as a residue instead of solving directly the boundary condition. We note that Eq. (27) is an exact formulation, but it is not practical to consider the x-integral and the sum over K without approximation. In order to obtain the final result, it is sufficient to use an asymptotic behavior of the quark energy spectrum. In the following manipulation, we discuss the K = 0 and K ≥ 1 components separately for convenience. First, we consider the K = 0 component, which is defined by B (K=0) q (m, F ) = 1 2 lim τ →0 + ∞ −∞ dx 2iπ e iτ x d dx ln S 0 (m, F ; ix) S * 0 (m, F ; ix) .(30) It is convenient to write the function S 0 (m, F ; ix) in a polar coordinate S 0 (m, F ; ix) = R 0 (x)e iΦ 0 (x) .(31) By using the asymptotic form j 1 (ix) ≃ i 2x e x + e −x ,(32) we obtain Φ 0 (x) ≃ arctan e 2x − e −2x e 2x + 2 −2x √ x 2 +m 2 sin F +m x cos F .(33) Then, it is straightforward to perform the integral, giving B (K=0) q (m, F ) ≃ F π ,(34) for any quark mass m. Second, for K ≥ 1, we use the Debye expansion for the modified Bessel function in estimation of Eq. (29). In our notation, the modified Bessel function I ν (x) is defined by j K (ix) = π 2x i K I ν (x),(35) with ν = K + 1/2. The Debye expansion is a uniform asymptotic expansion with no constraint between ν and x. The Debye expansion gives I ν (x) ≃ Γ(1/2) π √ 2t e fν (x) ,(36) in the lowest order, where we define t = √ ν 2 + t 2 and f ν (x) = t − ν sinh −1 (ν/x). Now, let us define B (K≥1) q (m, F ) = 1 2 lim τ →0 + ∞ −∞ dx 2iπ e iτ x K≥1 d dx ln S K (m, F ; ix) S * K (m, F ; ix) .(37) Here, we write S K (m, F ; ix) in a polar coordinate S K (m, F ; ix) = R K (x)e iΦ K (x) .(38) Then, we apply the Debye expansion (36). After a little tedious calculation, we pick up only the terms of the leading order ofm, and obtain Φ K (x) ≃ arctan − 1 2K + 1 νx 2t 3 sin 2F − 4νm t .(39) We consider that the quark mass m is smaller as compared with the momentum p. Indeed, this is a good approximation, since we are interested in the asymptotic behavior of the quark energy spectrum. The x-integral for Φ K (x) is integrated out and the integral is determined only by Φ K (∞) − Φ K (−∞). Concerning the term which has no quark mass, we use an identity K≥1 cos(τ νz) ν = − ln tan(τ z/4) − 2 cos(τ z/2), and lim τ →0 + ∞ −∞ dx 2πi e −iτ x K≥1 d dx −i νx 2t 3 = − 1 2π .(41) Concerning the term proportional tom, we use lim τ →0 + ∞ −∞ dx 2πi e −iτ x K≥1 d dx 4νm t = 0,(42) which indicates the mass term does not contribute to the integral. Consequently, we obtain the result B (K≥1) q (m, F ) = − sin 2F 2π .(43) Finally, by Eqs. Chiral Casimir energy The chiral Casimir energy arises as a result of the modification of quark energy levels for finite pion cloud. Following the regularization scheme in the baryon number, the chiral Casimir energy is defined by E C (m, F ) = 1 2 O\|[ψ † , Hψ]\|O = − 1 2 lim t→0 + n sign(E n )E n e −t\|En\| ,(44) as in Refs. [13,14,15,18,21,26,27]. We show the numerical procedure by using the Strutinsky's smearing method. By using the Gaussian function (20), we rewrite Eq. (44) as E C (m, F ) = ∞ −∞ dxxρ x (m, F ).(45) By choosing the reference point at F = 0, we obtain an alternative formulation ∆E C (m, F ) = E C (m, F ) − E C (m, 0) = ∞ −∞ dxxρ x (m, F ).(46) Restricting the range of integral in a finite range of [−x max , x max ] with sufficient convergence, we obtain, ∆E C (m, F ) = xmax −xmax dxxρ x (m, F ).(47) In order to achieve a rapid convergence of the integral, it is much more practical to use the partial integration 1 . We mention that we need x max ∼ 40 to obtain the convergence in the calculation. More states are necessary for the chiral Casimir energy as compared for the baryon number (x max ∼ 20). For massless quarks, it has been known that there is a logarithmic divergence term in the Casimir energy (44). It was shown that the divergence term is proportional to sin 2 F [13,14,15,19,20,21,26,27]. Here, we also derive the logarithmic divergence term for massive quark by using the Debye expansion developed in [19,20]. Following the procedure in [19,20], we rewrite the Casimir energy as ∆E C (m, F ) = 1 2R lim τ →0 + ∞ −∞ dx 2π xe iτ x ∞ K=0 (2K + 1) d dx ln S K (m, F ; ix) S K (m, 0; ix) .(48) Just as in the case of the baryon number, we consider the calculation for K = 0 and K ≥ 1, separately for convenience. 1 We use a relationship ∞ −∞ dxxρ x (m, F ) = − ∞ −∞ dx x 2 2ρ ′ x (m, F ) withρ x (m, F ) → 0 for \|x\| → ±∞. First, we investigate the K = 0 sector, ∆E (K=0) C (m, F ) = 1 2R lim τ →0 + ∞ −∞ dx 2π xe iτ x d dx ln S 0 (m, F ; ix) S 0 (m, 0; ix) .(49) We apply the asymptotic form of the spherical Bessel function (32). By expanding \|S 0 (m, F ; ix)/S 0 (m, 0; ix)\| in the lowest order of m, we obtain ∆E (K=0) C (m, F ) ≃ 1 2R lim τ →0 + ∞ −∞ dx 2π xe iτ x d dx cos 2 F + sin F tanh 2 (2x) + sin F 2m x tanh 2 (2x) .(50) The first two terms in the integrand, which does not contain quark mass, are given in [19,20]. The integral of the third term proportional tom is easily shown to be equal to zero. Consequently, we obtain the result ∆E (K=0) C (m, F ) ≃ F 2 4πR .(51) The divergence term does not appear in the K = 0 sector. Second, we investigate the sum over K ≥ 1, ∆E (K≥1) C (m, F ) = 1 2R lim τ →0 + ∞ −∞ dx 2π xe iτ x ∞ K=1 (2K + 1) d dx ln S K (m, F ; ix) S K (m, 0; ix) .(52) Then, introducing new variables h = t/ν and z = x/ν, we have ∆E (K≥1) C (m, F ) ≃ sin 2 F πR lim τ →0 + ∞ 0 dz 2z 4 h 6 ∞ K=1 cos(τ νz) + sin 2 F πR lim τ →0 + ∞ 0 dz 6 h 3 − 31 h 5 + 46 h 7 − 21 h 9 ∞ K=1 cos(τ νz) ν +m 2 8πR sin 2 F lim τ →0 + ∞ 0 dz − 1 h 2 + 6 h 4 − 4 h 6 ∞ K=1 cos(τ νz) +m 2 8πR sin 2 F lim τ →0 + ∞ 0 dz − 1 h 3 + 23 h 5 − 41 h 7 + 21 h 9 ∞ K=1 cos(τ νz) ν .(55) The summation over K ≥ 1 and the integral is achieved by the formula Eq. (40). Finally, we arrive at the result ∆E (K≥1) C (m, F ) ≃ − 3 16R + − 2 15πR ln τ + 12 ln 2 − 9 30πR +m 2 64R − 1 15m 2 8πR (ln τ + 1 − 4 ln 2) sin 2 F(56) Therefore, we obtain a logarithmic divergence term with coefficient of sin 2 F for a finite quark mass. We note that our result reproduces the previous result for the massless quarks inm → 0 [19,20]. Now, in order to remove the logarithmic divergence, we subtract the second derivative of the chiral Casimir energy with respect to the chiral angle at the reference point F = 0, and we obtain a finite contribution E f in C (m, F ) = ∆E C (m, F ) − 1 2 sin 2 F ∂ 2 E C (m, F ) ∂F 2 F =0 .(57) In Fig. 1, we show the chiral Casimir energy E f in C as a function of the chiral angle F . The lines are distinguished by the quark masses mR = 0 (dashed line), mR = 1 (solid line), respectively. It is a remarkable point that the Casimir energy for the massive quark takes a nonzero value at F = −π, while that of the massless quark becomes zero at F = −π. This is because that the massive quark has an asymmetric energy spectrum between F = 0 and −π, while the massless quark has a symmetric spectrum, as shown in Fig. 1 in [8]. Here, we recall the Cheshire Cat picture in the chiral bag model with massless quarks. There, the continuous transformation from the chiral bag to the Skyrmion was induced in the limit of small bag radius [13,14,15]. This picture seems not to be applied to the case of the massive quarks, since the chiral Casimir energy takes a finite value at the zero bag radius, or F = −π. However, this observation does not make us abandon our discussion. Indeed, in our formulation, the quark mass is a dynamical variable which should be determined by the energy variation. Our previous result suggested that the chiral symmetry was restored in the limit of small bag radius [8]. Therefore, the NJL chiral bag model holds the Cheshire Cat picture. Summary We discussed the chiral properties in the chiral bag model which contained massive quarks. Considering that the dynamical quark mass was generated by chiral symmetry breaking, the fractional baryon number and the chiral Casimir energy were investigated. We discussed the effects of the finite quark mass in the hedgehog configuration with an emphasis on the technical procedure. We showed the numerical calculation by the Strutinsky's smearing method and the analytical technique by the Debye expansion. It was shown numerically and analytically that the fractional baryon number carried by vacuum massive quarks inside the bag was canceled that of the π meson outside the bag. Therefore, the total baryon number is exactly conserved. By using the Debye expansion, it was shown that the chiral Casimir energy had a logarithmic divergence term. The chiral Casimir energy was obtained numerically by the Strutinsky's smearing method by removing the divergence term. It is a point different from massless quark that the chiral Casimir energy for massive quark had a nonzero value at the chiral angle F = −π. As further development, we plan to discuss a fully chiral symmetric equation of motion for a finite quark mass. In our present analysis of the NJL chiral bag model, we performed a mean field approximation only in the scalar channel in the four point interaction [8]. However, in the hedgehog configuration, it may be allowed to have a finite expectation value ofqq, not only in the sigma channel, but also in the pion channel. In the latter case, the expectation value ofqq with pion quantum number can be given as a finite value in a basis set of the hedgehog quark wave function. This study is now in progress. The quantization of the hedgehog configuration for massive quark is an important subject in order to obtain the state with definite spin and isospin. It is also an interesting subject to include the strangeness sector in our framework. These subjects are left as future works. (34) and (43), the fractional baryon number obtained in the analytical procedure coincides with the numerical result Eq.(24). Figure 1 : 1The Casimir energy as a function of the chiral angle at the bag surface. The dashed and solid lines indicate the quark masses mR = 0 and mR = 1, respectively. t 3 + 6ν 2 t 4 + 23ν 2 t 5 − 4ν 4 t 6 − 41ν 4 t 7 + 21ν 6 t 9 .(54) AcknowledgementThe author acknowledge to Prof. H. Hosaka and Prof. M. Oka for useful discussions. The author is also grateful to Prof. T. Kunihiro. Part of this study was proceeded when the author was belonging to Yukawa Institute for Theoretical Physics, Kyoto University. This work was also partially supported by Grant-in-Aid for Scientific Research for Priority Areas, MEXT (Ministry of Education, Culture, Sports, Science and Technology) with No. 17070002.After a little troublesome calculation by using the Debye expansion and expanding in the lowest order ofm, we obtainHere, we define L ± = I ν±1 /I ν for a short notation. The first term, which does not contain quark mass, coincides with the case of massless quarks[19,20]. The second term proportional tom is further analyzed as followings. 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[ "An Improved Text Sentiment Classification Model Using TF-IDF and Next Word Negation", "An Improved Text Sentiment Classification Model Using TF-IDF and Next Word Negation" ]	[ "Bijoyan Das ", "Student Member, IEEE Member, IEEE, KolkataSarit Chakraborty ", "India " ]	[]	[]	With the rapid growth of Text sentiment analysis, the demand for automatic classification of electronic documents has increased by leaps and bound. The paradigm of text classification or text mining has been the subject of many research works in recent time. In this paper we propose a technique for text sentiment classification using term frequency-inverse document frequency (TF-IDF) along with Next Word Negation (NWN). We have also compared the performances of binary bag of words model, TF-IDF model and TF-IDF with 'next word negation' (TF-IDF-NWN) model for text classification. Our proposed model is then applied on three different text mining algorithms and we found the Linear Support vector machine (LSVM) is the most appropriate to work with our proposed model. The achieved results show significant increase in accuracy compared to earlier methods.I.	null	[ "https://arxiv.org/pdf/1806.06407v1.pdf" ]	49,301,351	1806.06407	010209ce689fec6e31e83657f870ef6cb27b36d4	An Improved Text Sentiment Classification Model Using TF-IDF and Next Word Negation Bijoyan Das Student Member, IEEE Member, IEEE, KolkataSarit Chakraborty India An Improved Text Sentiment Classification Model Using TF-IDF and Next Word Negation With the rapid growth of Text sentiment analysis, the demand for automatic classification of electronic documents has increased by leaps and bound. The paradigm of text classification or text mining has been the subject of many research works in recent time. In this paper we propose a technique for text sentiment classification using term frequency-inverse document frequency (TF-IDF) along with Next Word Negation (NWN). We have also compared the performances of binary bag of words model, TF-IDF model and TF-IDF with 'next word negation' (TF-IDF-NWN) model for text classification. Our proposed model is then applied on three different text mining algorithms and we found the Linear Support vector machine (LSVM) is the most appropriate to work with our proposed model. The achieved results show significant increase in accuracy compared to earlier methods.I. INTRODUCTION In recent past there has been a good hike in the usage of micro-blogging websites. These platforms have brought the entire world under a single domain, where everyone is free to share their opinions. The current decade has become a digital book where each post one shares cumulates to the sentiment of a particular topic. A lot of research works have been done to gather and calculate the sentiment of posts/tweets and also a good number of text-mining algorithms have been designed to analyze the sentiments. The gradual growth of the number of users and the data related to them, has provided a good impetus to every company or organizations to mine these microblogging sites to collect information about people's opinion about their services or products. Due to this increase in user interaction, the future sales of any product or service depends a lot on the sentiments and perceptions of the previous buyers [1]. Therefore, it is necessary to have an efficient way to predict user sentiments about a product or service. The solution to this problem is to classify the text using a strong machine-learning algorithm. Humans face many decisions on a daily basis and sentiment analysis can automate the process of coming to a decision based on past outcomes of that decision. For example, if someone has to buy tickets for a movie, then rather than manually going through all the long reviews, a sentiment classifier can predict the overall sentiment of the movie. Based on positive or negative sentiment a decision can be taken whether or not to buy tickets. Although this is a very trivial problem, text classification can be used in many different areas as follows: In this paper we have demonstrated a study on the three different techniques to build models for text classification. The first two techniques which are simple binary bag of words model and TF-IDF model are common in text classification and we have tried to improve the accuracy by introducing next word negation. The performance of these techniques on several different machine learning algorithms is also shown at the end.  The methods to perform text classification can be broadly classified into supervised and unsupervised learning techniques [3]. The unsupervised learning techniques mainly use lexicon based approach where they use existing lexical resources like WordNet and language specific sentiment seed words to construct and update sentiment prediction [4]. Although unsupervised learning algorithms do not require a corpus of previously classified data and generates a general sentiment, they fail to capture context/domain specific information of the document. The supervised learning techniques use machine learning on a previously classified dataset which is considered to be almost accurate. These preclassified datasets are often domain specific, therefore the model it generate can work only for a particular domain. These datasets are first converted into intermediate models where documents are represented as vectors [6], and then the intermediate representations are fed to the machine learning algorithm. Through our research we have found out that Multinomial Naïve Bayes, Max Entropy Random Forest and Linear Support Vector Machines are the popular choice of algorithm for text classification. The documents are represented as a vector, where every word is converted into a number. This number can be binary (0 and 1) or it can be any real number in case of TF-IDF model. In case of binary bag of words model if a word appears in a document it gets a score 1 and if the word does not appear it gets a score 0. So, the document vector is a list of 1s and 0s. In case of TF-IDF the document vector can be a list of any numbers which are calculated using term frequencyinverse document frequency method. In our work we have used three datasets, the IMDB movie review dataset [12], Amazon Product review dataset [5] and SMS Spam Collection dataset [8]. Each of these datasets have textual data precategorized into classes. We have tried all the three approaches on these datasets starting with simple binary bag of words approach, then moving towards TF-IDF and TF-IDF with word negation approaches. In all the cases we have started with a base feature size and increased it gradually to produce better results. In the next section we have displayed a survey of the various sentiment analysis techniques used all over the world, and then move towards our own proposed method, experiments and results. II. PRIOR WORK Pang, Lee and Vaithyanathan were the first to propose sentiment classification using machine learning models. They analyzed the Naïve Bayes, Max Entropy and Support Vector Machine models for sentiment analysis on unigrams and bigrams of data [9]. In their experiment SVM paired with unigrams produced the best results. Mullen and Collier performed sentiment classification using SVM by collecting data from a lot of sources [10]. Their work showed that using hybrid SVM with features based on Osgood's theory [ref.] produced the best results. This method worked well but failed to give importance to more contextual classifications and because of this domain variability the overall result greatly diminished. The accuracy rate from their proposed method was 86.6% which needs to be greatly improved. Zhang constructed a computational model to explore reviews linguistics properties [14] to judge its usefulness. Support Vector Machine (SVM) algorithm was used for classification. Zhang concluded that the quality of review is good if it contains both subjective and objective information. However, the efficiency of the analysis was only 72% because of employing fuzzy search technique to opinion mining which resulted in occurrence of a major problem when confronted by any misspelled word. Efthymios et al. under-went sentiment analysis on Twitter messages using various features for classifications-N-gram feature, lexicon feature, POS feature. Their work was mainly subject specific and achieved an accuracy of nearly 80% and also concluded that POS feature diminishes accuracy level [16]. Farooq, et.al performed negation handling techniques in sentiment analysis [15]. They analyzed the effects of both syntactic and diminishing negation words in their experiment. They achieved an average accuracy rate of 83.3%. III. PROPOSED WORK The overview of our proposed model is displayed in Figure 1. In the first phase we have imported the data from the specific domain and preprocessed that data removing the different punctuations. In the next stage the specific model is prepared using the preprocessed data. We have tested the performance of three different text representation models. We have started with the simple binary bag of words of model where each document is represented as a fixed size vector of 0s and 1s where if a word appears in a document it gets a 1 and if it doesn't then it gets a 0. As an example, consider these four document below: D1: the movie was a very indulging cinematic experience. D2: standard of this movie is above its contemporaries. D3: director brought out the best of the pair. D4: moviegoers won't mind seeing the pair again. The binary bag of words representation for these four documents using 8 frequently occurring words is shown in the table 1. This model is represents only the existence of words but does not take into account the importance of specific words in a document, like in the first document "indulging" is a much more important word compared to the other words for measuring the polarity of the sentence. But in this model all the words appearing in document 1 get the value '1' and words not appearing get a value of '0'. Thus, it is a binary model/two dimensional sentiment analysis model. This led us to try out some other enhanced bag of words models. The second model we tested is the bag of words model with term frequency-inverse document frequency scores. Here the documents are also represented as vectors but instead of a vector of '0's and '1's, now the document contains scores for each of the words. These score are calculated by multiplying TF and IDF for specific words. So, the score of any word in any document can be represented as per the following equation: ( , ) = ( , ) * ( ) Therefore in this method, two matrices have to be calculated, one containing the inverse document frequency of a word in the whole corpus of documents and another containing the term frequency of each word in each document. The formulae to calculate both of them are as follows: ( , ) = ∈ ℎ . ∈ ℎ ( ) = (1 + . . ℎ ) The proposed example sentences can be converted into a TF-IDF model using the above method. Firstly, an IDF dictionary is created containing the 8 frequently occurring words and their IDF values. Then the TF dictionary is formulated containing the TF values for the corresponding words in each documents. The TFIDF model is shown in table 2. This model is different compared to the simple binary bag of words model as it does not represent the documents as vectors of '0's and '1's, rather assigns more precise values within 0 and 1 [11]. The simple TF-IDF model works well and gives importance to the uncommon words rather than treating all the words as equal in case of binary bag of words model. This model however fails to perform accurately when it encounters any sentence containing negations. This negation is a very common linguistic construction that affects word/sentence polarity. Therefore, the model should be framed in such a way that if presence of negations is considered then better result can be obtained. In the third model we have applied a negation strategy in which words are negated based on prior knowledge of polar expressions [13]. In this model whenever a negation word is tracked some changes are made to the words succeeding it. Many earlier proposed models [13] have also used this strategy before. Whenever a negation word is tracked all the words right after it are preceded with a 'not_' until a punctuation is received [13]. But this approach doesn't seem realistic to negate all the words as it will introduced a lot of unwanted words in the whole corpus. We have modified this strategy and instead of negating all the words till punctuation, we have negated the very next word following the negation word. So, if the sentence, "The bird is not flying in the sky" is received then instead of converting it to "The bird is not_flying not_in not_the not_sky", we have converted it to "The bird is not_flying in the sky". So, the job of tracking all the remaining punctuations after negation word is also excluded. In this approach meaning of the very next word followed by the negation is changed only and thus the meaning of the entire sentence gets changed. The algorithm for preprocessing in case of this model is shown in figure 1. The preprocessing phase of removing the punctuations, stop words is omitted in the algorithm and only the negation part is displayed to simplify the explanation. Our model therefore takes as input punctuation less documents. It then loops through the whole document and for each document performs the NWN technique. Figure 1. Proposed Algorithm for NWN After this preprocessing, a TFIDF model is formed in the same way as before. The proposed example sentences converted into the TFIDF NWN model is displayed in table 3. It can be seen in this model above the word "not_mind" has higher score than both "wont" and "mind" in the simple TFIDF model. After preparing the model for both training and testing using the text dataset, we have fitted the model in three popular classification algorithms as mentioned before i.e. Linear SVM [17], Multinomial Naïve Bayes [18] and Max Entropy Random Forest [19]. These models produces the various classifiers that can used to predict sentiment of new incoming data. In the next section the various experiments along with the obtained results are shown. IV. EXPERIMENTAL RESULTS We have used three different datasets and we have chosen the movie review dataset as the primary one for the experiments. For training and testing purposes we have split the dataset into two parts with split ratio of 0.8 (80% data for training and 20% data for testing). In the next sub sections we have conducted several experiments with different classification algorithms and also used various feature sizes. In the movie review dataset, we have used 40,000 reviews to train the classifier and 10,000 reviews to test the model performance. We have found out that accuracy of the classification increases as we increase the feature size. As shown in the graph below there is a steep increase in accuracy between the range of 2000-3000 and then it gradually slows down. Exp. 2 (10-Fold Cross Validation with SVM): After testing the accuracy of the model on the test set, we have used 10 fold cross validation technique to find out the average accuracy. Dataset was divided in 10 parts. At every fold 90% of the data are used to train the model and 10% of the data are used to check model performance. Therefore, after the process we get a list of accuracies of the different folds. So, we can calculate the mean of these different accuracies to better understand the accuracy of the model. Fig. 3 below shows the accuracies obtained in each fold of the training set with 5000 features for 10 fold cross validation. Exp. 3 (Accuracy comparison of three algorithms): We have used three of the most popular algorithms to train the classifier. The accuracy rate from each of the different algorithms for 10,000 features of data are displayed in the table 5 below. We found out that Linear Support Vector Machine produces the highest amount of accuracy among the three. Exp. 4 (10-Fold Cross Validation Comparison): In this experiment we have tested and compared the 10-Fold cross validation results for each of the algorithm for IMDB movie review dataset as follows. The figure 4 below displays the 10-Fold cross validation results for each of the algorithms. These results are from testing with 5000 features. { Explain how Graph- goes on } Exp. 5 (Performance in different Datasets): Apart from using the IMDB large movie review dataset [12] we have also tested the model performance on two other datasets from two different domain. While measuring the accuracies of each of our model in each of these datasets we have taken 8000 features into account. The accuracies from the different datasets are displayed in the In this experiment we have compared the performances of the three text representation models on the movie review dataset. In this experiment 8000 features are taken into consideration for maximum output accuracy. The accuracies for these models are displayed in the figure 5 below and it can be seen that our model (TFIDF with next word negation) outperforms both the basic binary bag of words representation and the simple TFIDF representation. CONCLUSION In this paper, we have conducted experiments on three datasets, IMDB movie reviews [12], Amazon product reviews [5] and SMS spam detection dataset [8]. After performing sentiment analysis on these datasets using binary bag of words model and TF-IDF model we found out the accuracy as XX.XX and XX.XX respectively [ Fig. 5]. But, after conducting experiment with our proposed model, i.e, using NWN with TF-IDF, we found out a good increase in the accuracy level. The accuracy percentages for IMDB review datasets, Amazon product review and SMS spam datasets came as 89.91%, 88.86%, 96.83% respectively[ Table 6]. So, from our experiments, we have concluded that when TF-IDF model is coupled with Next Word Negation then the performance of the sentiment classifier increases by a good percentage. Figure 2 . 2Accuracy Rate vs Feature Size Figure 3 . 3Accuracy Rate vs Fold Segment Figure 4 . 410-Fold Cross Validation Comparison Figure 5 . 5Model Performance Comparison V. 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[ "Nonclassical Properties and Anderson Localization of Quantum States in Coupled Waveguides", "Nonclassical Properties and Anderson Localization of Quantum States in Coupled Waveguides", "Nonclassical Properties and Anderson Localization of Quantum States in Coupled Waveguides", "Nonclassical Properties and Anderson Localization of Quantum States in Coupled Waveguides" ]	[ "Thais L Silva \nInstituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil\n\nInstituto de Física\nUniversidade Federal do Rio de Janeiro\n21941-972Rio de JaneiroRJBrazil\n", "Wesley B Cardoso \nInstituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil\n", "Ardiley T Avelar \nInstituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil\n", "Jorge M C Malbouisson \nInstituto de Física\nUniversidade Federal da Bahia\n40.210-340Salvador, BahiaBrazil\n", "Thais L Silva \nInstituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil\n\nInstituto de Física\nUniversidade Federal do Rio de Janeiro\n21941-972Rio de JaneiroRJBrazil\n", "Wesley B Cardoso \nInstituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil\n", "Ardiley T Avelar \nInstituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil\n", "Jorge M C Malbouisson \nInstituto de Física\nUniversidade Federal da Bahia\n40.210-340Salvador, BahiaBrazil\n" ]	[ "Instituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil", "Instituto de Física\nUniversidade Federal do Rio de Janeiro\n21941-972Rio de JaneiroRJBrazil", "Instituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil", "Instituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil", "Instituto de Física\nUniversidade Federal da Bahia\n40.210-340Salvador, BahiaBrazil", "Instituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil", "Instituto de Física\nUniversidade Federal do Rio de Janeiro\n21941-972Rio de JaneiroRJBrazil", "Instituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil", "Instituto de Física\nUniversidade Federal de Goiás\n74.690-900Goiânia, GoiásBrazil", "Instituto de Física\nUniversidade Federal da Bahia\n40.210-340Salvador, BahiaBrazil" ]	[]	We consider the propagation of light beams through disordered lattices of coupled waveguides searching for Anderson localization and investigating the evolution of nonclassical properties of injected quantum states. We assume that the beam is initially in a variety of states, such as the complementary coherent state, the reciprocal binomial state and the polynomial state. The statistical properties of the evolved states were analyzed numerically as functions of the localization/delocalization parameters averaged over many realizations of disorder. We also numerically reconstruct the Wigner function of the output state. Interestingly, we find that high values of disorder tend to preserve quantum properties of some input states when we look at the input waveguide despite of the coupling between it and the neighboring waveguides.	10.1103/physreva.105.023710	[ "https://arxiv.org/pdf/2108.04844v1.pdf" ]	236,975,887	2108.04844	f56274d2874412e8b168fb6afb5ce53e52c76230	Nonclassical Properties and Anderson Localization of Quantum States in Coupled Waveguides 10 Aug 2021 Thais L Silva Instituto de Física Universidade Federal de Goiás 74.690-900Goiânia, GoiásBrazil Instituto de Física Universidade Federal do Rio de Janeiro 21941-972Rio de JaneiroRJBrazil Wesley B Cardoso Instituto de Física Universidade Federal de Goiás 74.690-900Goiânia, GoiásBrazil Ardiley T Avelar Instituto de Física Universidade Federal de Goiás 74.690-900Goiânia, GoiásBrazil Jorge M C Malbouisson Instituto de Física Universidade Federal da Bahia 40.210-340Salvador, BahiaBrazil Nonclassical Properties and Anderson Localization of Quantum States in Coupled Waveguides 10 Aug 2021arXiv:2108.04844v1 [quant-ph] We consider the propagation of light beams through disordered lattices of coupled waveguides searching for Anderson localization and investigating the evolution of nonclassical properties of injected quantum states. We assume that the beam is initially in a variety of states, such as the complementary coherent state, the reciprocal binomial state and the polynomial state. The statistical properties of the evolved states were analyzed numerically as functions of the localization/delocalization parameters averaged over many realizations of disorder. We also numerically reconstruct the Wigner function of the output state. Interestingly, we find that high values of disorder tend to preserve quantum properties of some input states when we look at the input waveguide despite of the coupling between it and the neighboring waveguides. I. INTRODUCTION Nonclassical properties of quantized light field and its generation -i.e., the "quantum states engineering"are essential ingredients of quantum optics and quantum information. Indeed, nonclassical states perform a crucial role in many potential applications such as quantum teleportation [1][2][3], quantum cryptography [4], quantum computation and quantum communication [5], quantum internet [6], etc. In this direction, periodic photonic systems has emerged as a platform to engineer new light field structures, presenting numerous significant technological advances [7][8][9][10]. On the other hand, the study of nonperiodic photonic structures, by using small defects in periodic lattices [11][12][13] or disordered and quasi-periodic structures [14][15][16][17][18], has demonstrated a diversity of optical effects in the presence of Anderson Localization, such as engineering of random lasers [19] and structurally colored materials with precisely controllable wavelength and angular dependence of scattering [20]. Recently, it was demonstrated how to tune and freeze disorder in photonic crystals by using percolation lithography [21]. Anderson localization -i.e., the suppression of transport due to destructive interference of the many paths associated with coherent multiple scattering from the modulation of a disordered potential [22]-has been experimentally observed in microwaves [23], light diffusive media [14,24], photonic crystals [15,16], Bose-Einstein condensates [25,26], sound waves [27], optical fiber arrays [28], etc. Inspired by these experimental investigations, many theoretical studies have been proposed by considering the system in the presence of disordered potentials (see for example [29][30][31][32][33][34][35][36][37][38][39]). Recently, Anderson localization of nonclassical light was investigated for propagation in an array of waveguides in which neighboring waveguides are evanescently coupled and disorder can be added in a controlled manner [40]. Specifically, that work investigated the consequences of using both sub-Poissonian and super-Poissonian input light on the characteristics of Ander-son localization, verifying the enhancement in fluctuations of localized light and superbunching due to the medium's disorder. Also, an important consequence of sub-Poissonian statistics of the incoming light is to quench the total fluctuations at the output [40]. The system employed in Ref. [40] is similar to that used in Ref. [16] to experimentally investigate the evolution of linear and nonlinear waves in the presence of Anderson localization. Moreover, the disordered one-dimensional waveguide lattice was also used to experimentally investigate an extensive list of phenomena: the signature of a localization phase transition for light by directly measuring wave transport inside the lattice [41], quantum correlations between noninteracting particles evolving simultaneously in a disordered medium [42], Hanbury Brown and Twiss correlations of Anderson localized waves [43], the control of the polarization state of coherent light propagating through an optically thick multiple scattering medium by controlling only the spatial phase of the incoming field with a spatial light modulator [44], the coherent manipulation of two-photon path-entangled states by multimode interference in multimode waveguides [45], the observation of topological phase transitions in photonic quasicrystals [46], the observation of ensembleaveraged quantum correlations between path-entangled photons undergoing Anderson localization [47], the twophoton Anderson localization in a quadratic waveguide array with the emergence of off-diagonal disorder [48], etc. Here, inspired by the results obtained in Ref. [40], we numerically investigate the propagation of light beams, previously prepared in nonclassical states of the electromagnetic field, propagating in disordered lattices of waveguides and undergoing Anderson localization. Our goal is to verify the influence of Anderson localization on the statistical properties of previously prepared input light field (nonclassical states). To this end, we assume the beam in a variety of states, namely, the complementary coherent state [49], reciprocal binomial state [50], polynomial state [51], thermal state, coherent state, and squeezed state [52]. The rest of the paper is organized as follows. The theoretical model is considered in next section, where we present the dynamical model of the system in Subsec. II A, the characterization of the quantum states under consideration in Subsec. II B and the numerical methods in Subsec. II C. In Sec. III we present the numerical results and our analyzes. We conclude the paper in Sec. IV. II. LINEAR ARRAY OF WAVEGUIDES The system we consider is a one-dimensional finite array of monomodal waveguides where prescribed input states of the electromagnetic field can propagate. Classically, fields propagating in the array of waveguides are coupled through evanescent waves passing over their boundary-barriers; on the quantum level, one says that photons can coherently tunnel between neighboring waveguides so that the quantum state represents the overall-overlapping superposition of the modes of the waveguides [16]. The field can be injected into one or a few waveguides and disorder can be implemented in the array either by randomly adjusting the spacing among the parallel waveguides along the x-direction, the propagation being in the z-direction, or by randomly fixing the thicknesses of the waveguides. This kind of system has been constructed on an AlGaAs substrate [53,54] and direct identification and measurements of Andersonlocalization of states have been performed. A. Theoretical model The electromagnetic energy density in the array of waveguides, assuming that all media are linear and nonmagnetic, and that the relevant evanescent overlap occurs only between neighboring waveguides, is given by H = j 1 2 ǫ 0 n 2 E 2 j + 1 2µ 0 B 2 j + (j,j ′ ) 1 2 ǫ 0 n 2 E j · E j ′ + 1 2µ 0 B j · B j ′ ,(1) where n is the refraction index and (j, j ′ ) denote nextneighbor pairs. We consider monochromatic fields propagating along the z-direction with velocity c/n. At the quantum level, the field mode in the j-th waveguide is written in terms of photon-annihilation and photon-creation operators, a j and a † j respectively, and the evolution of the system is dictated by a Hamiltonian in the form [42] H = j β j a † j a j + j C j+1,j a † j+1 a j + C j,j+1 a † j a j+1 ,(2) where β j is the propagation constant associated with the j-th waveguide and C j+1,j and C j,j+1 are coupling coefficients between nearest neighbor waveguides. Notice that, we are considering fields with lower enough intensities to make non-linear effects negligible; also, along this paper unless stated in contrary, we use = 1 and c = 1. The creation and annihilation operators satisfy the commutation relations [a j , a l ] = 0; [a † j , a † l ] = 0; [a j , a † l ] = δ jl ,(3) and we assume the existence of eigenstates \|ψ , such that n j \|ψ = a † j a j \|ψ = n j \|ψ , where n j is the photon number of the j-th waveguide. We will analyse arrays where we can fix a constant coupling (tunneling rate) between neighbor waveguides, C j+1,j = C j,j+1 = C, and introduce disorder by taking the coefficients β j as random variables with zero-mean Gaussian distributions; the model then becomes isomorphic to the one-dimensional quantum tight-binding model used by Anderson [22], with β j being the on-site energy, and the system should then present localization of states. This assertion is experimentally feasible since these coefficients are related with the guide geometry, which can be appropriately adjusted [41,43] to spatially modulate the index of refraction n(x). In this case, the Heisenberg equations can be written as [16,41,43,47,55] i ∂a j ∂z = [a j , H] = β j a j + C(a j+1 + a j−1 ),(4) where z = ct/n, i.e. measurements of intensity distribution at position z give the time evolution along the array. Now we search for a solution of the Heisenberg equations (4), depending on the initial input state. Since the Heisenberg equations are linear in the annihilation (or for the creation) operators, it can be solved by finding the Green's function in such way that a j (z) = l G jl (z)a l (0),(5) where a l (0) correspond to the input state (at z = 0) into the l-th waveguide. The Green's function correlates fields in the j-th and the l-th waveguides at all positions z. By inserting Eq. (5) into Eq. (4), one gets the following set of first order differential equations for the Green's functions, i ∂G jl ∂z = β j G jl + C(G j+1,l + G j−1,l ).(6) From our assumptions, G jl (z) depend on the parameters β j , which vary randomly, and C that remains fixed; this is usually referred to as diagonal disorder. These equations can be solved numerically with great precision; taking specific distributions of {β j } and a given value of C, solutions are obtained just depending on the initial input state. We can work with a great simplification if we consider that the input field \|Ψ in is injected into only one waveguide [40], which we label by j 0 , that is, a j (0)\|Ψ in = 0 for all j = j 0 . Thus, considering the light injection only into the j 0 -th waveguide, the mean field intensity output by waveguide j as a function of the lattice length z is given by I j (z) = a † j (z)a j (z) = \|G j,j0 (z)\| 2 a † j0 a j0 ,(7) where the mean value of the Green's functions is a standard statistical mean over several realizations for different values of {β j }, while the mean a † j0 a j0 represents the quantum expectation value of the number operator a † j0 (0)a j0 (0), which depends only on the input state. Also, we can calculate the correlation of the intensities between the output of two waveguides j and l, to be given by I j (z)I l (z) = \|G j,j0 (z)\| 2 \|G l,j0 (z)\| 2 a †2 j0 a 2 j0 .(8) Another important quantity to qualify the statistics of photons propagating in the array is the second-order correlation function defined, for the j-th waveguide, by g (2) j (z) = a †2 j (z)a 2 j (z) a † j (z)a j (z) 2 = \|G j,j0 (z)\| 4 \|G j,j0 (z)\| 2 2 a †2 j0 a 2 j0 a † j0 a j0 2 .(9) The correlation function g (2) (z) indicates weather states evolve in a Poissonian way, or if they present either bunching or antibunching (g (2) (z) > 1 and g (2) (z) < 1, respectively) in their photon distribution. In any case, all these quantities depend on the initial state in the input of the array of waveguides. The expressions for other quantities used to characterize the output field are presented in the results section. B. Characterization of the input quantum states In Ref. [40], a theoretical study of Anderson localization of light in an array of waveguides was presented, using coherent, thermal and squeezed states at the input, to investigate the effects of nonclassicality. Here, we extend the work of Ref. [40] by considering different input states that show more general quantum statistics, and by presenting the evolution in the Wigner representation. We shall consider specifically the complementary coherent state (CCS) [49], the reciprocal binomial state (RBS) [50] and the polynomial state (P S) [51] as input states; these states present peculiar statistical properties and it is interesting to investigate how they evolve along the array. The complementary coherent state [49] is written in the number basis as \|Ψ CCS (α, N ) = ℵ CCS e −\|α\| 2 /2 2 −N √ N ! N k=0 √ k!α * N −k e ikπ/2 \|k ,(10) where the normalization constant ℵ CCS is given by ℵ 2 CCS (α, N ) = N !e \|α\| 2 2 2N N k=0 k!\|α\| 2(N −k) .(11) For these states we find the mean number of photons, a † j0 a j0 CCS = N k=0 k!k\|α\| 2(N −k) N k=0 k!\|α\| 2(N −k) ,(12) and the mean a †2 j0 a 2 j0 is given by a †2 j0 a 2 j0 CCS = N k=0 k!k(k − 1)\|α\| 2(N −k) N k=0 k!\|α\| 2(N −k) .(13) The reciprocal binomial state [50], written as \|Ψ RBS (φ, N ) = ℵ RBS N k=0 N k 1 2 exp ik φ − π 2 \|k ,(14) has normalization constant ℵ RBS given by ℵ 2 RBS = N k=0 N k −1 .(15) For the RBS, we find a † j0 a j0 RBS = ℵ 2 RBS N k=0 N k k (16) and a †2 j0 a 2 j0 RBS = ℵ 2 RBS N k=0 N k k(k − 1).(17) The polynomial state [51] is defined by \|Ψ P S (x, N ) = ℵ P S N k=0 N k − 1 2 H N −k (x/ √ 2)e ikπ 2 (2N − 2k − 1)!! \|k ,(18) where H N −k (y) is a Hermite polynomial, with the normalization constant ℵ P S written as ℵ 2 P S = N k=0 N k −1 H 2 N −k (x/ √ 2) (2N − 2k − 1)!! −1 .(19) For the polynomial state we find a † j0 a j0 P S = ℵ 2 P S N k=0 N k −1 H 2 N −k (x/ √ 2) (2N − 2k − 1)!! k (20) and a †2 j0 a 2 j0 P S = ℵ 2 P S N k=0 N k −1 H 2 N −k (x/ √ 2) (2N − 2k − 1)!! k(k − 1).(21) For sake of comparison, we also investigate the input states used in Ref. [40]: thermal states (T S), mixed states with density matrix ρ T S = 1 1 +n ∞ n=0 n 1 +n n \|n n\| ,(22) for which a † j0 a j0 T S =n and a †2 j0 a 2 j0 T S = 2n 2 ; coherent states (CS), \|α = e −\|α\| 2 /2 ∞ n=0 α n √ n! \|n ,(23) for which a † j0 a j0 CS = \|α\| 2 and a †2 j0 a 2 j0 CS = \|α\| 4 ; and squeezed-vacuum states, ψ SS (\|ζ\|e φ ) = 1 cosh \|ζ\| ∞ n=0 (−e iφ tanh \|ζ\|) n (2n)! 2 n n! \|2n ,(24) with squeezing parameter ζ = \|ζ\|e iφ , for which a † j0 a j0 SS = sinh 2 \|ζ\| and a †2 j0 a 2 j0 CS = sinh 2 \|ζ\|(1 + 3 sinh 2 \|ζ\|). C. Computational method Since we are assuming that the input light is injected only in the j 0 -th waveguide, all relevant quantities depend only on the Green's functions G j,j0 , which satisfy i ∂G j,j0 ∂z = β j G j,j0 + C(G j+1,j0 + G j−1,j0 ).(25) To get solutions numerically, these equations were discretized and solved by using the Crank-Nicholson method [56] with step size ∆z = 0.001. In the numerical analysis, we consider an array with 101 waveguides, which we label from 1 to 101 with light injected only in the waveguide j 0 = 51. Thus the initial condition is that G 51,51 (0) = 1 while G j,51 (0) = 0 for all j = 51; additionally, since the waveguide array is finite containing M waveguides (here 101), we add the boundary conditions G 0,l (z) = G M+1,l (z) = 0 for all l = 1, 2, ...., M . Following Ref. [40], we assume that the random coefficients β j are independent of each other and follow a zero-mean Gaussian probability distribution of the form P (β) = 1 √ 2π∆ 2 exp −β 2 2∆ 2 ,(26) with the variance ∆ 2 measuring the disorder in the waveguide array. For sake of simplicity but without loss of generality, we fix the interaction parameter C = 1 and take ∆/C quantifying the arrangement disorder. In order to get the β j coefficients, we have used the Box-Müller method [57] that requires the generation of two random numbers with uniform distribution, which was done using a congruence method. For each set {β j } of disorder parameters, Eqs. (25) are solved numerically (fixing C) to find the relevant Green's functions. The functions G j,j0 (z) depend on the disorder parameters β j and on the coupling parameter C, and completely describe the dynamical evolution of any state injected into the waveguide array, only through the 51 th waveguide. Naturally, to average over the sets of the random coefficients β j one has to consider a reasonable number of realizations of disorder, different sets {β j }; here we take 1000 realizations of disorder in each simulation. In our study, we take all the input states with the same energy, that is the same mean number of photons, specifically n in (0) = a † 51 (0)a 51 (0) = 10. To present the results obtained by our simulations, we shall choose five states from the families we have shown in Subsec. II B, namely: two CCS states, named CCS 1 (= \|Ψ CCS (0.1414, 10) ) and CCS 2 (= \|Ψ CCS (1.916, 11) ); two P S states, referred to as P S 1 (= \|Ψ P S (0.374, 12) ) and P S 2 (= \|Ψ P S (0.9345, 13) ); and, in the case of the RBS state, \|Ψ RBS (φ, N ) , we set N = 20 and φ = 0. For completeness and comparison, we also consider as input states a thermal state (T S) withn = 10, a coherent state (CS) with \|α\| 2 = 10 and a squeezed-vacuum state (SS) with sinh 2 \|ζ\| = 10. III. NUMERICAL RESULTS As explicitly shown in Eq. (7), which is valid whenever light is injected into the j 0 -th waveguide, the average output intensity of the waveguide array depends on the average number of photons of the input state (here fixed as n in = 10) and on the degree of disorder of the system, carried by the random β-coefficients, which is manifested by the Green's functions; the output intensity, that is the mean number of photons at the end of the array, does not depend on any other characteristics of the input state. The average output intensity, as distributed among the 101 waveguides of the array, is shown in Fig. 1 for some values of ∆/C and two values of propagation distance z. As mentioned before, ∆/C measures the arrangement disorder, i.e., as the value of ∆ increases the coefficients β j become more distinct from each other. In the absence of disorder, (Fig. 1(a)), the light suffers only the standard dispersion. However, as the disorder is increased ( Fig. 1(b-d)), a narrow peak of intensity around the input (j 0 -th, number 51) waveguide emerges, which characterizes the localization of the solution. For ratios ∆/C greater then 1.5, the output-intensity profile does not change anymore with the increasing of the propagation distance, which is shown by the exact overlapping of the solid-line (z = 5) and dashed-line (z = 20) in Fig. 2. In Fig. 3 we present, using a logarithmic scale, the mean number of photons for some values of disorder (∆/C) and considering two values of propagation distance, z = 5 and z = 20 respectively. The choice of using a logarithmic scale is justified by the fact that an ex- ponential decay of the light intensity, in the waveguides different of the j 0 -th one, would signalize the appearance of Anderson localization. We find that, in all cases, there exist two regions of decreasing exponentials and, except in case ∆/C = 1.0, the pattern holds for both propagation distances. It should be mentioned, however, that plots of the output intensities (mean number of photons) may not be sufficient to state whether the light beam presents Anderson localization, since the system might have yet a small diffusion, which would become more evident by increasing the propagation distance. In order to further investigate the occurrence of Anderson location, we can also calculate the participation number defined by, and given in our case by, P(z) = j a † j (z)a j (z) 2 j a † j (z)a j (z) 2 = 1 + j =k \|G j,51 (z)\| 2 \|G k,51 (z)\| 2 j \|G j,51 (z)\| 2 2 .(27) We see that the participation number is not only independent of the mean number of photons of the input state, but it is actually completely independent of the electromagnetic field mode injected in the j 0 -th waveguide of the array, the number 51; it depends only on the waveguide array itself and its disorder encoded in the Green's functions. The participation number indicates in how many waveguides there are photons as a function of propagation distance; thus, if P(z) increases, it means that the beam remains scattering among the waveguides along propagation. It should be emphasized that, in the case of absence of disorder, the dispersion of P(z) is linear, as indicated by Fig. 4(a). Also, for a very small amount of disorder, ∆/C = 0.1, we find that the diffusion occurs faster than in the absence of disorder, while for ∆/C = 0.5 dispersion still occurs, but with a growth rate smaller than that in the case of no disorder. In all other cases reported in Figs. 4(b-d), for larger values of ∆/C, the plots of P(z) stabilise as z increases, characterizing localization of the light in the array. Notice that some fluctuations are still observed, but on a small scale when compared to 1, which is the smallest value that can characterize a single waveguide. Figures 5 and 6 show the second order correlation function g (2) of the field output by waveguide number 51, as a function of the disorder degree for different input states, calculated with Eq. (9), taking propagation distances z = 5 and z = 20, respectively. We first notice that, for all input states analysed, taking measurements for a short propagation length (z = 5, Fig. 5), the second order correlation function increases, as the disorder degree (∆/C) raises, it reaches a maximum for ∆/C = 1 and then decreases to values greater than the value of g (2) without disorder. Similarly, for longer arrays (z = 20, Fig. 6), one finds peaks of g (2) somewhat higher than those of Fig. 5 but occurring at a much lower disorder degree and rapidly decaying for large disorder degree. Yet interestingly, for the propagation distance z = 20, we find that only the CCS 1 state returns to the g (2) < 1 regime for large values of disorder. Note also that the curves for the CCS 2 and the RBS states are nearly coincident due to the fact that, for these states, a †2 51 (0)a 2 51 (0) are very close, 94.78 and 95, respectively. The variance of the output field intensity, (∆I) 2 = I 2 − I 2 , of the 51 th waveguide, when the input field is injected only in it, is obtained directly from Eqs. (7) and (8) as (∆I 51 (z)) 2 = \|G 51,51 (z)\| 4 a †2 51 (0)a 2 51 (0) + \|G 51,51 (z)\| 2 a † 51 (0)a 51 (0) − \|G 51,51 (z)\| 2 2 a † 51 (0)a 51 (0) 2 . (28) In order to investigate the influence of the disorder on the intensity variance of the output states, in Figs. 7(a) and 7(b) we show these variances as functions of the disorder degree ∆/C, at z = 5 and z = 20 respectively, for different input states injected into the 51 th waveguide. We see that, for all states discussed, when the degree of disorder increases, the variances increase, some of them tending to stabilize, as for states P S 1 , P S 2 and CS, while others, like the cases of the states CCS 1 , CCS 2 and RBS, reach a maximum and then decrease for large values of the disorder parameter, the decreasing rate being bigger for the state CCS 1 . It is interesting to notice that the patterns of (∆I) 2 practically do not change when one compares the results for the propagation lengths z = 5 and z = 20, Figs. 7(a) and 7(b) respectively. Here, likewise Figs. 5 and 6, the curves for the CCS 2 and the RBS states are almost coincident. We display the states CCS1 in circles (blue), CCS2 in boxes (black), RBS in pentagon (cyan), CS in triangles (red), P S1 in down triangles (grey), and P S2 in diamond (green), respectively. A. Wigner representation We can also study the influence of disorder in the propagation of light in the waveguide array using the Wigner representation of the quantum states. The Wigner function (WF) of the output state in the 51 th waveguide is given by W 51 (α, z) = 1 π d 2 ξ exp(αξ * − α * ξ) χ 51 (ξ, z) ,(29) with the symmetrically ordered characteristic function given by χ 51 (ξ, z) = Tr ρ 51 (0)e ξa † 51 (z)−ξ * a51(z) ,(30) where a 51 (z) = G 51,51 (z)a 51 (0) and the input state is given by ρ 51 (0) = \|ψ 51 ψ 51 \|. We take \|ψ 51 = N n=0 c n \|n , which is the form of the states described in Subsec. II B. Then, the integral defining W 51 (α, z) can be performed analytically, and writing α = x + iy, we obtain W (N ) 51 (α, z) = 2e −2(x 2 +y 2 ) N t=0 t j=0 N n=0 n k=0 c * n c t [G 51,51 (z)] j G * 51,51 (z) k √ t!n! (n − k)! δ (n−k),(t−j) I jk (x, y),(31) where I jk (x, y) = k l=0 j s=0 (−1) 2j+k i −(l+s) (k − l)!l!(j − s)!s! I p (y)I q (−x),(32) with p = l + s and q = k − l + j − s and I r (u) = (2u) r + e(r) m=2 (−1) m 2 2 − m 2 r! m 2 !(r − m)! (2u) r−m ,(33) where the summation is over even integers and e(r) is the largest even integer not greatter than r, i.e. e(r) = r if r is even and e(r) = r − 1, for r odd. We can use the WF, given by Eq. (31) to investigate the propagation of states through the array, looking at the output state in the 51 th waveguide, when the input is a truncated state in the number basis, N n=0 c n \|n ; Figs. 8 and 9 show examples of this case. In Fig. 8, we present the WF when the input state is the CCS 1 , in Fig. 8(a) for the input state (z = 0), and in Figs. 8(b-d) for the output states in the 51 th waveguide, for the length z = 20, considering the disorder parameters ∆/C = 0, 1, and 7, respectively. We find that, in absence of disorder (∆/C = 0) ( Fig. 8(b)), the WF is similar to the one of the vacuum state; this is also confirmed by the photon number distribution, as shown below. But, for small values of the disorder parameter, Figs. 8(b-c), the WF profiles present very important differences when compared with that for the input state, showing that dispersion prevails. However, by increasing the disorder parameter, the WF of the output state becomes similar to that one for the input state; clearly this behavior is directly linked to a strong localization, but not necessarily of Anderson type. In Fig. 9(a), we present the WF for the input state P S 1 and, in Figs. 9(b-d), for the output state of the 51 th waveguide at z = 20 with the same values of the disorder parameter as in Fig. 8, ∆/C = 0, 1, and 7, respectively. Here, we observed that, similarly to the results obtained for the CCS 1 (Fig. 8 ( Fig. 9(b)) or a small disorder (here characterized by the value of the disorder parameter ∆/C = 1, Fig. 9(c)) the output state of the 51 th waveguide at z = 20 presents a configuration for the WF similar to that of the vacuum state. Clearly, the weak disorder is not enough to ensure that the WF remains in the same shape, due to the interaction of the main guide (51 th ) with the neighboring guides. On the other hand, as the disorder parameter increases (for example with ∆/C = 7 displayed in Fig. 9(d)) the WF is now preserved, which can be observed when comparing the panels (a) and (d) of Figs. 8 and 9. In other words, by increasing the disorder parameter, the WFs become more robust to changes due to variations of the coupling between neighbor waveguides. Also, it should be pointed out that, in all cases, due to the computational cost the number of realizations to produce the output state was reduced to 100. Finally, we can also use the WF (Eq. (31)) to obtain the photon number distribution (PND) of the output state of the 51 th waveguide, given by P (N ) 51 (n, z) = 1 π d 2 α W (N ) 51 (α, z) W n (α),(34) where W n (α) is the WF for the number state \|n . The integral in Eq. (34) is calculated numerically and, as before, the number of realizations to get the proper averages was reduced to 100. In Fig. 10 we show the PND in the 51 th waveguide when the input state is the CCS 1 and the length of the array is z = 20. In Fig. 10(a), we plot together the PND of the input CCS 1 state and the output PND (P (N ) 51 (n, 20)) for null disorder, for comparison. Interestingly, the PND of the state CCS 1 , which has n CCS 1 = 10, is very close to that of the number state Fig. 8, but now considering the P S1 as the input state. Fig. 10 but now considering the input state being the P S1. \|10 , P \|10 (n) = δ n, 10 . On the other hand, the output state of the 51 th waveguide (at z = 20), in absence of disorder, is close to that for the vacuum state; this fact reinforces the results for the Wigner functions presented in Figs. 8(b) and 9(b). In Fig. 10(b) we show the PND for the output state in the 51 th waveguide considering three different values of the disorder parameter, ∆/C = 0, 1, and 7. We clearly see that the increasing of disorder tend to favor the output PND to become closer to that of the input state, that is disorder tends to preserve the PND. In Fig. 11 we present the results of the PND, for the same parameters as in Fig. 10, but considering the input state given by the P S 1 . Now we see, from Fig. 11(a), that the PND of the input P S 1 state differs significantly from the PND for the number state, distinctly to the case of the CCS 1 , although we still have n P S1 = 10. However, also in agreement with the case of the CCS 1 , we see in Fig. 11(b) that the PND of the output state is clearly closer to that of the input state as greater is the value of ∆/C. IV. CONCLUSION We have discussed the evolution of quantum states of the electromagnetic field propagating through a disordered plane waveguide array. Specifically, we have analysed the propagation of three truncated states (in the number basis), namely the complementary coherent state, the reciprocal binomial state and the polynomial state; we also considered a thermal state, a coherent state and a squeezed-vacuum state, for comparison. In our numerical calculations, we considered arrays with a 101 waveguides focusing in the injection and detection of waves in the middle one, the 51 th -waveguide. The in site energy parameters β j were assumed to be independent of each other and randomly taken following a zero-mean Gaussian distribution with variance ∆ 2 , while the coupling between neighbor waveguides were fixed as constant, C; the disorder parameter was defined as ∆/C. We have investigated some quantities that qualify the nature of the statistics of the state. First, we verified that increasing the degree of impurity of the lattice, the mean photon-number distribution tends to concentrate around the waveguide where the input state is injected, with an exponential decay of the light intensity in the others, for any injected state with a given mean number of photons, here n in = 10; although this is not a definitive indication of localization, it does signalize it. We also analysed the participation number, indicating in how many waveguides there were photons, which shows the effects of disorder in the array and is totaly independent of the input state. We find, as the disorder parameter is increased, from no-disorder to a high disorder regime, the participation number changes from a linear increase with the propagation distance, characteristic of dispersion, to a flat behavior with very small fluctuations which represents a localization pattern. We investigated second order quantities as the g (2) function and the output intensity variance at the central waveguide. We observe for all input states that, although localized, the average output state presents a classical behavior relatively to the bunching feature. However, as the disorder increases, the g (2) function decreases and even presents antibunching again for the complementary coherent state and high disorder. While the variance in the output intensity at the central waveguide increases with disorder, and stabilizes to a final value for most of the input states investigated. In order to investigate how the propagation through the lattice changes the input state, it is not enough to analyse quantities quantum averaged over the input state. It can be noticed, for example, when we look at the results for CCS2 and RBS, two different states with different features, however presenting nearly the same values of g (2) and intensity variance at the output. To tackle this question we reconstructed the output state by means of its Wigner function, which could also be used to obtain the output probability distribution of number of photons. We observed a preservation of the characteristics of the Wigner function for high disorder parameter values, although we also notice a suppression of negative values in average. Moreover, as the output probability distribution of number of photons changes with the features of the array as well as with the input state, a well designed array of coupled waveguides could be used to produce new states of the electromagnetic field. FIG. 1 . 0 FIG. 2 . 102(Color online) Mean photon number distribution at the output of the array of waveguides (numbered as n = 1, 2, 3, . . . , 101) for two values of propagation distance z: solid (blue) curves are for outputs at z = 5, while dashed (black) curves correspond to outputs at z = 20. The disorder parameters were fixed as: (a) ∆/C = 0, (b) ∆/C = 0.1, (c) ∆/C = 0.5, and (d) ∆/C = 1. These plots were obtained by averaging over 1000 realizations, fixing the number of photons in the injected (input) state (z = 0, at the 51 th waveguide) nin = a † 51 ((Color online) The same as in Fig. 1, but now using (a) ∆/C = 1.5 and (b) ∆/C = 5. FIG. 3 . 3(Color online) The same as in Fig. 1, but now in logarithm scale and using the values (a) ∆/C = 1, (b) ∆/C = 3, (c) ∆/C = 5, and (d) ∆/C = 10. FIG. 4 . 4(Color online) Participation number as a function of propagation distance for various values of the disorder parameter: (a) ∆/C = 0, 0.1 and 0.5; (b) ∆/C = 1, 1.5 and 2; (c) ∆/C = 2.5, 2.8 and 3; (d) ∆/C = 3.5, 4 and 5; corresponding to solid-line (blue), dashed-line (black) and dotted-line (red), respectively, in all cases. Results were obtained after averaging over 1000 realizations. FIG. 5 .FIG. 6 . 56(Color online) Second order correlation function of the output state in the 51 th waveguide, at propagation distance z = 5, as a function of the disorder parameter for different input states. Panel (a) displays the curves corresponding to the states CCS1 in circles (blue), CCS2 in boxes (black), RBS in pentagon (cyan), CS in triangles (red), P S1 in down triangles (grey), and P S2 in diamond (green). For comparison, panel (b) also presents the states SS in circles (magenta) and T S in boxes (orange). All results were obtained with an average over 1000 realizations. (Color online) The same as inFig. 5, but now with propagation distance z = 20. FIG. 7 . 7(Color online) Variance of the output intensity of the 51 th waveguide as a function of the disorder parameter for the propagation distances (a) z = 5 and (b) z = 20. FIG. 8 . 8(Color online) Panel (a) shows the Wigner function of the CCS1 input state, while panels (b), (c) and (d) show the WF of the output state in the 51 th waveguide, at the propagation distance z = 20, considering the disorder parameters ∆/C = 0, 1 and 7 respectively. The plots consist on an average over 100 realizations considering the step-size of dx = dy = 0.1. FIG. 9 . 9(Color online) The same as in 10. (Color online) PND of the output state of the 51 th waveguide showing in (a) the input state CCS1 and output state (at z = 20) for ∆/C = 0 and, in (b), the output states for ∆/C = 1, 3, and 7. These results correspond to an average over 100 realizations. FIG. 11. (Color online) The same as in ACKNOWLEDGMENTSWe acknowledge financial support from the Brazilian agencies CNPq (#311408 . C H Bennett, G Brassard, C Crépeau, R Jozsa, A Peres, W K Wootters, Phys. Rev. Lett. 701895C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. 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[ "On the Representation of Partially Specified Implementations and its Application to the Optimization of Linear Algebra Kernels on GPU", "On the Representation of Partially Specified Implementations and its Application to the Optimization of Linear Algebra Kernels on GPU" ]	[ "Ulysse Beaugnoń [email protected] \nInria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n\n", "Basile Clément [email protected] \nInria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n\n", "Nicolas Tollenaere [email protected] \nInria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n\n", "Albert Cohen Google \nInria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n\n" ]	[ "Inria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n", "Inria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n", "Inria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n", "Inria andÉcole Normale Supérieure\nInria andÉcole Normale Supérieure\nEcole Normale Supérieure and Google\n" ]	[]	Traditional optimizing compilers rely on rewrite rules to iteratively apply program transformations. This iterative approach hides optimization opportunities behind intermediate transformation steps. For instance, vectorization can only be applied to the innermost loop in a nest: one must first perform a loop interchange before even considering vectorization of an outer loop. In contrast, we propose an implementation framework representing programs as sets of possible implementation decisions. Specifying one decision can have an impact on others in a bidirectional manner: specifying that a loop must be vectorized prevents other loops from being nested inside it; conversely, specifying a loop as an outer loop will prevent it from being vectorized. These optimization decisions commute, obviating the pass ordering problem. We present a constraint programming system to formally define, represent and explore such implementation spaces. We also propose an exploration strategy combining tree search and branch-and-bound; the strength and novelty of this strategy reside in an analytical model of the lower bound on the execution time of a set of possible implementations. We showcase our approach on the construction and exploration of an implementation space for linear algebra kernels running on GPUs. We show this search space is expressive enough to represent complex decisions that fundamentally change the structure of the generated code. We also present preliminary results competitive with the performance of native GPU libraries.	null	[ "https://arxiv.org/pdf/1904.03383v1.pdf" ]	102,350,660	1904.03383	207bcb267808fff95c77c2b0a34d49e5b256ecaf	On the Representation of Partially Specified Implementations and its Application to the Optimization of Linear Algebra Kernels on GPU Ulysse Beaugnoń [email protected] Inria andÉcole Normale Supérieure Inria andÉcole Normale Supérieure Ecole Normale Supérieure and Google Basile Clément [email protected] Inria andÉcole Normale Supérieure Inria andÉcole Normale Supérieure Ecole Normale Supérieure and Google Nicolas Tollenaere [email protected] Inria andÉcole Normale Supérieure Inria andÉcole Normale Supérieure Ecole Normale Supérieure and Google Albert Cohen Google Inria andÉcole Normale Supérieure Inria andÉcole Normale Supérieure Ecole Normale Supérieure and Google On the Representation of Partially Specified Implementations and its Application to the Optimization of Linear Algebra Kernels on GPU Traditional optimizing compilers rely on rewrite rules to iteratively apply program transformations. This iterative approach hides optimization opportunities behind intermediate transformation steps. For instance, vectorization can only be applied to the innermost loop in a nest: one must first perform a loop interchange before even considering vectorization of an outer loop. In contrast, we propose an implementation framework representing programs as sets of possible implementation decisions. Specifying one decision can have an impact on others in a bidirectional manner: specifying that a loop must be vectorized prevents other loops from being nested inside it; conversely, specifying a loop as an outer loop will prevent it from being vectorized. These optimization decisions commute, obviating the pass ordering problem. We present a constraint programming system to formally define, represent and explore such implementation spaces. We also propose an exploration strategy combining tree search and branch-and-bound; the strength and novelty of this strategy reside in an analytical model of the lower bound on the execution time of a set of possible implementations. We showcase our approach on the construction and exploration of an implementation space for linear algebra kernels running on GPUs. We show this search space is expressive enough to represent complex decisions that fundamentally change the structure of the generated code. We also present preliminary results competitive with the performance of native GPU libraries. Introduction General-purpose multicore processors encounter stiff competition from specialized hardware accelerators such as Graphic Processing Units (GPUs). Those accelerators usually offer massive parallelism, providing a raw processing power orders of magnitude above that of general-purpose processors. In some domains, adequately taking advantage of this parallelism can be game changing; for instance, the recent success of deep learning methods can largely be attributed to the performance of GPU-accelerated libraries [21,12]. Unfortunately, writing or generating efficient code for GPUs is a complex problem. It requires making careful decisions such as mapping computations to the appropriate level of parallelism, moving data across the memory hierarchy and through memory spaces of different sizes and properties, in addition to the many possible thread-local optimizations. While the set of potential implementation decisions is usually well known, complex interactions between them make it hard to find the best global implementation. Taking one decision can alter the low-level structure of the code or change the pressure on constrained resources (such as available memory or hardware threads), which can render other decisions invalid or inefficient. These interactions are critical for performance; for instance, parallelizing too many loops can be harmful if the threads end up doing too few computations. The problem of picking a good implementation in an interaction-heavy optimization space is not unique to GPUs and occurs on any architecture complex enough that exhaustive enumeration is impossible. Multiple approaches have thus already been proposed to represent and explore the space of potential implementations, such as SPIRAL [26] for DSP codes, LGen [29] for basic linear algebra, or LIFT [31] for more general computations expressed using combinators. These approaches all share a reliance on rewrite rules to iteratively apply transformations to the code. We argue that rewrite rules make it hard to anticipate possible downstream transformations and to derive profitability information from one point in the search space: while the transformations that can be directly applied are known, further ones may only become available after so-called "enabling" transformations have been applied. Moreover, transformations may not commute and different sequences of transformations may lead to the same result. These factors make it hard to find the best implementation and as a result, the production and the specialization to a given target or problem size of highly tuned libraries remain a critical challenge. • We propose an approach to formally define, represent and explore spaces of implementation candidates with partially instantiated decisions, and we model the interactions between such decisions as a constraint satisfaction problem. • Open decisions are listed explicitly and can be taken in any order, offering a well-behaved optimization space; in particular, the most performance-impacting decisions can be made first. • Precise information can be derived from sub-spaces even before all implementation decisions are made; this includes performance estimates and bounds. • We provide a method and tool to automatically generate efficient code to represent and manipulate partially specified implementations, starting from a high-level description; this tool could be applied to other classes of code optimization problems with different implementation decisions. • We demonstrate that the approach is expressive enough to represent complex decisions that fundamentally change the structure of the generated code, including compositions of stripmining, loop fusion, loop interchange, unrolling, vectorization, mapping to multiple levels parallelism, and orchestrating data movements across the memory hierarchy. • We present preliminary results on the generation of optimized linear algebra kernels for GPUs, competitive with native libraries. Section 2 explains how we represent implementation candidates, Section 3 presents the framework we use to describe potential decisions and their interactions and Section 4 applies this framework to linear algebra on GPUs. Then Section 5 shows its practical benefits and Section 6 compares our work with other approaches. Partially Specified Implementations Our first goal is to build a representation for implementation spaces that is suitable for search algorithms. We build this representation on the notion of implementation candidate-or candidate for short-which is a partially specified implementation. A search algorithm works on implementation candidates, which are further refined until they become fully specified implementations that can be executed. All candidates refer to a common semantic backbone describing the computation to implement. Partially Instantiated Decision Vector We represent implementation candidates as partially instantiated vector of decisions that specify how to implement computations. Decisions are variables of a Constraint Satisfaction Problem (CSP). Each variable has a small set of values it can take, called its domain. A constrained domain contains a single value. The objective of a search algorithm is to find an assignment of variables respecting the constraints by successively restricting the domain of each variable. The decision vector is the product of the domains and thus a Cartesian over-approximation of the set of implementations. Two domains may contain legal values that are incompatible with each other because of some constraint. A search algorithm typically builds a tree whose nodes are candidates. The root is the fully unconstrained candidate. From there, each node is a strictly more constrained version of its parent. Fully constrained nodes have no children and correspond to actual implementations for which the algorithm can generate code and measure execution time. After restricting a domain, constraint propagators must be run to remove incompatible values from other domains. In the general case, some incompatible values may not be detected before domains are further restricted, thus requiring to backtrack. Section 5.2 shows that the constraints we use are simple enough that this is rarely needed in practice compared to usual backtracking algorithms. This approach offers two main advantages. First, domains define a well behaved set of actions the algorithm can take to generate implementations. In particular, the set of available actions decreases when descending in the tree, and the order in which actions are selected does not matter. This allows the algorithm to make the most performance-impacting decisions first. Second, we can define functions on candidates to drive the search. For instance, we can compute an estimate of the execution time or a probability distribution on the decisions to take next. These functions can derive relevant information about the sets of potential implementations described by the domains instead of reasoning on a single implementation at a time. Constraint Satisfaction Problem The decision vector alone is not enough to generate code. It does not specify the semantics of the code to generate nor how to instantiate decisions for different kernels. This information is present in a separate structure describing the computation to implement. We call this structure a semantic backbone; it is mostly constant through the search, although some decisions such as introducing temporary variables can interact in a limited way with the backbone. Decisions can be understood as defining properties on the objects in the backbone: for instance, if the backbone defines a loop, it induces a decision for unrolling the loop. Let us now present the formalism to derive a CSP from a backbone instance. It describes generic choices and constraints to instantiate on each semantic backbone to define the decision vector. For that, we view the backbone as a set of objects that respect different properties. For example, properties can be x is an instruction or x depends on y. While objects vary from one kernel to the other, properties remain the same. They allow to describe the implementation space independently of the backbone instance. Formally, let Ω 0 be a set of identifiers representing basic objects. We define the set of objects Ω as the set of nested tuples of basic objects: for n > 0, we recursively define Ω n+1 = Ω n ∪ Ω * n where Ω * n is the set of tuples of arbitrary length, and Ω = ∪ n∈N Ω n . Note that this definition implies Ω k ⊆ Ω for all k. Let Π be a set of identifiers called properties; a : Π → N denotes their arity. A backbone instance is a function X mapping properties to the objects satisfying it: X : Π → P (Ω) p → X(p) ⊆ Ω a(p) Finally, let D be a set of domains, where each domain is a (distinct) set of identifiers. A generic choice is a tuple c : ((p 0 , . . . p k−1 ), D) ∈ Π * × D. It indicates a decision among the values of D for each tuple of objects verifying the properties. For instance, the decision could be how each loop should be implemented (regular loop, unrolled loop, parallel loop, etc.) or scheduling information for pairs of statements (recall that objects can be tuples of basic objects). We define an implementation space as the combination of a set C ⊆ Π * × D of generic choices, and a set of first-order formulae called constraints. The alphabet for the constraints contains one predicate symbol with arity a(p) for each property p ∈ Π, and one function symbol with arity a(p 0 ) × · · · × a(p k−1 ) for each choice c = ((p 0 , . . . , p k−1 ), D) ∈ C. From there, a partially instanciated decision for a choice c = ((p 0 , . . . , p k−1 ), D) is a function: χ c : X(p 0 ) × · · · × X(p k−1 ) → P (D) . The decision is partial in the sense that it restricts the possible values, but does not necessarily select one. A partially instantiated decision is invalid if its target domain contains the empty set, and is fully instanciated if its target domain only contains singletons. For a given implementation space and a backbone X, a decision vector or implementation candidate (candidate for short) is a vector with one partially instantiated decision for each choice. An implementation is an implementation candidate where • all decisions are fully instantiated; • all the implementation space constraints hold when interpreting the predicate symbol for property p with X(p) and the function symbol for choice c with χ c . This formalism allows us to dynamically extend the implementation space during search. Extending the set of objects respecting properties adds decisions and constraints without invalidating preexisting ones. This is useful to lower specific constructs upon reaching decisions opening for further refinement of the implementation choices. For example, allowing to insert the appropriate load and store instructions after deciding to allocate a temporary array in memory. Lowerings may only add new objects or add properties to existing ones. It is important to note that such lowerings only concern the CSP solver. We pre-compute all possible lowerings ahead of time for search heuristics. Lowerings allow to delay introducing variables and avoid cluttering the CSP with decisions and constraints that may not affect the generated code. The same properties can always be encoded with regular constraints instead, ensuring this does not contradict our claim that all potential decisions can be available upfront. Implementation Space Description We now present the details of the search space definition framework following the formalism from Section 2.2. It exposes a high-level language to define the different properties exposed by the semantic backbone (Section 3.1), the decisions (Section 3.2) and the constraints (Section 3.3). It also specifies when to add new objects to the semantic backbone (Section 3.4). A compiler reads this language and generates code to create, represent and manipulate the partially instantiated vector of decisions. In particular, it generates code to propagate the constraints, inserts new objects in the backbone when needed and extend the CSP with the corresponding new choices and constraints. This framework allowed us to experiment with multiple encodings for linear algebra kernels on GPUs with limited effort. The same framework can easily be reused for different application domains or hardware targets. Interface With the Kernel Representation The first step to specify an implementation space is to define sets of objects exposed by the semantic backbone. Each set corresponds to a property p ∈ Π and is instantiated into X(p) for each kernel instance X. Definitions indicates both the relation of the set to others and how to generate code that manipulates its objects. This way, we can easily apply our approach to kernel representations. Exposing a new property requires less than 10 lines of code. For example, Listing 1 defines a set Instructions that contains objects of type Instruction in the host language. The programmer specifies fields to indicate how to iterate on the set and to retrieve its objects. Programmers can express inclusion and disjointness relations between sets. They allow to declare a set structure that matches the subtyping relation in the host language. For instance, MemAccesses is a subset of Instructions and contains objects of type MemAccess, a subtype of Instruction. forall $b in Instructions : forall $c in Instructions : order ($a, $c) is AFTER \|\| order ($a, $b) is not AFTER \|\| order ($b, $c) is not AFTER Last we can parametrize sets with another set. For example, Listing 1 defines an AccessedRegions set for each memory access. This translates to a binary property containing tuples of AccessedRegions x MemAccess in our formalism. Choice Definition Next we define the set of generic choices C. Each c ∈ C has a list of sets corresponding to properties p 0 , . . . p k−1 ∈ Π and a domain D c ∈ D. When instantiated for a particular kernel X, the choice defines a function χ c from tuples of distinct objects in X(p 0 ) × X(p k−1 ) to D c . For example, Listing 2 defines a function cache from memory accesses to cache directives: cache : MemAccesses → {L1, L2, READ ONLY, NONE} This defines a CSP variable for each memory access exposed by the semantic backbone. The goal of the implementation space exploration is then to assign a value v ∈ D c to χ c (x 0 , . . . x n−1 ) for each (x 0 , . . . x n−1 ) ∈ X(p 0 ) × · · · × X (p n−1 ) with x 0 , . . . x n−1 distinct. In our example, this is assigning a cache directive to each memory access. Another example, Listing 3 defines a generic choice or der from pairs of instructions to orderings. It first illustrates why we only consider tuples of distinct objects: it is the behavior expected by the programmer when defining choices. The order between an instruction and itself would not make sense. It also illustrates how we define antisymmetric choices, allowing the code generator to generate stronger propagators and only store half of the domains of order variables. We currently support two kinds of domains: • enum choices, such as cache in Listing 2, that can take a small set of predefined values and • integer choices, that take values in a small universe specific to each instance of the choice. The choice definition includes a piece of code that retrieves the universe from the kernel representation. Our tool automatically generates types to represent domains and code to instantiate the generic choices for a semantic backbone instance. Constraints Next, we express constraints on variables domains to avoid invalid implementations. We use constraints to: • avoid incoherent decisions, e.g., non-transitive ordering of instructions; • enforce correctness constraints imposed by the semantic backbone, e.g., data dependencies; • and respect hardware-specific limitations, e.g., the size of the memory blocks allocated to a memory space cannot exceed the size of that memory space. Constraints are first order logic sentences on the choices, quantified over objects in specific sets. Currently, they are universally quantified disjunctions of conditions on zero, one or two choices: boolean constants, restrictions of a variable to specific values or comparisons between two variables. For example, Listing 4 ensure the order choice defined in Listing 3 is transitive with a constraint applied to all triples of distinct instructions a, b and c. Constraints implicitly assume quantified objects are distinct. This makes it easier to write concise constraints and matches the intended behavior in most cases. Our tool automatically inserts clauses to enforce this in the generated code. Constants can be pieces of code parametrized by the objects. This enables us to parametrize constraints at a fine grain without having to declare new sets. In particular, it allows to: • parametrize bounds on integer decisions size ($mem) <= "$mem. max_size " • selectively disable constraints for some objects: "$inst. is _v e ct or iz a bl e " \|\| < constraint > Our tool generates code to propagate constraints. It restricts domains to remove incompatible decisions at the initialization of the implementation space and after restricting any domain. The simple form of our constraints allow to easily generate efficient propagation code. For fixed object variables, a constraint only reference a few choices. This limits the amount of propagation to perform after each update. Often, hardware limitations impose constraints on sums or products of quantities, such as the sum of the memory region sizes (to ensure they fit in memory) or the product of the size of nested thread dimensions (to ensure we do not exceed the maximal number of threads). Simple constraints cannot express such values. Instead, one can define counters that track sums or products of values. For example, Listing 5 keeps track of the local memory usage. While remaining simple, our constraints are expressive enough to encode correct implementations. Because the propagation code is automatically generated, it is easy to experiment with and encode a wide range of optimization decisions. Kernel Representation Lowering Finally, we define triggers to lower backbone constructs when certain conditions are met. Triggers run a callback that either adds properties to existing objects or creates new objects. For example, it can add the property is stored in memory to a variable, creating new decisions to specify the memory layout. At the same time, the trigger can also add a store and a load instruction to refine placement decisions in the memory hierarcy. Callbacks may only add objects to properties, not remove them so that previous variables and constraints remain valid. Moreover, we require that callbacks commute. It is important to understand that triggers are not mandatory: one can add objects to properties upfront and condition decisions and constraints with the lowering condition. In our example, cache directives would be forced to None in candidates where memory accesses do not exist. The reasons to use triggers are: • to avoid cluttering the decision vector with decisions that will not affect the generated code, making constraint propagation more efficient; • and intentionally delay some decisions to focus the search algorithms on more meaningful choices first. As with constraints, we define lowering conditions as first order sentences, with universally quantified object variables. This allows definitions to be independent of backbone instances. Triggers can be used to define quotient sets: we often end-up needing to account for classes of equivalent objects such as fused loops. Quotient sets contain one representative for each class of objects respecting some condition. They are automatically maintained by our tool using triggers. For example, in Listing 6 we count the number of threads. This is done with a quotient set containing one representative for each class of fused iteration dimensions mapped to a hardware thread. This also define a boolean choice, is_thread_dim indicating which dimensions are the representative of their class (this is redundant, but helps when writing constraints). Overall, we built a simple, domain-specific constraint programming framework that facilitates the construction of search spaces, involving new choices and constraints. The framework also eases porting our approach to different architectures or application domains. Because we abstract the kernel representation as sets of objects, the definition is independent of the kernel instance. We could probably implement our tool on top of a standalone CSP framework; apart from the need for a domain-specific front-end to provide the above-mentioned kernel genericity, this would involve several optimizations and customizations to reproduce the search strategy and match the efficiency of our approach, motivating a custom design. In particular, we use the fact that the CSP definition is independent of the backbone to avoid storing constraints instances in memory. Instead we statically generate code that we call for different objects. This reduces the memory footprint of the CSP to the sole decision vector, making it compact, easy to serialize for logging and fast to copy. Partial Implementations for GPUs This section builds on the framework presented in Section 3 representing partial implementations for linear algebra kernels on GPU. It serves both as an illustration of our approach and as an effective way to generate competitive code on GPU. Since this representation targets dense linear algebra, we assume all loops are for-loops with dataindependent bounds (i.e., loop bounds only depend on the input size) and affine memory accesses to multidimensional tensors. We focus on optimizations that work well on GPUs. Other representations can be developped with similar ideas using the framework presented in Section 3. We highlight the set and choice definitions and omit counters and constraints for the sake of brevity. We highlight the main difficulties and original aspects of code generation, deferring more systematic coverage to a later, dedicated paper. Statements We represent computations to perform as a list of instructions to execute. Each instruction performs a single scalar operation: an arithmetic operation (addition, multiplication, cast, . . . ) or a memory access (load or store). We then specify how many time instructions are executed by nesting them into iteration dimensions. Iteration dimensions are akin to counted for-loops but may be implemented differently. Depending on decisions, they can be unrolled, vectorized or parallelized. Table 1: Outer Product of Two Vectors Kernel Table 1 shows an example of kernel that computes the outer product of two vectors A and B of sizes m and n and stores the result in a matrix C of size m × n. This kernel has two iteration dimensions d m and d n of respective size m and n and current index i m and i n . Note that this representation does not imply a particular order of instructions or nesting of dimensions. Instructions Dimensions a = load A[i_m] {d m } b = load B[i_n] {d n } c = a * b {d m , d n } store C[i_m * n + i_n] <-c {d m , d n } Listing 7 shows the definition of two sets listing the instructions and the dimensions. Together, they form the statements. The order choice dictates the control flow structure. It simultaneously encodes statements sequential ordering (BEFORE and AFTER), nesting (INNER and OUTER), and loops fusion (MERGED). In particular, it can: • move code inside or outside of loops, for example by setting order($stmt, $dim) to BEFORE or INNER, • interchange loops, by setting their order to INNER or OUTER. • fuse loops by setting their order to MERGED. • schedule loops and instructions by setting the order between two statements to BEFORE or AFTER. The order choice demonstrates the first benefit of our approach: the compiler is not limited to a set of predefined high-level transformations. It does not even have to be aware of them. Instead we expose many smaller decisions (here, the pairwise ordering between statements). High level transformations are implemented as specific combinations of decisions. We are free to pick any other assignment that respects the constraints. This approach makes it easier to understand the interaction between transformations (as decisions) and to combine them since they are all exposed in the same framework. Listing 7 also defines a quotient set IterationDims that contains the iteration dimensions nested outside each instruction. At the creation of the domain, it specifies the nesting imposed by the backbone. Afterward, new dimensions may be added when order gets constrained. It is grouped into Together with order, IterationDims demonstrates another benefit of our approach: the structure of the code does not needs to be coherent before the candidate is fully constrained. For example, in Table 1 Iteration Dimensions The dim_kind choice, shown in Listing 9, specifies how to implement dimensions. They can be parallel (at the BLOCK or THREAD levels), fully unrolled or imply vector instructions. We usually strip-mine dimensions of the original code into multiple ones when creating the search space. This allows both to apply different dim_kind decisions to the resulting dimensions and to tile computations to improve locality. Listing 10: Encoding of Strip-Mining and Tiling Listing 10 shows how we encode strip-mining. We split dimensions in two categories: StaticDims that have a statically known size, specified by the size choice, and dynamic dimensions (not exposed in a set) whose size depend on input parameters. Logical dimensions represent dimensions of the original code. They are formed of static dimensions, listed in TilingDims($logical_dim) and zero or one dynamic dimensions, listed in TiledDim($logical_dim), depending on whether the logical dimension has a static or dynamic size. Memory Accesses Listing 11 shows how we encode cache directives and memory placement. MemRegions are distinct pieces of memory in which we allocate arrays. While memory regions holding input arrays are always in RAM, others regions may also be placed in shared memory, local to a group of threads. Listing 11: Encoding Memory Placement Decisions MemInsts lists loads and stores and AccessedRegions the memory regions accessed by such instructions. cache indicates which caches to use (see Listing 2 for the full definition). Instruction Operands Instruction operands are either constants, kernel inputs, induction variables or values produced by instructions. Induction variables are linear combination of dimension indexes. We use them to compute memory accesses addresses. We handle them separately from regular instructions as it is clear how they should be implemented, and exposing them in the search space would only increase complexity. By default, operands can only take the last value produced by preceding instructions. To implement reductions, an instruction operand can also take the value produced by the same instruction at the previous iteration on a given set of dimensions. In that case, we specify another instruction to initialize the reduction. Additionally, we can specify point-to-point communication between dimensions: the value produced at iteration i of a dimension is consumed at iteration i of another dimension. For example, in Listing 12, the instruction nested in d0_b and d1_b reads the values produced in d0_a and d1_a. Point-to-point communications allows putting each instruction in its own loop nest. This is useful as it gives us more flexibility to schedule, fuse and implement dimensions. Point-to-point communication is either implemented by: • fusing both dimensions; • unrolling or vectorizing both dimensions, with the data stored in different registers for each iteration; • mapping the two dimensions to the same hardware thread dimension, as explained in Section 4.5; • or using a temporary array. A trigger automatically creates a temporary array and the associated load and store when other options are impossible. In particular, this allows copying data to a temporary array in a faster memory for improving locality. For example Listing 12 shows a possible implementation that stores chunks of A in an array TMP. Thread Mapping GPUs expose two levels of parallelism: threads and blocks of threads. Thread dimensions define a block of threads that share a fast memory and can synchronise. Block dimensions replicate blocks in parallel. Each block or thread dimension can be mapped to one of three hardware dimensions, hereafter referred to as levels to distinguish them from iteration dimensions. Blocks cannot easily communicate so block dimensions are outermost parallel dimensions that span the entire computation. On the other hand, synchronization within a thread block is critical to achieve good performance. We encode synchronisation barriers by creating multiple loop nests of thread dimensions that maps to the same hardware level. At code generation, we fuse the loop nests and insert a synchronisation instruction between them. In particular, this allows point-to-point communications between thread dimensions mapped to the same level. Listing 13: Thread Dimensions Mapping The thread-mapping choice in Listing 13 specifies how thread dimensions map to hardware levels. The nesting order of thread dimension is crucial as it determines memory coalescing. The number of threads may vary between thread nests: they might not map to the same levels. In that case, we use predicated instruction to disable some threads. Experiments Let us now evaluate our approach on concrete search and code generation problems. We implemented a search strategy that combines a statistical component with a performance model of implementation candidates (Section 5.1). Our goal is to show that: • we can generate code competitive with reference hand-tuned libraries (Section 5.2); • the formalism and performance model allows to extract pertinent information before specifying decisions (Section 5.4); • and we improve search performance by making decisions commute (Section 5.5). All experiments are run on a Linux machine equipped with a 12-core Xeon E5-2620v2 processor, 64GB of RAM and a Quadro K4000 GPU Kepler GPU running under CUDA 8.0. Search Strategy The search space exploration is driven by a Monte-Carlo Tree Search (MCTS) algorithm. We use a variant of Threshold Ascent on Graph (TAG), which was previously applied to Spiral [13], a code generator for fast Fourier transforms. In our case, the algorithm builds a tree whose nodes are candidates. It starts with only the root node, representing the full search space. It iteratively selects a leaf to expand by using the TAG formula and evaluation statistics from previous iterations. The leaf expansion creates one child per possible value of the decision. Then, the MCTS performs a Monte-Carlo simulation to set remaining decisions. It runs the resulting implementation on the GPU and adjusts the statistics along the selected path with the execution time. The order in which decisions are taken is fixed upfront. We manually select an order that specifies most important decisions first. Section 5.5 discusses the impact of the order on exploration performance. We complement the TAG algorithm with a performance model of the candidates [2]. The model provides a lower bound on the execution time of all implementations derivable from a candidate. We use the bound in two ways: • In the selection phase, we ignore children which have a lower bound higher than the execution time of the current best implementation. • During Monte-Carlo simulations, when choosing amongst candidates (X i ) i , we pick X i with probability: p(X i ) ∼ max(T − b(X i ), 0) where T is the execution time of the best implementation so far and b(X i ) is the lower bound given by the performance model to the candidate X i . Note that p(X i ) = 0 when b(X i ) > T. In both cases, the performance model prunes regions of the search space which cannot possibly improve on the current best implementation found. We show in Section 5.4 that this can eliminate large portions of the search space. Since this paper focuses on the definition, representation and exploration of the optimization space, we defer deeper treatment and analysis of the seartch strategy-involving MCTS and performance modeling-to a later paper. Generated Code Performance We first show the code we generate compares to hand-tuned reference implementations. We created implementation spaces for a few kernels and compare the execution time of the best implementation found by our exploration strategy with the reference implementation. We created all the implementation spaces using a similar procedure. Every instruction is placed in its own loop nest, with point to point communication between them. We also select a few stripmining factors dividing the input size, for each dimension. The search algorithm is free to reorder, fuse, unroll or vectorize loops or to map them to the different levels of parallelism. It can implement point-to-point communications using registers or by allocating temporary arrays in shared memory, and chooses the level of cache to use for each memory accesses. We consider the following kernels. Unless specified otherwise, the reference implementation calls CuBLAS, Nvidia's hand-tuned implementation of basic linear algebra kernels. Matrices are column major order. axpy : computes z = α.x + y, where α is a scalar and x, y and z vectors of size n = 2 26 . We strip-mine n twice, with factors in 2, 4 and 2, 1024 . matmul : computes C = A · B when A and B are matrices of respective size m × k and k × n. We strip-mine m and n twice, with factors in 2, 32 and 2, 4 . We try different values for m, n and k to show how the algorithm adapts. In the rest of the paper we refer to them as matmul m × n × k. strided matmul : is the same a matmul 1024 × 1024 × 1024 but with consecutive elements of A stored with stride of 32. CuBLAS does not support such strides. The reference is a naive implementation that computes one element of C per thread. Table 2: Implementation Space Exploration Results The benchmarks are representative of both compute-intensive (mm) and bandwidth-bound (axpy) kernels. They also span a variety of memory access patterns, including strides at different dimensions, transposed layouts, all of these being typical of higher dimensional tensor algebra in computational chemistry, simulation codes, and machine learning [1]. Table 2 summarises the characteristics of implementation spaces and explorations results. We ran 4 explorations of 4 hours on each kernel. For each exploration, we evaluated the best implementation 40 times. The 95% confidence interval on the average speedup was always within ±0.5%. We report the average runtime and speedup among explorations, along with the maximum variation of the speedup compared to the average among exploration. The Space Size and Dead Ends columns respectively provide estimations of the size of the search space (see Section 5.3) and of the probability to encounter a dead-end when randomly walking the tree from the root with a uniform sampling among the valid children of each node. Dead ends occur when constraints propagation is unable to detect incompatible decisions before more choices are specified. Both columns also indicate the 95% confidence intervals. For all experiments, the ratio of dead-ends is inferior to a third. This shows that finding valid implementations is easy. We only use the CSP formalism to encode the search space and not to search for valid implementations. axpy and matmul 256×256×32 show that our approach is able to outperform hand-tuned implementations by finding better implementation decisions. matmul 1024 × 1024 × 1024 compares against an implementation that almost reaches peak performance. We have no chance of beating it as it relies on features we do not support. In particular, it uses texture memory and manual allocation of registers to avoid bank conflicts [22], which is impossible with public Nvidia APIs. However, we still achieve reasonable performance. strided matmul shows the benefits of our approach for kernels not available in hand tuned libraries, with a 66× speedup over a naive implementation. While not reaching peak performance, it is within a factor 2.7 of CuBLAS non-strided version and thus can be useful. The high variance on the execution time of the best implementation among explorations on the same space is a real issue. We are working on developing better search algorithms and refining the performance model to mitigate the problem. This is to put in perspective with the large size of implementation spaces and with the fact that this paper is not about search algorithms themselves but about how to expose the implementation space to them. Implementation Space Size Estimation Computing the exact size of the search spaces is intractable due to the large number of possible choices. To estimate their size, we turn to probabilistic methods which have been used to reliably estimate the size of search trees in constraint solvers [19]. The precise method we use was described by Chen [7], which is a generalization of an earlier method by Knuth [20]. The Knuth algorithm starts at the root and performs a random descent until reaching a leaf. It then estimates the size of the tree by using the branching factor along the path. The Chen algorithm, called heuristic sampling, adds the concept of strata: classes of subtrees estimated by the algorithm designer to be structurally similar. The algorithm maintains a queue containing strata, represented by a single subtree in the stratum, along with an estimate its size. When a subtree is discovered, the corresponding stratum in the queue is updated with its count, and it can randomly be selected as the new representative for the stratum. The total size estimate is then the sum of estimates for the leaves encountered. A partial order on the strata which is strictly decreasing along the tree is required to ensure well-formedness. This algorithm was chosen as it strikes a balance between simplicity and performance. We use a lexicographic pair of the depth in the tree and number of remaining choices (assuming none get forced through propagation-this provides a simple proxy for the subtree size to guide the algorithm) as a stratifier. We computed the results in Tables 2 and 3 with either 1000 iterations of the Chen method or 100, 000 iterations of the Knuth method, whichever gave a better confidence interval. Each time, we report the 95% confidence interval. For some of the larger implementation spaces, we perform additional iterations to bring the confidence interval to the correct order of magnitude. Discriminant Information in Candidates We use the size estimates to justify that our representation allows extracting pertinent information early, with only a few decisions set. Table 3 reports the estimated size of the search space after pruning the first 10 levels with the performance model. We cut branches whose lower bound was higher than the average runtime obtained from the experiments of Section 5.2. Note that we use the execution time without the kernel launch time (obtained using performance counters) instead of the one reported in Table 2 for pruning. We also computed through exhaustive enumeration the number of nodes at depth 10 with and without cut. These experiments show that we are able to eliminate a large portions of search spaces early on. For most kernels, we reduce the size of the space by two or three orders of magnitudes by cutting on the first 10 levels. 3.5e21 ± 1.8e21 9.5e17 ± 4.6e17 161, 980 6, 738 (4.2%) strided matmul 6.0e20 ± 2.0e20 8.1e17 ± 4.5e17 142,780 9, 512 (6.62%) Table 3: Implementation Space Exploration Results Impact of the Decisions Order One core feature of our approach is that decisions commute. Here we show how this helps search algorithms. Experiments in Sections 5.2 and 5.4 first specify memory layout decisions, then size, then dim kind, then thread mapping, then mem space, then order and last cache. We selected this order to prioritise what we think are most important decisions. This allows both the performance model and the MCTS to discriminate higher in the tree and to focus on a fewer branches. This does not impact implementations, only the structure of the search tree. We ran experiments on matmul 1024 × 1024 × 1024 to compare this order with the reverse order. In both case, we computed the ratio of nodes that the performance model can prune in the first T levels of the tree, assuming the cut threshold is the average runtime reported in Table 2. The number of candidates with a depth ≤ d varies greatly with the order of decisions. To have comparable results, we took for each order the first d such that the number of candidates of depth d is above 10 5 . This resulted in d = 10 with 1.6e5 for the direct order and d = 16 with 1.1e5 nodes for the reverse. With the direct order, the performance model reduces the tree size by a factor of 24. This factor falls down to 1.8 for the reverse order. This is 13 times more branches to consider. We also tried running the search algorithm using the reverse order but it ran out of memory due to an explosion in the number of branches. While we illustrated the impact of the decisions order with the performance model, it also applies to other algorithms: picking the most discriminant decisions first helps focusing the search. Discussion and Related Work Traditional optimizers work on a single implementation that they iteratively improve using rewrite rules based on pattern matching. In constrast, we propose to work on classes of implementations modulo optimization. We first discuss how it obviates the problem of optimization ordering and enable global heuristics aware of all potential optimizations. Then we explain how our system differ from other encodings of compilation problems in logical frameworks. Partial Implementations Ordering decisions is a common problem in compilers: transformations may enable, disable or even undo others. Some domain-specific approaches avoid the issue with a carefully curated set of rules. They then build and explore a tree (or graph) whose nodes are implementations and edges rules applications. For example, Spiral [26] uses this approach for fast Fourier transforms, LGen [29] for linear algebra kernels and TCE [1] for large tensor contractions in computational chemistry and physics simulation. More recently, LIFT [31] applied rules to rewrite a functional representation down to low-level OpenCL constructs and generate efficient GPU code. However, these systems still suffer from the original problem: transformations may be hidden behind others. In contrast, our representation allows to see from the start which decisions are available in which branch and to make most important decisions first. An expert programmer can even manually set decisions upfront. An alternative to rewrite rules is to use algorithmic skeletons [10] and to map them to the hardware using a fixed strategy that leverages domain specific information. In its simplest form, this is just parametric libraries such as Thrust [3] and SkelCL [30]. Otherwise, it takes the form of high level functional operators in a domain specific language such as Delite [32], Copperhead [5], Accelerate [6] or NOVA [11]. While theses systems allow for optimizations, such as the fusion of operators, they rely on a fixed strategy that makes it hard to adapt to different hardware, different input sizes of new kind of computations. The Sea of Nodes approach [9] places instructions outside of basic blocks when possible, effectively representing classes of implementations modulo scheduling. However, this is limited to scheduling decision. Global Heuristics The partially instantiated vector offers a complete view of potential implementations. This allows to define heuristics aware of what a fully optimized implementation look like. The lower bound performance model mentioned in Section 5 could not work if it just had access to an intermediate implementation in the compilation process. A similar performance model relying on ad-hoc partial implementations was previously introduced [2]. We generalize the idea by encoding partial implementations as a CSP problem on top of a semantic backbone. Our approach is close to Equality Saturation [33], with similar benefits. Equality Saturation uses rewrite rules but keeps both the original and rewritten pattern. However the number of patterns can grow exponentially with the number of rules. In contrast, our decision vector has a fixed size. A fixed size vector also makes it easier to extract features for machine learning algorithms. Frameworks that dissociate the algorithm from the schedule, such as Halide [27] for image processing and TVM [8] for machine learning arguably also deal with partial implementations. The algorithm is akin to our semantics backbone and the schedule to our decision vector. However, they do not have an easy way to reason about partial schedules. TVM applies machine learning techniques, but only on fully specified implementations. An interesting idea would be to use our approach on top of their representation to explore schedules. This is also true for other script-based transformation tools such as UTF [18] or URUK [17] that start from a fully specified implementation but lift it into a mathematical representation that abstracts the initial schedule. Encoding as an Operation Research Problem The idea of encoding compilation problems in logical frameworks is not new. Polyhedral compilation encodes transformations on loop nests as Integer Linear Programming [15,16] and PPCG [35] applies it to generate code for GPUs. Super-optimization techniques also use CSP [23] and SAT [28] solvers to generate optimal sequences of instructions. The originality of our approach is to use CSP to expose potential decisions, not to find a solution (Section 5.2 shows that this is easy in our case). This allows us to manipulate whole sets of potential implementations, and is a core reason for using CSP: it is easier to guide the search through the domains, while SAT and ILP solvers often act more like black boxes. ILP-based approaches maximize a single metric (such as data locality in polyhedral schedules [4]) that does not reflect the full complexity of the architecture. Because we do not try to embed performance constraints in the logical framework, we have much more flexibility and can use a combination of custom heuristics, actual evaluations and statistical search. One way of solving this problem while staying in an ILP framework is to find schedules for a single loop level at a time, starting from the outermost [4]. This enables more complex heuristics and incremental evaluation at each level that goes beyond the expressive power of linear objective functions [36]. However, loop levels must still be considered in a fixed order. It would be interesting to use our approach with a similar encoding. Polyhedral compilers have been designed with search algorithms complementing or replacing linear programming. Pouchet et al. proposed custom genetic operators on affine schedules to search for dependence-preserving loop nest optimizations [24,25]. Vasilache et al. constructed a two-level optimization strategy where the lower tier is a gray box based on integer linear programming, exposing a fixed set of strategies and parameters to a higher tier genetic algorithm [34]. Diesel [14] is a recent framework from Nvidia instantiating an ILP-based scheduler for a specific domain, specializing it for each kernel with parameters specific to a GPU micro-architecture (tile sizes, mapping strategies), complemented with target-specific transformations (e.g., software pipelining, instruction-level and register-level optimizations). Code generated by Diesel reaches impressive performance levels, matching or outperforming native libraries. This involves deep knowledge of the target architecture encoded in the optimization strategy, and register-level optimizations not currently modeled in our search space. Overall, our domain-specific language for defining decisions and constraints proved useful for developing an implementation space for linear algebra, facilitating the design and experiments with numerous decisions and constraints. We are not aware of any other approach reaching the same level of automation while remaining competitive with native GPU libraries. Conclusion We presented a generic approach for program optimization based on partially specified implementations. In our approach, optimization decisions are available upfront and can be applied in any order, obviating the problem of optimization ordering. The most performance-impacting decisions can be made first, and can be specified manually through expert knowledge when available. At the core of our representation is a decision vector listing all possible decisions for each choice. Global heuristics can be defined which operate on sets of possible implementations. They can derive pertinent information, such as profitability estimates to guide the search and performance bounds to prune the space, long before all decisions have been made. Starting from a high level description of the implementation choices and their interaction, we built a tool leveraging constraint programming principles to derive efficient code for the manipulation of decision vectors. We applied this tool to the construction of a partial implementation search space for linear algebra kernels running on GPUs, generating code competitive with or outperforming carefully hand-tuned reference libraries. So far, we limited ourselves to relatively simple search algorithms with a few remaining hardwired decisions. For example, our representation allows to search for the best decision order, and we are working on more complex algorithms that fully exploit this potential. It also enables easily sharing information between branches of the search tree as choices have fixed positions in the decision vector, opening up more opportunities to prune the search space. , load A is nested in d m , load B in d n and c = a * b in d m and d n . As shown in Listing 8, such a loop structure is impossible: either d n and load B are nested inside d m or d m and load A inside d n . Here it is legal because ordering decisions are still open. We can delay choosing the nesting while a regular control flow representation would need to pick one. Listing 3: Choice Definition for Ordering Decisionsrequire forall $a in Instructions :set Instructions : type = " Instruction " iterator = ... item_getter = ... ... // Elided for brevity end set MemAccesses subsetof Instructions : type = " MemAccess " ... // Elided for brevity end set A cc es s ed Re gi o ns ($inst in MemAccess ): ... // Elided for brevity end Listing 1: Set Definitions choice enum cache ($inst in MemAccesses ): value L1 : // Use L1 + L2 caches value L2 : // Use L2 cache value READ_ONLY : // Use read -only cache value NONE : // Do not use caches end Listing 2: Choice Definition for Cache Directives choice enum order ($lhs in Instructions , $rhs in Instructions ): value BEFORE : value AFTER : antisymmetric : BEFORE -> AFTER end Listing 7: Control-Flow Related Defintions equivalence classes by the order is MERGED relation so that fused dimensions are only accounted for once.set Statements : ... set Insts subsetof Statements : ... set Dimensions subsetof Statements : ... choice enum order ($lhs in Statements , $rhs in Statements ): // $lhs is executed before $rhs value BEFORE : // $rhs is executed before $lhs value AFTER : // $lrhs is nested inside $rhs value INNER : // $rhs is nested inside $lhs value OUTER : // $lhs and $rhs are fused dimensions value MERGED : antisymmetric : BEFORE -> AFTER INNER -> OUTER end quotient IterationDims ($inst in Insts ) of $dim Dimensions : is_outer_dim = order ($dim, $inst) is OUTER / order is MERGED ... end choice enum cache ($inst in MemInsts ): ...set MemRegions : ... set MemInsts subsetof Insts : ... set A cc es s ed Re gi o ns ($mem in MemInsts ): ... choice enum mem_space ($mem in MemRegions ): // $mem is stored in RAM . value GLOBAL : // $mem is stored in the memory shared // by a group of threads . value SHARED : end Synthesis of high-performance parallel programs for a class of ab initio quantum chemistry models. 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[]	[ "Jitendra Bajpai " ]	[]	[]	We study the examples mentioned in [2, Tables A& C] and establish the arithmeticity of four examples of symplectic hypergeometric groups of degree six. Following [2] we know that there are 458 inequivalent symplectic hypergeometric groups of degree six, and combining the results of this article with the work of [1, 2, 11], we now know that at least 384 are arithmetic and at least 63 are thin whereas the arithmeticity and thinness of remaining 11 examples are still unknown.	null	[ "https://arxiv.org/pdf/2203.05529v1.pdf" ]	247,362,908	2203.05529	2f80af960c3824559be13e705b763ada445bdca3	10 Mar 2022 Jitendra Bajpai 10 Mar 2022arXiv:2203.05529v1 [math.GR] COMMENTARY ON Sp(6) HYPERGEOMETRIC GROUPS We study the examples mentioned in [2, Tables A& C] and establish the arithmeticity of four examples of symplectic hypergeometric groups of degree six. Following [2] we know that there are 458 inequivalent symplectic hypergeometric groups of degree six, and combining the results of this article with the work of [1, 2, 11], we now know that at least 384 are arithmetic and at least 63 are thin whereas the arithmeticity and thinness of remaining 11 examples are still unknown. Introduction A hypergeometric differential equation of order n is an ordinary differential equation of order n with three regular singular points. It is defined on the thrice punctured Riemann sphere P 1 (C)\{0, 1, ∞}. Let θ = z d dz and the parameters α = (α 1 , . . . , α n ), β = (β 1 , . . . , β n ) ∈ C n . We define the hypergeometric differential equation of order n by (1.1) [z(θ + α 1 ) · · · (θ + α n ) − (θ + β 1 − 1) · · · (θ + β n − 1)]u(z) = 0 . This has n linearly independent solutions which can be explicitly expressed as hypergeometric functions of type n F n−1 around any point z ∈ P 1 (C)\{0, 1, ∞}. For α and β, we define n F n−1 (α 1 , . . . , α n ; β 1 , . . . , β n−1 \|z) = ∞ k=0 (α 1 ) k . . . (α n ) k (β 1 ) k . . . (β n−1 ) k the hypergeometric group corresponding to the parameters α, β ∈ C n is the subgroup of GL n (C) generated by the companion matrices A and B of the polynomials f (x) = n j=1 (x − e 2πiα j ), g(x) = n j=1 (x − e 2πiβ j ) respectively, and the monodromy is defined by (1.2) g ∞ → A, g 0 → B −1 , g 1 → A −1 B, where g 0 , g 1 , g ∞ are, respectively, the loops around 0, 1, ∞, which generate the fundamental group of P 1 (C)\{0, 1, ∞} modulo the relation g ∞ g 1 g 0 = 1. Any other hypergeometric group having the same parameters is a conjugate of this one. Note that the condition α j − β k / ∈ Z for all 1 ≤ j, k ≤ n ensures that the polynomials f and g do not have any common root. Let Γ(f, g) denote the hypergeometric group generated by the companion matrices of the polynomials f, g; as said above, it is a subgroup of GL n (C). We consider the cases where the coefficients of f, g are integers with f (0) = ±1, g(0) = ±1 (for example, take f, g as products of cyclotomic polynomials); in these cases, Γ(f, g) ⊂ GL n (Z). In addition, we assume that f, g form a primitive pair [3,Definition 5.1], are self-reciprocal and do not have any common root. Beukers and Heckman [3,Theorem 6.5] have determined the Zariski closures G of the hypergeometric groups Γ(f, g), which can be described as follows. • If n is even and f (0) = g(0) = 1, then the hypergeometric group Γ(f, g) preserves a non-degenerate integral symplectic form Ω on Z n and Γ(f, g) ⊂ Sp Ω (Z) is Zariski dense, that is, G = Sp Ω . • If Γ(f, g) is infinite and f (0) g(0) = −1, then Γ(f, g) preserves a non-degenerate integral quadratic form Q on Z n and Γ(f, g) ⊂ O Q (Z) is Zariski dense, that is, G = O Q . • It follows from [3,Corollary 4.7] that Γ(f, g) is finite if and only if either α 1 < β 1 < α 2 < β 2 < · · · < α n < β n or β 1 < α 1 < β 2 < α 2 < · · · < β n < α n . In this case, we say that the roots of f and g interlace on the unit circle. Definition 1. We call a hypergeometric group Γ(f, g) ⊆ G(Z) arithmetic if it is of finite index in G(Z), and thin if it has infinite index in G(Z), where G is the Zariski closure of Γ(f, g) inside GL n (C). We take this opportunity to remind the readers that this is not the most general definition of an arithmetic group. However, the groups Γ(f, g) under consideration are simply subgroups of GL n (Z) and therefore the above definition to define an arithmetic group is quite natural. For quick introduction on the theory of arithmetic groups we refer the interested reader to [4]. Sarnak's question [8] about classifying the pairs of polynomials f, g for which the associated hypergeometric group Γ(f, g) is arithmetic or thin, has witnessed many interesting developments over the past 10 years, and this article, with no exception, is a small addition to these developments. More precisely, this article is an extension of the work carried out by the author and his collaborators in [2] and [1] about determining the arithmeticity and thinness of symplectic hypergeometric groups of degree six. In [2], the authors showed that there are in total 458 pairs of polynomials f, g (up to scalar shifts) which define symplectic hypergeometric groups Γ(f, g) of degree six. For 211 of them, the absolute value of the leading coefficients of the difference polynomials f − g are at most 2 and therefore the arithmeticity of these 211 examples simply follows from Theorem 1.1 of [11]. The arithmeticity of 164 examples follows from Theorems 2 and 3 of [2]. Note that among these 164 arithmetic examples, arithmeticity of one group, that is the example [2, Table B-145], was proved in [6] by computing explicitly the index of this particular group inside Sp 6 (Z). In [2], the arithmeticity of these 164 examples were shown by finding an element γ satisfying the hypotheses of Proposition 1 from [2, Page 260]. Further, in [1] the authors manage to show the arithmeticity of 5 more examples by finding an element γ satisfying the hypotheses of this Proposition 1 in [2]. Moreover, the authors also proved the thinness of 63 examples in [1] by using a version of the well-known "pingpong lemma" from geometric group theory. At this stage, there were in total 15 examples left whose arithmeticity and thinness were still unknown. For these 15 remaining cases, in [1], the authors also pointed out that by all means our approach to play ping-pong will remain inconclusive. Moreover, there are 6 examples where the arithmeticity can not be concluded by Prop. 1 of [2], since it will be impossible to find an element γ which satisfies the hypotheses of Prop. 1. This is due to the gcd of the coordinates of vector v = (A −1 B − I)e 6 , where e 6 = (0, 0, 0, 0, 0, 1), which is larger than 2 in these 6 cases, see the entries of the last column in Tables 1 and 2. These 6 cases are labelled as A-1, C-1, C-10, C-42, C-59, C-61 in the Tables 1 and 2 below, to which we focused on in this article to try to prove their arithmeticity. At the end, we are successful in establishing the arithmeticity of 4 out of these 6 examples by using the method used to prove one of the main results of [11,Thm. 1.2]. More precisely, the work of this article provide the following result. Theorem 2. The hypergeometric groups associated to the four pairs of parameters α, β of Table 1 are arithmetic. label α β v C-1 0, 0, 0, 0, 1 2 , 1 2 1 3 , 1 3 , 2 3 , 2 3 , 1 6 , 5 6 (−3 , −3 , 3 , −3 , −3 , 0) C-10 0, 0, 0, 0, 1 3 , 2 3 1 9 , 2 9 , 4 9 , 5 9 , 7 9 , 8 9 (−3 , 3 , −3 , 3 , −3 , 0) C-42 (0, 0, 1 4 , 1 4 , 3 4 , 3 4 ) ( 1 3 , 2 3 , 1 12 , 5 12 , 7 12 , 11 12 ) (−3 , 3 , −3 , 3 , −3 , 0) C-59 (0, 0, 1 12 , 5 12 , 7 12 , 11 12 ) 1 3 , 2 3 , 1 4 , 3 4 , 1 4 , 3 4 (−3 , −3 , 0 , −3 , −3 , 0) Remark 1. We will label the hypergeometric groups discussed in this article according to how they appear in Table A and Table C of [2]: "X-Y" will represent the entry Y from At the end, we are only left with 11 examples of degree six hypergeometric groups whose arithmeticity and thinness are still unknown. We list these remaining 11 cases in Table 2 below. Label α β v A-15 (0, 0, 0, 0, 0, 0) 1 3 , 1 3 , 1 3 , 2 3 , 2 3 , 2 3 (−9 , 9 , −27 , 9 , −9 , 0) A-16 (0, 0, 0, 0, 0, 0) 1 3 , 1 3 , 2 3 , 2 3 , 1 4 , 3 4 Note that 9 out of these 11 open cases, are still available to try to show the arithmeticity by finding such an element γ, if we can. However, the method which we adapt in this article to show the arithmeticity of 4 examples, can simply be attempted to establish the arithmeticity of all the remaining 11 cases. We remind the reader that the thinness of the 63 examples mentioned in Corollary 3 above was achieved by adapting the approach of Brav and Thomas [5] from dimension n = 4 to n = 6. Note that the first examples of higher rank thin hypergeometric groups were found in [5] in the case of symplectic hypergeometric groups Γ(f, g) ⊂ Sp 4 (Z) by playing ping-pong. Preliminaries Let f, g be a pair of degree 6 polynomials that are products of cyclotomic polynomials, do not have any common roots, form a primitive pair (that is, there do not exist polynomials f 1 , g 1 ∈ Z[x] so that f (x) = f 1 (x k ), g(x) = g 1 (x k ) for k ≥ 2), and f (0) = g(0) = 1. Observe that f, g are self-reciprocal and the product of all the roots of f , as well as of g, is 1. These conditions ensure that the corresponding hypergeometric group Γ(f, g) preserves a nondegenerate symplectic form Ω, and Γ(f, g) is a Zariski dense subgroup of the corresponding symplectic group Sp Ω (cf. [3, Theorem 6.5]). Since the polynomials f, g have integer coefficients and their constant terms are 1, it follows that Γ(f, g) ⊆ Sp Ω (Z). For given polynomials f (x) = x 6 + a 5 x 5 + · · · + a 1 x + 1, g(x) = x 6 + b 5 x 5 + · · · + b 1 x + 1, the corresponding companion matrices are A =         0 0 0 0 0 −1 1 0 0 0 0 −a 1 0 1 0 0 0 −a 2 0 0 1 0 0 −a 3 0 0 0 1 0 −a 3 0 0 0 0 1 −a 5         , B =         0 0 0 0 0 −1 1 0 0 0 0 −b 1 0 1 0 0 0 −b 2 0 0 1 0 0 −b 3 0 0 0 1 0 −b 3 0 0 0 0 1 −b 5         respectively, and C = A −1 B =         1 0 0 0 0 a 1 − b 1 0 1 0 0 0 a 2 − b 2 0 0 1 0 0 a 3 − b 3 0 0 0 1 0 a 4 − b 4 0 0 0 0 1 a 5 − b 5 0 0 0 0 0 1         . 2.1. Structure of the unipotent groups. We will now briefly describe the structure of the unipotent groups corresponding to the roots of Sp 6 (Ω). Let D = {ǫ 1 , ǫ 2 , ǫ 3 , ǫ * 3 , ǫ * 2 , ǫ * 1 } be a basis of Q 6 over Q, such that the matrix form Ω ′ of Ω with respect to this basis, is Ω =         0 0 0 0 0 λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 −λ 3 0 0 0 0 −λ 2 0 0 0 0 −λ 1 0 0 0 0 0         with Ω(ǫ i , ǫ * i ) = λ i ∈ Q * for 1 ≤ i ≤ 3. Let T be the maximal torus in Sp 6 (Ω) given by the group of diagonal matrices, T =                        t 1 0 0 0 0 0 0 t 2 0 0 0 0 0 0 t 3 0 0 0 0 0 0 t −1 3 0 0 0 0 0 0 t −1 2 0 0 0 0 0 0 t −1 1         \|t i ∈ Q * , for1 ≤ i ≤ 3                then T defines a root system and, as T is a Q-split torus, this root system is also a Q-root system for Sp 6 . More precisely, we may now define the root system Φ := Φ(T ) for Sp 6 (Ω) by simply taking, for 1 ≤ i ≤ 3, t i be the character of T defined by         t 1 0 0 0 0 0 0 t 2 0 0 0 0 0 0 t 3 0 0 0 0 0 0 t −1 3 0 0 0 0 0 0 t −1 2 0 0 0 0 0 0 t −1 1         → t i . Then the set of all roots is defined by Φ = Φ + ∪ Φ − , the union of the set of all positive and negative roots denoted by φ + respectively Φ − . Hence, for a fix set of simple roots Π = {t 1 t −1 2 , t 2 t −1 3 , t 2 3 }, we have Φ + = t 2 1 , t 1 t 2 , t 1 t 3 , t 1 t −1 3 , t 1 t −1 2 , t 2 2 , t 2 t 3 , t 2 t −1 3 , t 2 3 , Φ − = t −2 1 , t −1 1 t −1 2 , t −1 1 t −1 3 , t −1 1 t 3 , t −1 1 t 2 , t −2 2 , t −1 2 t −1 3 , t −1 2 t 3 , t −2 3 , and therefore t 2 1 , t 1 t 2 are the highest respectively the second highest roots in Φ + . We can now describe that the unipotent groups U t 2 1 and U t 1 t 2 corresponding to the highest and second highest roots as follows: U t 2 1 =                        1 0 0 0 0 y 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1         \|y ∈ Q                , U t1t2 =                        1 0 0 0 x 0 0 1 0 0 0 λ1 λ2 x 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1         \|x ∈ Q, λ 1 , λ 2 as in M                . 2.2. Methodology. Note that to prove the arithmeticity of four examples in Table 1, we follow the method used to prove Theorem 1.2 of [11]. However, the computations are quite intricate and involved. We start the process by computing the symplectic form Ω ′ (up to scalar multiples) preserved by the Γ := Γ(f, g) = A, B . Then, we show that there exists a basis D = {ǫ 1 , ǫ 2 , ǫ 3 , ǫ * 3 , ǫ * 2 , ǫ * 1 } of Q 6 such that the matrix form of Ω ′ , with respect to this new basis D, denoted by Ω ′′ = X t Ω ′ X is anti-diagonal. Here, X denote the change of basis matrix: from the standard basis E = {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } into the new basis D, see Section 3.1 for details. In this basis, the diagonal matrices, the group of upper triangular matrices and group of unipotent upper triangular matrices in Sp 6 (Ω ′′ ) form a maximal torus, a Borel subgroup B and the unipotent radical U of B respectively. We find that U is a nilpotent subgroup of GL 6 (R). Now, following [7, Thm. 2.1], if Γ ∩ U(Z) is a Zariski dense subgroup of U then U/Γ ∩ U(Z) is compact, and hence Γ ∩ U(Z) has finite index inside U(Z). Hence, to show that Γ ∩ U(Z) is of finite index in U(Z), it is enough to show that Γ ∩ U(Z) is Zariski dense in U, and for this it is enough to show that Γ contains nontrivial unipotent elements corresponding to each of the positive roots, and the arithmeticity of Γ follows from [12]. However, following [13,Thm. 3.5], if Γ is a Zariski dense subgroup of Sp 6 (Ω)(Z), and intersects the highest and second highest root groups non-trivially, then Γ has finite index in Sp 6 (Ω)(Z). Hence, to show the arithmeticity of the examples under consideration in this article, following [13,Thm. 3.5], it is enough to find the unipotent elements corresponding to the highest and second highest roots of Sp 6 (Ω ′′ ). In particular, we show that the Zariski dense subgroup Γ ′ = X −1 ΓX of Sp 6 (Ω ′′ ) contains some non-trivial elements of the unipotent groups U t 2 1 and U t 1 t 2 which establishes the arithmeticity of Γ ′ in Sp 6 (Ω ′′ ) following [13,Thm. 3.5]. Consequently, this proves the arithmeticity of Γ in Sp 6 (Ω ′ ). 2.3. Symplectic form preserved by Γ(f, g). We know that Γ(f, g) preserves a nondegenerate integral symplectic form Ω on Z 6 and Γ(f, g) ⊂ Sp Ω (Z) is Zariski dense by Theorem 6.5 in [3]. For our task, computations of the symplectic form Ω (which is unique up to scalar) is one of the important steps, and in fact we compute them explicitly in all the examples discussed in this article. For this purpose, we closely follow the notations and method explained in [11,9,10] to prove the arithmeticity of various hypergeometric groups of type Sp(4). we simply adapt the method to perform the computations for the examples under consideration . Let A and B be the companion matrices of f and g respectively. Let E = {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } be the standard basis vectors of Q 6 over Q, and v be the last column vector of C −I, where I is the identity matrix. Then, Cv = v. Therefore, using the invariance of Ω under the action of C, we get that v is Ω-orthogonal to the vectors e 1 , e 2 , e 3 , e 4 , e 5 and Ω(v, e 6 ) = 0 (since Ω is non-degenerate). We may now assume that Ω(v, e 6 ) = 1. It can be easily checked that the set B = {v, Bv, B 2 v, B 3 v, B 4 v, B 5 v} (similarly the set {v, Av, A 2 v, A 3 v, A 4 v, A 5 v}) is linearly independent over Q. Since Ω is invariant under the action of A, that is, Ω(A i v, A j v) = Ω(A i+1 v, A j+1 v), for any i, j ∈ Z, to determine the symplectic form Ω on Q 6 , it is enough to compute Ω(v, A j v), for j = 0, 1, 2, 3, 4, 5. Also, since v is Ω-orthogonal to the vectors e 1 , e 2 , e 3 , e 4 , e 5 and Ω(v, e 6 ) = 1 (say), we get that Ω(v, A j v) is the coefficient of e 6 in A j v. Since the companion matrix A (resp. B) of f (resp. g) maps e i to e i+1 for 1 ≤ i ≤ 5, to know the symplectic forms Ω preserved by the symplectic hypergeometric groups it is enough to find the scalars Ω(e 1 , e j ) for 1 ≤ j ≤ 6 , since Ω(e i , e j ) = Ω(Ae i , Ae j ) = Ω(e i+1 , e j+1 ) for 1 ≤ i, j ≤ 5. Combining, all the above information about the symplectic form preserved by Γ(f, g), we can now describe the matrix form of Ω, which we denote by the same letter. we can write the matrix form of Ω associated to the group Γ(f, g) as follows: (2.1) Ω =         0 b c d e f −b 0 b c d e −c −b 0 b c d −d −c −b 0 b c −e −d −c −b 0 b −f −e −d −c −b 0         . Proof of Theorem 2 3.1. Arithmeticity of C-1. In this case, the parameters are α = 0, 0, 0, 0, 1 2 , 1 2 and β = 1 3 , 1 3 , 2 3 , 2 3 , 1 6 , 5 6 . The corresponding polynomials are f ( x) = x 6 − 2 x 5 − x 4 + 4 x 3 − x 2 − 2 x + 1 and g(x) = x 6 + x 5 + 2 x 4 + x 3 + 2 x 2 + x + 1. Therefore, f (x) − g(x) = −3 x 5 − 3 x 4 + 3 x 3 − 3 x 2 − 3 x. Let A and B be the companion matrices of f (x) and g(x) respectively. Then A =         0 0 0 0 0 −1 1 0 0 0 0 2 0 1 0 0 0 1 0 0 1 0 0 −4 0 0 0 1 0 1 0 0 0 0 1 2         , B =         0 0 0 0 0 −1 1 0 0 0 0 −1 0 1 0 0 0 −2 0 0 1 0 0 −1 0 0 0 1 0 −2 0 0 0 0 1 −1         , C = A −1 B =         1 0 0 0 0 −3 0 1 0 0 0 −3 0 0 1 0 0 3 0 0 0 1 0 −3 0 0 0 0 1 −3 0 0 0 0 0 1         . The hypergeometric group Γ(f, g) = A, B is a subgroup of SL 6 (Z), preserves a symplectic form Ω which we now compute following the discussion of Section 2.3. Let E = {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } be the standard basis of Q 6 over Q, and let v = (C − I)e 6 . Then v = −3e 1 − 3e 2 + 3e 3 − 3e 4 − 3e 5 , Bv = −3e 2 − 3e 3 + 3e 4 − 3e 5 − 3e 6 , B 2 v = 3e 1 + 3e 2 + 3e 3 + 9e 5 , B 3 v = 3e 2 + 3e 3 + 3e 4 + 9e 6 , B 4 v = −9e 1 − 9e 2 − 15e 3 − 6e 4 − 15e 5 − 9e 6 , B 5 v = 9e 1 + 9e 3 − 6e 4 + 12e 5 − 6e 6 . As mentioned, the set B = {v, Bv, B 2 v, B 3 v, B 4 v, B 5 v} forms a basis of Q 6 , and we find that with respect to the basis B, the matrix of the symplectic form preserved by Γ is given by Ω =            0 −3 0 9 −9 −6 3 0 −3 0 9 −9 0 3 0 −3 0 9 −9 0 3 0 −3 0 9 −9 0 3 0 −3 6 9 −9 0 3 0            Now, let Z be the change of basis matrix from the basis B to the standard basis E. Then, following a simple computation we obtain that Z =           − 1 9 − 1 27 − 2 27 0 2 27            and with respect to the standard basis E, the matrix, up to scalar multiplication, of the symplectic form preserved by the hypergeometric group Γ is given by Ω ′ = 27Z t ΩZ =            0 2 1 3 −4 −5 −2 0 2 1 3 −4 −1 −2 0 2 1 3 −3 −1 −2 0 2 1 4 −3 −1 −2 0 2 5 4 −3 −1 −2 0            where Z t denotes the transpose of the matrix Z and multiplication by 27 is purely aesthetic purposes, that is to have cleaner entries in the matrix of Ω. We now find that A t Ω ′ A = Ω ′ = B t Ω ′ B. We now change the standard basis E to D for which the change of basis matrix takes the following shape X =            0 0 0 0 0 −3 −3 −6 0 0 0 −3 −3 −6 0 0 6 3 3 12 − 27 2 0 −3 −3 −3 −9 − 27 2 27 2 6 −3 −3 0 0 0 0 0            . Now, with respect to this new basis D, we find that Ω ′ takes the anti diagonal form denoted by Ω ′′ = X t Ω ′ X, that is, Ω ′′ =                      . With respect to the basis D, we write the generators a = X −1 AX, b = X −1 BX and c = a −1 b as follows: a =            3 3 9/2 −9/2 −2 1 0 −3/2 −9/4 9/4 1 0 0 0 −1 1 0 0 0 1 −5/2 3/2 0 0 1 −1 0 0 0 0 −1 0 0 0 0 0            b =            0 3 9/2 −9/2 −2 1 0 −3/2 −9/4 9/4 1 0 0 0 −1 1 0 0 0 1 −5/2 3/2 0 0 1 −1 0 0 0 0 −1 0 0 0 0 0            , c =                      . Now, note that the elements c = a −1 b and ba −1 in these cases are always unipotent elements. Therefore, in this particular case, we write q 1 = ba −1 and consider the elements w 1 = [a b −1 ], w 2 = [a b 2 ], w 3 = w 2 q −8 1 , w 4 = q 1 c w 5 = [w 1 w 3 ], w 6 = [w 4 w 3 ], w 7 = [w 6 w 5 ]. Following simple computation, we find that the q 1 and q 2 = w 2916 3 w −1 7 are the desired unipotent elements corresponding to the highest and the second highest roots of Sp 6 : q 1 = ba −1 =                , q 2 =         1 0 0 0 52488 0 0 1 0 0 0 −26244 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1         . Hence, following the discussion in Section 2.2, the arithmeticity follows by the existence of the unipotents q 1 and q 2 . The corresponding polynomials are f (x) = x 6 − 3 x 5 + 3 x 4 − 2 x 3 + 3 x 2 − 3 x + 1 and g(x) = x 6 + x 3 + 1. Therefore, f (x) − g(x) = −3 x 5 + 3 x 4 − 3 x 3 + 3 x 2 − 3 x and A =                      , C = A −1 B =           0 0 0 0 0 1            . Now, we write down the symplectic form Ω ′ (up to scalar multiple) preserved by Γ = A, B , with respect to the basis D the change of basis matrix X, and symplectic form Ω ′′ preserved by the group Γ ′ = X −1 ΓX = a = X −1 AX, b = X −1 BX : Ω ′ =           0 1 2 2 1 −1 −1 0 1 2 2 1 −2 −1 0 1 2 2 −2 −2 −1 0 1 2 −1 −2 −2 −1 0 1 1 −1 −2 −2 −1 0           , X =            0 0 0 0 0 −1 −1 1 0 0 0 1 1 0 0 0 −1 −1 −1 −2 −1 0 2 1 1 2 0 1 −1 −1 −1 0 0 0 0 0            , Ω ′′ =            0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 0            , where a =            2 −2 0 −1 1 1 0 −2 0 −1 1 0 0 0 0 1 0 0 0 1 −1 2 0 0 1 −3 0 −1 1 0 −1 0 0 0 0 0            , b =            −1 −2 0 −1 1 1 0 −2 0 −1 1 0 0 0 0 1 0 0 0 1 −1 2 0 0 1 −3 0 −1 1 0 −1 0 0 0 0 0            , c = a −1 b =           −3 0 0 0 0 1            . In this particular case, we write q 1 = ba −1 which forms a unipotent element with respect to highest root. Then consider the elements w 1 = [a b 2 ], w 2 = c −1 b 2 a −3 q 1 , w 3 = cw 2 w 4 = [w 1 w 3 ], w 5 = [w 4 w 2 ] , w 6 = w 59 3 w 5 . This gives us the desired unipotent element q 2 = w 6 c with respect to the second highest root. Hence, the arithmeticity follows by the existence of the unipotents q 1 and q 2 . The corresponding polynomials are q 1 = ba −1 =           f (x) = x 6 − 2 x 5 + 3 x 4 − 4 x 3 + 3 x 2 − 2 x + 1 and g(x) = x 6 + x 5 − x 3 + x + 1. Therefore, f (x) − g(x) = −3 x 5 + 3 x 4 − 3 x 3 + 3 x 2 − 3 x and A =                      , C = A −1 B =           0 0 0 0 0 1            . Now, we write down the symplectic form Ω ′ (up to scalar multiple) preserved by Γ = A, B , with respect to the basis D the change of basis matrix X, and symplectic form Ω ′′ preserved by the group Γ ′ = X −1 ΓX = a = X −1 AX, b = X −1 BX : Ω ′ =            0 0 −1 −1 0 −1 0 0 0 −1 −1 0 1 0 0 0 −1 −1 1 1 0 0 0 −1 0 1 1 0 0 0 1 0 1 1 0 0            , X =            0 0 0 −3 0 0 0 3 −3 3 0 3/2 0 −3 3 −3 0 3/2 0 −6 −3 3 6 0 −3/2 3 3 −3 −3 −3/2 0 0 −3 0 0 0            , Ω ′′ =                      , where a =            1/2 3 1 0 −3 −1/2 1/2 −3/2 0 0 1 1/4 1/2 −1 1 1 1 1/2 0 0 −1 0 0 0 3/4 −5/2 −1/2 0 3/2 3/4 0 1 0 0 0 1/2            , b =            1/2 3 1 0 −3 −1/2 1/2 −3/2 0 0 1 1/4 1/2 −1 −2 1 1 1/2 0 0 −1 0 0 0 3/4 −5/2 −1/2 0 3/2 3/4 0 1 0 0 0 1/2            , c = a −1 b =                      . Then consider the elements w 1 = [a b −1 ] w 2 = a 3 ca −3 w 3 = a 4 ca −4 w 4 = b 3 cb −3 w 5 = w 2 c −1 w 6 = [w 1 w 5 ] w 7 = w 4 5 w 8 = b 2 cb −2 w 9 = [w 4 w 5 ] w 10 = w 4 9 w −           . Hence, the arithmeticity follows by the existence of the unipotents q 1 and q 2 . 3.4. Arithmeticity of C-59. In this case, the parameters are 0, 0, 1 12 , 5 12 , 7 12 , 11 12 and 1 3 , 2 3 , 1 4 , 3 4 , 1 4 , 3 4 . The corresponding polynomials are f (x) = x 6 − 2 x 5 + 2 x 3 − 2 x + 1 and g(x) = x 6 + x 5 + 3 x 4 + 2 x 3 + 3 x 2 + x + 1. Therefore, f (x) − g(x) = −3x 5 − 3x 4 − 3x 2 − 3x, and This will give us the desired unipotent elements q 1 = c and q 2 = w 180 3 w 5 . Hence, the arithmeticity follows by the existence of the unipotents q 1 and q 2 . q 1 = c =            Corollary 3 . 3There are 458 degree six symplectic hypergeometric groups up to equivalence. Among them, at least 384 are arithmetic and 63 are thin. 1 , e 1 ), b = Ω(e 1 , e 2 ), c = Ω(e 1 , e 3 ), d = Ω(e 1 , e 4 ), e = Ω(e 1 , e 5 ), f = Ω(e 1 , e 6 ), 3. 2 . 2Arithmeticity of C-10. In this case, the parameters are . Arithmeticity of C-42. In this case, the parameters are w 16 = [w 3 w 15 ] w 17 = w 9 15 w 16 w 18 = w 3 w 15 w −1 3 w −117 . This will give us the desired unipotent elementsq 1 = (w 8 15 w 18 ) . Now, we write down the symplectic form Ω ′ (up to scalar multiple) preserved by Γ = A, B , with respect to the basis D the change of basis matrix X and symplectic form Ω ′′ preserved by the groupΓ ′ = X −1 ΓX = a = X −1 AX, b = X −1 BX : = [a b] , w 2 = [a b −1 ] , w 3 = [b 2 a −1 ] , w 4 = [w 1 w 2 ] , w 5 = [w 3 w 4 ] . Table 1 . 1Arithmetic hypergeometric groups in Sp(6) Table of this article, we describe the current state of the problem about determining arithmeticity and thinness of symplectic hypergeometric groups of degree six in the form of following corollary.X of [2]. Now, following [11, Thm. 1.1], [2, Thms. 2 & 3], [1, Thms. 2, 3, 5 & 6], along with Theorem 2 Table 2 . 2Open cases AcknowledgementsThe author would like to thank the Max Planck Institute für Mathematics (MPIM), Bonn where much of the work on this article was accomplished, for the hospitality and support. Arithmetic and thin monodromy in Sp. J Bajpai, D Dona, M Nitsche, arXiv:2112.1211113J. Bajpai, D. Dona, and M. Nitsche. Arithmetic and thin monodromy in Sp(6). arXiv:2112.12111, 2021. 1, 2, 3 Symplectic hypergeometric groups of degree six. J Bajpai, D Dona, S Singh, S V Singh, J. Algebra. 5753J. Bajpai, D. Dona, S. Singh, and S. V. Singh. Symplectic hypergeometric groups of degree six. J. Algebra, 575:256-273, 2021. 1, 2, 3 Monodromy for the hypergeometric function nFn−1. F Beukers, G Heckman, Invent. Math. 9527F. Beukers and G. Heckman. Monodromy for the hypergeometric function nFn−1. Invent. Math., 95(2):325-354, 1989. 1, 2, 5, 7 Translated from the 1969 French original. A Borel, University Lecture Series. American Mathematical Society. Dave Witte Morris732Introduction to arithmetic groupsA. Borel. Introduction to arithmetic groups, volume 73 of University Lecture Series. American Math- ematical Society, Providence, RI, 2019. Translated from the 1969 French original [ MR0244260] by Lam Laurent Pham, Edited and with a preface by Dave Witte Morris. 2 Thin monodromy in Sp(4). C Brav, H Thomas, Compos. Math. 1503C. Brav and H. Thomas. Thin monodromy in Sp(4). Compos. Math., 150(3):333-343, 2014. 4 Experimenting with symplectic hypergeometric monodromy groups. A S Detinko, D L Flannery, A Hulpke, 10.1080/10586458.2020.1780516Exp. Math. 3to appearA. S. Detinko, D. L. Flannery, and A. Hulpke. Experimenting with symplectic hypergeometric mon- odromy groups. Exp. Math., to appear (https://doi.org/10.1080/10586458.2020.1780516). 3 Discrete subgroups of Lie groups. M S Raghunathan, Ergebnisse der Mathematik und ihrer Grenzgebiete. New York-HeidelbergSpringer-Verlag68M. S. Raghunathan. Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzge- biete, Band 68. Springer-Verlag, New York-Heidelberg, 1972. 6 Notes on thin matrix groups. P Sarnak, Thin groups and superstrong approximation. CambridgeCambridge Univ. Press61P. Sarnak. Notes on thin matrix groups. In Thin groups and superstrong approximation, volume 61 of Math. Sci. Res. Inst. Publ., pages 343-362. Cambridge Univ. Press, Cambridge, 2014. 2 Arithmeticity of four hypergeometric monodromy groups associated to Calabi-Yau threefolds. S Singh, Int. Math. Res. Not. IMRN. 18S. Singh. Arithmeticity of four hypergeometric monodromy groups associated to Calabi-Yau threefolds. Int. Math. Res. Not. IMRN, (18):8874-8889, 2015. 7 Arithmeticity of some hypergeometric monodromy groups in Sp(4). S Singh, J. Algebra. 4737S. Singh. Arithmeticity of some hypergeometric monodromy groups in Sp(4). J. Algebra, 473:142-165, 2017. 7 Arithmeticity of certain symplectic hypergeometric groups. S Singh, T N Venkataramana, Duke Math. J. 16337S. Singh and T. N. Venkataramana. Arithmeticity of certain symplectic hypergeometric groups. Duke Math. J., 163(3):591-617, 2014. 1, 3, 6, 7 Systèmes générateurs de groupes de congruence. J Tits, C. R. Acad. Sci. Paris Sér. A-B. 2839AiJ. Tits. Systèmes générateurs de groupes de congruence. C. R. Acad. Sci. Paris Sér. A-B, 283(9):Ai, A693-A695, 1976. 7 Zariski dense subgroups of arithmetic groups. T N Venkataramana, J. Algebra. 1082T. N. Venkataramana. Zariski dense subgroups of arithmetic groups. J. Algebra, 108(2):325-339, 1987. 7	[]
[ "ONE-SIDED GRP SOLVER AND NUMERICAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLUID FLOWS", "ONE-SIDED GRP SOLVER AND NUMERICAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLUID FLOWS", "ONE-SIDED GRP SOLVER AND NUMERICAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLUID FLOWS", "ONE-SIDED GRP SOLVER AND NUMERICAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLUID FLOWS" ]	[ "Jiequan Li ", "And Qinglong Zhang ", "Jiequan Li ", "And Qinglong Zhang " ]	[]	[]	In the computation of compressible fluid flows, numerical boundary conditions are always necessary for all physical variables at computational boundaries while just partial physical variables are often prescribed as physical boundary conditions. Certain extrapolation technique or ghost cells are often employed traditionally for this issue but spurious wave reflections often arise to cause numerical instability. In this paper, we associate this issue with the one-sided generalized Riemann problem (GRP) solver motivated by the accelerated piston problem in gas dynamics so that the extrapolation technique can be actually avoided. In fact, the compatibility arguments naturally requires to formulate the one-sided generalized Riemann problem and incorporate it into the numerical procedure of boundary conditions. As far as the interaction of nonlinear waves with physical boundaries, such a one-sided GRP solver shows significant effects, as numerical experiments demonstrate, on avoiding spurious wave reflections at the computational boundaries.	10.1016/j.jcp.2022.111138	[ "https://arxiv.org/pdf/2107.05927v1.pdf" ]	235,815,932	2107.05927	64339031bfefa907adf98c71f95670d6835ae4ae	ONE-SIDED GRP SOLVER AND NUMERICAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLUID FLOWS 13 Jul 2021 Jiequan Li And Qinglong Zhang ONE-SIDED GRP SOLVER AND NUMERICAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLUID FLOWS 13 Jul 2021arXiv:2107.05927v1 [math.NA]Compressible fluid flowsnumerical boundary conditionsone-sided generalized Riemann problem (GRP) solver In the computation of compressible fluid flows, numerical boundary conditions are always necessary for all physical variables at computational boundaries while just partial physical variables are often prescribed as physical boundary conditions. Certain extrapolation technique or ghost cells are often employed traditionally for this issue but spurious wave reflections often arise to cause numerical instability. In this paper, we associate this issue with the one-sided generalized Riemann problem (GRP) solver motivated by the accelerated piston problem in gas dynamics so that the extrapolation technique can be actually avoided. In fact, the compatibility arguments naturally requires to formulate the one-sided generalized Riemann problem and incorporate it into the numerical procedure of boundary conditions. As far as the interaction of nonlinear waves with physical boundaries, such a one-sided GRP solver shows significant effects, as numerical experiments demonstrate, on avoiding spurious wave reflections at the computational boundaries. Introduction The issue on boundary conditions for hyperbolic problems and particularly for compressible fluid flows is a classic topic and so is the corresponding numerical treatment. There are a number of contributions via various approaches in literature, which roughly consist of three types of concerns: physical considerations [27], mathematical justifications (well-posedness arguments) [16] and numerical treatment. Physical considerations prescribe boundary data for a part of physical variables based on specific problems such as the solid-wall boundary condition; mathematical wellposedness justifies the validity of modelings subject to the prescribed boundary conditions; while numerical boundary conditions are prescribed for all physical variables so that discrete (approximate) equations can be implemented practically. These concerns, though with different objectives, have the common goal on correctly describing the underlying problems, for which the compatibility among the governing equations, prescribed boundary conditions and the initial data is a fundamental issue. As far as the numerical treatment is concerned, extrapolation technique is often employed, particularly for high order accurate numerical methods. For example, in [17,18] a lagrangian interpolation is performed to achieve a second order accurate approximation to the boundary data in space. However, it just gives the first order accurate approximation in time. Other works can be found, e.g. in [7,15], in the finite volume framework, and even in complex geometries [19]. In [27], characteristic method is used but restricted to first order accuracy for smooth flows. Although the resulting schemes may be well-implementable, the corresponding validation is not clear both from rigorous mathematical analysis and numerical performance. Improper extrapolation may lead to numerical instability such as spurious oscillations [27]. From the viewpoint of numerical analysis, it is questionable whether the numerical boundary conditions are compatible with the discretized governing equations even though the underlying PDE models are well-posed. Hence it is worth addressing issue even though there are lots of studies available [7,11,12,13,14,15,16,17,18,19,28,29,30]. We associate this issue with the so-called one-sided generalized Riemann problem (GRP). As motivation, we take a look at the initial-boundary value problem for the Burgers equation [2,1], u t + (u 2 /2) x = 0, x ∈ (0, L), t > 0, u(x, 0) = u 0 (x), x ∈ (0, L), (1.1) and focus on the left boundary x = 0. We assume that u 0 (x) ≡ 1 for example and inspect various situations upon the boundary requirement on x = 0. There are three typical cases: (i) u(0, t) = a, 0 ≤ a ≤ 1. For this case, the solution contains a rarefaction wave u(x, t) =          a, 0 ≤ x/t < a, x/t, a ≤ x/t ≤ 1, 1, x/t > 1. (1.2) (ii) u(0, t) = b, b > 1. For this case, we have a shock solution, u(x, t) = b, 0 ≤ x/t < (b + 1)/2, 1, x/t > (b + 1)/2,(1.3) (iii) u(0, t) = c, c < 0. There exists no physically admissible solution for such a case and so the boundary condition is not well prescribed. This example shows the subtlety of nonlinear problems as investigated in [2], unlike linear hyperbolic problems. In fact, for linear problems, physical boundary conditions depend on characteristic propagations. While for nonlinear compressible fluid flows, many physical boundary conditions are prescribed upon surroundings and cannot be even given a priori, such as the interaction of shock with solid boundaries [6] and the solid body floating in the air [13]. Numerically, situations become more complicated. First, numerical boundary conditions should be given for all variables in order to be suitable for the computation so that proper extrapolations have to be used. For the strong interaction of waves with physical boundaries, the nonlinearity actually prevents the validity of extrapolations that may result in factitious phenomena. Second, high order approximations of boundary conditions are often made independently of the discretization of the governing equations, which may lead to incompatibility and loss of accuracy. The GRP formulated here is different from the traditional GRP [3,4] and more suitably called one-sided GRP. It is an initial-boundary value problem rather than a purely initial value problem. Correspondingly, a numerical method to solve this problem is called a one-sided GRP solver. Such a study has two-fold goals: (i) It is used to the compatibility of prescribed boundary condition with the governing equations; (ii) It proposes a family of high order numerical boundary conditions effectively compatible with the discretized governing equations. In fact, accelerated piston problems [8,13] are the one-sided GRP formulated here and they have been refined to put into the simulation of fluid flows with moving boundaries [14,13]. In [12], the similar idea was employed if the flow is smooth and consistent with the inverse Lax-Wendroff method [29]. Note that since the two-stage fourth order framework [23,24,12] can be used to develop high order methods, there is no need to compute derivatives of order more than second. Hence this paper just focuses on second order GRP solvers, which provides a reliable tool no matter whether the solution is smooth or not. We organize this paper as follows. In Section 2, we formulate the one-sided GRP problem. In Section 3 we discuss numerical boundary conditions for compressible Euler equations via the passage of one-sided GRP. We implement the resulting scheme in Section 4 and particularly display numerical results in Subsection 4.2 to demonstrate the performance. High Order Numerical boundary conditions and one-sided GRP solver for hyperbolic balance laws In this section we formulate one-sided Riemann problems for hyperbolic balance laws in a general framework and discuss related numerical solvers for the construction of numerical boundary conditions. This is different from the classical initial-boundary value problem for compressible fluid flows in [25] since initial and boundary values are generally not compatible in a continuous way. Such a setting is proposed for practical request. For example, the flow is discontinuous at the reflection point of a shock on a solid boundary. Correspondingly, the associated initial-boundary value problem is formulated below as the one-sided generalized Riemann problem (OS-GRP). Consider hyperbolic balance laws u t + f(u) x = h(x, u), x ∈ (0, L), t > 0, u(x, 0) = u 0 (x), x ∈ (0, L), (2.1) where h(x, u) is a source term representing external forces or geometrical effects, f(u) is the flux function. This system includes the compressible Euler equations we specified in the next section and many other models [10]. It is assumed to be hyperbolic in the sense that the Jacobian A(u) of f(u) has m real eigenvalues λ k with a complete set of associated eigenvectors r k , A(u)r k = λ k r k , λ 1 ≤ · · · ≤ λ m . (2.2) Each λ k is genuinely nonlinear or linearly degenerate in the sense of Lax [20]. We focus on the left boundary x = 0. The right boundary x = L is treated similarly. We emphasize that the free boundary problem can be studied too [13]. On the boundary x = 0, the data is imposed as Bu = g(t) ∈ Σ ⊂ R m−k(u) ,(2. 3) where the operator B : Ω ⊂ R m → Σ ⊂ R m−k(u) projects the solution onto the boundary x = 0, 0 ≤ k(u) ≤ m is the number of negative eigenvalues. If (2.1) is a linear problem, i.e., f(u) = Au, A is a constant matrix, then the operator B can be expressed in the matrix form Bu = g(t), rank(B) = m − k,(2.4) if A has m − k positive eigenvalues, where B is a ℓ × m matrix. For nonlinear problems the eigenvalues depend on the solution u and thus the integer k(u) may vary depending on the solution u too. Hence the precise meaning of the operator is determined together with the solution of (2.1), as shown for the Burgers equation. Denote γ : R m → R m the trace operator on the boundary x = 0, u(x, t)\| x=0 = γu(x, t). (2.5) Then we propose the following assumption. Assumption. The problem (2.1) -(2.3) is well-posed at least locally so that B(γu) = g(t) (2.6) in some "appropriate" sense. This assumption is very "rough" and understood in certain intuitive way. A primary judgement of the well-posedness boils down to the solvability of the following one-sided Riemann problem, while the dynamics is dependent on the one-sided generalized Riemann problem (GRP). Numerically each component of u should be given a value on the boundary so that the corresponding numerical code can be implemented. Even with extrapolation, the approximation should be consistent with the solution of this one-sided GRP up to some desired accuracy order. In this section, we denote by u b (t) = u(0, t) the boundary value for the solution u, and (∂u/∂t) b (t) = ∂u/∂t(0, t) the derivative of u along the boundary x = 0. 2.1. Linear equations with constant coefficients. Let's first get motivation from linear equations. Consider with the assumption as above u t + Au x = h(u, t), x > 0, t > 0. (2.7) The characteristic decomposition tells that ∂v i ∂t + λ i ∂v i ∂x = L i h(u, x), i = 1, · · · , m, (2.8) where v i = L i u, L i is a left-eigenvector associated with the eigenvalue λ i . The solution formula is v i (x, t) = v i (x − λ i t, 0) + t 0 L i h(u(x − λ i (t − s), s), x − λ i (t − s))ds =: K i (x, t), i = 0, · · · , k. (2.9) To obtain the solution values on the boundary x = 0, we have from (2.8) v i (0, t) = v i (−λ i t, 0) + t 0 L i h(u(−λ i (t − s), s), −λ i (t − s))ds =: K i (0, t), i = 0, · · · , k. (2.10) Hence the boundary value of u can be obtained by solving the following system v i = L i u = K i (t), i = 1, · · · , k, Bu(0, t) = g(t). (2.11) Indeed, the well-posedness of (2.7) depends on the solvability of (2.11). That is, rank{L 1 , · · · , L k , B 1 , · · · , B ℓ } = m,(2.12) where B i , i = 1, · · · , ℓ, are the row vectors of the matrix B. Such a solution formula in turn helps to develop high order schemes. We can refer to [12,29] and next sections for the practical implementation in gas dynamics, corresponding to the acoustic case of one-sided GRP problem. The above discussion is of course made in the theoretical viewpoint. Numerically, we implement at each time level t = t n , as follows. (i) First order approximation. The initial data is assumed to be constant u R . Then we derive all components of u b by solving the following system Bu b (t n ) = g(t n ) v i (0, t n ) = (v i ) R , i = 1, · · · , k. (2.13) Obviously, this is exactly the same as the usual extension from the neighboring interior point using the characteristic method. (ii) Second order approximation. As high order approximations are concerned, we not only need to know the value in the first order approximation, but we have to approximate the value (∂u/∂t) b (t n ) = (∂u/∂t)(0, t n ) as well. We denote by u ′ R := u ′ 0 (0 + 0) and subsequently v ′ R = v ′ 0 (0 + 0). Then we have B(∂u/∂t) b (t n ) = g ′ (t n ) (∂v i /∂t) b (t n ) = −λ i (v i ) ′ R + L i h(u 0 (0), 0), i = 1, · · · , k. (2.14) Solving this system yields the value (∂u/∂t) b (t n ). This second order approximation shows clearly that the source term h is input into the numerical boundary condition, unlike some direct extrapolation technique. Moreover, this characteristic method allows to deal with discontinuities at the origin. Indeed, these two approximations correspond to the one-sided Riemann problem and one-sided generalized Riemann problem, respectively. 2.2. One-sided Riemann problem. The one-sided Riemann problem is motivated from the piston problem [8] and formulated in [13]. Here we formulate this problem for hyperbolic conservation laws u t + f(u) x = 0, x ∈ (0, ∞), t > 0, u(x, 0) = u R , x ∈ (0, ∞). (2.15) The boundary data is prescribed as B(u) = v * ∈ R m−k ,(2.16) for some k ≥ 0, where the operator B prescribes certain physically meaningful values to partial state components. Corresponding to (2.1)-(2.3), u R = u 0 (0 + 0) and v * = g(0). In order to solve this problem, we can mimick the method for the standard Riemann problem in the state space [9,20,31]. At least for Euler equations, we will show how to solve it in the next section. The solvability of this one-sided Riemann problem depends on the compatibility of the prescribed boundary conditions with the initial data. Generally speaking, as shown for the Burgers equation, this problem may not have to be well-posed. Hence this one-sided Riemann problem plays a role in checking whether the boundary conditions are correctly prescribed. Another role of the one-sided Riemann problem is to supplement all state variables for the practical calculation because the boundary conditions just prescribe partial components of them. For instance, consider the linear case, as indicated in (2.4), with m − k characteristics leaving the boundary x = 0 so that the rank of the boundary operator B is m − k. Then we use the characteristic decomposition to obtain other k equations, as shown above. Assume that we are able to solve this problem and obtain the solution u(x, t) with the trace on the boundary x = 0 such that, u(x, t)\| x=0 = u * , B(u * ) = v * . (2.17) Then it is necessary to check whether there are exactly m − k characteristics leaving from the boundary x = 0, similar to the linear case. 0 < λ k+1 (u * ) ≤ · · · ≤ λ m (u * ). (2.18) That is, the dimension of manifold dim{B(u) = v * } = m − k. Just like the case for the Burgers equation, this is not necessary true. Hence the solvability of one-sided Riemann problem is a necessary to judge the well-posedness of initial-boundary value problem for (2.4). 2.3. One-sided GRP. As (2.1) includes a source term or/and the initial condition is not uniform (typically consists of piecewise polynomials), one has to consider a one-sided generalized Riemann problem (GRP). From the numerical point of view, one needs to have high order accurate prescription of all components of u on the boundary x = 0 as well as the construction of spatial variation near the boundary when high order methods are sought. For completeness, the one-sided generalized Riemann problem (GRP) is reformulated here as the initial and boundary value problem, u t + f(u) x = h(x, u), x ∈ (0, ∞), t > 0, u(x, 0) = u 0 (x), x ∈ (0, ∞), B(γu)(0, t) = v * (t) ∈ R m−k(u) , t > 0, (2.19) where u 0 (x) is smooth, and v * (t) is measurable. This is associated with the one-sided Riemann problem above. Similar to the interrelation between the standard generalized Riemann problem and the associated Riemann problem, we have the following proposition [4]. Proposition 2.1. Assume that (2.19) is well-posed and let u(x, t) be its solution. Denote that u A (x/t; u R , v * ) be the solution of the associated one-sided Riemann problem (2.15)-(2.16). Then for every direction α = x/t > 0, lim t→o + u(αt, t) = u A (α; u R , v * ). (2.20) This implies the wave configuration of (2.19) is the same as that of (2.15)-(2.16) asymptotically. Note that we assume that the associated one-sided Riemann problem is uniquely solvable. Since the current paper is mainly concerned with a numerical algorithm for high order numerical boundary conditions, we leave aside for the moment the investigation of the rigorous mathematical theory. 2.4. One-sided Riemann solver and one-sided GRP solver. So-called solvers refer to the processes numerically solving the corresponding problems. Standard numerical Riemann solvers can be found in [31] and the generalized Riemann problem (GRP) solver in [3,4]. The one-sided solvers proposed here are associated with the Riemann solver [31] and the GRP solver [3,4]. These solvers aims (i) to provide all physical variables on the boundary x = 0; (ii) to apply the inverse GRP to inspect the interaction of boundary and initial data. Note that the boundary value u(0, t) that we obtain is not necessary to be continuous with the initial data u(x, 0) at the origin (x, t) = (0, 0). If so, the solution is discontinuous. Such observation is heuristic when dealing with the interaction between shocks and solid boundaries. Besides, such a process provides several indications: (i) The compatibility of the resulting boundary data u(0, t) and the initial data u(x, 0) determines the regularity of flows (solutions) around the origin locally. The one-sided Riemann solution is a key clue to the well-posedness. The one-sided Riemann solver aims to find u b (0) numerically. (ii) The one-sided GRP solution depends on the associated Riemann solution, and the corresponding GRP solver aims to find the value (∂u/∂t) b (0) and helps to build high order numerical schemes. 2.5. One-sided GRP solver in two dimensions. We extend the one-sided GRP solver to two dimensions in this part. Suppose we have a boundary L : Γ(x, y) = 0 which is independent of time. For the boundary conditions that depend on the time such as a piston problem, we refer to [13] for the associated GRP solver. Our strategy includes the following steps: we first solve a normal one-sided Riemann problem at any fixed point on the boundary L along the normal direction, namely, ∂u ∂t (2.21) where the boundary value is prescribed to be v * (x, y, t). Denote by n(x, y, t) the unit normal vector of L . For the presentation simplicity, the boundary is set along the y-axis, thanks to the Galilean invariance for fluid dynamical systems. Then (2.21) can be transformed to solving the following normal generalized Riemann problem along the y-axis, (2.22) where u gal , u gal R and v * ,gal are the Galilean transform of u, u R and v * , respectively. After resolving u gal , we transform back to obtain u. The same as the 1-D case, we solve the normal conservation law at (0, y * ) (2.23) to obtain the normal Riemann solution u N . Then we solve the following IBVP, + ∂ f (u) ∂x + ∂g(u) ∂y = 0, Γ(x, y) > 0, t > 0, u(x, y, 0) = u R (x, y), Γ(x, y) > 0, B(γu)(x, y, t) = v * (x, y, t), Γ(x, y) = 0, t > 0,∂u gal ∂t + ∂ f (u gal ) ∂x + ∂g(u gal ) ∂y = 0, x > 0, t > 0, u gal (x, y, 0) = u gal R (x, y), x > 0, B(γu gal )(x, y, t) = v * ,gal (x, y, t), x = 0, t > 0,∂u N ∂t + ∂ f (u N ) ∂x = 0, u N (x, t = 0) = u gal R (0, y * ), x > 0, B(γu N )(0, t) = v * ,gal (0, y * , 0), t > 0,∂u gal ∂t + ∂ f (u gal ) ∂x = − ∂g(u) ∂y N , u gal (x, y, 0) = u gal R (x, y), x > 0, y ∈ R, B(γu gal )(0, y, t) = v * ,gal (0, y, t), y ∈ R, t > 0, (2.24) to obtain ∂u gal ∂t b = lim t→0 ∂u gal ∂t (0, y * , t) (2.25) at (0, y * ), where the term ∂g(u) ∂y N = ∂g ∂u (u N ) ∂u ∂y N is a fixed value with the instantaneous value u * obtained from (2.23) and ∂u ∂y N interpolated from u gal R , reflecting the tangential effect along the boundary [24]. Then the 2-D one-sided GRP solver follows exactly the same as the 2-D GRP solver, one can find more details in [24]. 2.6. High order numerical boundary conditions. Once the one-sided GRP solver is available, the boundary data can be approximated with second order accuracy and the boundary volume can be dealt with as the ordinary control volume. That is, if at moment t = t n , u(0, t n ) and (∂u/∂t)(0, t n ) are known, then the boundary flux is approximated in a common way, u(0, t n + ∆t/2) = u N + ∆t 2 ∂u gal ∂t b , 1 ∆t t n+1 t n f(u(0, t))dt = f(u(0, t n + ∆t/2)) + O(∆t 2 ). (2.26) Furthermore the integral of source term can be evaluated using the interface method too, 1 ∆t∆x t n+1 t n ∆x 0 h(u(x, t))dxdt = 1 2 (h(u(0, t n + ∆t/2)) + h(u(∆x, t n + ∆t/2))) + O(∆t 2 + ∆x 2 ). (2.27) Analogously, we deal with the 2-D case. Application to Gas Dynamical Systems In this section we discuss the one-sided Riemann problem (RP) and the one-sided generalized Riemann problem (GRP) for gas dynamical systems. We first discuss one-dimensional case in the form (2.1) with u = (ρ, ρv, ρE) ⊤ , f(u) = (ρv, ρv 2 + p, v(ρE + p)) ⊤ , (3.1) and h(u, x) is a problem-dependent source term. In particular, for nozzle flows h(u, x) takes the form h(u, x) = − a ′ (x) a(x) ρv, ρv 2 , v(ρE + p) ⊤ . (3.2) where ρ, v, p are the density, velocity and pressure of the fluids, respectively. a(x) is the crosssection area of the duct. E = v 2 2 + e is the total energy, the internal energy e is given by the equation of state (EOS) e = e(ρ, p). Note that (3.2) also includes the case of radially symmetric flows [22]. The reactive Euler flows [30] can be treated similarly. System (3.1) has three eigenvalues λ − = v − c, λ 0 = v, λ + = v + c, (3.3) where c is the local sound speed. All other properties of this system can be found in any textbook about gas dynamics, e.g. [8,9,3,31]. The one-sided Riemann solver for Euler equations. As we pointed out in the last section, the one-sided Riemann problem plays a role in the justification of local well-posedness besides its numerical value. This problem is formulated as u t + f(u) x = 0, x > 0, t > 0, u(x, 0) = u R , x > 0, Bu(0, t) = w b . (3.4) The method solving this problem (3.4) follows the one for the classical Riemann problem. We fix the wave curve W R associated with λ + = v + c from the state u R in the phase space, (ρ, v, p)space, and then investigate the solvability for the prescribed data w b . It is easily checked that the solvability of such a problem is up to the following two conditions: (i) There is an intersection point u * of W R and Bu = w b ; (i) Prescribed velocity v b . Then we find a point on W R so that p b is fixed. Certainly as v b > 0, λ 0 = v b > 0 and λ + = v b + c b > 0 so that one additional condition is needed. For instance, we can supplement the gas density ρ b as the gas property on the boundary x = 0. However, as v b < 0, we need to check the Mach number M b = \|v b \|/c b . If M b > 1, the boundary condition is not suitably prescribed. (ii) The dimension dim{Bu = w b } = #{λ i (u * ) > 0, i = −, 0, +}. b p v W R u R ( ) a v=v b ( ) v ,p b b W R v p v+c=M b u v ,p R ( ) b b ( ) (ii) Given upstream Mach number M b . The given value M b = v b /c b actually implies v + c = M b . (3.5) We look for its intersection point with W R to find (v b , p b ) and then ρ b using the equation of state (EOS). In summary, we can investigate the one-sided Riemann problem to identify that whether the upstream flow is supersonic or not as well as make clear the correct prescription of boundary conditions. One-sided GRP solver. The one-sided GRP solver serves to solve (2.19) numerically. Assume that u 0 (x) is (or approximated by) a smooth function with regular limiting values u R = lim x→0+0 u 0 (x), u ′ R = lim x→0+0 u ′ 0 (x). (3.6) Based on the corresponding one-sided Riemann problem, we can obtain the limiting value (∂u/∂t) b (0) on the boundary x = 0. Essentially there are two versions in analogy with the standard GRP solver: An acoustic version and a nonlinear version. (i) Acoustic GRP. As u b − u R ≪ 1, we can use the acoustic approximation, i.e., the linear method in Subsection 2.1. (ii) Nonlinear GRP. As strong waves emit from the corner (0, 0) (i.e., u b − u R ≫ 1), we have to develop a genuinely nonlinear GRP solver, similar to the standard GRP solver for general hyperbolic balance laws [4] Specified to the Euler equations, the one-sided GRP solver is implemented as follows. (i) Judge from the associated one-sided Riemann solution whether there emit strong waves in order to determine to use the acoustic or nonlinear GRP solver. (ii) The acoustic GRP solver is the same as the linear case above. (iii) The nonlinear GRP solver consists of two cases: a supersonic upstream flow and a subsonic upstream flow. (a) A supersonic upstream flow. All conditions are given at boundary x = 0. (b) A subsonic upstream flow. We apply the same procedure of the standard GRP solver [5] and naturally derive the one-sided relation a R ∂v ∂t b + b R ∂p ∂t b = d R ,(3.7) where the coefficients a R , b R and c R are fully determined by the values u R (0), u * R and the slope value u ′ R (0), the detailed expressions can be found in [4]. We are in position to compute the partial derivative values (∂v/∂t) b and (∂p/∂t) b from (3.7). If the boundary condition is given as v b (t) = g(t) and subsequently (∂v/∂t) b = g ′ (t), then (∂p/∂t) b follows by the linear relation (3.7). As for the density derivative (∂ρ/∂t) b , we have ∂ρ ∂t b = 1 (c * R ) 2 ∂p ∂t b (3.8) on the boundary from the EOS. Here c * R is the local sound speed. If the upstream boundary condition is given in terms of Mach number M b (t) = g(t), then one has ∂v ∂t b + c p ∂p ∂t b + c ρ ∂ρ ∂t b = g ′ (t), (3.9) where c p = ∂c ∂p and c ρ = ∂c ∂ρ . This, together with (3.7) and the relation (3.8), provides the boundary condition. 2-D one-sided GRP solver for Euler. The 2-D compressible Euler equations can be written as ∂u ∂t + ∂f(u) ∂x + ∂g(u) ∂y = 0, u =               ρ ρv x ρv y ρE               , f(u) =               ρv x ρ(v x ) 2 + p ρv x v y v x (ρE + p)               , g(u) =               ρv y ρv x v y ρ(v y ) 2 + p v y (ρE + p)               , (3.10) where ρ, v x , v y , p and E represent the density, x−velocity, y−velocity, pressure and total energy, respectively. The 2-D one-sided GRP solver is the practical combination of the above 1-D onesided GRP solver and the 2-D GRP solver [4,12]. A key point is that the transversal effect is included in the solver development [21]. 4. Implementation of the one-sided GRP scheme 4.1. Brief summary of the one-sided GRP scheme. So far, the one-sided GRP solver is developed to suit the GRP scheme near the boundary. The boundary control volume is then treated the same as the interior control volumes in the finite volume framework. We discretize the domain by equally computation mesh size ∆x = x j+ 1 2 − x j− 1 2 and set I j = (x j− 1 2 , x j+ Since a standard finite volume method, such as the GRP method in [5], can be applied over computational cells I j ( j = 1, ..., M − 1) in the interior domain, we only focus on the boundary cell I 0 . The one-sided GRP scheme at the boundary cell I 0 assumes the piecewise linear data u(x, t n ) = u n 0 + σ n 0 (x − x 0 ), x ∈ (x − 1 2 , x1 2 ). (4.1) The vector σ n 0 is the constant slope of u(x, t n ) over cell I 0 at time t n = n∆t, n ∈ N with ∆t the time step size. To obtain the second order accuracy, the mid-point value is used u n+ 1 2 − 1 2 = u(x − 1 2 , (n + 1/2)∆t) (4.2) in the resolution of numerical flux and the source term discretization. We apply the 1-D one-sided GRP solver in the following steps. Step 1. Given the piecewise linear initial data (4.1), approximate the mid-point value u n+ 1 2 − 1 2 as follows, u n+ 1 2 − 1 2 = u n − 1 2 + ∆t 2 ∂u ∂t n − 1 2 . (4.3) The computation of (∂u/∂t) n −1/2 is the main ingredient of the one-sided GRP scheme. The value u n − 1 2 is the local solution at (x − 1 2 , t n ) to the following one-sided Riemann problem:                  ∂u ∂t + ∂f(u) ∂x = 0, Bu = w b , x = x − 1 2 , u R := u n 0 − (x 0 − x − 1 2 )σ n 0 , x > x − 1 2 ,(4.4) which can be solved by an exact or approximate one-sided Riemann solver [31]. Here we apply the one-sided GRP procedure (3.7)-(3.9) to obtain the instantaneous value (∂u/∂t) n −1/2 on the boundary x = x − 1 2 , and then approximate u n+ 1 2 −1/2 using (4.3). Step 2. Evaluate the next time values u n+1 0 by using the following formula u n+1 0 = u n 0 − ∆t ∆x f(u n+ 1 2 1 2 ) − f(u n+ 1 2 − 1 2 ) + ∆t 2 h(x1 2 , u n+ 1 2 1 2 ) + h(x − 1 2 , u n+ 1 2 − 1 2 ) ,(4.5) where the source term h(x, u) is discretized with the mid-point rule in time and the trapezoidal rule in space. Step 3. In order to suppress local oscillations as discontinuities are present near the boundary, we update the slope σ n+1 0 by using the following monotonicity algorithm limiter σ n+1 0 = minmod           u n+1,− 1 2 − u n+1 − 1 2 ∆x , u n+1 1 − u n+1 0 ∆x           . (4.6) More details about the minmod function can be found in [5,31]. In two-dimensional computations, we take rectangular meshes ∪I j,k , j = 0, ..., M, k = 0, ..., N, as an example for simplicity, here I j,k =(x j−1/2 , x j+1/2 ) × (y k−1/2 , y k+1/2 ) centered at the grid point (x j , y k ). The finite volume formula is applied over all cells I j,k , u n+1 j,k = u n j,k − ∆t ∆x f (u n+ 1 2 j+ 1 2 ,k ) − f (u n+ 1 2 j− 1 2 ,k ) − ∆t ∆y g(u n+ 1 2 j,k+ 1 2 ) − g(u n+ 1 2 j,k− 1 2 ) . (4.7) The initial data at time t = t n is expressed as bilinear functions u(x, y, t n ) = u n j,k + (σ x ) n j,k (x − x j ) + (σ y ) n j,k (y − y k ), j = 0, 1, ..., M, k = 0, 1, ..., N. Example 1. The scalar equation. We first use the Burgers equation to test the performance of the one-sided GRP solver. Consider the following initial-boundary value problem for the Burgers equation v t + v 2 2 x = 0, x ∈ (0, 2), t > 0, v(x, 0) = −x, 0 < x < 1, −1, 1 < x < 2, v(0, t) = 0, 0 < t < 1, 2, t > 1. (4.9) The solution v(x, t) has an explicit formula v(x, t) =                  x t − 1 , 0 < x < 1 − t, 0 < t < 1, 2, 0 < x < t/2, t > 1, −1, x > 1 − t, 0 < t < 1, −1, x > t/2, t > 1. (4.10) A compressible wave propagates to the left and forms a shock at (0, 1) on the boundary. As t ≥ 1, a shock from (0, 1) propagates to the right. We compute the solution using the GRP scheme with the reflective boundary condition and the one-sided GRP solver, respectively. The solution v(x, t) is plotted from time t = 0 to time t = 2 in Fig. 4.3, from which it is observed that the one-sided GRP solver gives very sharp resolution of the singularity point (0, 1), compared with the reflective boundary condition treatment. Example 2. A single shock interaction with a solid boundary We test the example that a single shock wave interacts with a solid wall to verify the numerical performance of the one-sided GRP solver. The computational domain is [0,10] where the boundary is at x = 0. A left-propagating shock wave is initially positioned at x = 2. We take γ = 1.4 and the initial data is set to be A reflected shock wave is observed when the output time is set to t = 2.0. We compare the results by two different boundary condition treatments: the traditional reflective boundary condition and the one-sided GRP solver. From Fig. 4.4, one can observe that the result obtained by the one-sided GRP solver is more stable and has less oscillations near the boundary. We further plot 30 equallydistributed density contours from time t = 0 to time t = 2 at every time interval 0.01 in Fig. 4.5, one can see again that the one-sided GRP solver gives very sharp resolution at the interaction point of the shock with the boundary. Example 3. The Woodward-Colella problem. This is a classical interacting blast wave problem with the gas initially at rest and γ = 1.4. The density is everywhere unit, the pressure is p = 1000 for 0 ≤ x < 0.1 and p = 100 for 0.9 < x ≤ 1.0, while it is only p = 0.01 for 0.1 < x < 0.9. The solid-wall boundary conditions are prescribed at both ends. We compare the results of the reflective boundary condition treatment with that of the one-sided GRP solver. The CFL number is 0.6. The output time is set to t = 0.038. The numerical results for both boundary condition treatments are shown in Fig. 4.6 with 400 cells and 800 cells, respectively. It can be seen that the one-sided GRP solver is effective and robust for the blast wave problem. (ρ, v, p)(0, x) =        cross-sectional area function A(x) of the duct is given by A(x) =            A in exp −log(A in )sin 2 (2πx) , 0 ≤ x ≤ 0.25, A ex exp −log(A ex )sin 2 ( 2π(1 − x) 3 ) , 0.25 < x ≤ 1 (4.12) with A in = 4.864317646 and A ex = 4.234567901. The governing equations are the Euler equations with geometric source term (3.1), (3.2). Set x = 0 as the entrance of the duct and x = 1 as the exit. We are concerned with the present boundary treatment to attain the steady state solution. Two types of steady states are discussed: A continuous steady state and a discontinuous steady state containing a standing shock wave. The initial data for both cases can take as u(0, x) =        (ρ 0 , 0, p 0 ), x < 0.25, (ρ 0 , 0, ρ 0 (p ex /p 0 ) γ ), x > 0.25,(4.13) where γ = 1.4 and ρ 0 , p 0 are parameters to be determined, p ex is a constant value determined by the steady solution at x = 1. In the previous study [3,4], the inflow density, velocity and pressure are assigned to the inflow boundary condition, the outflow pressure is assigned as the outflow boundary condition. Here we apply the one-sided GRP solver to test its ability of attaining steady solutions. For the first case, we set ρ 0 = p 0 = 1 and p ex = 0.0272237 in (4.13). This produces an isentropic continuous steady solutions which is defined by 14) in which the Mach number M(x) = v(x)/c(x) is determined by A(x) through the algebraic relation ρ(x) = ρ 0 1 + γ − 1 2 M 2 (x) − 1 γ−1 , p(x) = p 0 1 + γ − 1 2 M 2 (x) − γ γ−1 , v(x) = M(x) γp(x)/ρ(x),(4.A 2 (x) = 1 M 2 (x) 2 γ + 2 1 + γ − 1 2 M 2 (x) γ+1 γ−1 . (4.15) In this case, the flow is transonic across the throat at the position x = 0.25. Thus the inflow boundary condition at the entrance x = 0 should be prescribed by p in := p 0 1 + γ − 1 2 M 2 (0) − γ γ−1 , ρ in := ρ 0 1 + γ − 1 2 M 2 (0) − 1 γ−1 . (4.16) While at the exit x = 1, the flow is supersonic and no boundary condition is needed. The computational result is given in Fig. 4.7 where 22 cells are used. The CFL number is 0.6 and the output time is t = 5. The solution obtained by implementing the one-sided GRP solver converges to the exact steady one and is comparable with the result obtained in [4]. For the other case, where the steady solution contains a standing shock wave, we set ρ 0 = p 0 = 1 and p ex = 0.4 in (4.13) to get the initial data. In this case, the flow jumps from supersonic to subsonic after passing the standing shock wave. As the outflow is subsonic in this case, both inflow boundary condition and outflow boundary condition should be imposed. The inflow boundary condition is ρ 0 = p 0 = 1 at the entrance x = 0 and the outflow boundary condition is p ex = 0.4 at the exit x = 1. The computational result with 22 cells is given in Fig. 4.8. The CFL number is 0.6 and the output time is t = 5. The solution obtained by taking the one-sided GRP solver matches well with the exact solution. Example 5. The spherical symmetric shock interaction problem. We test the one-sided GRP solver for the simulation of the spherical symmetric flows where a spherical shock wave interacts with the symmetric center. The initial data is taken to be such that a left-going spherical shock moves toward the center. The output time is t = 5.0 with the CFL=0.5. One can see from Fig. 4.9 that near the symmetric center, the one-sided GRP solver has much better numerical performance compared with the reflective boundary condition. For more details about the GRP solver of radially symmetric flows, we refer to [22] and references cited therein. Example 6. Noh problem. The Noh problem [26] is a typical radially symmetric compressible flows problem. The governing equations include source term, which can be used to test the performance of the one-sided GRP solver. We consider the spherically converging flow of zero-pressure gas with γ = 5/3. The initial data has the uniform form [ρ, v, p] = [1, −1, 0], 0 < r ≤ 100, (4.18) here r is the radius. The exact solution consists of an expanding shock wave which begins from the center r = 0. Here the initial pressure is set to be 10 −6 instead of zero. The boundary condition at the rightmost cell is given by (ρ, v, p)(0, x) =       [ρ, v, p] n+1 (r) = [(1 + t n+1 /r) 2 , −1, 10 −6 ], r ∈ [r K−1/2 , r K+1/2 ],(4.19) which is the exact solution at t = t n+1 . On the left boundary one has v(0, t) = 0. The one-sided GRP solver is implemented on both boundaries. The result is shown in Fig. 4.10. The discrepancies near the center is caused by the "startup" of the captured shock wave, as pointed out in [22]. The result obtained here has less oscillations near the boundary compared with that in [22]. Example 7. The spherical explosion problem. This is another problem of radially symmetric compressible flows. The initial gas is at rest with ρ = 21.7333, p = 15.514 for 0 ≤ r ≤ 5 and ρ = 2.0, p = 1.0 for 5 ≤ r ≤ 50. The spherical explosion is quite complex and a complete analysis can be found in [22]. The numerical results are shown in Fig. 4.11, where we implement the GRP with two boundary condition treatments : the one-sided GRP solver and the method developed in [22]. From Fig. 4.11 one see that the one-sided GRP solver has a good agreement with the method that proposed in [22]. Figure 11. The numerical results for Noh problem with 400 cells by using the one-sided GRP solver (squares) and the traditional reflective boundary treatment (dots), respectively, the solid line is given as the exact solution. . Initially a Mach 10 shock wave is moving to the right which is at the position x = 1 6 , y = 0 and makes π 3 angle with the x-axis. More details about the problem can be seen in [32]. We compute the problem by using the traditional boundary condition treatment and the onesided GRP solver, respectively, to deal with the reflective boundary condition along the bottom wall {(x, y) : 1 6 < x < 4, y = 0}. The results are displayed in Fig. 4.12 with 30 contours of the density at time t = 0.2 where 720 × 180 cells are used here. The CFL number is 0.6. From the figure we see that the one-sided GRP solver works well for the two-dimensional solid-wall boundary condition. Example 9. The forward facing step problem. This is another classical test problem for the two-dimensional equations. The wind tunnel is 1 length unit wide and 3 length units long. The step is 0.2 length units high and is located 0.6 length units from the left-hand end of the tunnel. Initially a unit right moving Mach 3 shock wave with (ρ 0 , v x 0 , v y 0 , p 0 ) = (1.4, 3, 0, 1) in the tunnel. The reflective boundary conditions are applied along all the walls. Again, we compute the problem by using the traditional boundary condition treatment and the one-sided GRP solver, respectively. The CFL number is 0.6. The results are displayed in Fig. 4.13 with 900×300 cells at time t = 4. A three-shock Mach reflection configuration is formed. According to [32], the correct Mach stem is located at x = 0.6. We can see that the results obtained by the one-sided GRP solver has the shock at the correct position, compared with that obtained by the reflective boundary condition treatment. Appendix A. The useful one-sided GRP coefficients. The coefficients of the one-sided GRP solver are collected in Table 1. In this table, the 1-shock (3-shock, resp.) refers to the shock associated with the v − c characteristic family (v + c, resp.). The same for the 1-rarefaction wave and 3-rarefaction wave. We deal with both the left boundary case and the right boundary case. For the left boundary case, the one-sided Riemann problem (2.15) has ) = (a rare R , b rare R ), d R = d rare R 3-shock wave (a R , b R ) = (a shock R , b shock R ), d R = d shock R the solution which consists of a single 3-shock wave or a 3-rarefaction wave. Similarly, when there exists a right boundary, the solution of (2.15) consists of a single 1-shock wave or a 1-rarefaction wave. We denote J = L or R in the rest part of the paper. The one-sided Riemann solution is denoted as u * J which can be obtained through solving (2.15). The Riemann invariants φ and ψ are introduced by ψ = v + 2c γ − 1 , φ = v − 2c γ − 1 . (A.1) For the second law of thermodynamics, we have T dS = dp (γ − 1)ρ − c 2 (γ − 1)ρ dρ. (A.2) These are useful to derive the coefficients below. More details can be found in [4]. Remember that we always have (∂v/∂t) * = g ′ (t) on the boundary, then the expected instantaneous values (∂u/∂t) * can be obtained directly through solving (3.7) with (A.9) A.2. (Sonic case) When the left boundary is located inside the 3-rarefaction wave, we have ∂v ∂t * = g ′ (t), ∂p ∂t * = ρ * R v * R ∂v ∂t * − θ 2γ γ−1 T R S ′ R − a ′ (0) a(0) (v * R ) 2 . (A.10) Similarly, when the right boundary locates inside the 1-rarefaction wave, we just replace u * R , T R S ′ R by u * L , T L S ′ L in (A.10). A.3. (Acoustic case) Assume that on the right boundary, u * L = u L , (u * L ) ′ u ′ L , or on the left boundary, u * R = u R , (u * R ) ′ u ′ R , we have the acoustic case. (∂v/∂t) * and (∂p/∂t) * can be given by ∂v ∂t * = g ′ (t), ∂p ∂t Figure 1 . 1The one-sided Riemann problem for two typical boundary conditions: (a) Prescribed velocity v b . (b) Given upstream Mach number M b . The following are two typical examples, see Fig. 3.1. 1 2 ) 2. The cell I 0 = (x − 1 2 , x1 2 ) represents the left boundary cell centered at x 0 and the cell I M = (x M− 1 2 , x M+ 1 2 ) represents the right boundary cell centered at x M , as shown in Fig. 4.1. L . . . Figure 2 . 2The computational domain (0, L). Set x 0 = ∆x/2 and x M = 1 − ∆x/2, The one-sided GRP is solved at the boundaries x = 0 and x = L, respectively. analytically derived by the resolution of a local quasi 1-D GRP solver at each interface. The one-sided GRP solver is applied on the boundary. Then we can take the same procedure as that for 1-D case to implement the finite volume scheme.4.2. NumericalExamples. We will present several numerical examples to validate the performance as the one-sided GRP solver is used. The examples include the interaction of shocks with solid boundaries, the radially symmetric flows, the nozzle flows, the Mach reflection of shock and the forward facing step problem. Figure 3 . 3The boundary is initially along the y− axis. Set x 0 = ∆x/2 and the one-sided GRP is solved on the boundary x = 0. Figure 4 . 4The contours of the solution v(x, t) of the Burgers equation obtained by the one-sided GRP solver (left) and the traditional boundary condition treatment (right), respecticely. 100 cells are used and 30 contours are drawn. Figure 5 . 5A shock wave interacts with a solid wall. We compare the density profile obtained with the onesided GRP solver (squares) with that obtained with the traditional reflective boundary condition (dots) with 400 cells (200 are shown). Figure 6 . 6The contours of the solution of Example 2 obtained by the one-sided GRP solver (left) and the traditional boundary condition treatment (right). Thirty contours are drawn. Figure 7 . 7The Woodward-Colella problem computed with the one-sided GRP solver (squares) and the traditional reflective boundary condition treatment (dots) with 400 cells (left) and 800 cells (right). The numerical scheme used in the interior domain is the GRP scheme. The solid lines are the reference solution computed with 4000 cells. Example 4 . 4The nozzle flow. The nozzle flow problem is a classical quasi one-dimensional problem. Consider a flow in a converging-diverging nozzle occupying the domain x ∈ [0, 1]. The Figure 8 . 8The computation of nozzle flow equations with continuous steady solutions by using the onesided GRP solver. The pressure and Mach number at t = 5 are shown with 22 cells. The solid line represents the exact solution given by (4.14). Figure 9 . 9The computation of nozzle flow equations with a standing shock wave by using the one-sided GRP solver. The pressure and Mach number at t = 5 are shown with 22 cells. The solid line represents the exact solution given by(4.14). Figure 10 . 10A spherical shock wave interacts with the symmetric center. We compute the density profile by the one-sided GRP solver (squares) and the traditional boundary condition treatment (dots). 200 cells are used. Figure 12 . 12The comparison of the results of the spherical explosion problem with two boundary condition treatments: one-sided GRP solver (squares) and the effective boundary condition treatment (dots) proposed in[22]. Figure 13 . 13The numerical results of the double Mach reflection problem. The upper is the GRP scheme with the one-sided GRP solver. The lower is the GRP scheme with the traditional reflective boundary condition treatment. Figure 14 . 14The numerical results of the forward facing step problem. The upper is the GRP scheme with the one-sided GRP solver. The lower is the GRP scheme with the traditional reflective boundary condition treatment. L , b L ) = (a rare L , b rare L ), d L = d rare L 1-shock wave (a L , b L ) = (a shock L , b shock L ), d L = d shock L 3-rarefaction wave (a R , b R A. 1 ., 1(Nonsonic case) The coefficients for rarefaction waves are given by(a rare L , b rare L ) (a rare R , b rare R ) = 1, − Example 8. The double Mach reflection problem. We turn to two-dimensional example. The computational domain is [0, 4] × [0, 1], and [0, 3] × [0, 1] is shown. A solid-wall is at the bottom of the domain starting from x = 1 6 Table 1 . 1The coefficients in (3.7) for all possible cases. Denote D/Dt = ∂/∂t + v∂/∂x. Then we have1 ρ * R c * ∂v ∂t = Dv Dt + v ρc 2 Dp Dt + a ′ (0) a(0) v 2 , ∂p ∂t = Dp Dt + ρv Dv Dt . (A.9) .(A.4)Heresgn(J) = −1, if J = L, 1, if J = R, η(J) = ψ L , if J = L, φ R , if J = R. 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[ "Possible unconventional pairing in (Ca,Sr) 3 (Ir,Rh) 4 Sn 13 superconductors revealed by controlling disorder", "Possible unconventional pairing in (Ca,Sr) 3 (Ir,Rh) 4 Sn 13 superconductors revealed by controlling disorder" ]	[ "E H Krenkel \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n", "M A Tanatar \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n", "M Kończykowski \nLaboratoire des Solides Irradiés\nCEA/DRF/lRAMIS\nÉcole Polytechnique\nCNRS\nInstitut Polytechnique de Paris\nF-91128PalaiseauFrance\n", "R Grasset \nLaboratoire des Solides Irradiés\nCEA/DRF/lRAMIS\nÉcole Polytechnique\nCNRS\nInstitut Polytechnique de Paris\nF-91128PalaiseauFrance\n", "E I Timmons \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n", "S Ghimire \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n", "K R Joshi \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n", "Y Lee \nAmes Laboratory\n50011AmesIowaUSA\n", "Liqin Ke \nAmes Laboratory\n50011AmesIowaUSA\n", "S Chen \nCondensed Matter Physics\nMaterials Science Department\nBrookhaven National Laboratory\n11973UptonNew YorkUSA\n\nDepartment of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNYUSA\n", "C Petrovic \nCondensed Matter Physics\nMaterials Science Department\nBrookhaven National Laboratory\n11973UptonNew YorkUSA\n\nDepartment of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNYUSA\n", "P P Orth \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n", "M S Scheurer \nInstitute for Theoretical Physics\nUniversity of Innsbruck\nA-6020InnsbruckAustria\n", "R Prozorov \nAmes Laboratory\n50011AmesIowaUSA\n\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA\n" ]	[ "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA", "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA", "Laboratoire des Solides Irradiés\nCEA/DRF/lRAMIS\nÉcole Polytechnique\nCNRS\nInstitut Polytechnique de Paris\nF-91128PalaiseauFrance", "Laboratoire des Solides Irradiés\nCEA/DRF/lRAMIS\nÉcole Polytechnique\nCNRS\nInstitut Polytechnique de Paris\nF-91128PalaiseauFrance", "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA", "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA", "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA", "Ames Laboratory\n50011AmesIowaUSA", "Ames Laboratory\n50011AmesIowaUSA", "Condensed Matter Physics\nMaterials Science Department\nBrookhaven National Laboratory\n11973UptonNew YorkUSA", "Department of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNYUSA", "Condensed Matter Physics\nMaterials Science Department\nBrookhaven National Laboratory\n11973UptonNew YorkUSA", "Department of Physics and Astronomy\nStony Brook University\n11794-3800Stony BrookNYUSA", "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA", "Institute for Theoretical Physics\nUniversity of Innsbruck\nA-6020InnsbruckAustria", "Ames Laboratory\n50011AmesIowaUSA", "Department of Physics and Astronomy\nIowa State University\n50011AmesIowaUSA" ]	[]	We study the evolution of temperature-dependent resistivity with added point-like disorder induced by 2.5 MeV electron irradiation in stoichiometric compositions of the "3-4-13" stannides, (Ca,Sr)3(Ir,Rh)4Sn13. Three of these cubic compounds exhibit a proposed microscopic coexistence of charge-density wave (CDW) order and superconductivity (SC), while Ca3Rh4Sn13 does not develop CDW order. As expected, the CDW transition temperature, TCDW, is universally suppressed by irradiation in all three compositions. The superconducting transition temperature, Tc, behaves in a more complex manner. In Sr3Rh4Sn13, it increases initially in a way consistent with a direct competition of CDW and SC, but quickly saturates at higher irradiation doses. In the other three compounds, Tc is monotonically suppressed by irradiation. The strongest suppression is found in Ca3Rh4Sn13, which does not have CDW order. We further examine this composition by measuring the London penetration depth, λ(T ), from which we derive the superfluid density. The result unambiguously points to a weak-coupling, full single gap, isotropic superconducting state. Therefore, we must explain two seemingly incompatible experimental observations: a single isotropic superconducting gap and a significant suppression of Tc by non-magnetic disorder. We conduct a quantitative theoretical analysis based on a generalized Anderson theorem which points to an unconventional multiband s +− -pairing state where the sign of the order parameter is different on one (or a small subset) of the smaller Fermi surface sheets, but remains isotropic and overall fully-gapped.	10.1103/physrevb.105.094521	[ "https://arxiv.org/pdf/2110.02025v3.pdf" ]	238,353,835	2110.02025	5c968f8f2686b0f48365c82d7ee0992b40e6013a	Possible unconventional pairing in (Ca,Sr) 3 (Ir,Rh) 4 Sn 13 superconductors revealed by controlling disorder E H Krenkel Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA M A Tanatar Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA M Kończykowski Laboratoire des Solides Irradiés CEA/DRF/lRAMIS École Polytechnique CNRS Institut Polytechnique de Paris F-91128PalaiseauFrance R Grasset Laboratoire des Solides Irradiés CEA/DRF/lRAMIS École Polytechnique CNRS Institut Polytechnique de Paris F-91128PalaiseauFrance E I Timmons Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA S Ghimire Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA K R Joshi Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA Y Lee Ames Laboratory 50011AmesIowaUSA Liqin Ke Ames Laboratory 50011AmesIowaUSA S Chen Condensed Matter Physics Materials Science Department Brookhaven National Laboratory 11973UptonNew YorkUSA Department of Physics and Astronomy Stony Brook University 11794-3800Stony BrookNYUSA C Petrovic Condensed Matter Physics Materials Science Department Brookhaven National Laboratory 11973UptonNew YorkUSA Department of Physics and Astronomy Stony Brook University 11794-3800Stony BrookNYUSA P P Orth Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA M S Scheurer Institute for Theoretical Physics University of Innsbruck A-6020InnsbruckAustria R Prozorov Ames Laboratory 50011AmesIowaUSA Department of Physics and Astronomy Iowa State University 50011AmesIowaUSA Possible unconventional pairing in (Ca,Sr) 3 (Ir,Rh) 4 Sn 13 superconductors revealed by controlling disorder (Dated: Submitted: 5 October 2021; Accepted: 26 February 2022) We study the evolution of temperature-dependent resistivity with added point-like disorder induced by 2.5 MeV electron irradiation in stoichiometric compositions of the "3-4-13" stannides, (Ca,Sr)3(Ir,Rh)4Sn13. Three of these cubic compounds exhibit a proposed microscopic coexistence of charge-density wave (CDW) order and superconductivity (SC), while Ca3Rh4Sn13 does not develop CDW order. As expected, the CDW transition temperature, TCDW, is universally suppressed by irradiation in all three compositions. The superconducting transition temperature, Tc, behaves in a more complex manner. In Sr3Rh4Sn13, it increases initially in a way consistent with a direct competition of CDW and SC, but quickly saturates at higher irradiation doses. In the other three compounds, Tc is monotonically suppressed by irradiation. The strongest suppression is found in Ca3Rh4Sn13, which does not have CDW order. We further examine this composition by measuring the London penetration depth, λ(T ), from which we derive the superfluid density. The result unambiguously points to a weak-coupling, full single gap, isotropic superconducting state. Therefore, we must explain two seemingly incompatible experimental observations: a single isotropic superconducting gap and a significant suppression of Tc by non-magnetic disorder. We conduct a quantitative theoretical analysis based on a generalized Anderson theorem which points to an unconventional multiband s +− -pairing state where the sign of the order parameter is different on one (or a small subset) of the smaller Fermi surface sheets, but remains isotropic and overall fully-gapped. I. INTRODUCTION Extensive studies over the past few decades have identified a number of characteristics that are common in unconventional superconductors. First, unconventional superconductivity (SC) often develops in cooperation, competition, or close proximity to other electronic longrange orders. Second, non-Fermi-liquid behavior is often observed in the normal state around the superconducting "dome". This behavior can be associated with proximity to a putative quantum critical point (QCP) inside the dome [1][2][3][4][5][6][7][8][9]. A QCP occurs when a continuous second-order phase transition is driven at T = 0 by a non-thermal parameter, such as composition [10][11][12], pressure [1,13,14], magnetic field [15][16][17] or disorder [2,[18][19][20]. It has been suggested that the fluctuations of the coexisting order parameter may act as a "glue" for Cooper pairing of conduction electrons [1,3,5,9,13]. This approach is actively discussed for high−T c cuprates [4,5,7,21], heavy-fermion materials [14,22], and it is particularly relevant in iron based superconductors where there is a significant range of microscopic coexistence of * Corresponding author: [email protected] antiferromagnetic and superconducting phases [8,9,[23][24][25][26][27][28][29]. In most known cases, the order parameter tuned to a QCP is spin-density wave (SDW). However, chargedensity wave (CDW) ordering is another candidate if it can be continuously suppressed [30][31][32]. While pressure or magnetic field tuning is particularly useful for singular compositions, it is desirable to find superconducting systems tuneable through QCP by doping, allowing for a wider range of different types of measurements. Unfortunately, in most known CDW/SC systems, CDW ordering appears only in single compositions. A CDW is formed when electronic energy is sufficiently lowered by opening an energy gap on parts of the Fermi surface [33][34][35]. Usually this leads to the formation of a spatially-modulated charge-density state. In a onedimensional case, a straightforward nesting determines the modulation wave-vector [33], as observed in onedimensional organic materials [35,36]. In two dimensional systems such as transition metal dichalcogenides, 2H-NbSe 2 [37], 2H-TaSe 2 [38] and 2H-TaS 2 [39]), the nesting mechanism is not so obvious. It is even more complicated in three dimensions, such as our 3-4-13 cubic practically isotropic compounds. The charge density wave in series [40] was studied by a variety of the techniques, and has a arXiv:2110.02025v3 [cond-mat.supr-con] 26 Feb 2022 number of anomalous features. Modulation of the crystal lattice with the wavevector q = (1/2, 1/2, 0) was found in Ca 3 Ir 4 Sn 13 [41] and Sr 3 Ir 4 Sn 13 [42], which does not seem to correspond to nesting conditions. Similarly, in (Sr) 3 (Rh) 4 Sn 13 computational mode decomposition has revealed the same q-vector (1/2, 1/2, 0) [31]. In a closely structurally related compound Yb 3 Co 4 Ge 13 , charge density modulation was found to depend on sample stoichiometry [43]. The EXAFS phase derivative analysis supports the CDW-like formation by revealing different bond distances between two tin sites [Sn1(2)-Sn2] below and above T CDW in the (110) plane in Sr 3 Ir 4 Sn 13 [44]. XANES spectra at the Ir L3-edge and Sn K-edge demonstrated an increase (decrease) in the unoccupied (occupied) density of Ir 5d-derived states and a nearly constant density of Sn 5p-derived states. A close relationship was suggested to exist between local electronic and atomic structures and the CDW-like phase in the Sr 3 Ir 4 Sn 13 single crystal [44]. Inelastic neutron scattering data point towards a displacive structural transition in the Ca 3 Ir 4 Sn 13 compound arising from the softening of a low-energy phonon mode with an energy gap of ∆ =120 K [41]. Softening of the acoustic phonon modes was also suggested by ultrafast spectroscopy study in Sr 3 Ir 4 Sn 13 revealing also a correlation of optical phonons with the transition [45]. Reduction of the magnetic susceptibility and a sign change of the Hall resistivity could be due to transformation of the Fermi surface below T CDW in Ca 3 Ir 4 Sn 13 and Sr 3 Ir 4 Sn 13 [46]. This conclusion is supported optical reflection study [47] and by the anomalies in the NMR Knight shift [48]. Splitting of the NMR lines in the CDW phase imply local distortions of the Sn2 icosahedra [48]. On the other hand, the detailed structure of Remeika series compounds may be much more complicated than usually assumed primitive cubic cell [49][50][51]. The influences of different structural models on the calculated electronic structures of some 3:4:13 compounds were discussed in Ref. 52. Furthermore, unconventional character of CDW and second order phase transition have been found by X-ray structural studies in a related to this work compositions, (La,Ce) 3 (Ir,Rh) 4 Sn 13 [53]. Various mechanisms of CDW formation in these materials are suggested [54]. Importantly, the 3-4-13 compounds, specifically (Ca,Sr) 3 (Ir,Rh) 4 Sn 13 seem to exhibit a putative QCP under the dome of superconductivity [30]. Here we study the influence of controlled disorder on CDW and superconductivity in 3-4-13 superconductors, to uncover the connection between the two quantum orders and the nature of the superconducting state. Intuitively, the opening of the CDW gap should decrease the density of states at the Fermi surface and thus lower the superconducting transition temperature in materials where CDW and superconductivity coexist [55]. This is indeed frequently observed [56]. In the YBa 2 CuO 6−δ , the CDW transition is enhanced when superconductivity is suppressed by magnetic field, and the superconducting transition temperature increases when CDW or-dering is suppressed with pressure [57]. In the transition metal dichalcogenides, 2H-NbSe 2 , 2H-TaS 2 , and 2H-TaSe 2 , 2.5 meV electron irradiation experiments suggested that long-range ordered CDW directly competes with SC so that superconducting transition temperature, T c , increases with the introduction of disorder [18]. However, this simple competition between CDW and SC is only part of the story. Further irradiation experiments showed that as soon as the long-range CDW order breaks down above approximately 6 × 10 18 electrons per cm 2 , T c starts to decrease rapidly, initially in a step like fashion [20]. This implies that CDW also helps superconductivity which benefits from softening of the phonon modes due to long-range CDW order [20]. Phonon softening near the T CDW transition is also observed in Sr 3 Ir 4 Sn 13 [45], Sr 3 Rh 4 Sn 13 [58], and Ca 3 Ir 4 Sn 13 [41]. Furthermore, later studies of 2H-NbSe 2 showed that in systems with electron-phonon pairing mechanism, the largest superconducting gaps occur in the regions of the Fermi surface connected by the CDW nesting vector. [59] The 3-4-13 family of compounds is well-suited for studying the relationship between CDW and superconductivity. Their CDW transition can be tuned through a broad range of temperatures by the selection of different elements or by the application of pressure. The suppression of CDW ordering extrapolates to a region where the resistivity exhibits non-Fermi liquid behavior, suggesting the existence of a QCP in the phase diagram. This QCP was first discovered in (Ca x Sr 1−x ) 3 Ir 4 Sn 13 compounds at the pressure of about 20 kbar [31], and was later found to be accessible via doping in the (Ca x Sr 1−x ) 3 Rh 4 Sn 13 series at around x = 0.9 [30,58]. The structural nature of the QCP was confirmed using x-ray diffraction, showing the continuation of the CDW ordering inside of the superconducting dome [32]. The summary phase diagram as determined from these measurements, with the location of our samples marked, is shown in Fig. 1(a). Experimentally, it is determined that CDW materials exhibit mostly conventional electron-phonon mechanism of superconductivity [60]. In the 3-4-13 compounds most studies, including this work, are consistent with a single isotropic gap weak-coupling superconductivity. Thermal conductivity measurements of Ca 3 Ir 4 Sn 13 found a vanishing residual linear term and a weak increase with applied magnetic field, consistent with a full gap with small or no anisotropy [61]. Heat capacity measurements show exponential decrease at low temperatures [62] and a linear magnetic field dependence [63], which also agree with a full-gap superconducting state. Temperature dependence of the London penetration depth, λ(T ), determined from lower critical field measurements [46] as well as this work discussed later in the text provide strong evidence of a fully-gaped superconducting state. Even more so we found a perfect agreement of the data with λ(T ), expected from the weak-coupling isotropic BCS theory, parameter-free, both close to T → 0 and in the full temperature range. On the other hand, an apparent enhancement of the electronic specific-heat jump at T c in (Sr) 3 (Ir) 4 Sn 13 and (Sr) 3 (Rh) 4 Sn 13 was interpreted as a sign of a strongcoupling nature of superconductivity in these compounds [64]. Furthermore, there are signs of strong coupling superconductivity in heat capacity measurements around the QCP region [65][66][67][68], which could also be due to the contribution of critical quantum fluctuations. Muon-spin rotation (µSR) experiments under pressure find that the superfluid density strongly increases when the system is tuned closer to the QCP in Ca 3 Ir 4 Sn 13 [67]. While µSR measurements of both Ca 3 Ir 4 Sn 13 [69] and Sr 3 Ir 4 Sn 13 [70] agree with a single isotropic gap, they also could not rule out possible two-gap superconductivity with two very different gaps on different Fermi surface sheets. The same group discusses possible multi-band physics from the nuclear magnetic resonance (NMR) measurements [71]. We note that in 2D CDW/SC 2H-NbSe 2 , angleresolved photoemission spectroscopy (ARPES) [72], specific heat [73] and London penetration depth [74] measurements found strong evidence for multi-gap superconductivity. In Sr 3 Ir 4 Sn 13 , possible importance of multiband effects was identified in electronic band-structure study where at least four sheets of the Fermi surface with sizes differing by a factor of nearly 20 were found [75]. Regarding the superconducting gap(s) anisotropy, most measurements are consistent with a fully-gapped isotropic superconducting state described by a weakcoupling Bardeen-Cooper-Schrieffer (BCS) theory [76], which is natural for a phonon-mediated attractive pairing potential. In the case of SDW antiferromagnetic fluctuations as in the cuprates, a sign-changing d−wave pairing is favored [77]. In the present case of CDW/SC compounds, the pairing type is an open question and our present work strongly suggests a possibility of an unconventional multiband s +− -pairing state where the sign of the order parameter is different on one (or a small subset) of the smaller Fermi surface sheets, but remains isotropic and overall fully-gapped. Such a state will manifest itself only in select experiments, such as the response to a non-magnetic disorder. On a general note, there is currently significant revived interest in superconductivity in seemingly conventional compounds, such as elemental niobium where the response to disorder has helped to reveal anisotropic strong-coupling superconductivity [78], or in the case of a Dirac semi-metal from our earlier work [79]. It should be noted that thermodynamic measurements are not sensitive to the sign of the order parameter. On the other hand, studying the variation of T c when changing the non spin-flip (non magnetic) scattering rate is a phase-sensitive method that provides insights into the nature of the order parameter and pairing mechanisms [79,80]. In the well-known limit of an isotropic singleband s−wave superconductor, T c is not affected by weak non-magnetic disorder, known as the "Anderson theorem" [81,82]. In a stark contrast, the transition temperature in materials with anisotropic gap(s) [78,83,84], or sign-changing d−wave superconductivity in the cuprates [85], as well as s +− pairing states in iron-based superconductors [86], is strongly affected by non-magnetic disorder. A generalized treatment extending the original Abrikosov-Gor'kov theory [82] for anisotropic order parameters is given by Openov [87,88], and it can be easily extended to a multiband case with different gap amplitudes [89]. In multi-band and multi-orbital systems, particularly in the presence of spin-orbit coupling, the suppression of T c is expected to be somewhere in between these two limits [79,[90][91][92]. Importantly, combined with independent measurements of the superfluid density and theoretical calculations that take into account particular crystal and electronic structure, the evolution of T c with disorder is a powerful tool to extract important information about the superconducting order parameter [78,79]. In this work, we use artificial point-like disorder to study the relationship between superconductivity and CDW ordering in the stoichiometric compounds of the (Ca,Sr) 3 (Ir,Rh) 4 Sn 13 "3-4-13" Remeika series. The lowtemperature (20 K) 2.5 MeV electron irradiation produces vacancy-interstitials "Frenkel pairs", which leave a metastable population of vacancies upon warming up to room temperature due to very different rates of diffusion of vacancies and interstitials [93][94][95]. This leads to a residual resistivity increase which is monotonic with the irradiation dose, reflecting the increase in the scattering rate. We find that in 3-4-13 compounds, the CDW transition is universally suppressed by disorder. We also observe a weak increase of the superconducting transition temperature, T c , in Sr 3 Rh 4 Sn 13 , and a nonlinear scattering-rate dependence of T c in Sr 3 Ir 4 Sn 13 and Ca 3 Ir 4 Sn 13 . Contrary to the expectations for conventional superconductivity, T c is rapidly suppressed with disorder in Ca 3 Rh 4 Sn 13 , which does not exhibit any longrange CDW order. This behavior became puzzling when precision London penetration depth measurements found a full and isotropic single superconducting gap in this compound. This apparent contradiction is resolved by a detailed theoretical analysis of possible pairing states, which provides strong argument in favor of unconventional multiband s +− -pairing state where the sign of the order parameter is different on one (or a small subset) of the smaller Fermi surface sheets, but remains overall fully-gapped. We note that the influence of atomic defects produced by rapid quenching from high temperatures in Ca 3 Rh 4 Sn 13 was studied thirty years ago using x-ray spectroscopy [95]. The observed reduction of T c [96] was attributed to the creation of Sn-Ca ions exchange anti-sites. Unfortunately, no physical properties, for example conductivity, were measured, hence the dimensionless scattering rate was not determined. The authors of Ref. [95] speculated that T c decreased due to the suppression of the density of states at the Fermi level due to the disturbance of the Ca-Ca bond length. However, we believe that it is more likely that they dealt with the same unconventional mechanism as proposed in our report here. As discussed below, our electron irradiation creates roughly one atomic defect per thousand formula [30] and (CaxSr1−x)3Ir4Sn13 (top axis, square symbols) [31]. The phase diagram for (CaxSr1−x)3(Rh, Ir)4Sn13 was mapped using a combination of doping and pressure. The positions of the samples used in this study are shown by yellow-red stars. (b) Temperature-dependent resistivity of (Sr, Ca)3(Rh, Ir)4Sn13 samples selected for electron irradiation in this study. The inset zooms at the superconducting transition. units, which has no appreciable effect on the density of states. The paper is organized as follows: details of sample preparation and methods are provided in Section II. The experimental results for all compounds can be found in Section III and theoretical analysis in Section IV. Finally, Section V summarizes our findings. II. EXPERIMENTAL METHODS Single crystals of (Ca,Sr) 3 (Rh,Ir) 4 Sn 13 were grown using a high temperature self-flux method, as described in Ref. [63]. X-ray diffraction (XRD) data were taken with Cu K α (λ = 0.15418 nm) radiation of a Rigaku Miniflex powder diffractometer, and the elemental analysis was performed using an energy-dispersive x-ray spectroscopy (EDX) in a JEOL JSM-6500 scanning electron microscope. Electrical resistivity measurements were conducted in a Quantum Design PPMS using a conventional fourprobe method. The contacts to the crystal surface were made by soldering silver wires with tin [97,98]. The contact resistance is below 100 µΩ, and they are sufficiently mechanically stable to withstand electron irradiation [99]. The samples for resistivity measurements were cut and polished from single crystals, with typical sample sizes of (1-2)×0.3×0.1 mm 3 . The long sample axis was arbitrary with respect to the cubic structure of these crystals. Standard resistivity runs were made on both cooling and heating, with negligible hysteresis. The variation of the in-plane London penetration depth, ∆λ(T ), was measured using a sensitive selfoscillating tunnel-diode resonator (TDR) described in detail elsewhere [100][101][102][103]. In brief, the TDR circuit resonates at approximately 14 MHz, and the frequency shift, which is proportional to the sample magnetic susceptibility, is measured with precision better than one part per billion (ppb). The coefficient of proportionality that includes the demagnetization correction is measured directly by pulling the sample out of the resonator at base temperature [103]. This technique was developed specifically to detect minute changes in the London penetration depth and is now considered one of the sensitive tools for studying the anisotropy of the superconducting order parameter [104][105][106]. We use this technique to determine the superconducting gap structure, as well as to show that we do not induce magnetic states with disorder, and that our crystals are very homogeneous. Point-like disorder was introduced at the SIRIUS facility in the Laboratoire des Solides Irradiés atÉcole Polytechnique, Palaiseau, France. Electrons accelerated in a pelletron-type accelerator to 2.5 MeV knock out ions creating vacancy -interstitial Frenkel pairs [93,94]. During irradiation the sample is held in liquid hydrogen at around 20 K. The low-temperature environment is needed not only to remove the significant amount of heat produced by sub-relativistic electrons upon collisions, but also to prevent the immediate recombination and migration of produced atomic defects. The acquired irradiation dose is determined by measuring the total charge collected by a Faraday cage located behind the sample. As such, the acquired dose is measured in the "natural" units of C/cm 2 , which is equal to 1 C ≡ 1/e ≈ 6.24 × 10 18 electrons per cm 2 . Upon warming the sample to room temperature, the interstitials, which have a lower barrier of diffusion [93,94], migrate to various sinks (dislocations, surfaces etc). This leaves a metastable population of vacancies. The resultant vacancy density is determined by the highest temperature the sample was exposed to. In most materials, including 3-4-13, vacancies are stable as verified by the transport measurements of the same samples years apart and even if the density would slowly change, the resistivity measurement provides a snapshot of the current scattering rate in a particular sample. This is the point-like disorder discussed in this paper [107,108]. Practically, the level of disorder induced by the irradiation is gauged experimentally by the change of resistivity well above the CDW transition, at the room temperature, where the carrier density is roughly constant across all compositions and the only change in resistivity comes from the difference in the residual resistivity. We also calculated the number of defects per formula unit (dpf) numerically using specialized "SECTE" software developed inÉcole Polytechnique (Palaiseau, France) specifically to describe ionresolved knock-out cross-sections for MeV-range electron irradiation. The summary of the results for our four compositions is given in Table I. The first three columns show partial cross-sections of the defects created upon head-on collision of a 2.5 MeV electron with an indicated ion, assuming the same value of the barrier for ion displacement from its position, E d = 25eV. This is a "generic" number for intermetallic compounds, usually in the range of tens of eV, and it can be calculated using methods of molecular dynamics [93,94]. However, its exact value is not very important for our rough estimates. The fourth column shows the total cross-section of knocking out any ion by using molecular weight averaging of the partial crosssections. The last column shows the estimated number of defects per formula unit ignoring possible annealing upon warming up after irradiation at 20 K. The realistic percentage lost in that process varies from almost no annealing to about 10 % − 30 %, for example measured by in-situ resistivity in iron pnictides [107]. Our SECTE calculations show that electron irradiation of the 3-4-13 compounds creates less than 1 defect of any kind per 1000 formula units, which cannot alter the chemical or electronic nature of the material. This also means that the defects are well-separated and can be treated as pointlike in the dilute limit. This disorder is much "softer" than that induced by rapid quenching from high temperatures used in earlier experiments [95]. Importantly, electron irradiation does not "dope" the system as was shown directly by Hall resistivity measurements [107]. In the present case, even if there was some induced variation of stoichiometry, T c (x) of 3-4-13 compounds is practically flat and could not result in the systematic shift observed. We note that chemical inhomogeneity and disorder may lead to the significant spread of T c [109]. This, however, would change the observed superfluid density from exponential to a power law at low temperatures. The comparison of the total cross-sections as function of electron energy for the studied compounds is shown in Fig. 2. There is practically negligible differences between Ca-(Ir/Rh)-Sn and Sr-(Ir/Rh)-Sn compounds and quite small differences between (Ca/Sr)-Ir-Sn and (Ca/Sr)-Rh-Sn, where in Ir compounds the cross-sections are larger by about 30 barn. The resulting numbers at our operating frequency of 2.5 MeV are summarized in Table I. In our experiments, the same physical samples were measured before and after electron irradiation, thus avoiding uncertainties from possible variation of stoichiometry within each batch, geometric factors and other parameters unique to each sample. For most compositions, measurements were performed on at least three samples to obtain as objective results as possible, see Table II. III. EXPERIMENTAL RESULTS We now discuss the experimental results obtained in our irradiation studies for the following compounds . This way, we are moving from left to right towards and beyond the quantum critical point in the generic phase diagram shown in Fig. 1(a). The trend in the superconducting transition temperature, T c , is non-monotonic in this sequence, with Sr 3 Rh 4 Sn 13 having the lowest transition at T c = 4.2 K, and overall representing a typical for unconventional superconductors shallow "dome" of superconductivity. The characteristic transition temperatures and resistivity values at room temperature in the pristine state were determined by averaging the measurements of multiple samples as summarized in Table II: The overall resistivity decreases with decreasing T CDW , which is particularly obvious from the measurements on the samples selected for electron irradiation, shown in Fig. 1(b). That comparison also reveals similar slopes of the temperature-dependent resistivity near room temperature. In the full temperature range, the temperature dependence of the resistivity, ρ (T ), is quite unusual. In all compounds, the resistivity in the metallic phase, above T CDW , extrapolates to a very high residual resistivity, similarly to the tantalum dichalcogenides [110]. The resistivity "bump" when crossing into the CDW phase (therefore, lowering carrier density, hence increasing ρ) barely reaches 10 % of the resistivity value at T CDW , and a significant decrease in the resistivity is observed on further cooling down to low temperatures. This behavior suggests that the loss of the carrier density due to the opening of the CDW gap is small, as would naturally be expected for a three-dimensional CDW material. This is in line with NMR measurements of Sr 3 Rh 4 Sn 13 , which found that only approximately 13% of the total density of states is lost in the CDW transition [111]. The very high values of ρ obtained by linear extrapolation from high temperatures to T = 0, and the quick loss of resistivity upon CDW ordering suggest significant contribution of charge-disorder scattering, similar to that suggested by Naito and Tanaka for the transition metal dichalcogenides [112,113]. Interestingly, a similar type of ρ(T ) behavior is observed in Ca 3 Rh 4 Sn 13 , in which long-range CDW is not observed. This may be indicate that despite the total suppression of the long-range CDW ordering in that compound, short-range correlations may persist similarly to the case of CDW suppression by disorder in 2H-NbSe 2 [20] and in doped ZrTe 3 [114]. A. Sr3Ir4Sn13 The temperature dependent electrical resistivity for Sr 3 Ir 4 Sn 13 and its evolution with electron irradiation are shown in the main panel of Figure 3. The resistivity value for the selected sample of Sr 3 Ir 4 Sn 13 is in reasonable agreement with previous reports of 120 µΩ· cm [63,66]. Irradiation shifts the ρ(T ) curves upward at high temperatures, but they remain nearly parallel to each other above T CDW . This can be seen in the plot of the difference between the two curves ∆ρ = ρ(4.4C/cm 2 ) − ρ(0C/cm 2 ), which is shown as the green line in Fig. 3. Matthiessen's rule is largely obeyed above the transition temperature, suggesting that we are in a normal metallic state, albeit one with very high residual resistivity. The minimum in the difference plot is caused by the shift in the CDW transition temperature as irradiation disrupts the long-range order. The suppression of that transition temperature is shown in the left inset via a plot of the derivative of the resistivity, dρ/dT , with arrows indicating the location of T CDW . The superconducting transition temperature is monotonically suppressed with disorder, and sharpens after irradiation. B. Sr3Rh4Sn13 In Sr 3 Rh 4 Sn 13 , similarly to Sr 3 Ir 4 Sn 13 , the CDW transition is monotonically suppressed with the increase of disorder. However, Sr 3 Rh 4 Sn 13 is the only compound in which the expected increase of the superconducting transition temperature, T c , with the suppression of CDW is actually observed. The response of T c to disorder is distinctly non-linear, with a significant initial increase which becomes smaller at higher doses. Also, we found a larger variation of T c between the samples from the same batch when performed initial screening, suggesting that the superconducting state is sensitive to disorder either directly or via the disruption of CDW order. It is possible that in Sr 3 Ir 4 Sn 13 the incipient superconductivity is too weak and T c (x) is too shallow to show any response to the suppressed CDW. In other words, CDW is too strong. Then next in line, Sr 3 Rh 4 Sn 13 , has just right ratio of CDW and SC phases strength to see the effect. Of course, the T c is always monotonically suppressed if CDW is not considered. The full range of resistivity is shown for a selected representative sample in Fig. 4(b). Mattheissen's rule is largely obeyed above the CDW transition, similar to Sr 3 Ir 4 Sn 13 . C. Ca3Ir4Sn13 Ca 3 Ir 4 Sn 13 is the compound with the lowest T CDW . As shown in Fig. 5, a clear feature in the temperature-FIG. 4. Temperature-dependent resistivity of Sr3Rh4Sn13 before irradiation (blue curve), after 1.14 C/cm 2 irradiation (yellow), and after 4.44 C/cm 2 (red) irradiations. The green dashed line shows the resistivity difference between the pristine and 4.44 C/cm 2 curves, finding the Matthiessen rule valid above TCDW, but violated below. The small cartoon in the top left corner indicates the sample position on generic phase diagram. The left inset shows the derivative of the resistivity in a region around the CDW transition with arrows pointing to TCDW. This emphasizes the transition shift between the pristine state and after 4.44 C/cm 2 dose of irradiation. The right inset zooms into a region around the superconducting transition showing a non-monotonic behavior of Tc: a small initial Tc increase after 1.14 C/cm 2 irradiation, but only minimal changes in the behavior between 1.14 and 4.44 C/cm 2 . dependent resistivity is observed at ∼ 40 K in the pristine sample (arrow in the derivative plot, left inset). It is also the closest CDW composition to the structural quantum critical point. The suppression of CDW with irradiation is clear for 2.17 C/cm 2 irradiation. The transition feature in the derivative plot cannot be resolved at 5.47 C/cm 2 , suggesting that the CDW order has been completely suppressed. Similar to Sr 3 Ir 4 Sn 13 , but unlike Sr 3 Rh 4 Sn 13 increasing the disorder in this compound only decreases the superconducting transition temperature Furthermore, unlike the Sr 3 (Ir,Rh) 4 Sn 13 compounds, Matthiessen's rule is weakly violated in this material for all temperatures, below and above T CDW . Matthiessen's rule holds in good metals, where the introduction of disorder affects only the residual resistivity (scattering off the defects and impurities) and appears as a constant offset. Since the change of resistivity under increasing disorder in Ca 3 Ir 4 Sn 13 is more complex than just a constant offset, it may suggest the presence of a short-range order consistent with the above discussion and similarity with 2H-NbSe 2 [20]. D. Ca3Rh4Sn13 Ca 3 Rh 4 Sn 13 is our only compound which does not have a long-range CDW ordering, and is positioned to the right of the quantum critical point in the generic phase diagram in Fig. 1(a). Still, the evolution of the temperaturedependent resistivity with disorder, Fig. 6, reveals that Matthiessen's rule is conspicuously not obeyed in the "normal" state. Similar to Sr 3 Ir 4 Sn 13 , this suggests that there is some other type of (short-range) electronic order which is affected by the introduction of point-like disorder. One potential candidate is the residual shortrange CDW order which persisted across the QCP, as observed in 2H-NbSe2 [20]. We note that second-order structural phase transition in a 3-4-13 family, specifically La,Ce 3 Rh 4 Sn 13 , has been discussed in the context of unconventional chiral CDW based on structural X-ray studies [53]. The superconducting transition temperature in Ca 3 Rh 4 Sn 13 is significantly affected by electron irradiation. T c is suppressed from T c,0 = 8.2 K by more than 2 K after 2.1 C/cm 2 of irradiation using the zeroresistance offset criterion, and the transition broadens. Potential scattering is not expected to suppress T c in conventional isotropic single-band s−wave superconductors, so we must consider the possibility of nodal superconductivity, or at least a strong variation of the superconducting gap magnitude on the Fermi surfaces. This will be addressed in detail in the next section. IV. DISCUSSION The increase of a sample's residual resistivity as a function of irradiation dose is an intrinsic measure of the disorder introduced by irradiation. However, because the resistivity in the CDW state depends on the size of the gapped part of the Fermi surface, which is compounddependent, a direct comparison across chemical compositions is not very informative. A better proxy for the quantification of the effect of disorder across compounds is the resistivity in the metallic state above T CDW . As can be seen from Fig. 1(b), the ρ(T ) curves for all compounds are nearly parallel approaching room temperature, and the overall resistivity variation does not exceed 30 % or so. The values of ρ (300 K) are listed in Table II. This validates the assumption of practically negligible differences in the carrier density between the different compounds in the normal state at elevated temperatures. By measuring the change in the resistivity in the normal state, we can thus determine the change in the disorder scattering. Combined with the numerical estimates of the defect density as described in Section II, these measures allow for a direct comparison between different samples. A. Interplay of charge-density wave and superconductivity In the following, we summarize critical temperatures extracted from figures 3-6 as a function of irradiation. Ideally, such summary plots would show the error bars in both X and Y directions. However, it cannot be done in our case because we did not measure many samples that would allow statistical analysis. On the other hand, each critical temperature is determined with such precision that the corresponding uncertainty error bar is smaller than the symbol size. This is also true for estimating the X axis values that involve measured total irradiation dose and residual resistivity. Figure 7(a) shows the evolution of the CDW ordering temperature, T CDW , with the defects per formula unit (dpf). The observed dependence is striking. While the CDW is suppressed at the same rate in Sr 3 (Ir,Rh) 4 Sn 13 compounds, the closer to QCP Ca 3 Ir 4 Sn 13 shows a much larger suppression rate. Intriguingly, this is the composition where the Matthiessen's rule is violated above the CDW transition, as it is expected that quantum fluctuations affect the properties near QCP. A similar graph of the normalized T c /T c,0 in Fig. 7(b) shows complex behavior. The rate of suppression is similar in samples [A] FIG. 8. Normalized suppression of the superconducting transition temperature, Tc/Tc0, as function of the dimensionless scattering rate γ λ evaluated from resistivity and London penetration depth, λ (0) using Eq.(1). To set the scale, grey dashed line shows the predictions of Abrikosov-Gor'kov theory for a line-nodal d-wave superconductor with non-magnetic impurities [82,87,88]. Purple stars are experimental data for the nodal iron-based superconductor BaFe2(As1−xPx)2 [10], redyellow squares are for the superconducting Dirac semimetal PdTe2, maroon cross-pentagons are for CDW superconductor 2H-NbSe2. The data for 3-4-13 compounds are shown by red rhombi for Ca3Rh4Sn13, blue squares for Ca3Ir4Sn13, yellow-blue circles for Sr3Rh4Sn13 and yellow triangles for Sr3Ir4Sn13. and [C] , while T c increases in sample [B]. Such increase is expected when superconducting pairing and chargedensity wave inter-band interaction energies are comparable and the enhancement of superconductivity due to CDW suppression over-weights the natural suppression of T c by disorder. For Ca 3 Rh 4 Sn 13 which is away from CDW phase, the suppression of T c is dramatic, despite the fact that its T c,0 is similar to [B] and [C]. Interestingly, and consistent with this picture as soon as CDW is completely suppressed in sample [C], the T c suppression becomes much faster and similar to sample [D]. Our measurements establish that in 3-4-13 stannides, there is a direct competition of CDW and superconductivity, in addition to quantum fluctuations around QCP that affect even normal-state properties. Of course, despite similarities, we are dealing with four distinctly different compounds and some unique structural and/or electronic features may certainly contribute to the results. B. Matthiessen's rule The temperature-dependent resistivity in the normal state of the 3-4-13 system is anomalous and reveals notable Matthiessen's rule violations in the vicinity of the QCP, but not too far away. Comparison of the four compounds finds some similarity at high temperatures. At temperatures above massive downturn in the resistivity on cooling, coinciding with T CDW in Sr 3 Ir 4 Sn 13 and Sr 3 Rh 4 Sn 13 , the ρ(T ) curves extrapolate to quite high values in T = 0 limit. This feature is known to be caused by spin-disorder scattering in magnetic materials [115]. It is also observed above T CDW in tantalum dichalcogenides, TaS 2 and TaSe 2 [112,113] and was suggested to be scattering on charge fluctuations above the transition. In Ca 3 Ir 4 Sn 13 and particularly strongly in Ca 3 Rh 4 Sn 13 , the low-temperature downturn in ρ(T ) does not coincide with long range CDW ordering. This type of response may be suggestive of the scenario realised in NbSe 2 [20]. Here long range charge density wave ordering is suppressed with irradiation, however short range ordering remains unaffected. It is interesting to compare this behavior with another fully gapped system where doping-dependent spin density (SDW) wave coexists with superconductivity, for example electron-doped Ba(Fe 1−x Co x ) 2 As 2 (BaCo122) [12,107] and iso-electron-substituted iron-based superconductor BaFe 2 (As 1−x P x ) 2 [19], both showing proven SDW/QCP under the dome of superconductivity [10][11][12]. In these compounds, Matthiessen's rule is obeyed near the QCP, as well as it is in the cuprates if the sample is not in the regime of weak localization [85]. On the other hand, the observed behavior of 3-4-13 bears some similarity to the hole doped Ba 1−x K x Fe 2 As 2 (BaK122) in which Matthiessen's rule is also strongly violated [107]. C. Dimensionless scattering rate To put our data in a broader perspective, we compare the T c suppression rate in the 3-4-13 compounds with other known cases. For this, we will use a dimensionless scattering rate defined as defined as [79,108] γ λ = ∆τ −1 2πk B T c,0 = 2πk B µ 0 ∆ρ 0 λ 2 clean (0)T c,0 .(1) Here, ∆ρ 0 is the change of the residual resistivity after irradiation compared to the pristine state value, and λ clean (0) is the zero temperature penetration depth in the pristine sample. Note that we obtain ∆ρ 0 by extrapolation to T = 0. Inserting the dimensional constants and using units of µΩcm for ∆ρ 0 , 10 −7 m for the penetration depth λ clean (0), and K for T c,0 , Eq. 1 takes the form γ λ = 0.97∆ρ 0 / λ 2 clean (0)T c,0 . To arrive at Eq. (1) we used the simple Drude model for resistivity, ρ = m * /(ne 2 τ ), and the London model for the penetration depth, λ 2 clean (0) = m * / µ 0 ne 2 [116] (see also Appendix D of Ref. [79]). Note that we have used that the superfluid density equals the total carrier density at zero temperature. This allows expressing the (change of the) normal-metal scattering time via measurable parameters, ∆τ −1 = ∆ρ 0 /µ 0 λ 2 clean (0). We note that λ clean (0) and the normal-state scattering time, τ , do not depend on parameters of superconductivity and Eq. (1) can thus be used for different gap symmetries. Now we can compare the results of 3-4-14 stannides with various theoretical expectations as well as other superconductors in which the effect of disorder was studied. Figure 8 shows normalized T c suppression for our four systems as a function of γ λ . The data are compared with nodal s ± BaFe 2 (As 1−x P x ) 2 , [117], Dirac semi-metal PdTe 2 [79], and CDW superconductor 2H-NbSe 2 [20]. The expectation from the Abrikosov-Gor'kov theory for a single-band d-wave superconductor with non-magnetic scattering [82,87,88], shown by the dashed line, provides the scale for the largest suppression rate possible. While in three CDW/SC 3-4-13 compounds, it can be argued that anything is possible due to cooperation and/or competition between these two quantum orders, the significant T c suppression rate in Ca 3 Rh 4 Sn 13 is shown to be intermediate between nodal and nodeless superconductors. In fact, it is comparable to the T c suppression rate in the nodeless sign-changing s +− state of the optimally doped Ba(Fe 1−x Ru x ) 2 As 2 [108], and is significantly higher than that of a two-gap s ++ 2H-NbSe 2 after the suppression of CDW order. This relatively high T c suppression rate naturally raises questions about the superconducting gap structure of Ca 3 Rh 4 Sn 13 and to get an insight into the momentum dependence of the order parameter, we measured the London penetration depth in Ca 3 Rh 4 Sn 13 . D. London penetration depth of Ca3Rh4Sn13 To examine the anisotropy of the energy gap, we used a sensitive tunnel-diode resonator technique, described in the experimental methods, Sec. II, to measure the lowtemperature variation of the London penetration depth in Ca 3 Rh 4 Sn 13 . Figure 9 shows the variation of the superfluid density, ρ s = λ 2 (0)/λ 2 (T ), calculated from the measured variation of the London penetration depth, ∆λ(T ) = λ(T ) − λ(0), over the whole temperature range. This is important to detect possible signatures of two-gap superconductivity. The top right inset in Fig. 9 shows a full-range variation of ∆λ(T ) and the lower left inset zooms on the characteristic low-temperature range, approximately T < T c /3, where the order parameter amplitude is practically constant and any changes in λ(T ) come from the quasiparticles generated due to angular variation of the gap function. The red line in the bottom-left inset shows an excellent fit to the isotropic single-gap s-wave function with λ(0) = 330 nm and ∆ 0 /k B T c = 1.764. There are no reported measurements of λ(0) in Ca 3 Rh 4 Sn 13 , however µSR measurements report λ(0) = 291 nm in Sr 3 Ir 4 Sn 13 [70] and λ(0) = 351 nm in Ca 3 Ir 4 Sn 13 [69]; so our measurement is perfectly in range considering that λ(0) is a normal-state property that depends only on the parameters of electronic bandstructure. The superfluid density calculated from the obtained λ(T ) = λ(0) + ∆λ(T ) (main panel, symbols) is in very good agreement with the parameter-free prediction for an isotropic full-gap s-wave superconducting state (main panel, thick orange line). For comparison, the expectation for a d−wave superconductor is shown by the dashed line. This nearly perfect and robust agreement with the simplest isotropic BCS is at odds with the significant rate of the disorder-induced reduction of the T c . As we show below, these conclusions are impossible to reconcile without invoking unconventional pairing in Ca 3 Rh 4 Sn 13 . Measurements of λ(T ) allow us to address the question of whether the defects induced by electron irradiation become magnetic. In principle, non-magnetic ions may become magnetic when their ionization changes. Such magnetic defects can cause pair-breaking due to spin-flip scattering, resulting in a T c suppression even in isotropic s−wave fully-gapped superconductors [82,87,88]. Our precision measurements of the London penetration depth in this system exclude this scenario. Due to the sensitivity of these measurements, even a minute paramagnetic signal coming from magnetic defects would be detected. In particular, in the presence of magnetic impurities, the London penetration depth estimated from the magnetic susceptibility measurements (such as tunnel-diode resonator) is renormalized as λ m (T ) = µ (T )λ (T ), where λ (T ) is the London penetration depth of a superconducting sample without magnetic impurities and µ (T ) is the normal-state magnetic permeability due to dilute non-interacting magnetic moments (ions, impurities etc). We refer to Ref. [118], which shows what the measured penetration depth looks like when this effect is relevant. Here we do not see any trace of the paramagnetic upturn expected if we had magnetic impurities. From the concentration of defects induced by irradiation, which is up to 5 × 10 −3 dpf, the volume of the conventional unit cell, 9.7Å 3 , one obtains with Z = 2 formulas for the the concentration of defects in conventional units n d ≈ 1 × 10 25 m −3 = 18 mol/m 3 . Now we can evaluate the Curie constant. Assuming the simplest case, that each scattering center is a two-level system with the magnetic moment of one Bohr magneton, µ = µ B = 9.27 × 10 −24 J/T. With the estimated n d we obtain, C = µ 0 n d µ 2 B /k B ≈ 7.8 × 10 −5 K. This is a very small number even for such a large moment. It gives a correction to our penetration depth, λ (0) = 330 nm, of approximately ∆λ (0.4 K) ≈ 0.32Å at the minimum temperature of 0.4 K. This is a negligible correction. Of course, when T → 0, it will grow large, but in this paper we are mostly examining what happens at T c , and such a dilute system will not be able to shift T c in any appreciable way. If for some reason the magnetic moment is larger or more defects are generated, the measurements of London penetration depth are capable of resolving sub-nm variation and would pick up such a signal. We can therefore say with confidence that magnetism of the defects induced by electron irradiation does not play a role in the obtained results. Finally, together with a very sharp resistive and magnetic transitions in pristine sample, the behavior of λ(T ) also rules out possible chemical and structural inhomogeneities that were shown to lead to a significant spread of the observed T c in polycrystalline Ca 3 Rh 4 Sn 13 [109]. E. Candidate pairing states for Ca3Rh4Sn13 Since Ca 3 Rh 4 Sn 13 does not exhibit a transition into a CDW phase, the normal state symmetries out of which superconductivity emerges are expected to be those of the room-temperature symmetry group of the 3-4-13 seriesthe space group P m3n (no. 223) with associated point group O h ; this is confirmed by XRD measurements [119]. Both in the literature and in our measurements, there are no indications of multiple consecutive superconducting transitions. Therefore, we can use the irreducible representations (IRs) of the normal-state symmetry group [120] to classify the superconducting order parameters. In the absence of magnetic fields, it is further natural to assume that the pairing order parameter transforms trivially under lattice translations and we can focus on the IRs of the point group O h . Note that the involved atoms are moderately heavy and we thus expect spin-orbit coupling to be sufficiently strong such that the symmetries of O h should be thought of as acting on the spatial coordinates (three-dimensional momentum k) and spin simultaneously. Since O h contains inversion, i, all bands are doublydegenerate despite the presence of spin-orbit coupling. We label the degeneracy with a pseudospin quantum number. Another consequence of i ∈ O h is that all IRs decay into even, g, and odd, u, representations under i, associated with pseudospin singlet and triplet. For each µ = g, u, O h has two 1D IRs, A 1µ and A 2µ , one 2D IR, E µ , and two 3D IRs, T 1µ and T 2µ , leading to a total number of 10 IRs. This gives rise to a large number (26) of possible pairing states [120]. However, most of these states necessarily have nodes which is not consistent with the observed temperature dependence of the penetration depth in Fig. 9. As summarized in Table III, only six states are left that can be fully gapped. When specifying the superconducting order parameter, ∆ k , in Table III, we focus on generic momenta on the Fermi surfaces without additional degeneracies between different bands. Therefore, ∆ k can be taken to be a 2 × 2 matrix in pseudospin-space, which we have expanded in terms of Pauli matrices σ j in Table III. These six candidate states can be further divided into two categories: (i) four states that will be fully gapped right below T c since their primary order parameters are associated with a full gap: these are the regular BCS swave, spin singlet state, transforming under A 1g , a helical triplet (A 1u ), and two triplets transforming under E u and T 2u , respectively; (ii) two states where the primary order parameter has line nodes but which, once non-zero, can induce secondary superconducting orders that have a full gap: these are two singlets, one transforming under E g and one under T 2g . The states (ii) are not consistent with experiment for the following reasons: they will have line nodes in a finite range below T c , which together with the temperaturedependent admixture of a secondary order parameter, is generically expected to lead to a more unconventional temperature dependence of the penetration depth than what is seen in Fig. 9. Further, the admixture of the secondary component has to be extremely large to not only remove the nodes but also lead to an approximately isotropic gap function (see also Appendix A 1). Among the remaining four states of type (i) in Table III, we can further distinguish between (ia) states that can have a fully isotropic gap function and (ib) states which are, by symmetry, forced to have a momentumdependent order parameter that generically leads to a significantly momentum-dependent gap. The ratio of the maximum to minimum gap size, ∆ max /∆ min , on the Fermi surface is expected to be of the order of 2 for the (ib) states. Based on the penetration-depth data, the (ia) states thus seem more natural candidates. We therefore focus for the following analysis of the irradiation data on the A 1g singlet and A 1u triplet states. . The first four states above the horizontal line can be fully gapped right below the superconducting critical temperature Tc. The two states below the horizontal line exhibit line nodes right below Tc but can, in principle, be fully gapped at sufficiently low T . The column dn denotes the dimensionality of the IR and the column TRS states whether the pairing state has time-reversal symmetry. We use the short-hand notation X = X k , Y = Y k , Z = Z k to denote real-valued Brillouin-zone-periodic functions that transform as x, y, and z under O h . We also indicate the ratio of the maximal to minimal value of the superconducting gap, ∆max/∆min, for an isotropic Fermi surface around k = 0 and with (X, Y, Z) = (kx, ky, kz). As discussed in the text, only the states of "type" (ia) are natural candidates consistent with the temperature dependence of the superfluid density in Fig. 9. IR pairing dn TRS order parameter ∆ k iσy ∆max/∆min type A1g s wave 1 a + b(X 2 + Y 2 + Z 2 ) 1 (ia) A1u p wave 1 Xσx + Y σy + Zσz 1 (ia) Eu e u(0,1) wave 2 2Zσz − Xσx − Y σy 2 (ib) T2u t 2u(1,1,1) wave 3 (Y + Z)σx + (X + Z)σy + (X + Y )σz 2 (ib) Eg e g(0,1) wave 2 2Z 2 − X 2 − Y 2 ∞ (line nodes) (ii) T2g t 2g(1,1,1) wave 3 Y Z + ZX + XY ∞ (line nodes) (ii) F. Constraints on pairing from sensitivity to disorder scattering To quantitatively analyze the measured impact of impurities on T c in Ca 3 Rh 4 Sn 13 , we use the general expression derived in Ref. 79 for the sensitivity parameter ζ that describes the disorder-induced reduction of the superconducting critical temperature according to T c,0 − T c (τ −1 ) T c,0 ∼ π 4T c,0 τ −1 ζ.(2) This expression holds in the limit of weak scattering rates, τ −1 → 0, where ζ corresponds to the linear slope of the T c reduction as a function of τ −1 . With the normalization in Eq. (2), we have ζ = 1 for magnetic impurities in a single-band, isotropic, spin-singlet superconductor and ζ = 1/2 for a single-band d-wave superconductor in the presence of non-magnetic impurities (see grey dashed line in Fig. 8). Comparison of the slopes in Fig. 8 We begin with the A 1g singlet and assume a general momentum-dependent order parameter, ∆ k = Ψ k iσ y where Ψ k is invariant under all symmetries of O h . Considering point-like, non-magnetic disorder without any momentum dependence in the pseudospin basis, Eq. (A7) readily yields ζ = \|Ψ k \| 2 FS − \| Ψ k FS \| 2 2 \|Ψ k \| 2 FS ;(3) here . . . FS denotes the average over all momenta k on the Fermi surfaces of the system (normalized such that 1 FS = 1). Note that our assumption of disorder neglects the fact that the wavefunctions at the Fermi surfaces are composed of k-dependent superpositions of spin and different orbitals, which is expected [79,[90][91][92] to reduce the impact of disorder on T c further. Therefore, the following values of ζ should technically be viewed as upper bounds. It holds ζ = 0 in Eq. (3) if Ψ k is independent of k, recovering the well-known Anderson theorem [81]. Therefore, to obtain finite ζ in Eq. (3) for the A 1g singlet, we need to allow for momentum dependent Ψ k . Let us first assume that this momentum dependence arises from Ψ k varying within a closed Fermi sheet. To illustrate the consequences for ζ, we will for concreteness focus on a single Fermi surface enclosing the Γ point. Let us approximate it to be spherical, and only include the lowest-order lattice harmonic (g-wave in this case) correction to Ψ k = Ψ 0 that transforms under the trivial IR A 1g of O h , Ψ k = Ψ 0 1 + δ[k 4 x + k 4 y + k 4 z ] .(4) Here the parameter δ determines the strength of the momentum-dependent perturbation and has to be real as a gauge has to exist where Ψ k ∈ R (due to timereversal symmetry in the normal state). Note that the superconductor will be nodal if and only if −3 < δ < −1. From Eq. (3), it is straightforward to evaluate ζ which is found to be ζ(δ) = 8 δ 2 5δ(41δ + 126) + 525 . As expected, we have ζ(δ = 0) = 0 since the order parameter is momentum independent when δ = 0. The maximal value of 1/2 is reached when δ = −5/3 for which the Fermi surface average of Ψ k vanishes. For large \|δ\|, the order parameter approaches that of the subleading, g-wave basis function, associated with a value lim \|δ\|→∞ ζ(δ) = 8/205 ≈ 0.039. x − k 2 y ), cases are shown by dashed lines. Clearly, the computed superfluid density is far from the experimental data shown in Fig. 9. (b) Two-band A ++ 1g superconductor with two different gap magnitudes, ∆1 and ∆2. To reproduce the observed Tc suppression, the gap ratio should be ∆1/∆2 = 2.78, see text. There are many sets of the interaction matrix elements to obtain that value at Tc but with different temperature dependencies of ∆1/∆2 below Tc (see inset). We show the computed ρs(T ) for several choices, but none of them is consistent with the experimental data. See Appendix B for more details of the computations. Most importantly, for the experimental value ζ = 1/9, Eq. (5) is only consistent with two possible values of δ: either δ ≈ −1.08, which leads to a superconductor with nodal lines, or δ ≈ −3.66, for which the superconductor almost exhibits nodal lines; the associated anisotropy is quite large, ∆ max /∆ min ≈ 12. For both values of δ, we have computed the temperature dependence of the superfluid density ρ s , see Fig. 10(a) for the results and Appendix B for more details. As can be clearly seen, the strong anisotropy or presence of nodes leads to a ρ s (T ) profile that differs significantly from the observed s-wavelike behavior and more closely resembles that of a d-wave state. Since none of these two values of δ are consistent with our data, we conclude that the momentum dependence of Ψ k on one (or several) Fermi sheets is not a possible cause of the observed suppression of T c . Next, we consider the possibility that the order parameter of the A 1g state varies between different sheets. As can be seen in Fig. 11, the normal state of Ca 3 Rh 4 Sn 13 has eight bands crossing the Fermi level, giving rise to very complex Fermi surfaces. Assuming that Ψ k is constant on each Fermi sheet, we write Ψ k = ∆ n if k belongs to the nth sheet. Denoting the density of states at the Fermi level of the nth Fermi surface by ρ n , we find ζ = 1 2 1 − \| n ρ n ∆ n \| 2 ( n ρ n \|∆ n \| 2 ) n ρ n(6) from Eq. (3). We note in passing that it is also possible that the order parameter on different, symmetry unrelated pockets exhibits non-trivial complex phases, due to frustration [123,124], ∆ * n ∆ n / ∈ R. However, this can only happen via two (or more) consecutive superconducting transitions, as a result of time-reversal symmetry. As there are no indications of multiple transitions in Ca 3 Rh 4 Sn 13 we will assume ∆ n ∈ R. Since ∆ * n ∆ n < 0 is expected to be impossible for a conventional phonon-mediated pairing mechanism [125,126], we first focus on the case where ∆ * n ∆ n > 0 for all n, n , which we refer to as the A ++ 1g state. In the simplest case of only two different gap magnitudes in Eq. (6), it is straightforward to show via optimization of the respective density of states that the maximum possible ζ for given ∆ 1 /∆ 2 reads as ζ max ∆ 2 ∆ 1 = ζ max ∆ 1 ∆ 2 = (∆ 2 /∆ 1 − 1) 2 2(1 + ∆ 2 /∆ 1 ) 2 .(7) From this, it is easy to see that ζ = 1/9 can only be reached when ∆ 2 /∆ 1 > 2.78 (or ∆ 1 /∆ 2 < 0.35), which is not consistent with the penetration depth measurement, see Fig. 10(b). For reference, ζ max ≈ 0.004 1/9 assuming a maximal imbalance of 20%, ∆ 2 /∆ 1 = 1.2. In Appendix A 3, we show that this conclusion is not altered by allowing for three or more independent gaps. We also use values of ρ n determined from first-principle calculations to show that, irrespective of how two different gap magnitudes are distributed among the various Fermi surface sheets, the minimal gap anisotropy consistent with ζ = 1/9 is ∆ max /∆ min ≈ 2.86. Since our data cannot be explained by the A ++ 1g , we now allow for ∆ * n ∆ n < 0. Such a multiband s +− state, which we denote by A +− 1g , cannot be stabilized by electron-phonon coupling alone and, hence, requires an unconventional pairing mechanism. As a consequence of the sign change, two different gap values ∆ 1 and ∆ 2 with ∆ * 1 ∆ 2 < 0 are sufficient to cause much larger ζ in Eq. (6): the maximum possible value of ζ = 1/2 is reached when ρ 1 \|∆ 1 \| = ρ 2 \|∆ 2 \|. Consequently, for the A +− 1g states, the crucial question is whether ζ = 1/9 is too small. In Appendix A 3, we show that there are multiple different ways of distributing ∆ 1 and ∆ 2 with ∆ 1 /∆ 2 ≈ −1, i.e., with almost identical (and isotropic) gaps, among the various Fermi sheets. Consequently, the unconventional A +− 1g state is thus far the only option consistent with our measurements. Finally, let us look into the A 1u triplet. As readily follows from the general expression for ζ derived in [79], it holds ζ = 1/2 for the A 1u triplet state. In fact, ζ = 1/2 holds for any other unconventional pairing state such as the E g and T 2g singlets in Table III; as already discussed, these latter two, are less natural candiates for Ca 3 Rh 4 Sn 13 since their gap function is expected to have an anisotropy of about 2, while the gap of the A 1u triplet state can be completely isotropic. The value of ζ = 1/2 is still too large by about a factor of four. However, as alluded to above, assuming impurities that have k-independent, pseudospin trivial matrix elements on the Fermi surfaces may not be such a good approximation in a complex multi-orbital material such as Ca 3 Rh 4 Sn 13 . In particular, the presence of spin-orbit coupling can further reduce ζ significantly as discussed in several previous works [79,[90][91][92]. Therefore, the A 1u triplet cannot be excluded based on our observations, but requires the additional, yet not implausible, assump-tion of spin-orbit-coupling-induced suppression of disorder matrix elements between relevant states at the Fermi surface (see, e.g., Ref. 79 for a general discussion of this aspect). One observation that further disfavors the A 1u triplet, however, is that the amount of spin-orbit coupling in the Bloch states at the Fermi surface should vary significantly among the four stannides studied and yet the suppression of T c with disorder is of the same order in Ca 3 Rh 4 Sn 13 and Ca 3 Ir 4 Sn 13 after CDW order has been fully suppressed in the latter, see Fig. 7, right panel. Taken together, the A 1u triplet cannot be rigorously excluded based on our data but requires more fine-tuning and additional assumptions than the A +− 1g superconductor. V. CONCLUSIONS We have studied the impact of controlling the number of non-magnetic defects on the transition temperatures of the superconducting, T c , and CDW, T CDW , phases in the four stoichiometric 3-4-13 stannides Sr 3 Ir 4 Sn 13 , Sr 3 Rh 4 Sn 13 , Ca 3 Ir 4 Sn 13 , and Ca 3 Rh 4 Sn 13 . While T CDW is suppressed with increasing defect concentrations in the three compounds that exhibit CDW order, the behavior of superconductivity is more complex, see Fig. 7, and reveals non-trivial microscopic physics. The suppression of T c with weak disorder is by far the strongest in Ca 3 Rh 4 Sn 13 , which does not exhibit any long-range CDW. Furthermore, T c increases with weak disorder in Sr 3 Rh 4 Sn 13 . All of these findings are consistent with a picture where CDW and superconductivity compete. Quantitatively, the suppression of T c with disorder in Ca 3 Rh 4 Sn 13 is about only 4.5 times weaker than the theoretical expectation for a nodal superconducting gap function, such as d-wave, with a vanishing average order parameter on the Fermi surface, see Fig. 8. However, the measured temperature dependence of the London penetration depth, Fig. 9, indicates a full isotropic gap. Based on the symmetries of the normal state, we classified the pairing states in Ca 3 Rh 4 Sn 13 and list those which can have a full gap in Table III. Among those, only the A 1g singlet and A 1u triplet are naturally consistent with the nearly isotropic gap. Based on a quantitative comparison [79] of theory and the measured disorder-induced change of T c in Ca 3 Rh 4 Sn 13 , a conventional A 1g singlet, where the sign of the order parameter is the same on all Fermi surfaces, is not consistent with the data. Instead, the A +− 1g singlet, a multiband s +− state, where the sign of the order parameter is different on one (or a small subset) of the smaller Fermi surfaces, naturally reproduces the observed suppression of T c . While we cannot rigorously exclude the A 1u triplet, further assumptions about the matrix elements of the disorder potential on the Fermi surfaces are required to reduce the impact of disorder on its critical temperature. In either case, the pairing mechanism giving rise to the A +− 1g or A 1u superconductor cannot [125,126] be based entirely on electron-phonon cou-pling, and thus must be unconventional. Similarly, since regular time-reversal-invariant CDW fluctuations cannot induce unconventional pairing [125], our work motivates further investigations into a possible microscopic origin of unconventional pairing. While this conclusion about unconventional pairing only directly applies to Ca 3 Rh 4 Sn 13 , it is natural to expect that the superconductivity has a very similar nature in all of the studied stannides. We observed an extremely similar superfluid density in Ca 3 Ir 4 Sn 13 (a separate study to be published), which indicates that it is also a fully gapped superconducting state. As shown in Fig. 7, T c is only weakly suppressed in Ca 3 Ir 4 Sn 13 when CDW is present, but is suppressed at a similar rate to Ca 3 Rh 4 Sn 13 as soon as CDW is suppressed. Therefore, it is reasonable to conclude that the underlying T c suppression rate is the same in Ca 3 Rh 4 Sn 13 as in Ca 3 Ir 4 Sn 13 , implying similar unconventional pairing. In this sense, Ca 3 Rh 4 Sn 13 could be the key compound to unravel the microscopic physics of superconductivity in the 3-4-13 series. To understand why the superconducting order parameters in the last two lines of Table III allow for admixture of a secondary order parameter that can be fully gapped, let us first focus on the e g (0, 1) state. When the order parameter ∆ k = η Eg 2 (2Z 2 k − X 2 k − Y 2 k )iσ y becomes non-zero at T c , it reduces the point symmetries not only in charged but also in charge-0 observables, such as the spectrum E k of the Bogoliubov quasi-particles. Formally, this means that for some g ∈ O h , no ϕ g ∈ R exists such that ∆ gk = e iϕg ∆ k , ∀ k. (A1) For the e g (0, 1) state, the maximal set of g ∈ O h for which a ϕ g in Eq. (A1) exists forms the subgroup D 4h of O h ; therefore, E k will only be invariant under these symmetries, while E gk = E k for g ∈ O h \D 4h (such as three-fold rotational symmetry). Since the order parameter of the e g (0, 1) state transforms under the trivial representation, A 1g , of D 4h [⇔ ϕ g = 0 in Eq. (A1) for all g ∈ D 4h ], it can couple linearly to the A 1g singlet in Table III. This coupling requires O h to be broken due to η Eg 2 = 0 and is thus a higher-order process in η Eg 2 . As such, we expect the admixed component to have a temperature dependence ∝ (T c − T ) n/2 , with n > 1, close to T c . We note that this would be different in case of the e g (1, 0) superconductor with order parameter ∆ k = η Eg 1 (X 2 k − Y 2 k )iσ y ; while it will also reduce O h to D 4h , we will have ϕ C z 4 = ϕ σ d = π in Eq. (A1) such that the order parameter will transform as B 1g under D 4h . Being odd under the mirror planes σ d of D 4h , any B 1g singlet will necessarily have line nodes. To demonstrate the admixture for E g pairing more explicitly and determine the exponent n in the temperature dependence of the secondary order parameter, we will next discuss it on the level of a Ginzburg-Landau expansion. To this end, we expand the order parameter in the E g and A 1g representation of O h as ∆ k iσ y = η Eg 1 √ 3(X 2 k − Y 2 k ) + η Eg 2 (2Z 2 k − X 2 k − Y 2 k ) + η A1g . (A2) As they transform under different IRs of O h , there cannot be a quadratic coupling of the form (η Eg j ) * η A1g , but upon noting that E g ⊗E g ⊗E g = A 1g ⊕A 2g ⊕3E g it is clear that quartic terms of the form (η Eg j ) * (η Eg k ) * η Eg l η A1g exist. As η † σ x η and η † σ z η, with η = (η Eg 1 , η Eg 2 ) T , transform as √ 3(x 2 − y 2 ) and 2z 2 − x 2 − y 2 under O h , the following coupling is allowed in the free energy: κ (η A1g ) * η † σ x η η Eg 1 + η † σ z η η Eg 2 + c.c.,(A3) where κ ∈ R as a consequence of time-reversal symmetry. In agreement with our discussion above, we find that the coupling vanishes for the e g (1, 0) superconductor, where η Eg 2 = 0; the same holds for the time-reversal-symmetrybreaking e g (1, i) state for which η = (1, ±i). On the other hand, it is non-zero and given by − 2κ\|η Eg 2 \| 2 Re[(η A1g ) * η Eg 2 ](A4) for the e g (0, 1) pairing phase. We thus see that η Eg 2 = 0 will induce a finite η A1g ∝ \|η While the behavior of η A1g (T ) and η Eg 2 (T ) further below T c cannot be captured by the leading-order Ginzburg-Landau expansion and will depend on microscopic details, we can estimate the gap anisotropy as a function of the ratio η = η A1g /η Eg 2 . Using, as in the main text, (X, Y, Z) = (k x , k y , k z ), the gap anisotropy of the e g (0, 1) state on a spherical Fermi surface reads as ∆ max /∆ min = 2+η η−1 η > 1, ∞ η ≤ 1.(A5) For instance, if we want ∆ max /∆ min < 1.1, we need η > 31, i.e., the secondary order parameter has to be about 30 times larger than the primary one, which does not seem to be a natural assumption. The analysis for the t 2g (1, 1, 1) singlet is similar. In this case, the coupling analogous to Eq. (A3) is associated with the A 1g term in T 2g ⊗ T 2g ⊗ T 2g = A 1g ⊕ A 2g ⊕ 2E g ⊕ 3T 1g ⊕ 4T 2g . General expression for ζ To be self-contained, we here provide the general expression for the disorder sensitivity parameter ζ in Eq. (2) derived in [79]. The central quantity is C k,k = ∆ k T † W k,k − t W W k,k ∆ k T † ,(A6) which is either a commutator or an anti-commutator depending on whether we consider time-reversal-even (t W = +1) or -odd (t W = −1) disorder, respectively; it also appeared in the generalized Anderson theorem of [121,122]. In Eq. (A6), ∆ k is the superconducting order parameter at T c , in our case a 2 × 2 matrix in pseudospin space, and T is the unitary part of the timereversal operator (T = iσ y for the states in Table III). Finally, W k,k are the matrix elements of the impurity potential W with respect to the Bloch states, \|k, s , at the Fermi surface, i.e., (W k,k ) s,s = k, s\|W \|k , s , with s labeling all bands including spin. Defining the Fermi-surface Frobenius norm according to \|\|C\|\| 2 F := 1 N 2 FS k,k ∈FS tr C † k,k C k,k , where k ∈ FS indicates that the sum involves all momenta in a finite vicinity around the Fermi surfaces and N FS = k∈FS , we can write [79] ζ = \|\|C\|\| 2 F 2 tr [W † W ] tr[∆ † k ∆ k ] FS ,(A7) where f k FS := 1 NFS k∈FS f k denotes the normalized Fermi surface average, as also used in the main text, see Eq. (3). Due to the generality of Eq. (A7), it can be readily applied in many different systems, see, e.g., Refs. 124 and 127 for two recent applications. Most importantly for us here, Eq. (3) in the main text is readily derived by focusing on ∆ k ∈ C 2×2 , k-dependent pseudospin-singlet pairing, ∆ k = Ψ k iσ y and scalar non-magnetic (t W = +1) disorder of the simple form W = W k,k = W 0 σ 0 , W 0 ∈ R. Different gaps on different Fermi sheets Finally, we discuss in more details which order parameter ratios ∆ n /∆ 1 in Eq. (6) are consistent with the observed ζ = 1/9. In our DFT calculations for Ca 3 Rh 4 Sn 13 (with details in Appendix C) we identify eight bands that give rise to Fermi surfaces, see Fig. 11. Their respective density of states at the Fermi level, ρ n , in decreasing order of magnitude are listed in Table IV. In principle, the order parameter can be different for any of these bands. For simplicity, we will first assume that there are only two different values, ∆ 1 and ∆ 2 , and the three bands with smallest ρ n are combined into one, i.e., we take them to exhibit the same ∆ n ; this amounts to studying the effective six band problem with respective density of states ρ n = ρ n , 1 ≤ n ≤ 5, ρ 6 = ρ 6 + ρ 7 + ρ 8 . (A8) There are still many (31) inequivalent ways of distributing two gaps on the six Fermi surfaces, as listed in Table V together with the associated anisotropy ratio consistent with ζ = 1/9. We can see that the smallest possible anisotropy ratio for the A ++ 1g state is 2.86. We have checked that this value does not change when allowing for ∆ 1 and ∆ 2 to be distributed arbitrarily on all eight Fermi surfaces in Table IV. As it should be, this value is larger than the theoretical lower bound (for ζ = 1/9) of (11 + 6 √ 2)/7 ≈ 2.78 based on Eq. (7) for arbitrary ratio of the density of states; due to the multitude of different Fermi surfaces, it is also natural that the A ++ 1g state can almost reach this theoretical bound. For the A +− 1g state, there are several solutions with \|∆ 1 \|/\|∆ 2 \| very close to 1 already in the six-band model, see Table V. As can also be seen in the table, this is possible for distributions of order parameters where the sign change happens between a set of Fermi surfaces and its complement exhibiting a ratio of density of states of about 6-7%. One might wonder whether more than two different values of ∆ n in Eq. (6) will allow for an A ++ 1g state with smaller gap anisotropy, A ∆ := max n,n ∆ n ∆ n ,(A9) for given ζ (in our case ζ = 1/9). Instead of systematically studying all possible ways of distributing N > 2 different order parameters, ∆ n > 0, n = 1, 2, . . . N , on the eight different Fermi surfaces in Table IV, we here derive a lower bound on A ∆ . To this end, let us assume we are given ∆ n > 0 which we order, without loss of generality, such that ∆ n > ∆ n+1 . It is not difficult to show that the maximum value of ζ in Eq. (6) is reached when ρ n = 0 for all n = 1, N . Consequently, only the smallest and largest ∆ n enter and we are back to the case with only two gaps, which we have already analyzed in Section IV F of the main text; the maximum value ζ max thus only depends on ∆ 1 /∆ N = A ∆ with form given in Eq. (7), i.e., ζ max ({∆ n }) = (A ∆ − 1) 2 2(1 + A ∆ ) 2 ,(A10) irrespective of how many different ∆ n are taken into account. Specifically, the lower bound for ζ = 1/9, A ∆ > (11 + 6 √ 2)/7 ≈ 2.78, still applies and the A ++ 1g state with three or more different gaps is not a natural candidate state either. Appendix B: Superfluid density in different models Having established in Section IV F which fully gapped conventional singlets are consistent with the observed suppression of T c with impurity concentration, we next investigate more quantitatively how the respective temperature dependence of the penetration depth or superfluid density compares with that measured experimentally (see Fig. 9). Anisotropic, single Fermi surface We first consider the anisotropic singlet on a single, isotropic Fermi surface as defined in Eq. (4). As discussed in the main text, only the values of δ = −1.08 and δ = −3.66 reproduce the observed T c suppression. The former is nodal and cannot possibly explain the exponential attenuation of the penetration depth. The latter is not nodal, but highly anisotropic. To see whether this anisotropy is consistent with the superfluid density ρ s of Fig. 9, we computed ρ s (T ) for this model. The calculations followed the Eilenberger formalism with a common ansatz that temperature and angular parts of the order parameter can be separated, ∆ (T, k F ) = Ψ (T ) Ω (k F ), where k F is Fermi wave vector and the angular part obeys the normalization condition for the Fermi surface average, Ω 2 FS = 1 [128]. Specifically, for the anisotropic A 1g state in Eq. (4), the angular part in spherical coordinates, k F = k F (sin θ cos ϕ, sin θ sin ϕ, cos θ), reads as Ω (θ, ϕ) = 1 + δ (sin θ cos ϕ) 4 + (sin θ sin ϕ) 4 + cos 4 θ 1 + (6/5)δ + (41/105)δ 2 . (B1) The temperature-dependent order parameter magnitude, Ψ (T ), is then obtained by solving the Eilenberger self-consistency equation and after that any thermodynamic quantity, including superfluid density, is calculated. The result for both values of δ is shown in Fig. 10(a) along with the curves for a weak-coupling isotropic s-wave BCS (Ω = 1) and d-wave (Ω = √ 2 cos 2ϕ) order parameters. The inset shows the angular dependence of the gap magnitude, \|Ω(k F )\|, for the same two values of δ. Clearly, ρ s (T ) differs strongly from s-wave behavior and, hence, from the data in Fig. 9 for all of these models. Isotropic self-consistent two-band model Another way to obtain substantial T c suppression in a conventional superconductor is to consider a two band system with two isotropic s-wave bands of different amplitude but same sign, denoted as A ++ 1g in the main text. To compute the superfluid density ρ s for this scenario, we use the self-consistent Eilenberger scheme, called the γ-model, which is detailed in Ref. 129. Starting with an interaction matrix containing two intra-band and one inter-band interaction constants, a system of 2 × 2 selfconsistency equations yields two order parameters from which the total ρ s can be calculated. Note that the tem-perature dependencies of the order parameters no longer follow the standard isotropic single-band curve, implying that the gap ratio is temperature-dependent; furthermore, its precise temperature evolution depends crucially on the interaction parameters while the amount of T c suppression is dictated by the gap ratio at T c (see Appendix A 2). Therefore, we selected several combinations of the interaction parameters, varying intra-and interband contributions, with the constraint that the gap ratio at T c is ∆ 1 /∆ 2 = 2.78, needed to obtain the measured T c suppression. In Fig. 10(b), we present the resulting temperature dependence of ρ s (main panel) and of the gap ratio ∆ 1 /∆ 2 (inset) for three different sets of interaction parameters, with roughly constant, increasing, and decreasing ∆ 1 /∆ 2 . While the low-temperature behavior exhibits saturation, it occurs below the temperature where the small gap saturates, much lower that T c /3 of isotropic s-wave. Most importantly, as before, none of these models of conventional pairing agree with the measure superfluid density. We therefore have a strong case in favor of unconventional pairing. . Appendix C: Density functional theory calculations We carry out first principles calculations to investigate the electronic structures in Ca 3 Rh 4 Sn 13 . Ca 3 Rh 4 Sn 13 crystallizes in the cubic Yb 3 Rh 4 Sn 13 -type (P m3n, space group no. 223) structure. The primitive cell contains two formula units. Ca atoms occupy the 6c (4m2) site, and Rh atoms occupy the 8e (32) site. The Sn atoms are divided into two sublattices; out of 13 Sn atoms in one formula unit, 12 Sn 1 atoms occupy the 24k (m) site, and one Sn 2 atom occupies the 2a (m3) site. We adopt the experimental crystal structure parameters [130] in all calculations. Density functional theory (DFT) calculations are performed using a full-potential linear augmented plane wave (FP-LAPW) method, as implemented in wien2k [131]. The generalized gradient approximation of Perdew, Burke, and Ernzerhof [132] is used for the exchange-correlation potentials. To generate the selfconsistent potential and charge, we employed R MT · K max = 8.0 with Muffin-tin radii R MT = 2.2, 2.4, and 2.4 a.u., for Ca, Rh, and Sn, respectively. The k-point integration is performed using a tetrahedron method with Blöchl corrections [133] with 119 k-points in the irreducible Brillouin zone (BZ). The calculations are iterated until the charge difference between consecutive iterations is smaller than 10 −4 e and the total energy difference is lower than 0.01 mRy. Figure 12 shows the DFT band structure along the Γ-X-M -Γ-R high-symmetry path and band-resolved partial density of states (PDOS) near the Fermi level. There are eight bands across the Fermi level. Figure 11 shows the Fermi surface contours calculated at various k z . We use the same color scheme to denote the eight bands FIG. 1 . 1(a) Combined phase diagram of 3-4-13 compounds as determined from measurement (CaxSr1−x)3Rh4Sn13 (bottom axis, cross symbols) FIG. 2 . 2Total knock-out cross-sections for studied compounds as function of electron energy. Our operating energy of 2.5 MeV is marked by a dotted line. The difference between Ca/Sr pairs is negligible and is not large between Ir/Rh, being about 30 barn larger for Ir compounds. , ordered by a decreasing value of T CDW : Sr 3 Ir 4 Sn 13 [A], Sr 3 Rh 4 Sn 13 [B], Ca 3 Ir 4 Sn 13 , [C] and Ca 3 Rh 4 Sn 13 [D] FIG. 3 . 3The evolution of temperature-dependent resistivity of Sr3Ir4Sn13 in pristine (blue line) and after electron irradiation of 1.14 C/cm 2 (yellow), and 4.4 C/cm 2 (red). The green dashed line shows the resistivity difference between the pristine and 4.4 C/cm 2 curves, finding the Matthiessen's rule to be valid above TCDW, but, expectantly, grossly violated below. The small cartoon in the top left corner indicates the sample's position on the generic phase diagram. The left inset shows the resistivity derivative dρ/dT in the vicinity of the CDW transition, where the arrows show the positions of the sharp features used to determine TCDW. The right inset zooms into the region around the superconducting transition. FIG. 5 . 5Temperature-dependent resistivity of Ca3Ir4Sn13 before irradiation (blue line), after receiving 2.17 C/cm 2 of irradiation (yellow line), and then an additional 3.3 C/cm 2 for a total dose of 5.47 C/cm 2 of electron irradiation. The green dashed line shows the difference between the pristine and 5.47 C/cm 2 curves, showing deviation from Matthiessen's rule below the transition temperature TCDW. The small cartoon in the top left corner indicates sample position on generic phase diagram. The left inset shows the derivative of the resistivity showing suppression and blurring of the CDW phase transition with irradiation. The right inset shows the shift in the superconducting transition temperature. FIG. 6 . 6Temperature-dependent resistivity of Ca3Rh4Sn13 in the pristine state before irradiation (blue curve) and after 2.1 C/cm 2 irradiation (red curve). The green dashed line shows the difference between the curves, finding strong Matthiessen's rule violation at all temperatures above the superconducting transition. The small cartoon in the top left corner indicates the position of the compound on the generic phase diagram, -the right-most, with no CDW transition. The inset shows resistivity in the vicinity of the superconducting transition, revealing substantial Tc suppression by electron irradiation. FIG. 7 . 7(a) The evolution of the CDW transition temperature normalized by the value before irradiation, TCDW/TCDW,0, and (b) similar plot of the superconducting transition temperature normalized by the value in pristine samples, Tc/Tc,0 with the induced disorder in units of defects per formula unit (dpf), see text for details. The X and Y scales on both graphs are the same for easy comparison. Note a significant increase of Tc suppression in samples where CDW does not coexist with superconductivity. FIG. 9 . 9Main panel: superfluid density in Ca3Rh4Sn13 calculated from the London penetration depth, λ (T ), measured using the tunnel-diode resonator technique. The thick orange line shows standard isotropic s−wave BCS behavior, while the dashed line shows the expectation for a nodal d−wave order parameter. The lower inset shows a BCS fit of ∆λ (T ) to the single-gap expression shown. The only fitting parameter was λ(0) = 330 nm, while the Tc and the weak-coupling gap ratio, ∆0/Tc = 1.764 was fixed. The obtained λ(0) = 330 nm was used to construct the ρs(T ) = (1 + ∆λ(T )/λ(0)) −2 shown in the main panel. The upper inset shows the sharp transition of our high-quality sample and λ(Tc) consistent with the expected skin depth of the normal state. FIG. 10 . 10Calculated normalized superfluid density to examine the influence of gap anisotropy. (a) g-wave correction to swave pairing, as defined in Eq. (4), with relative strengths δ = −3.66 (blue) and −1.08 (black), needed to reproduce the observed suppression of Tc. The 3D structure of the gap in the reciprocal space is shown as insets. The standard s-wave, Ψ k = Ψ0, and d-wave, Ψ k = Ψ0(k 2 FIG. 11 . 11Fermi surface contours at various kz in Ca3Rh4Sn13 obtained within density functional theory (DFT). Different colors denote different bands crossing the Fermi energy, (see Appendix C for details). 2 \| 3 ∝ (T c − T ) 3/2 close to T c (yielding n = 3). FIG. 12 . 12(Top) Band structures and (bottom) partial density of states of the eight bands across the Fermi level in Ca3Rh4Sn13. TABLE I . IHead-on knock out partial cross-sections by 2.5 MeV electron irradiation (1 barn = 1 × 10 −24 cm 2 ). The last column shows the number of defects created per formula unit, per 1 C/cm 2 . Roughly 1 defect per 1000 formula units is created. This is sufficiently close to the dilute limit to avoid significant compositional or electronic change.compound Sr/Ca Ir/Rh Sn total σ dpf×10 −3 barn barn barn barn per 1 C/cm 2 Sr3Ir4Sn13 139 261 148 181 1.13 Sr3Rh4Sn13 139 158 145 147 0.92 Ca3Ir4Sn13 79 258 145 177 1.11 Ca3Rh4Sn13 84 155 143 142 0.89 TABLE II . IIParameters of studied compositions in the pristine state, including CDW and superconducting transition temperatures, and the resistivity at room temperature averaged over indicated number of samples, N .compound TCDW (K) Tc (K) ρRT (µΩcm) N Sr3Ir4Sn13 145.2 ± 0.5 5.11 ± 0.03 168 ± 29 3 Sr3Rh4Sn13 135.76 ± 0.14 4.59 ± 0.1 129 ± 17 9 Ca3Ir4Sn13 39.0 ± 0.59 7.17 ± 0.02 120 ± 16 3 Ca3Rh4Sn13 no CDW 8.29 ± 0.01 112 ± 3.87 3 TABLE III . IIIPossible fully gapped pairing states for Ca3Rh4Sn13 as constrained by the point group O h TABLE IV . IVDensity of states of the bands per conventional unit cell (Z = 2) of Ca3Rh4Sn13 at the Fermi level ordered by decreasing magnitude.n band ρn ( states /eV cell) 1 #272 13.73 2 #273 9.534 3 #274 0.847 4 #275 0.826 5 #271 0.732 6 #276 0.670 7 #270 0.135 8 #269 0.105 TABLE V . VRatio of order parameters consistent with ζ = 1/9 in Eq. (6) for all possible independent distributions of the two different values, ∆1 and ∆2, among the six sets of Fermi sheets defined in Eq. (A8). Here S defines the set of sheets with order parameter ∆1, while the order parameter is ∆2 on the complementS = {1, 2, 3, 4, 5, 6} \ S. The relative fraction of the density of states of S is denoted by νS := n∈S ρn/ n∈S ρn. For a more clear representation of the gap anisotropy, we define ∆a/∆ b := max{∆1/∆2, ∆2/∆1}.S νS (∆a/∆ b )1 (∆a/∆ b )2 {1} 1.07 3.39 3.21 {2} 0.56 2.86 4.91 {3} 0.03 4.37 −0.56 {4} 0.03 4.41 −0.55 {5} 0.03 4.59 −0.50 {6} 0.04 4.27 −0.60 {1,2} 7.02 −0.35 3.03 {1,3} 1.21 3.61 3.09 {1,4} 1.21 3.61 3.09 {1,5} 1.19 3.58 3.10 {1,6} 1.23 3.63 3.08 {2,3} 0.64 2.92 4.30 {2,4} 0.64 2.91 4.31 {2,5} 0.63 2.91 4.37 {2,6} 0.65 2.92 4.26 {3,4} 0.07 3.55 −0.93 {3,5} 0.06 3.61 −0.99 {3,6} 0.07 3.51 −0.88 {4,5} 0.06 3.63 −0.99 {4,6} 0.07 3.52 −0.90 {5,6} 0.07 3.57 −0.95 {1,2,3} 9.77 −0.57 3.22 {1,2,4} 9.68 −0.57 3.22 {1,2,5} 9.29 −0.54 3.19 {1,2,6} 10.05 −0.59 3.24 {1,3,4} 1.38 3.91 2.99 {1,3,5} 1.36 3.87 3.00 {1,3,6} 1.4 3.94 2.98 {1,4,5} 1.35 3.86 3.00 {1,4,6} 1.39 3.93 2.98 {1,5,6} 1.37 3.89 2.99 . 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[ "Fast Modeling Methods for Complex System with Separable Features", "Fast Modeling Methods for Complex System with Separable Features" ]	[ "Chen Chen \nInstitute of Mechanics\nState Key Laboratory of High Temperature Gas Dynamics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Engineering Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "Changtong Luo \nInstitute of Mechanics\nState Key Laboratory of High Temperature Gas Dynamics\nChinese Academy of Sciences\n100190BeijingChina\n", "Zonglin Jiang \nInstitute of Mechanics\nState Key Laboratory of High Temperature Gas Dynamics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Engineering Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n" ]	[ "Institute of Mechanics\nState Key Laboratory of High Temperature Gas Dynamics\nChinese Academy of Sciences\n100190BeijingChina", "School of Engineering Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Institute of Mechanics\nState Key Laboratory of High Temperature Gas Dynamics\nChinese Academy of Sciences\n100190BeijingChina", "Institute of Mechanics\nState Key Laboratory of High Temperature Gas Dynamics\nChinese Academy of Sciences\n100190BeijingChina", "School of Engineering Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina" ]	[]	Data-driven modeling plays an increasingly important role in different areas of engineering. For most of existing methods, such as genetic programming (GP), the convergence speed might be too slow for large scale problems with a large number of variables. Fortunately, in many applications, the target models are separable in some sense. In this paper, we analyze different types of separability of some real-world engineering equations and establish a mathematical model of generalized separable system (GS system). In order to get the structure of the GS system, two concepts, namely 'block' and 'factor' are introduced, and a special method, block and factor detection is also proposed, in which the target model is decomposed into a number of blocks, further into minimal blocks and factors. Compare to the conventional GP, the new method can make large reductions to the search space. The minimal blocks and factors are optimized and assembled with a global optimization search engine, low dimensional simplex evolution (LDSE). An extensive study between the proposed method and a state-of-the-art data-driven fitting tool, Eureqa, has been presented with several man-made problems. Test results indicate that the proposed method is more effective and efficient under all the investigated cases.	10.1109/iscid.2017.144	[ "https://arxiv.org/pdf/1708.04583v1.pdf" ]	33,971,908	1708.04583	cee97db95f6161e07e654c97118d813a3eca9705	Fast Modeling Methods for Complex System with Separable Features Chen Chen Institute of Mechanics State Key Laboratory of High Temperature Gas Dynamics Chinese Academy of Sciences 100190BeijingChina School of Engineering Sciences University of Chinese Academy of Sciences 100049BeijingChina Changtong Luo Institute of Mechanics State Key Laboratory of High Temperature Gas Dynamics Chinese Academy of Sciences 100190BeijingChina Zonglin Jiang Institute of Mechanics State Key Laboratory of High Temperature Gas Dynamics Chinese Academy of Sciences 100190BeijingChina School of Engineering Sciences University of Chinese Academy of Sciences 100049BeijingChina Fast Modeling Methods for Complex System with Separable Features data-driven modelinggenetic programminggen- eralized separable systemblock and factor Data-driven modeling plays an increasingly important role in different areas of engineering. For most of existing methods, such as genetic programming (GP), the convergence speed might be too slow for large scale problems with a large number of variables. Fortunately, in many applications, the target models are separable in some sense. In this paper, we analyze different types of separability of some real-world engineering equations and establish a mathematical model of generalized separable system (GS system). In order to get the structure of the GS system, two concepts, namely 'block' and 'factor' are introduced, and a special method, block and factor detection is also proposed, in which the target model is decomposed into a number of blocks, further into minimal blocks and factors. Compare to the conventional GP, the new method can make large reductions to the search space. The minimal blocks and factors are optimized and assembled with a global optimization search engine, low dimensional simplex evolution (LDSE). An extensive study between the proposed method and a state-of-the-art data-driven fitting tool, Eureqa, has been presented with several man-made problems. Test results indicate that the proposed method is more effective and efficient under all the investigated cases. I. INTRODUCTION Data-driven modeling has become a powerful technique in different areas of engineering, such as industrial data analysis [6], circuits analysis and design [10], signal processing [11], system identification [2], etc. For a concerned datadriven modeling problem, we aim to find a performance function that best explains the relationship between input variables and the target system (or constrained system) based on a given set of sample points. Among the existing methods, genetic programming (GP) [4] is a classical approach. Theoretically, GP can get an optimal solution provided that the computation time is long enough. However, the computational cost of GP for a large scale problem with a large number of input variables is still very expensive. In many scientific or engineering problems, the target model are separable. Luo et al. [5] have presented a divideand-conquer (D&C) method for GP. The authors indicated that the solving process could be accelerated by detecting the correlation between each variable and the target function. In [5], a special method, bi-correlation test (BiCT), was pro-posed to divide a concerned target function into a number of sub-functions. Compared to conventional GP, D&C method could reduce the computational effort (computational complexity) by orders of magnitude. In this paper, different types of separability of some practical engineering problems are analyzed, and a mathematical model of generalized separable system (GS system) is established. In order to get the structure of the GS system, a block and factor detection method is proposed, where the target model is decomposed into a number of block, further into minimal blocks and factors. The new method is an improved version of the BiCT method [5]. The performance of the proposed method is compared with the results of Eureqa, which is a state-of-the-art data-driven fitting tool. Numerical results show that the proposed method is effective, and is able to recover all the investigated cases rapidly and reliably. II. OBSERVATION AND DISCUSSION OF THE TYPES OF SEPARABILITY A. Observation In this section, three examples of real-world problems are given as follows to illustrate several common types of separability in practical problems. Example 1. When developing a rocket engine, it is crucial to model the internal flow of a high-speed compressible gas through the nozzle. The closed-form expression for the mass flow through a choked nozzle iṡ m = p 0 A * √ T 0 γ R 2 γ + 1 (γ+1)/(γ−1) . (1) In Eq. (1), the five independent variables, p 0 , T 0 , A * , R and γ are all separable. The equation can be called a multiplicatively separable function, which can be re-expressed as followṡ m = f (p 0 , A * , T 0 , R, γ) = ϕ 1 (p 0 ) × ϕ 2 (A * ) × ϕ 3 (T 0 ) × ϕ 4 (R) × ϕ 5 (γ) .(2C L = C Lα (α − α 0 ) + C Lδe δ e S HT S ref ,(3) where the variable C Lα , C Lδe , δ e , S HT and S ref are separable. The variable α and α 0 are not separable, but their combination (α, α 0 ) can be considered separable. Hence, Eq. (3) can be re-expressed as C L =f (C Lα , α, α 0 , C Lδe , δ e , S HT , S ref ) =ϕ 1 (C Lα ) × ϕ 2 (α, α 0 ) + ϕ 3 (C Lδe ) × ϕ 4 (δ e ) × ϕ 5 (S HT ) × ϕ 6 (S ref ) .(4) Example 3. The flow past a circular cylinder is a classical problem in fluid dynamics. A valid stream function for the inviscid, incompressible flow over a circular cylinder of radius R is ψ = (V ∞ r sin θ) 1 − R 2 r 2 + Γ 2π ln r R ,(5) which can be re-expressed as ψ =f (V ∞ , sin θ, R, r, Γ) =ϕ 1 (V ∞ ) × ϕ 2 (sin θ) × ϕ 3 (r, R) + ϕ 4 (Γ) × ϕ 5 (r, R) .(6) Note that the variable r and R appear twice in Eq. (5). In other words, variable r and R have two sub-functions, namely ϕ 3 (r, R) = 1 − R 2 r 2 · r and ϕ 5 (r, R) = ln (r/R). Although Eq. (6) is not a strictly separable function, the variables of Eq. (6) also have separability. B. Discussion As seen from the above subsection, many practical problems have the feature of separability. Luo et al. suggested using the separability to accelerate the conventional GP for data-driven modeling. The separable function introduced in [5] could be described as follows. Definition 1 (Separable function). A scalar function f (X) with n continuous variables X = {x i : i = 1, 2, · · · , n} (f : R n → R, X ⊂ Ω ∈ R n , where Ω is a closed bounded convex set, such that Ω = [a 1 , b 1 ] × [a 2 , b 2 ] × · · · × [a n , b n ]) is said to be separable if and only if it can be written as f (X) = c 0 ⊗ 1 c 1 ϕ 1 (X 1 ) ⊗ 2 c 2 ϕ 2 (X 2 ) ⊗ 3 · · · ⊗ m c m ϕ m (X m ) , (7) where the variable set X i is a proper subset of X, such that X i ⊂ X with m i=1 X i = X, m i=1 X i = ∅, and the cardinal number of X i is denoted by card (X i ) = n i , for m i=1 n i = n and i = 1, 2, · · · , m. Sub-function ϕ i is a scalar function such that ϕ i : R ni → R. The binary operator ⊗ i could be plus (+) and times (×). Note that binary operator, minus (−) and division (/), are not included in ⊗ for simplicity. This does not affect much of its generality, since minus (−) could be regarded as (−) = (−1) · (+), and sub-function could be treated as ϕ i (·) = 1/ϕ i (·) if only ϕ i (·) = 0. We can see that Example 3 is inconsistent with the above definition of the separable function. It is because that some variables (e.g., variable V ∞ , sin θ and Γ of Eq. (5)) appears only once in a concerned target model, while the other variables (e.g., variable r and R of Eq. (5)) appears more than once. This feature motivates us to generalize the mathematical form of the separable function, namely Eq. (7), and establish a more general model. III. THE MATHEMATICAL MODEL OF GENERALIZED SEPARABLE SYSTEM Definition 2 (Generalized separable system). The mathematical model of a generalized separable system f (X) with n continuous variables X = {x i : i = 1, 2, · · · , n}, (f : R n → R, X ⊂ Ω ∈ R n , where Ω is a closed bounded convex set, such that Ω = [a 1 , b 1 ] × [a 2 , b 2 ] × · · · × [a n , b n ]) is defined as f (X) = f X r ,X r = c 0 + m i=1 c i ϕ i X r i ,X r i = c 0 + m i=1 c iωi (X r i )ψ i X r i = c 0 + m i=1 c i pi j=1 ω i,j X r i,j qi k=1 ψ i,k X r i,k ,(8) where the variable set X r = {x i : i = 1, 2, · · · , l} is a proper subset of X, such that X r ⊂ X, and the cardinal number of X r is card (X r ) = l.X r is the complementary set of X r in X, i.e.X r = X X r , where card X r = n − l. X r i is the subset of X r , such that X r i ⊆ X r , where card (X r i ) = r i . X r i,j ⊆ X r i , such that pi j=1 X r i,j = X r i , pi j=1 X r i,j = ∅, where card X r i,j = r i,j , for i = 1, 2, · · · , m, j = 1, 2, · · · , p i and pi j=1 r i,j = r i .X r i ⊂X r (X r i = ∅), such that m i=1X r i =X r , m i=1X r i = ∅, where card X r i = s i , for s i 1, m i=1 s i = n − l. X r i,k ⊆X r i , such that qi k=1X r i,k =X r i , qi k=1X r i = ∅, where card X r i,k = s i,k , for k = 1, 2, · · · , q i and qi k=1 s i,k = s i . Sub-functions ϕ i ,ω i ,ψ i , ω i,j and ψ i,k are scalar functions, such that ϕ i : R ri+si → R,ω i : R ri → R, ψ i : R si → R, ω i,j : R ri,j → R and ψ i,k : R s i,k → R, respectively. c 0 , c 1 , · · · , c m are constant coefficients. Definition 3 (Repeated variable, non-repeated variable, block and factor). In Eq. (8), the variables belong to X r andX r are called repeated variables and non-repeated variables, respectively. The sub-function ϕ i (·) is called the i-th minimal block of f (X), for i = 1, 2, · · · , m. Any combination of the minimal blocks is called a block of f (X). The sub-functions ω i,j (·) and ψ i,k (·) are called the j-th and k-th factors of the repeated variables and nonrepeated variables in i-th minimal block ϕ i (·), respectively, for j = 1, 2, · · · , p i and k = 1, 2, · · · , q i . IV. MODEL DETECTION AND DETERMINATION In order to detect the separability of the GS system f (X), we aim to divide f (X) into a suitable number of minimal blocks, and further into factors as the typical Example 3. This technique can be considered as a generalized bicorrelation test (BiCT) method. The BiCT [5] is developed to detect the separability of a certain additively or multiplicatively separable target function, i.e. the Eq. (7). The modeling process of GS-system mainly includes two parts, namely inner optimization and outer optimization. The inner optimization will be invoked to determine the function model and coefficients of the factors ω i,j and ψ i,k . Fortunately, many state-of-the-art optimization techniques, e.g., parse-matrix evolution [8], low dimensional simplex evolution [7], artificial bee colony programming [3], etc. can all be easily used to optimize the factors. Then, the optimized factors of each minimal block are multiplied together to produce minimal blocks. The outer optimization aims at combining the minimal blocks together with the proper global parameters c i . The whole process for modeling a GS system can be briefly described as follows: 1) (Minimal block detection) Partition a GS system into a number of minimal blocks with all the repeated variables fixed; 2) (Factor detection) Divide each minimal block into factors; 3) (Factor determination) Determine the factors by employing an optimization engine; 4) (Global assembling) Combine the optimized factors into minimal blocks multiplicatively, further into an optimization model linearly with proper global parameters. The flowchart of the modeling process could be briefly illustrated in Fig. 1. The proposed technique is described with functions with explicit expressions. While in practical applications, no explicit expression is available. In fact, for data-driven modeling problems, a surrogate model [1] of black-box type could be established as the underlying target function in advance. V. NUMERICAL RESULTS AND DISCUSSION In our implementation, a kind of global optimization method, low dimensional simplex evolution (LDSE) [7], is chosen as the optimization engine. LDSE is a hybrid evolutionary algorithm for continuous global optimization. The performances including 'structure optimization' and 'coefficient optimization' capabilities of the proposed method are tested by comparing with a state-of-the-art software, Eureqa [9], which is a data-driven fitting tool based on genetic programming (GP). Eureqa was developed at the Computational Synthesis Lab at Cornell University by H. Lipson. 10 test cases are taken into account. The calculation conditions are set as follows. The number of sampling points for each independent variable is 200. The regions for cases 1-5 and 7-10 are chosen as [−3, 3], while case 6 is [1,3]. The control parameters in LDSE are set as follows. The upper and lower bounds of fitting parameters is set as −50 and 50. The population size N p is set to N p = 10 + 10d, where d is the dimension of the problem. Sequence search and optimization method is suitable for global optimization strategy. The search will exit immediately if the mean square error is small enough (MSE ε target ), and the tolerance (fitting error) is ε target = 10 −6 . In order to reduce the effect of randomness, each test case is executed 20 times. The computing time (CPU time) consists three parts, t = t 1 + t 2 + t 3 , where t 1 is for the separability detection, t 2 for factors modeling, and t 3 for global assembling. In [5], authors have demonstrated that both the separability detection and function recover processes are double-precision operations and thus cost much less time than the factor determination process. That is, t ≈ t 2 . It is very easy to see that the computational efficiency of the proposed method is higher than Eureqa's. Note that our method is executed on a single processor, while Eureqa is executed in parallel on 8 processors. VI. CONCLUSION We have analyzed the different types of separability of some practical engineering problems and have established the mathematical model of the generalized separable system (GS system). In other to get the structure of the GS system, Figure 1 . 1Flowchart of modeling process. Table I 10 TEST CASES.No.Target model 1f (x) = 0.5 * e x 1 * sin 2x2 2f (x) = 2 * cos x1 + sin (3x2 − x3) 3f (x) = 1.2 + 10 * sin 2x1 − 3 * x 2 2 * cos x3 4f (x) = x3 * sin x1 − 2 * x3 * cos x2 two types of variables in GS system have been identified, namely repeated variable and non-repeated variable. A new method, block and factor detection, has also been proposed to decompose the GS system into a number of block, further into minimal blocks and factors. The minimal blocks and factors are optimized and assembled with a global optimization search engine, low dimensional simplex evolution (LDSE). The proposed method is tested on several manmade test cases. Remarkable performance is concluded after comparing with a state-of-the-art data-driven fitting tool, Eureqa. Numerical results show the algorithm is effective, and can get the target function more rapidly and reliably.ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (Grant No. 11532014). 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[ "RIGID AND SCHURIAN MODULES OVER CLUSTER-TILTED ALGEBRAS OF TAME TYPE", "RIGID AND SCHURIAN MODULES OVER CLUSTER-TILTED ALGEBRAS OF TAME TYPE" ]	[ "Robert J Marsh ", "Idun Reiten " ]	[]	[]	We give an example of a cluster-tilted algebra Λ with quiver Q, such that the associated cluster algebra A(Q) has a denominator vector which is not the dimension vector of any indecomposable Λ-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra Λ, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid Λ-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid Λ-modules in this case.	10.1007/s00209-016-1668-z	[ "https://arxiv.org/pdf/1412.6405v2.pdf" ]	119,147,191	1412.6405	9dd94508d53999efb609814b079d1259a6c848cd	RIGID AND SCHURIAN MODULES OVER CLUSTER-TILTED ALGEBRAS OF TAME TYPE 19 Dec 2014 Robert J Marsh Idun Reiten RIGID AND SCHURIAN MODULES OVER CLUSTER-TILTED ALGEBRAS OF TAME TYPE 19 Dec 2014 We give an example of a cluster-tilted algebra Λ with quiver Q, such that the associated cluster algebra A(Q) has a denominator vector which is not the dimension vector of any indecomposable Λ-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra Λ, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid Λ-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid Λ-modules in this case. Introduction In the theory of cluster algebras initiated by Fomin and Zelevinsky, the authors introduced some important kinds of vectors, amongst them the d-vectors (denominator vectors) [13] and the c-vectors [14]. These vectors have played an important role in the theory. In particular, they have been important for establishing connections with the representation theory of finite dimensional algebras. Let Q be a finite quiver with n vertices, without loops or two-cycles, and let A(Q) be the associated cluster algebra with initial cluster {x 1 , . . . , x n }. Each non-initial cluster variable is known to be of the form f /m, where m = x d1 1 · · · x dn n for nonnegative integers d i and f is not divisible by any x i . Then the associated d-vector is (d 1 , . . . , d n ). For the definition of c-vector we refer to [14]. On the other hand, we have the dimension vectors of the finite dimensional rigid indecomposable KQ-modules. Assume first that Q is acyclic. Then there are known interesting connections between the d-vectors and the c-vectors on the one hand and the dimension vectors of the indecomposable rigid KQ-modules on the other hand. More specifically, there is a bijection between the non-initial cluster variables and the indecomposable rigid KQ-modules such that the dvector of a cluster variable coincides with the dimension vector of the corresponding module (see [9,10,11]). Furthermore, the (positive) c-vectors of A(Q) and the dimension vectors of the indecomposable rigid KQ-modules coincide (see [12,24]). However, when the initial quiver Q is not acyclic, we do not have such nice connections (see [2,5,6] for work in this direction). Answering a question posed to us by Nakanishi, we found an example showing the following: () There is a cluster-tilted algebra Λ with quiver Q such that A(Q) has a denominator vector which is not the dimension vector of any indecomposable Λ-module. Since we know that there are denominator vectors which are not dimension vectors, it is natural to ask if the denominator vectors can be written as a sum of a small number of dimension vectors of indecomposable rigid Λ-modules. We consider this question for clustertilted algebras associated to tame hereditary algebras. In this case we show that it is possible to use at most 3 summands. We do not know if it is always possible with 2 summands. In order to prove the results discussed in the previous paragraph we need to locate the indecomposable rigid Λ-modules in the AR-quiver of Λ-mod. This investigation should be interesting in itself. Closely related is the class of indecomposable Schurian modules, which we also describe. If H is a hereditary algebra, then every indecomposable rigid (equivalently, τ -rigid) module is Schurian. So one might ask what the relationships are between the rigid, τ -rigid and Schurian Λ-modules. In general there are τ -rigid (hence rigid) Λ-modules which are not Schurian. However, it turns out that every indecomposable Λ-module which is rigid, but not τ -rigid, is Schurian. In Section 1, we recall some basic definitions and results relating to cluster categories. In Section 2 we discuss tubes in general. In Section 3 we fix a cluster-tilting object T in a cluster category associated to a tame hereditary algebra and investigate its properties in relation to a tube. Section 4 is devoted to identifying the rigid and Schurian End C (T ) oppmodules. In Section 5, we investigate an example in the wild case which appears to behave in a similar way to the tame case. In Section 6, we give the example providing a negative answer to the question of Nakanishi. Finally, in Section 7, we also show that for clustertilted algebras associated to tame hereditary algebras each denominator vector is a sum of at most 3 dimension vectors of indecomposable rigid Λ-modules. We refer to [3,4] for standard facts from representation theory. We would like to thank Otto Kerner for helpful conversations about wild hereditary algebras. Setup In this section we recall some definitions and results related to cluster categories and rigid and τ -rigid objects. We also include some lemmas which are useful for showing that a module is Schurian or rigid. For a modulus N , we choose representatives Z N = {0, 1, . . . , N − 1}, writing [a] N for the reduction of an integer a mod N . If N = 0, we take Z N to be the empty set. We fix an algebraically closed field K; all categories considered will be assumed to be Kadditive. For an object X in a category X , we denote by add(X) the additive subcategory generated by X. Suppose that X is a module category with AR-translate τ . Then we say that X is rigid if Ext 1 (X, X) = 0, τ -rigid if Hom(X, τ X) = 0, Schurian if End(X) ∼ = K, or strongly Schurian if the multiplicity of each simple module as a composition factor is at most one. Note that any strongly Schurian module is necessarily Schurian. If X is a triangulated category with shift [1] and AR-translate τ , we define rigid, τ -rigid and Schurian objects similarly, where we write Ext 1 (X, Y ) for Hom(X, Y [1]). For both module categories and triangulated categories, we shall consider objects of the category up to isomorphism. For modules X, Y in a module category over a finite dimensional algebra, we write Hom(X, Y ) for the injectively stable morphisms from X to Y , i.e. the quotient of Hom(X, Y ) by the morphisms from X to Y which factorize through an injective module. We similarly write Hom(X, Y ) for the projectively stable morphisms. Then we have the AR-formula: (1.1) DHom(X, τ Y ) ∼ = Ext 1 (X, Y ) ∼ = DHom(τ −1 X, Y ), where D denotes the functor Hom(−, K). Let Q = (Q 0 , Q 1 ) and Q ′ = (Q ′ 0 , Q ′ 1 ) be quivers with vertices Q 0 , Q ′ 0 and arrows Q 1 , Q ′ 1 . Recall that a morphism of quivers from Q to Q ′ is a pair of maps f i : Q i → Q ′ i , i = 0, 1, such that whenever α : i → j is an arrow in Q, we have that f 1 (α) starts at f 0 (i) and ends at f 0 (j). In order to describe the modules we are working with, it is convenient to use notation from [22], which we now recall. Definition 1.1. Let Q be a quiver with vertices Q 0 . A Q-coloured quiver is a pair (Γ, π), where Γ is a quiver and π : Γ → Q is a morphism of quivers. We shall always assume that Γ is a tree. As Ringel points out, a Q-coloured quiver (Γ, π) can be regarded as a quiver Γ in which each vertex is coloured by a vertex of Q and each arrow is coloured by an arrow of Q. In addition, if an arrow γ : v → w in Γ is coloured by an arrow α : i → j in Q then v must be coloured with i and w must be coloured with j. We shall draw Q-coloured quivers in this way. Thus we shall write π(v) ∈ Q 0 instead of each vertex v of Γ, and label each arrow a in Γ with its image π(a) in Q. But note that if Q has no multiple arrows then we can omit the arrow labels, since the label of an arrow in Γ is determined by the labels of its endpoints. We shall also omit the orientation of the arrows in Γ, adopting the convention that the arrows always point down the page. As in [22,Remark 4], a Q-coloured quiver (Γ, π) determines a representation V = V (Γ, π) of Q over K (and hence a KQ-module) in the following way. For each i ∈ Q 0 , let V i be the vector space with basis given by B i = π −1 (i) ⊆ Γ 0 . Given an arrow α : i → j in Q and b ∈ π −1 (i), we define (1.2) ϕ α (b) = b α − →c in Γ c, extending linearly. If A = KQ/I, where I is an admissible ideal, and V satisfies the relations coming from the elements of I then it is an A-module. Note that, in general, not every A-module will arise in this way (for example, over the Kronecker algebra). Also, a given module may be definable using more than one Q-coloured quiver (by changing basis). As an example of a coloured quiver, consider the quiver Q: (1.3) 1 / / ) ) 2 / / 3 / / 4. Then we have the following Q-coloured quivers and corresponding representations: Figure 1. A quiver Q, a Q-coloured quiver, together with the redrawing according to Remark 1.2 and the corresponding representation of Q. (1.4) T 2 = 1 2 3 2 1 3 2 1 b22 b12 b31 b21 b11 b22 b12 b31 b21 b11 K 2 K 2 K ( 1 0 0 1 ) ( 1 0 ) ( 0 1 ) To aid with calculations, we may also redraw Γ, placing all of the basis elements b ij (for fixed i) close together (according to a fixed embedding of Q in the plane). In this case, we must include the arrowheads on the arrows so that this information is not lost. For an example, see Figure 1. Definition 1.3. If (Γ, π), (Γ ′ , π ′ ) are Q-coloured quivers then we call a map ϕ : Γ → Γ ′ a morphism of Q-coloured quivers if it is a morphism of quivers and π = π ′ ϕ. If Γ ′′ is a full subquiver of Γ and π ′′ is the restriction of π to Γ then Γ ′′ is called a Q-coloured subquiver of (Γ, π); note that it is again a Q-coloured quiver. Remark 1.4. If (Γ ′ , π ′ ) is a Q-coloured subquiver of (Γ, π) with the property that every arrow between a vertex in Γ ′ and a vertex in Γ not in Γ ′ points towards the vertex in Γ ′ , then it is easy to see that there is a corresponding embedding of modules V (Γ ′ , π ′ ) ֒→ V (Γ, π). Similarly, if every such arrow points towards Γ ′ , there is a corresponding quotient map V (Γ, π) ։ V (Γ ′ , π ′ ). Let (Γ(1), π(1)) and (Γ(2), π(2)) be Q-coloured quivers. Suppose that there is a Qcoloured quiver (Γ, π) which is isomorphic to a Q-coloured subquiver of (Γ(1), π(1)) with the second property above. Suppose in addition that it is isomorphic to a Q-coloured subquiver of (Γ(2), π(2)) with the first property above. Then there is a KQ-module homomorphism V (Γ(1), π(1)) → V (Γ(2), π(2)) given by the composition of the quotient map and the embedding given above. We fix a quiver Q such that the path algebra KQ has tame representation type. For example, we could take Q to be the quiver (1.3). We denote by KQ-mod the category of finite-dimensional KQ-modules, with AR-translate τ . We denote by D b (KQ) the bounded derived category of KQ-mod, with AR-translate also denoted by τ . For objects X and Y in D b (KQ), we write Hom(X, Y ) for Hom D b (KQ) (X, Y ) and Ext(X, Y ) for Ext D b (KQ) (X, Y ). Note that if X, Y are modules, these coincide with Hom KQ (X, Y ) and Ext KQ (X, Y ) respectively. The category D b (KQ) is triangulated. Let C = C Q denote the cluster category corresponding to Q, i.e. the orbit category C Q = D b (KQ)/F , where F denotes the autoequivalence τ −1 [1] (see [7]). The category C is triangulated by [17, §4]. Note that an object in D b (KQ)-mod can be regarded as an object in C; in particular this applies to KQ-modules, which can be identified with complexes in D b (KQ) concentrated in degree zero. If X, Y are indecomposable objects in D b (KQ) regarded as objects in C, then by [7,Prop. 1.5]. We write Hom H C (X, Y ) = Hom(X, Y ) and refer to elements of this space as H-maps from X to Y , and we write Hom(X, F Y ) = Hom F C (X, Y ) and refer to elements of this space as F -maps from X to Y . So, we have: Hom C (X, Y ) = Hom(X, Y ) ⊕ Hom(X, F Y )Hom C (X, Y ) = Hom H C (X, Y ) ⊕ Hom F C (X, Y ). Note that Hom F C (X, Y ) = Hom(X, F Y ) = Hom(X, τ −1 Y [1]) ∼ = Ext(X, τ −1 Y ) ∼ = D Hom(τ −1 Y, τ X) ∼ = D Hom(Y, τ 2 X), (1.6) where D = Hom(−, K). If χ is an additive subcategory of C, we write: Hom H C/χ (X, Y ) , Hom F C/χ (X, Y ) for the quotients of Hom H C (X, Y ) and Hom F C (X, Y ) by the morphisms in C factoring through χ. A rigid object T in C is said to be cluster-tilting if, for any object X in C, we have Ext 1 C (T, X) = 0 if and only if X lies in add(T ). We fix a cluster-tilting object T in C. We make the following assumption. As explained in the proof of Theorem 4.10, to find the rigid and Schurian modules for any cluster-tilted algebra arising from C, it is enough to find the rigid and Schurian modules in this case. Assumption 1.5. The cluster-tilting object T is induced by a KQ-module (which we also denote by T ). Furthermore, T is of the form U ⊕ T ′ , where U is preprojective and T ′ is regular. Note that the module T is a tilting module by [7]. Example 1.6. For example, if Q is the quiver in (1.3), we could take T to be the tilting module: (1.7) T = P 1 ⊕ T 2 ⊕ T 3 ⊕ P 4 , where T 2 and T 3 are the KQ-modules defined in (1.4), (1.5). Note that T can be obtained from P 1 ⊕ P 2 ⊕ P 3 ⊕ P 4 by mutating (in the sense of [15,21]) first at P 2 and then at P 3 . The modules T 2 and T 3 lie in a tube of rank 3 in KQ-mod; see Figure 2. We define Λ = Λ T = End CQ (T ) to be the corresponding cluster-tilted algebra. For Example 1.6, Λ is given by the quiver with relations shown in Figure 3 (we indicate how to compute such a quiver with relations explicitly for a similar example in Section 5). Note that this quiver can obtained from Q by mutating (in the sense of [13]) first at 2 and then at 3. There is a natural functor Hom C (T, −) from C to Λ-mod. We have: Theorem 1.7. [8, Thm. A] The functor Hom C (T, −) induces an equivalence from the additive quotient C/ add(τ T ) to Λ-mod. We denote the image of an object X in C under the functor Hom C (T, −) by X. We note the following: Proposition 1.8. Let X be an object in C and X the corresponding Λ-module. Then (a) X is Schurian if and only if Hom C/ add(τ T ) (X, X) ∼ = K. (b) X is rigid if and only if Hom C/ add(τ T ⊕τ 2 T ) (X, τ X) = 0. Proof. Part (a) follows from the equivalence in Theorem 1.7. Part (b) follows from this combined with the AR-formula (1.1), noting that the injective modules in Λ-mod are the objects in the subcategory add Hom C (T, τ 2 T ) (see [8], [18, §2]). The following statement follows from [1,Thm. 4.1]. Theorem 1.9. [1] The functor Hom C (T, −) induces a bijection between isomorphism classes of indecomposable rigid objects in C which are not summands of τ T and isomorphism classes of indecomposable τ -rigid Λ-modules. Since a KQ-module is rigid if and only if the induced object of C is rigid (by [7, Prop. 1.7]), we have: Corollary 1.10. If X is a KQ-module not in add(τ T ) then X is rigid in KQ-mod if and only if X is τ -rigid in Λ-mod. Since (for modules over any finite-dimensional algebra) every τ -rigid module is rigid, we have that X is a rigid Λ-module for any rigid KQ-module X. Remark 1.11. Suppose that X is an indecomposable object of D b (KQ) which is either a preprojective KQ-module, a preinjective KQ-module or the shift of a projective KQmodule. Assume also that X is not a direct summand of τ T . Then X is rigid in D b (KQ), hence (by [7,Prop. 1.7]) rigid in C. By Theorem 1.9, X is τ -rigid in Λ-mod. Furthermore, X is Schurian in D b (KQ). We also have Hom F C (X, X) ∼ = D Hom(X, τ 2 X) = 0 by (1.6), so X is a Schurian object of C. It follows that X is a Schurian Λ-module by Proposition 1.8(a). Thus we see that, for any indecomposable transjective object of C (not a summand of τ T ), the corresponding Λ-module is Schurian and τ -rigid, hence rigid. Thus the main work in classifying indecomposable Schurian and (τ -)rigid Λ-modules concerns those which arise from tubes in KQ-mod. Finally, we include some lemmas which will be useful in checking whether a given Λmodule is Schurian or rigid. Lemma 1.12. Let X, Y, Z be indecomposable objects in D b (KQ), regarded as objects in C. Let f ∈ Hom F C (X, Y ) = Hom(X, F Y ). Then f factorizes in C through Z if and only if it factorizes in D b (KQ) through Z or F (Z). Proof. Since f is an F -map, it can only factorize through Z in C as an H-map followed by an F -map or an F -map followed by an H-map. The former case corresponds to factorizing through Z in D b (KQ) and the latter case corresponds to factorizing through F (Z) in D b (KQ). B γ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ Hom(B,τ α)(γ) / / τ C τ A τ α = = ③ ③ ③ ③ ③ ③ ③ ③ A α ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ Hom(α,B[1])(δ) / / B[1] C δ = = ④ ④ ④ ④ ④ ④ ④ ④ (b) Let β : C → BC β ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ Hom(β,τ A)(γ) / / τ A B γ > > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ A δ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ Hom(A,β[1])(δ) / / B[1] C[1] β[1] < < ② ② ② ② ② ② ② ② Proof. Part (a) follows from the commutative diagram: Proof. We first consider part (a). Note that, since the quasilength of M i,l is assumed to be at most r, the rays starting at M i+p,l−p for 0 ≤ p ≤ l − 1 do not intersect each other. It is then easy to see that, up to mesh relations, there is exactly one path in the AR-quiver of T from M i,l to the objects in these rays and no path to any other object in T . The result then follows from the fact that T is standard. A similar proof gives part (b). Hom(C, B[1]) ∼ / / M 0,1 M 1,1 M 2,1 M 0,1 M 0,2 M 1,2 M 2,2 M 2,3 M 0,3 M 1,3 M 2,3 Fix M i,l ∈ T with l ≤ r. It follows from Lemma 2.2 that if the quasisocle of X ∈ T does not lie in W M i,l then Hom(M i,l , X) = 0. Similarly, if the quasitop of X does not lie in W M i,l then Hom(X, M i,l ) = 0. This implies the following, which we state here as we shall use it often. Proof. The formulas are easily checked using the fact that T is standard. dim End(M i,l ) = 1, 1 ≤ l ≤ r; 2, r + 1 ≤ l ≤ 2r; dim Hom(M i,l , τ M i,l ) = 0, 1 ≤ l ≤ r − 1; 1, r ≤ l ≤ 2r − 1; dim Hom(M i,l , τ 2 M i,l ) = 0, 1 ≤ l ≤ r − 2; 1, r − 1 ≤ l ≤ 2r − 2. The last lemma in this section also follows from the fact that T is standard (since the mesh relations are homogeneous). Lemma 2.5. Let X, Y be indecomposable objects in T , and let π 1 (X, Y ), . . . , π t (X, Y ) be representatives for the paths in T from X to Y up to equivalence via the mesh relations. Then the corresponding maps f 1 (X, Y ), . . . , f t (X, Y ) form a basis for Hom(X, Y ). Properties of T with respect to a tube In this section, we collect together some useful facts that we shall use in Section 4 to determine the rigid and Schurian Λ-modules. Recall that we have fixed a tube T in KQ-mod of rank r. Let T T be the direct sum of the indecomposable summands of T lying in T (we include the case T T = 0). Let T k , k ∈ Z s be the indecomposable summands of T T which are not contained in the wing of any other indecomposable summand of T T , numbered in order cyclically around T . The indecomposable summands of T T are contained in ∪ k∈Zs W T k , where W T k denotes the wing of T k . Note that if T T = 0 then s = 0 and Z s is the empty set. A key role is played by the modules τ T k . Let i k ∈ {0, 1, . . . , r − 1} and l k ∈ N be integers such that τ T k ∼ = M i k ,l k . Note that l k ≤ r − 1, since T k is rigid. Then we have the wings W T k = {M i,l : i k + 1 ≤ i ≤ i k + l k , 1 ≤ l ≤ l k + i k + 1 − i}, (3.1) W τ T k = {M i,l : i k ≤ i ≤ i k + l k − 1, 1 ≤ l ≤ l k + i k − i}, (3.2) W τ 2 T k = {M i,l : i k − 1 ≤ i ≤ i k + l k − 2, 1 ≤ l ≤ l k + i k − 1 − i}, (3.3) For k ∈ Z s , the quasisimple objects in W τ T k are the Q i for i k ≤ i ≤ i k + l k − 1. Note that, since Ext 1 (T, T ) = 0, we have [i k+1 − (i k + l k − 1)] r = 0, 1 (by Lemma 2.2 and the AR-formula). In other words, two successive wings W τ T k and W τ T k+1 are always separated by at least one quasisimple module. For k ∈ Z s , we define Top k to be the module M i k ,r+l k . Note that Top k is the module of smallest quasilength in the intersection of the ray through the injective objects in W τ T k and the coray through the projective objects in W τ T k . Let H k be the part of W Top k consisting of injective or projective objects in W Top k of quasilength at least r. So (3.4) H k = {M i k ,l : r ≤ l ≤ r + l k } ∪ {M i k +p,r+l k −p : 0 ≤ p ≤ l k }. The unique object in both of these sets is Top k = M i k ,r+l k , the unique projective-injective object in W Top k . Let R k (respectively, S k ) be the part of W Top k consisting of non-projective, non-injective objects in W Top k of quasilength at least r (respectively, at least r − 1). Note that R k ⊆ S k . We have: R k = {M i,l : i k + 1 ≤ i ≤ i k + l k − 1, r ≤ l ≤ r + l k + i k − i − 1}; (3.5) S k = {M i,l : i k + 1 ≤ i ≤ i k + l k , r − 1 ≤ l ≤ r + l k + i k − i − 1}. (3.6) An example is shown in Figure 6. R k (respectively, S k ) are the Q i where i k + 1 ≤ i ≤ i k + l k − 1 (respectively, i k + 1 ≤ i ≤ i k + l k ) and the quasitops are the Q i where i k ≤ i ≤ i k + l k − 2 (respectively, i k − 1 ≤ i ≤ i k + l k − 2). In particular, the quasisocle of an indecomposable object in R k lies in W T k ∩ W τ T k (respectively, in W T k ). The quasitop of an indecomposable object in R k (respectively, S k ) lies in W τ T k ∩ W τ 2 T k (respectively, W τ 2 T k ). Proof. The first statement follows from (3.5). The quasitop of M i,l is Q i+l−1 . Hence, the quasitops of the indecomposable objects in R k are the Q i with (i k + 1) + r − 1 ≤ i ≤ (i k + l k − 1) + (r + l k + i k − (i k + l k − 1) − 1) − 1, i.e. i k + r ≤ i ≤ r + l k + i k − 2. i.e. the Q i with i k ≤ i ≤ i k + l k − 2, since we are working mod r. Similarly, the quasitops of the indecomposable objects in S k are the Q i with (i k + 1) + (r − 1) − 1 ≤ i ≤ (i k + l k ) + (r + l k + i k − (i k + l k ) − 1) − 1, i.e. i k + r − 1 ≤ i ≤ r + l k + i k − 2, i.e. the Q i with i k − 1 ≤ i ≤ i k + l k − 2. The last statements follow from the descriptions of the wings W T k , W τ T k and W τ 2 T k above (3.1). It easy to observe the result in this lemma in Figure 6, where the regions R k and S k are indicated. Recall that U denotes the maximal preprojective direct summand of T . Lemma 3.2. Let k ∈ Z s and let X be an indecomposable object in W τ T k . Then Hom(U, X) = 0. Proof. Since U is preprojective, Hom(U, −) is exact on short exact sequences of modules in T , so dim Hom(U, −) is additive on such sequences. This includes, in particular, almost split sequences in T , and it follows that: Figure 6. The wings W τ T k shown as shaded regions in two copies of T in the case r = 11. The elements in the H k are drawn as filled dots. The elements in the regions R k and S k are enclosed in triangles. (3.7) dim Hom(U, X) = Y ∈WX ,Y quasisimple dim Hom(U, Y ). • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • τ T0 τ T0 τ T1 τ T1 Top 1 Top 1 Top 0 Top 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Since Hom(U, τ T k ) = 0, we must have Hom(U, Y ) = 0 for all quasisimple modules in W τ T k , and the result now follows from (3.7). Lemma 3.3. Suppose that Y ∈ ind(T ) satisfies Hom C (T T , Y ) = 0 and Hom(U, Y ) = 0. Then Y ∈ ∪ k∈Zs W τ T k . Proof. Suppose Y satisfies the assumptions above. Then, if V is an indecomposable summand of T in a tube distinct from T , we have Hom(V, Y ) = 0 and Hom(Y, τ 2 V ) = 0, so Hom C (V, Y ) = 0. We also have that Hom(Y, τ 2 U ) = 0, since τ 2 U is preprojective, so Hom C (U, Y ) = 0. Hence, we have Hom C (T, Y ) = 0, so Ext C (Y, τ T ) = 0. Since T (and hence τ T ) is a cluster-tilting object in C, this implies that Y lies in add τ T and therefore in ∪ k∈Zs W τ T k as required. Proposition 3.4. Let X be an indecomposable object in T not lying in ∪ k∈Zs W τ T k . Then Hom(U, X) = 0. Proof. Since dim Hom(U, −) is additive on T , we can assume that X is quasisimple. We assume, for a contradiction, that Hom(U, X) = 0. If we can find a module Y ∈ T \ ∪ k∈Zs W τ T k such that Hom C (T T , Y ) = 0 and Hom(U, Y ) = 0 then, by Lemma 3.3, we have a contradiction. We now construct such a module Y , considering various cases for X. Case 1: Assume that X ∼ = Q i k −1 and X ∼ = Q i k +l k for any k ∈ Z s , i.e. that X is not immediately adjacent to any of the wings W τ T k , k ∈ Z s . There is a single module of this kind in the example in Figure 6; this is denoted by X 1 in Figure 7. In this case we take Y = X. If V is an indecomposable summand of T T , then V ∈ W T k for some k ∈ Z s . Since the quasisimple module X does not lie in ∪ k∈Zs W T k , we have Hom(V, X) = 0 by Corollary 2.3. Similarly, τ 2 V ∈ W τ 2 T k for some k ∈ Z s . Since the quasitop of X (i.e. X) does not lie in ∪ k∈Zs W T k , we have Hom(X, τ 2 V ) = 0 by Corollary 2.3. Hence Hom C (T T , X) = 0, completing this case. Figure 7. Proof of Proposition 3.4. For the quasisimple module X 1 , we take Y = X 1 ; for the module X 2 , we take Y = Y 2 , and for the module • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • τ T0 τ T1 Y3 Y2 X1 X1 X2 X3X 3 , we take Y = Y 3 . We next suppose that X ∼ = M i k +l k ,1 for some k ∈ Z s (the case X ∼ = M i k −1,1 is similar). Recall that there is always at least one quasisimple module between two wings W τ T k and W τ T k±1 . Case 2: Assume first that there are at least two quasisimple modules between the wings W τ T k and W τ T k+1 , so that X is not adjacent to the wing W τ T k+1 . In the example in Figure 7, the object X 2 is an example of this type (with k = 1). In this case, we take Y = M i k ,l k +1 (indicated by Y 2 in Figure 7). By Lemma 3.2, Hom(U, M i k +l k −1,1 ) = 0. By assumption, Hom(U, X) = 0. Since dim Hom(U, −) is additive on T , we have Hom(U, M i k +l k −1,2 ) = 0. If l k = 1 then Y = M i k +l k −1,2 and Hom(U, Y ) = 0. If l k > 1 then, since dim Hom(U, −) is additive on the short exact sequence: 0 → τ T k → T ′ k ⊕ M i k +l k −1,1 → M i k +l k −1,2 → 0, it follows that Hom(U, Y ) = 0 in this case also. Since the quasisocle Q i k of Y does not lie in ∪ k ′ ∈Zs W T k ′ , we have Hom(V, Y ) = 0 for all indecomposable summands V of T T by Corollary 2.3. Since there are at least two quasisimple modules between the wings W τ T k and W τ T k+1 , the quasitop of Y does not lie in ∪ k ′ ∈Zs W τ 2 T k ′ . Hence Hom(Y, τ 2 V ) = 0 for all summands V of T T , by Corollary 2.3. So Hom C (T T , Y ) = 0, completing this case. Case 3: We finally consider the case where there is exactly one quasisimple module between the wings W τ T k and W τ T k+1 . In the example in Figure 7, the object X 3 is an example of this type. In this case, we take Y = M i k ,l k +l k+1 +1 (indicated by Y 3 in Figure 7). The quasisimples in W Y are the quasisimples in W τ T k , the quasisimples in W τ T k+1 and X. For a quasisimple module Q in one of the first two sets, Hom(U, Q) = 0 by Lemma 3.2. By assumption, Hom(U, X) = 0. Hence, arguing as in Lemma 3.2 and using the additivity of dim Hom(U, −) on T , we have Hom(U, Y ) = 0. Since the quasisocle of Y is Q i k , which does not lie in ∪ k ′ ∈Zs W T k ′ , we see that Hom(V, Y ) = 0 for any indecomposable summand of T T by Corollary 2.3. Similarly, the quasitop of Y is Q i k+1 +l k+1 −1 , which does not lie in ∪ k ′ ∈Zs W τ 2 T k ′ . Hence Hom(Y, τ 2 V ) = 0 for any indecomposable summand of T T by Corollary 2.3. So Hom C (T T , Y ) = 0, completing this case. Lemma 3.5. Let P be an indecomposable projective KQ-module, and suppose that Hom(P, X 0 ) = 0 for some indecomposable module X 0 on the border of T . Then dim Hom(P, X) ≤ 1 for all indecomposable modules X on the border of T . Furthermore, if there is some indecomposable module X 1 on the border of T such that Hom(P, X) = 0 for all indecomposable modules X ∼ = X 1 on the border of T , then dim Hom(P, X 1 ) = 1. Proof. This can be checked using the tables in [23, XIII.2]. Proposition 3.6. Suppose that T T = 0 and let X ∈ T \ ∪ k∈Zs W τ T k be an indecomposable module on the border of T and V an indecomposable summand of U . Then dim Hom(V, X) ≤ 1. Furthermore, if k = 0 and T 0 has quasilength r − 1, then dim Hom(V, X) = 1. Proof. By applying a power of τ if necessary, we can assume that V is projective. By assumption, T contains a summand of T , so there is at least one quasisimple module X 0 in ∪ k∈Zs W τ T k . By Lemma 3.2, we have that Hom(V, X 0 ) = 0. The first part of the lemma then follows from Lemma 3.5. If k = 0 and T 0 has quasilength r − 1, then Hom(V, X 0 ) = 0 for every quasisimple in W τ T0 . Since X is the unique quasisimple in T not in W τ T0 , the second part now follows from Lemma 3.5 also. Abusing notation, we denote the down-arrows in T by x and the up-arrows by y. So, for example, x r means the composition of r down-arrows from a given vertex. Proposition 3.7. Let X = M i,l be an indecomposable module in T . (a) Suppose that r+1 ≤ l. Let u X = y r x r : X → X. Then u X factors through add(τ T T ) if and only if X ∈ ∪ k∈Zs H k ∪ R k . (b) Suppose that r ≤ l. Let v X = y r−1 x r−1 : X → τ X be the unique nonzero map (up to a scalar), as in Lemma 2.4. Then v X factors through add(τ T T ⊕ τ 2 T T ) if and only if X ∈ ∪ k∈Zs (H k ∪ R k \ {Top k }). (c) Suppose that r ≤ l. Let w X = y r−2 x r−2 : τ −1 X → τ X. Then w X factors through add(τ T T ) if and only if X ∈ ∪ k∈Zs S k . Proof. We start with part (a). Note that u X lies in the basis for Hom(X, X) given in Lemma 2.5. Also, by the mesh relations, u X = x r y r . Let D X be the diamond-shaped region in T bounded by the paths x r y r and y r x r starting at X. It is clear that u X factors through any indecomposable module in D X . For an example, see Figure 8, where part of one copy of D X has been drawn. If Y lies outside D X , then any path from X to X in T via Y must contain more than r downward arrows. By Lemma 2.5 it is a linear combination of basis elements distinct from u. So u cannot factor through the direct sum of any collection of objects outside this region. Hence u X factors through add(τ T T ) if and only if some indecomposable summand of τ T T lies in D X . Since the indecomposable summands of τ T T lie in ∪ k∈Zs W τ T k , we see that u X factors through τ T T if and only if M i+r,l−r (the module in D X with minimal quasilength) lies in ∪ k∈Zs W τ T k . The corners of the triangular region H k ∪ R k are M i k ,r , Top k = M i k ,r+l k and M i k +l k ,r . The part of H k ∪ R k consisting of modules with quasilength at least r + 1 is the triangle with corners M i k ,r+1 , M i k ,r+l k and M i k +l k −1,r+1 . Hence, X = M i,l lies in H k ∪ R k if and only if M i+r,l−r lies in the triangular region of T with corners M i k ,1 , M i k ,l k and M i k +l k −1,1 , i.e. W τ T k . The result follows. For part (b), we consider the diamond-shaped region E X bounded by the paths y r−1 x r−1 and x r−1 y r−1 starting at X. We have, using an argument similar to the above, that v X factors through add(τ T T ⊕ τ 2 T T ) if and only if some indecomposable direct sum- Figure 8. Proof of Proposition 3.7: the shaded region indicates part of (one copy of) the diamond-shaped region D X . In this case, u X does not factor through add(τ T T ). mand of τ T T ⊕ τ 2 T T lies in E X . Hence, v X factors through add(τ T T ⊕ τ 2 T T ) if and • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • X X τ T0 τ T0 τ T1 τ T1only if M i+r−1,l−r+1 lies in ∪ k∈Zs (W τ T k ∪ W τ 2 T k ). The corners of the trapezoidal region H k ∪R k \{Top k } are M i k ,r , M i k ,r+l k −1 , M i k +1,r+l k −1 , M i k +l k ,r . Hence X ∈ H k ∪R k \{Top k } if and only if M i+r−1,l−r+1 lies in the trapezoidal region with corners M i k +r−1,1 , M i k +r−1,l k , M i k +r,l k , M i k +l k +r−1,1 , i.e. M i k −1,1 , M i k −1,l k , M i k ,l k , M i k +l k −1,1 which is the intersection W τ T k ∪ W τ 2 T k . Part (b) follows. For part (c), we consider the diamond-shaped region F τ −1 X bounded by the paths y r−2 x r−2 and x r−2 y r−2 starting at τ −1 X. We have, using an argument similar to the above, that w X factors through add(τ T T ) if and only if some indecomposable direct summand of τ T T lies in F τ −1 X . Hence, w X factors through add(τ T T ) if and only if M i+1+r−2,l−r+2 = M i+r−1,l−r+2 lies in ∪ k∈Zs W τ T k . The corners of the triangular region S k are M i k +1,r−1 , M i k +1,r+l k −2 and M i k +l k ,r−1 . Hence X ∈ S k if and only if M i+r−1,l−r+2 lies in the the triangular region with corners M i k +1+r−1,1 , M i k +1+r−1,l k and M i k +l k +r−1,1 , i.e. M i k ,1 , M i k ,l k and M i k +l k −1,1 , which is the wing W τ T k . Part (c) follows. Rigid and Schurian Λ-modules We determine which objects X in T give rise to Schurian and rigid Λ-modules X. (a) If r + 1 ≤ l and X ∈ ∪ k∈Zs H k ∪ R k then X is not Schurian. (b) If r ≤ l and X ∈ ∪ k∈Zs H k ∪ R k \ {Top k } then X is not rigid. Proof. Firstly note that in both (a) and (b), X cannot be a summand of τ T . For part (a), let u X = y r x r : X → X. Since U is preprojective, any composition of maps in C from X to X factoring through U is zero. By Proposition 3.7(a) and Lemma 1.12, u X does not factor through add(τ T T ). It follows that u X does not factor through add(τ T ) and hence Hom H C/ add(τ T ) (X, X) ∼ = K, so X is not Schurian. A similar argument, using Proposition 3.7(b), gives part (b). Lemma 4.2. Let X be an indecomposable object in T which is not a summand of τ T . Then: (a) X is a τ -rigid Λ-module if and only if the quasilength of X is at most r − 1; (b) If the quasilength of X is at most r − 2, then X is Schurian. Proof. It is well-known (and follows from the fact that T is standard) that X is a rigid KQmodule if and only if its quasilength is at most r − 1, so part (a) follows from Corollary 1.10. If the quasilength of X is at most r − 2, then Hom(X, X) ∼ = K and Hom(X, τ 2 X) = 0 by Lemma 2.4, so Hom C (X, X) ∼ = Hom(X, X) ⊕ D Hom(X, τ 2 X) ∼ = K, giving part (b). We need the following. ∪ k∈Zs W τ T k then ε factors in D b (KQ) through U [1]. (b) If there is a map ϕ ∈ Hom(B, τ A) such that im(ϕ) has an indecomposable direct summand which does not lie in ∪ k∈Zs W τ 2 T k then ε factors in D b (KQ) through τ U [1]. Proof. We write ϕ as ϕ 2 ϕ 1 where ϕ 1 : B → im(ϕ) and ϕ 2 : im(ϕ) → τ A. We have the short exact sequence: (4.1) 0 / / ker(ϕ) / / B ϕ1 / / im(ϕ) / / 0 For part (a), we apply Hom(U, −) to this sequence (noting that, since U is preprojective, it is exact on T ), to obtain the exact sequence: 0 / / Hom(U, ker(ϕ)) / / Hom(U, B) Hom(U,ϕ1) / / Hom(U, im(ϕ)) / / 0 Since im ϕ has an indecomposable direct summand which does not lie in ∪ k∈Zs W τ T k , it follows from Proposition 3.4 that Hom(U, im ϕ) = 0. Hence, the epimorphism Hom(U, ϕ 1 ) is nonzero. Since ϕ 2 is a monomorphism, Hom(U, ϕ) = 0, so there is a map β ∈ Hom(U, B) such that Hom(U, ϕ)(β) = ϕβ = 0. Hence Hom(β, τ A)(ϕ) = ϕβ = 0, so Hom(β, τ A) = 0. Part (a) now follows from Proposition 1.14(b), taking C = U , and Lemma 1.12. For part (b), we apply Hom(τ U, −) to the sequence (4.1). Note that Hom(τ U, im(ϕ)) ∼ = Hom(U, τ −1 im(ϕ)) = 0 by Proposition 3.4, and the argument goes through as in part (a). Lemma 4.4. Fix k ∈ Z s and let X ∈ H k \ {Top k }. Then X is rigid. Proof. Firstly note that X cannot be a direct summand of τ T . By the assumption, the quasilength of X lies in the interval [r, 2r − 1], so, by Lemma 2.4, Hom(X, τ X) ∼ = K. Let u = y r−1 x r−1 be a nonzero element of Hom(X, τ X). Then by Proposition 3.7(b), u factors through add(τ T T ⊕ τ 2 T T ), so Hom H C/ add(τ T ⊕τ 2 T ) (X, τ X) = 0. Suppose that X ∼ = M i k ,l where r ≤ l ≤ r + l k − 1. We have Hom F C (X, τ X) = Hom(X, X[1]) ∼ = D Hom(τ X, τ 2 X) ∼ = D Hom(X, τ X) ∼ = K. We apply Lemma 4.3(a) in the case A = X, B = X and C = U . We take ϕ = u and ε to be a nonzero element of Hom(X, X [1]). Then im(ϕ) ∼ = M i k −1,l−r+1 ∈ ∪ k∈Zs W τ T k . By Lemma 4.3(a), we have that ε factors through U [1]. Hence, regarded as an F -map in C, ε factors through τ U . It follows that Hom F C/ add(τ T ⊕τ 2 T ) (X, τ X) = 0. Suppose that X ∼ = M i k +p,r+l k −p where 1 ≤ p ≤ l k . We have Hom F C (X, τ X) = Hom(X, X[1]) ∼ = D Hom(τ X, τ 2 X) ∼ = D Hom(X, τ X) ∼ = K. We apply Lemma 4.3(b) in the case A = X, B = X and C = τ U . We take ϕ = u and ε to be a nonzero element of Hom(X, X [1]). Then im(ϕ) = M i k +p−1,l k −p−1 ∈ ∪ k∈Zs W τ T k . By Lemma 4.3(b), ε factors through τ U [1]. Hence, regarded as an F -map in C, ε factors through τ 2 U . It follows that Hom F C/ add(τ T ⊕τ 2 T ) (X, τ X) = 0. In either case, we have shown that Hom C/ add(τ T ⊕τ 2 T ) (X, τ X) = 0, and it follows that X is rigid by Proposition 1.8(b). If T T contains an indecomposable direct summand of quasilength r − 1 then s = 1, l 0 = r − 1 and, by (3.4), (4.2) H 0 = {M i0,l : r ≤ l ≤ 2r − 1} ∪ {M i0+p,2r−1−p :, 0 ≤ p ≤ r − 1}. In particular, Top k = M i0,2r−1 has quasilength 2r − 1. In all other cases, Top k has smaller quasilength. Lemma 4.5. Fix k ∈ Z s . Suppose that X is an indecomposable object of T which is not a summand of τ T and satisfies either (a) X ∈ H k and ql(X) ≤ 2r − 2, or (b) ql(X) ∈ {r − 1, r} and X ∈ ∪ k∈Zs H k ∪ S k . Then X is Schurian. Proof. In case (a), r ≤ ql(X) ≤ 2r − 2, and in case (b), r − 1 ≤ ql(X) ≤ r. If ql(X) ≤ r then Hom(X, X) ∼ = K by Lemma 2.4. If ql(X) > r then Hom(X, X) ∼ = K 2 . A basis is given by the identity map and the map u X in Proposition 3.7(a). By Proposition 3.7(a), u X factors through add(τ T ). Hence, in either case, Hom H C/ add(τ T ) (X, X) ∼ = K. Since the quasilength of X lies in [r − 1, 2r − 2], we have, by Lemma 2.4, that Hom F C (X, X) = Hom(X, τ −1 X[1]) ∼ = Ext(X, τ −1 X) ∼ = D Hom(τ −1 X, τ X) ∼ = K. We apply Lemma 4.3(a) in the case A = X, B = τ −1 X. We take ϕ to be the map w τ −1 X in Proposition 3.7(c), the unique nonzero element of Hom(τ −1 X, τ X) up to a scalar, and ε to be a nonzero element of Hom(X, τ −1 X[1]). In case (a), there are two possibilities. If X ∼ = M i k ,l where r ≤ l ≤ r + l k − 1, then im(ϕ) ∼ = M i k +r−1,l−r+2 ∈ ∪ k∈Zs W τ T k . If X ∼ = M i k +p,r+l k −p where 1 ≤ p ≤ l k , then im(ϕ) ∼ = M i k +p+r−1,2+l k −p ∈ ∪ k∈Zs W τ T k . In case (b), there are also two possibilities. If ql(X) = r − 1, then X ∼ = M i,r−1 where i ∈ ∪ k∈Zs [i k + 1, i k + l k ]. Then im(ϕ) ∼ = M i+1+(r−2),r−1−(r−2) = M i−1,1 ∈ ∪ k∈Zs W τ T k . If ql(X) = r, then X ∼ = M i,r where i ∈ ∪ k∈Zs [i k , i k + l k ]. Then im(ϕ) ∼ = M i+1+(r−2),r−(r−2) = M i−1,2 ∈ ∪ k∈Zs W τ T k . Applying Lemma 4.3(a), we see that ε factors through U [1]. Hence, regarded as an F -map in C, ε factors through τ U . It follows that Hom F C/ add(τ T ⊕τ 2 T ) (X, X) = 0. We have shown that Hom C/ add(τ T ) (X, X) ∼ = K, and it follows that X is Schurian by Proposition 1.8(a). Lemma 4.6. Fix k ∈ Z s , and let X ∈ R k . Then X is not rigid. Proof. Since X ∈ R k , we have r ≤ ql(X) ≤ r + l k − 2 ≤ 2r − 3. In particular, this implies that X is not a direct summand of τ T . By Lemma 2.4, we have Hom(X, τ X) ∼ = K. Let u be a nonzero map in Hom(X, τ X), unique up to a nonzero scalar. We have Hom F C (X, τ X) = Hom(X, X[1]) ∼ = D Hom(X, τ X) ∼ = K. Let v ∈ Hom(X, X[1] ) be a nonzero map, unique up to a nonzero scalar. We show first that v cannot factor through V for any indecomposable summand V of τ T or τ 2 T . If Hom(X, V ) = 0 then we are done, so we may assume that Hom(X, V ) = 0. In particular, we may assume that V lies in T . By Lemma 3.1, the quasitop of X lies in W τ T k ∩ W τ 2 T k , so V must lie in W τ T k ∩ W τ 2 T k by Corollary 2.3. By Lemma 2.2, we have that Hom(X, V ) ∼ = K. Let f ∈ Hom(X, V ) be any nonzero map. Then the number of downward arrows in a path for f (and hence for τ f ) is at least ql(X) − ql(V ) ≥ ql(X) − l k ≥ ql(X) − r + 1. The number of downward arrows in a path for u is r − 1, so the number of downward arrows in a path for τ f • u is ql(X), so τ f • u = 0 by Lemma 2.1. Since u is a basis for Hom(X, τ X), it follows that Hom(X, τ f ) = 0. Therefore, by Proposition 1.14(a), v cannot factor through V . We next show that v cannot factor through τ −1 V [1] for any indecomposable summand V of τ T . If Hom(τ −1 V, X) = 0 then Hom(τ −1 V [1], X[1]) = 0 and we are done. Therefore, we may assume that Hom(τ −1 V, X) = 0. Suppose first that V ∈ T . By Lemma 3.1, the quasisocle of X lies in W T k ∩W τ T k , so τ −1 V must lie in W T k ∩ W τ T k by Corollary 2.3. By Lemma 2.2, we have that Hom(τ −1 V, X) ∼ = K. Let g ∈ Hom(τ −1 V, X) be any nonzero map. The number of downward arrows in a path for u is r − 1, hence the number of downward arrows in a path for ug is at least r − 1. As ql(τ −1 V ) ≤ l k ≤ r − 1, it follows that ug = 0. Since u is a basis for Hom(X, τ X), it follows that Hom(g, τ X) = 0. Secondly, suppose that V is an indecomposable direct summand of τ U or τ 2 U . Let h ∈ Hom(τ −1 V, X). By Proposition 3.7(b), we have that u factors through τ T k ⊕ τ 2 T k , so uh = 0 as τ −1 V is a direct summand of T ⊕ τ T . Since u is a basis for Hom(X, τ X), it follows that Hom(h, τ X) = 0. Applying Proposition 1.14(b) to the triple A = B = X, C = τ −1 V and β = g or h, we obtain that v does not factor through τ −1 V [1]. Hence, v does not factor through V or τ −1 V [1] for any indecomposable summand V of τ T ⊕τ 2 T . Since Hom(X, X[1]) ∼ = K, it follows that v does not factor through add(τ T ⊕T [1]). By Lemma 1.12, the morphism v, regarded as a morphism in Hom F C (X, τ X), does not factor through add(τ T ⊕ τ 2 T ). Hence Hom C/ add(τ T ⊕τ 2 T ) (X, τ X) = 0. Therefore X is not rigid by Proposition 1.8. Note that the objects in S k (see (3.5)) have quasilength at least r − 1, so if T has no indecomposable direct summand in T of qausilength r − 1, the objects in S k are not summands of τ T . It is easy to check directly that this holds in the case where T has an indecomposable direct summand T 0 in T of quasilength r − 1, since all the indecomposable direct summands of τ T in T lie in W τ T0 (see Figure 22). Lemma 4.7. Fix k ∈ Z s , and let X ∈ S k . Then X is not Schurian. Proof. Firstly note that, by the above, X is not an indecomposable direct summand of τ T . Since X ∈ S k , we have r − 1 ≤ ql(X) ≤ r + l k − 2 ≤ 2r − 3, so by Lemma 2.4, we have Hom(τ −1 X, τ X) ∼ = K. Let u be a nonzero map in Hom(τ −1 X, τ X), unique up to a nonzero scalar. We have Hom F C (X, X) = Hom(X, τ −1 X[1]) ∼ = D Hom(τ −1 X, τ X) ∼ = K. Let v ∈ Hom(X, τ −1 X[1] ) be a nonzero map, unique up to a nonzero scalar. We will first show that v cannot factor through V for any indecomposable summand V of τ T . If Hom(X, V ) = 0 then we are done, so we may assume that Hom(X, V ) = 0. In particular, we may assume that V lies in T . By Lemma 3.1, the quasitop of X lies in W τ 2 T k , so V must lie in W τ 2 T k by Corollary 2.3, and hence in W τ 2 T k ∩ W τ T k . By Lemma 2.2, Hom(X, V ) ∼ = K. Let f ∈ Hom(X, V ) be a nonzero map, unique up to a nonzero scalar. If V ∼ = τ T k then the number of downward arrows in a path for f (and hence for τ f ) is at least ql(X) − ql(V ) ≥ ql(X) − (l k − 1) ≥ ql(X) − r + 2. If V ∼ = τ T k then, since no object in S k is in the coray through τ T k , the number of downward arrows in a path for f (and hence for τ f ) is at least ql(X) − ql(V ) + 1 ≥ ql(X) − r + 2. The number of downward arrows in a path for u is r − 2. Hence in either case the number of downward arrows in a path for τ f • u is ql(X), so τ f • u = 0 by Lemma 2.1. Since u is a basis for Hom(X, τ X), it follows that Hom(X, τ f ) = 0. Therefore, by Proposition 1.14(a), v cannot factor through V . We next show that v cannot factor through τ −1 V [1] for any indecomposable summand V of T . If Hom(τ −1 V, τ −1 X) = 0 then Hom(τ −1 V [1], τ −1 X[1]) = 0 and we are done, so we may assume that Hom(τ −1 V, τ −1 X) = 0. Suppose first that V ∈ T . By Lemma 3.1, the quasisocle of τ −1 X lies in W τ −1 T k , so τ −1 V must lie in W τ −1 T k by Corollary 2.3. Hence V must lie in W T k and thus in W T k ∩ W τ T k . By Lemma 2.2, Hom(τ −1 V, τ −1 X) ∼ = K. Let g ∈ Hom(τ −1 V, τ −1 X) be a nonzero map, unique up to a nonzero scalar. The number of downward arrows in a path for u is r − 2. If V ∼ = τ T k , this is at least the quasilength of τ −1 V , so ug = 0 by Lemma 2.1. If V ∼ = τ T k then, since no element of τ −1 S k lies in the ray through τ −1 V ∼ = T k , a path for g has at least one downward arrow. It follows that a path for ug has at least r−1 = ql(τ −1 V ) downwards arrows, so ug = 0 in this case also. Since u is a basis for Hom(X, τ X), it follows that, in either case, Hom(g, τ X) = 0. Secondly, suppose that V is an indecomposable direct summand of τ U . Let h ∈ Hom(τ −1 V, τ −1 X). By Proposition 3.7(c), u factors through τ T k , since X ∈ S k . Hence, uh = 0 as τ −1 V is a direct summand of T . Since u is a basis for Hom(X, τ X), it follows that Hom(h, τ X) = 0. Applying Proposition 1.14(b) to the triple A = B = X, C = τ −1 V , we obtain that v does not factor through τ −1 V [1]. Hence, v does not factor through V or τ −1 V [1] for any indecomposable summand V of τ T . Since Hom(X, τ −1 X[1]) ∼ = K, it follows that v does not factor through add(τ T ⊕ T [1]). By Lemma 1.12, the morphism v, regarded as a morphism in Hom F C (X, X), does not factor through τ T . Hence Hom C/ add(τ T ) (X, X) ∼ = K. Therefore X is not Schurian by Proposition 1.8. Recall (equation 4.2) that if T T contains an indecomposable direct summand of quasilength r − 1 then H 0 = {M i0,l : r ≤ l ≤ 2r − 1} ∪ {M i0+p,2r−1−p :, 0 ≤ p ≤ r − 1}. and Top k = M i0,2r−1 . The following lemma shows, in particular, that Top k is Schurian. Lemma 4.8. Suppose that T T contains an indecomposable direct summand T 0 of quasilength r − 1. Let X ∈ H 0 . Then X is a strongly Schurian, and hence Schurian, Λ-module. Proof. Firstly note that ql(X) ≥ r, so X is not a summand of τ T . Let V be an indecomposable direct summand of T . Note that the entry in the dimension vector of X corresponding to V is equal to dim Hom C (V, X). Suppose first that V is an indecomposable summand of U . Then by Lemma 3.2, we have that Hom(V, Y ) = 0 for all objects Y in W τ T0 . By Proposition 3.6, dim Hom(V, Y ) ≤ 1 if Y = M i0−1,1 is the unique object on the border of T not in W τ T0 . Using the additivity of dim Hom(V, −) on T , we see that dim Hom(V, X) ≤ 1. Since V is preprojective, dim Hom(X, τ 2 V ) = 0, so, since Hom C (V, X) ∼ = Hom(V, X) ⊕ D Hom(X, τ 2 V ), we have dim Hom C (V, X) ≤ 1. If V lies in a tube other than T then Hom C (V, X) = 0. So we are left with the case where V lies in T . If X ∼ = M i0,l for some l with r ≤ l ≤ 2r − 1 then the quasisocle of X is Q i0 , which does not lie in W T . So, by Corollary 2.3, Hom(V, X) = 0. Since ql(V ) ≤ r − 1, it follows from Lemma 2.2 that dim Hom(X, τ 2 V ) ≤ 1. Hence dim Hom C (V, X) ≤ 1. If X ∼ = M i0+p,2r−1−p for some p with 0 ≤ p ≤ r − 1 then the quasitop of X is Q i0+p+2r−1−p−1 = Q i0−2 , which does not lie in W τ 2 T . So, by Corollary 2.3, we have that Hom(X, τ 2 V ) = 0. Since ql(V ) ≤ r − 1, it follows from Lemma 2.2 that dim Hom(V, X) ≤ 1. Hence dim Hom C (V, X) ≤ 1. We have shown that X is strongly Schurian as required. Since any strongly Schurian module is Schurian, we are done. Corollary 4.9. Let X ∈ ∪ k∈Zs H k . Then X is Schurian. Proof. Firstly note that, since ql(X) ≥ r, X is not a direct summand of τ T . Suppose k ∈ Z s and X ∈ H k . If ql(X) ≤ 2r − 2 then this follows from Lemma 4.5. The maximal quasilength of an object in H k is ql(Top k ) = ql(M i k ,r+l k ) = r + l k . This is only greater than 2r − 2 when l k is maximal, i.e. equal to r − 1. Then s = 1 (i.e. there is only one indecomposable direct summand of T T not contained in the wing of another indecomposable direct summand of T T ). We must have k = 0 and the result follows from Lemma 4.8. We have now determined whether X is rigid or Schurian for all indecomposable modules X in T which are not direct summands of τ T . We summarize this with the following theorem. Note that, by Theorem 1.7, every indecomposable Λ-module is of the form X for X an indecomposable object in C which is not a direct summand of τ T . Note also that part (a) of the following is a consequence of Lemma 4.2(a), which was shown using [1]. Theorem 4.10. Let Q be a quiver of tame representation type, and C the corresponding cluster category. Let T be an arbitrary cluster-tilting object in C. Let X an indecomposable object of C which is not a summand of τ T and let X the corresponding Λ-module. (a) The Λ-module X is τ -rigid if and only if X is transjective or X is regular and ql(X) ≤ r − 1. (b) The Λ-module X is rigid if and only if either (i) X is transjective, or (ii) X is regular and ql(X) ≤ r − 1 or (iii) X is regular and X ∈ ∪ k∈Zs H k \ {Top k }. (c) The Λ-module X is Schurian if and only if either (i) X is transjective, or (ii) X is regular and ql(X) ≤ r − 2, or (iii) X is regular, ql(X) ∈ {r − 1, r} and X ∈ ∪ k∈Zs S k , or (iv) X is regular, ql(X) ≥ r + 1 and X ∈ ∪ k∈Zs H k . Proof. If X is transjective, the result follows from Remark 1.11, so we may assume that X lies in a tube T . Let r be the rank of T . Replacing T with τ mr T for some m ∈ Z if necessary, we may assume that T is of the form U ⊕ T ′ where U is a preprojective module and T ′ is regular, i.e. that Assumption 1.5 holds (note that τ is an autoequivalence of C). For part (b), note that if ql(X) ≤ r − 1, then X is τ -rigid by (a), hence rigid. If ql(X) ≥ r and X ∈ ∪ k∈Zs H k ∪ R k \ {Top k } then X is not rigid by Lemma 4.1. If ql(X) ≥ r and X ∈ R k then X is not rigid by Lemma 4.6. And if ql(X) ≥ r and X ∈ ∪ k∈Zs H k \ {Top k } then X is rigid by Lemma 4.4. For part (c), note that if ql(X) ≤ r − 2 then X is Schurian by Lemma 4.2. If ql(X) ≥ r + 1 and X ∈ ∪ k∈Zs H k ∪ R k then X is not Schurian by Lemma 4.1. If ql(X) ≥ r + 1 and X ∈ R k then X ∈ S k so X is not Schurian by Lemma 4.7. If ql(X) ≥ r + 1 and X ∈ H k then X is Schurian by Corollary 4.9. If ql(X) ∈ {r − 1, r} and X ∈ ∪ k∈Zs H k ∪ S k then X is Schurian by Lemma 4.5. If ql(X) ∈ {r − 1, r} and X ∈ H k then X is Schurian by Corollary 4.9. If ql(X) ∈ {r − 1, r} and X ∈ S k then X is not Schurian by Lemma 4.7. Corollary 4.11. Let Q be a quiver of finite or tame representation type and Λ a clustertilted algebra arising from the cluster category of Q. Then every indecomposable Λ-module which is rigid, but not τ -rigid, is Schurian. Proof. If Q is of finite representation type, then it is known that every indecomposable object in D b (kQ) is rigid. Hence, by Theorem 1.10, every indecomposable Λ-module is τ -rigid and the statement is vacuous in this case. Suppose that Q is of tame representation type. Let Λ = End C (T ) opp , where T is a cluster-tilting object in the cluster category C of Q. Let X be an indecomposable object in C which is not a summand of τ T . If X is rigid, but not τ -rigid, then by Theorem 4.10, we have that X is regular and X ∈ ∪ k∈Zs H k \ {Top k }. If ql(X) = r, then X is Schurian by Theorem 4.10(c)(iii), since ∪ k∈Zs H k ∩ ∪ k∈Zs S k is empty. If ql(X) ≥ r + 1, then X is Schurian by Theorem 4.10(c)(iv). In Figure 9, we show part of the AR-quiver of Λ-mod for Example 1.6. The part shown consists of modules coming from the tube in KQ-mod shown in Figure 2. We give a Q Λcoloured quiver for each module, where Q Λ is the quiver of Λ. Note that we need to distinguish between the two arrows between vertices 1 and 4. We do this by decorating the arrow which is involved in the relations with an asterisk. Recall that this then has the following interpretation (see the text after Definition 1.1). Suppose that ϕ is the linear map corresponding to the decorated (respectively, undecorated) arrow in Q Λ . Then the image of a basis element b ∈ B 1 (the basis of the vector space at the vertex 1) under ϕ is the sum of the basis elements c ∈ B 4 which are at the end of an arrow starting at b labelled with (respectively, without) an asterisk. The diagram on the right shows which of these modules are τ -rigid, rigid and Schurian. In Figure 10, we illustrate the τ -rigid, rigid and Schurian Λ-modules given by Theorem 4.10 for the example in Figure 6 (choosing specific indecomposable summands of T in the wings of the T i ). . The left hand diagram shows part of the AR-quiver of Λ-mod. The right hand diagram shows the same objects. The symbol for a module is circular if it is Schurian, filled-in with gray if it is rigid but not τ -rigid, and filled-in with black if it is τ -rigid. The symbol × represents a gap in the AR-quiver (corresponding to an indecomposable direct summand of τ T ) . Figure 10. Schurian and rigid Λ-modules for a particular choice of tilting module T . The notation is as in Figure 9. Figure 11. Part of the AR-quiver of KQ-mod. • • • • • • • • • • • • • • • • × × × × × × • • • • • • • • • • • • • • • • • • • • × × • • • • • • • • • • • • • • • • • • × × × × • • • • • • • • • • • • • • • • • • • • × × • • τ T0 τ T0 τ T1 τ T1 Top 1 Top 1 Top 0 Top 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Wild case In this section we determine whether some modules are rigid or Schurian for a specific quiver of wild representation type. We will see that there are some similarities with the tame case. Let Q be the quiver: 0 / / 1 / / ( ( 2 / / 3 / / 4 and KQ the corresponding path algebra, of wild representation type. Let P 0 , . . . , P 4 be the indecomposable projective KQ-modules (with Q-coloured quivers as in (5.1)), and S 0 , . . . , S 4 their simple tops. The simple module S 2 is a quasisimple object in a regular component R of type ZA ∞ in the AR-quiver of KQ-mod. Figure 11 depicts part of this component. It is easy to check using the AR-formula that the modules S 2 , of quasilength 1, and We mutate (in the sense of [15,21]) the tilting module KQ at P 2 , via the short exact sequence: 0 → P 2 → P 1 → T 2 → 0, where T 2 = 1 4 . We obtain the tilting module P 0 ⊕ P 1 ⊕ T 2 ⊕ P 3 ⊕ P 4 . We mutate this tilting module at P 3 , via the short exact sequence 0 → P 3 → P 1 → T 3 → 0, where T 3 = 2 1 4 . This gives the tilting module T = P 0 ⊕ P 1 ⊕ T 2 ⊕ T 3 ⊕ P 4 , which induces a cluster-tilting object in C. We define maps a, b, c, d, e, f in C as follows (see Figure 12). Let a be the embedding of P 1 into P 0 , b a surjection of P 1 onto T 3 . We have Hom(T 3 , P 4 ) = 0, while Figure 11). We take c to be a non-zero element of Hom F C (T 3 , P 4 ). Hom F C (T 3 , P 4 ) ∼ = D Hom(P 4 , τ 2 T 3 ) ∼ = K, since τ 2 T 3 = 4 3 1 0 (see There are two embeddings of the simple module P 4 = S 4 into P 1 (see (5.1)). We choose d to be the map whose image is the lower 4 in the Q-coloured quiver for P 1 in (5.2), and e to be the map whose image is the upper 4. We take f to be the map from T 3 to T 2 factoring out the simple S 2 in the socle of T 3 . Let g : P 4 → P 1 be equal to d or e. Applying Proposition 1.13(b) with A = T 3 , B = τ −1 P 1 , C = τ −1 P 4 , and β = τ −1 g we see that Hom(T 3 , τ −1 g[1]) = 0 if and only if Hom(τ −1 g, τ T 3 ) = 0, which holds if and only if the map Hom(g, τ 2 T 3 ) : Hom(P 1 , τ 2 T 3 ) → Hom(P 4 , τ 2 T 3 ) is zero. We have dim Hom(P 1 , τ 2 T 3 ) = 1 (see Figure 11), so let h : P 1 → τ 2 T 3 be a nonzero map. From the explicit definition of the maps d and e, we see that hd = 0, while he = 0. Hence Hom(d, It follows that Λ = End C (T ) opp is given by the quiver Q Λ with the relations shown in Figure 12 (where we have labelled the arrows with the corresponding maps between indecomposable projectives in KQ-mod -note that these go in the opposite direction). As in the tame case (see Figure 9), we shall draw modules for Λ as Q Λ -coloured quivers, decorating the arrow between vertices 1 and 4 which is involved in the relations (corresponding to the map d) with an asterisk. Note that the AR-quiver of Λ-mod is the image of the AR-quiver of C under Hom C (T, −) by [8,Prop. 3.2], with the indecomposable summands of τ T deleted; we will denote them by filled-in vertices. Let P Λ 0 , . . . , P Λ 4 denote the indecomposable projective modules over Λ, S Λ 0 , . . . , S Λ 4 their simple tops and I Λ 0 , . . . , I Λ 4 the corresponding indecomposable injective modules. We have: Figure 16 illustrates part of the AR-quiver of Λ-mod, including the image of the part of the AR-quiver of C shown in Figure 11. (5.2) P Λ 0 = Proof. Firstly, note that Hom C (T, T i ) ∼ = P Λ i and Hom C (T, τ 2 T i ) ∼ = I Λ i , so applying Hom C (T, −) to the first two rows in Figure 11 gives the first two rows in Figure 16 except for X 1 . If α : P → P ′ is a map between projective Λ-modules, we denote by α the corresponding map between injective modules, α * : D Hom Λ (P, Λ) → D Hom Λ (P ′ , Λ). A projective presentation of S Λ 2 is: P Λ 3 α / / P Λ 2 / / S Λ 2 / / 0, where α is the embedding. So τ S Λ 2 is the kernel of α * : I Λ 3 → I Λ 2 . Let β be the nonzero map from P Λ 1 to P Λ 3 . Since αβ = 0, we have α * β * = 0, so α * must be the map factoring out the lower 2. It follows that τ S Λ 2 = X 1 , completing the verification of the first two rows in Figure 16. The irreducible maps from I Λ 3 have targets given by the indecomposable direct summands of I Λ 3 /S Λ 3 , i.e. I Λ 2 and X 3 . The irreducible map with target P Λ 3 must be the inclusion of its (indecomposable) radical X 3 . We have: Ext(X 3 , τ X 3 ) ∼ = D Hom(τ X 3 , τ X 3 ) ∼ = K, so there is a unique non-split short exact sequence ending in X 3 , which must be as shown. Next, we compute τ X 6 . From its Q Λ -coloured quiver, we see that the projective cover of X 6 is given by ϕ : P Λ 0 ⊕ P Λ 1 ⊕ P Λ 2 → X 6 . We need to compute the the kernel L of ϕ. Let B = ∪ i∈{0,1,2,3,4} B i be the basis of P Λ 0 ⊕ P Λ 1 ⊕ P Λ 2 coming from the Q Λ -coloured quiver given by the disjoint union of the Q Λ -coloured quivers of P Λ 1 , P Λ 2 and P Λ 3 in (5.2). As in Figure 14. Computation of a basis for L i , i a vertex of Q Λ . Remark 1.2, we will write the basis elements in B i as b i1 , b i2 , . . . (in an order taking first the basis elements for P Λ 0 , then P Λ 1 and P Λ 2 ). We shall also redraw each connected component of this Q Λ -coloured quiver as in Remark 1.2. We do the same for X 6 , using the notation c ij . The result is shown in Figure 13. L i 0 b 01 → c 01 empty b 11 → c 11 1 b 12 → c 12 b 11 − b 13 b 13 → c 11 2 b 21 → c 21 empty b 31 → 0 3 b 32 → c 32 b 31 − b 34 , b 34 b 33 → c 31 b 34 → 0 b 41 → c 41 b 42 → 0 5 b 43 → c 42 b 41 − b 45 , b 42 , b 44 − b 45 b 44 → c 41 b 45 → c 41 Let L = ker ϕ, regarded as a representation with the vector space L i at vertex i of Q Λ . We describe a basis for each L i in the table in Figure 14. This basis is carefully chosen to allow us to give an explicit description of L as a direct sum of indecomposable modules. Using Figure 13, we can compute the restriction of the linear maps defining P to the submodule L to get the description of L in Figure 15. We obtain a Q Λ -coloured quiver for this module, and we obtain that L = ker ϕ ∼ = P Λ 1 ⊕ P Λ 4 . Let ψ : P Λ 1 ⊕ P Λ 4 → P Λ 0 ⊕ P Λ 1 ⊕ P Λ 2 be the embedding of ker ϕ into P Λ 0 ⊕ P Λ 1 ⊕ P Λ 2 . We can write ψ as a 3 × 2 matrix ψ = (ψ ij ), and the components ψ ij can be read off from the above explicit description of ker ϕ. We have ψ * = (ψ * ij ) : I Λ 1 ⊕ I Λ 4 → I Λ 0 ⊕ I Λ 1 ⊕ I Λ 2 . Since ψ 11 : P Λ 1 → P Λ 0 is nonzero, ψ * 11 is a surjection onto I Λ 0 ∼ = S Λ 0 . Since ψ 21 : P Λ 1 → P Λ 1 is the zero map, so is ψ * 21 . Since ψ 31 : P Λ 1 → P Λ 2 is nonzero, ψ * 31 is a surjection onto I Λ 2 ∼ = S Λ 2 . Since ψ 12 : P Λ 4 → P Λ 0 is the zero map, so is ψ * 12 . . Figure 15. The kernel of the projective cover of X 6 Let γ : P Λ 1 → P Λ 2 be a nonzero map (unique up to a scalar). Then γψ 22 = 0, so γ * ψ * 22 = 0. Hence ψ * 22 is the map from I Λ 4 to I Λ 1 whose image is the submodule 1 0 . Since ψ 32 = 0, so is ψ * 32 : I Λ 4 → I Λ 2 , so it must be a surjection onto I Λ 2 ∼ = S Λ 2 . We thus have an explicit description of the map ψ * : I Λ 1 ⊕ I Λ 4 → I Λ 0 ⊕ I Λ 1 ⊕ I Λ 2 . Using a technique similar to the above, we can compute the kernel τ X 6 of ψ * and verify that it is X 5 . A similar technique can be used to show that τ −1 X 3 ∼ = X 4 . We have Ext(X 4 , X 3 ) ∼ = DHom(X 3 , τ X 4 ) ∼ = DHom(X 3 , X 3 ) ∼ = K. Let ϕ be the embedding of X 3 into X 7 , mapping it to the submodule of this form appearing on the right hand side of the displayed Q Λ -coloured quiver of this module. Then a computation similar to the above can be done to show that coker ϕ i ∼ = X 4 , where i is the embedding of X 3 into P Λ 3 . This gives a non-split short exact sequence 0 → X 3 → X 7 ⊕ P Λ 3 → X 4 → 0 which must be almost split. This completes the proof. Note that the modules in the τ ±1 -orbits of I Λ 2 , I Λ 3 , P Λ 2 , P Λ 3 are all τ -rigid (and hence rigid) by Lemma 5.1 and Corollary 1.10. Proposition 5.3. The Λ-modules X 2 , X 3 , X 5 and X 7 are all rigid, while X 6 is not rigid. Proof. We will use Remark 1.4 throughout. We have Ext(X 2 , X 2 ) ∼ = DHom(τ −1 X 2 , X 2 ) ∼ = DHom(X 3 , X 2 ). We have Hom(X 3 , X 2 ) ∼ = K, and any nonzero map from X 3 to X 2 has image 4 1 and so factors through the embedding of X 3 into P Λ 2 . See Figure 17, where we highlight in bold the images of the map from X 3 to X 2 and the map from X 3 to P Λ 2 . It follows that X 2 is rigid. We have Ext(X 3 , X 3 ) ∼ = DHom(X 3 , τ X 3 ) ∼ = DHom(X 3 , X 2 ). In this case, any nonzero map from X 3 to X 2 factors through the embedding of X 3 into I Λ 3 (see Figure 17). It follows that X 3 is rigid. We have: Ext(X 5 , X 5 ) ∼ = DHom(τ −1 X 5 , X 5 ) ∼ = DHom(X 6 , X 5 ). From the Q Λ -coloured quivers of X 5 and X 6 in Figure 16, we see that S Λ 1 is a quotient of X 6 and is embedded into X 5 . Let f 1 : X 6 → X 5 be the composition of these two maps. Figure 16. Part of the AR-quiver of Λ-mod From the Q Λ -coloured quiver of X 6 , we see that the module 4 1 is a quotient of X 6 , and is embedded into X 5 ; let f 2 be the composition of the two maps. Then it is easy to check that {f 1 , f 2 } is a basis of Hom(X 6 , X 5 ). Furthermore, f 1 factors through P Λ 3 : we take the composition of the map from X 6 to P Λ 3 with image isomorphic to X 3 and the map from P Λ 3 to X 5 whose image is the submodule ; see Figure 18. 1 I Λ 2 P Λ 2 I Λ 3 P Λ 3 X 2 X 3 X 4 X 5 X 6 X 7 Note that the image of the map f 1 + f 2 has basis given by the sum of the basis elements of X 5 corresponding to the two copies of 1 in the Q Λ -coloured quiver of X 5 and the basis element corresponding to the 4; we indicate the basis elements involved in the right hand diagram in Figure 18. The map f 1 + f 2 factors through P Λ 2 : we take the composition of the map from X 6 to P Λ 2 with image isomorphic to X 3 and the map from P Λ 2 to X 5 taking the Figure 18. Rigidity of X 5 . f 1 + f 2 X 6 X 5 P Λ 2 basis element corresponding to the 2 in P Λ 2 to the basis element corresponding to the 2 in X 5 . See Figure 18. Since {f 1 , f 1 + f 2 } is a basis for Hom(X 6 , X 5 ), it follows that X 5 is rigid. We have: Ext(X 7 , X 7 ) ∼ = DHom(X 7 , τ X 7 ) ∼ = DHom(X 7 , X 6 ). From the Q Λ -coloured quivers of X 6 and X 7 in Figure 16, we see that each of the modules is a quotient of X 7 and a submodule of X 6 ; we set g 1 , g 2 to be the maps from X 7 to X 6 given by the composition of the quotient map and the embedding in the first and second case respectively. Then it is easy to check that {g 1 , g 2 } is a basis of Hom(X 7 , X 6 ). Furthermore, g 1 factors through I Λ 3 : we take the composition of the map from X 7 to I Λ 3 with image 3 4 1 2 and the map from I Λ 3 to X 6 with image isomorphic to X 2 (the composition of the irreducible maps from I Λ 3 to X 2 and from X 2 to X 6 ); see Figure 19. The map g 2 also factors through I Λ 3 : we take the composition of the map from X 7 to I Λ 3 with image and the map from I Λ 3 to X 6 with image isomorphic to X 2 considered above. See Figure 19. Since {g 1 , g 2 } is a basis for Hom(X 6 , X 5 ), it follows that X 6 is rigid. Finally, we have: Ext(X 6 , X 6 ) ∼ = DHom(τ −1 X 6 , X 6 ) ∼ = DHom(X 7 , X 6 ). Consider the nonzero map g 1 : X 7 → X 6 . The projective cover of X 6 is P (X 6 ) ∼ = P Λ 0 ⊕ P Λ 1 ⊕ P Λ 2 , so if g 1 factors through a projective, it must factor through P (X 6 ). It is easy to check directly that Hom(X 7 , P Λ 0 ) = 0, Hom(X 7 , P Λ 1 ) = 0 and Hom(X 7 , P Λ 2 ) = 0, so Hom(X 7 , P (X 6 )) = 0. Hence g 1 does not factor through a projective and Hom(X 7 , X 6 ) = 0. It follows that X 6 is not rigid. It is easy to check that X i is Schurian for i ∈ {1, 2, 3, 5, 7} and not Schurian for i ∈ {4, 6}, and that I Λ 2 and P Λ 2 are Schurian, while I Λ 3 and P Λ 3 are not. This gives the picture of Schurian and rigid modules shown on the left hand side of Figure 20 (using the same notation Figure 19. Rigidity of X 6 . Figure 20. τ -rigid, rigid and Schurian Λ-modules in part of a wild example (left hand diagram). In the right hand diagram we recall the τ -rigid, rigid and Schurian modules from the tame case in Example 1.6 shown in Figure 9. • • × • × • • • • Tame case Wild case • • × • × • • • • • • as Figure 10), corresponding to the modules in Figure 16 (with X 4 omitted, as we have not checked if it is rigid). In a tube of rank 3, a module is rigid if and only if it has quasilength at most 2, which is also the case in the regular component R. On the right hand side of Figure 20, we show the pattern of τ -rigid, rigid and Schurian modules corresponding to the indecomposable objects in a tube of rank 3. This is from the tame case in Example 1.6, which was shown in Figure 9. It is interesting to note the similarity of the pattern of τ -rigid, rigid and Schurian Λmodules in these two cases, and to ask what the pattern is for the whole of R. A counter-example In this section, we give the counter-example promised in the introduction. This concerns the relationship with cluster algebras. For background on cluster algebras, we refer to [13,14]. We fix a finite quiver Q with no loops or 2-cycles and label its vertices 1, 2, . . . , n. Let F = Q(x 1 , . . . , x n ) be the field of rational functions in n indeterminates over Q. Then the associated cluster algebra A(Q) is a subalgebra of F. Here, cluster variables and clusters play a key role. The initial cluster variables are x 1 , . . . , x n . The non-initial cluster variables can be written in reduced form f /m, where m is a monomial in the variables x 1 , . . . , x n and f ∈ F. Writing m = x d1 1 · · · x dn n , where d i ≥ 0, we obtain a vector (d 1 , . . . , d n ), which is called the d-vector associated with the cluster variable f /m. On the other hand, let M be an indecomposable finite dimensional KQ-module, and let S 1 , . . . , S n be the nonisomorphic simple KQ-modules. Then we have an associated dimension vector (d ′ 1 , . . . , d ′ n ), where d ′ i denotes the multiplicity of the simple module S i as a composition factor of M . It was then of interest to investigate a possible relationship between the denominator vectors and the dimension vectors of the indecomposable rigid KQ-modules. In the case where Q is acyclic, the two sets coincide (see [9,10,11]). When Q is not acyclic, we do not have such a nice correspondence in general, but there are results in this direction in [2,5,6]. We have found the following example of a d-vector which is not the dimension vector of an indecomposable KQ-module. Example 6.1. Let Q be the acyclic quiver from Example 1.6: (6.1) 1 / / ) ) 2 / / 3 / / 4. and let Λ be the cluster-tilted algebra from this example. The quiver Q Λ of Λ is shown in Figure 3, and can be obtained from Q by mutating at 2 and then at 3. Recall that the AR-quiver for the largest tube in KQ-mod (which has rank 3) is shown in Figure 2 and the corresponding part of the AR-quiver for Λ-mod is shown in Figure 9. Let M be the KQ-module , has dimension vector (1, 2, 2, 1). Then we know from [5, Thm. A] that the denominator vector of the corresponding cluster variable in the cluster algebra A(Q Λ ) is (1, 2, 1, 1) = (1, 2, 2, 1) − (0, 0, 1, 0). It is then easy to see that (1, 2, 1, 1) cannot occur as the dimension vector of any indecomposable KQ Λ -module, by looking at an arbitrary representation with this dimension vector: K K K 2 K Here, a nonzero summand of K 2 has to split off, so that M cannot be indecomposable. Hence we have found a d-vector which is not the dimension vector of any indecomposable KQ Λ -module. Note that it cannot be the dimension vector of any indecomposable Λ-module either, by the same argument. There is another interesting class of vectors occurring in the theory of cluster algebras, known as the c-vectors. They were introduced in [14] (see [14] for the definition). In the case of an acyclic quiver Q it is known that the set of (positive) c-vectors coincides with the set of real Schur roots (see [12,24]), that is, the dimension vectors of the indecomposable rigid KQ-modules. But the relationship between c-vectors and d-vectors is not so nice in the general case. It is known for any finite quiver Q without loops or two-cycles that each positive c-vector of Q is the dimension vector of a finite dimensional Schurian rigid module over an appropriate Jacobian algebra with quiver Q ( [19]; see [12,Thm. 14]). As pointed out in [20], this implies that every positive c-vector of Q is a Schur root of Q, hence a root of Q. Then we get the following: Proposition 6.2. There is a finite quiver Q without loops or 2-cycles for which the set of d-vectors associated to A(Q) is not contained in the set of positive c-vectors of A(Q). Proof. We consider the quiver Q Λ in Example 6.1. In this case, the set of d-vectors is not contained in the set of dimension vectors of the indecomposable KQ Λ -modules. If the set of d-vectors of Q Λ was contained in the set of positive c-vectors of Q Λ , then we would have a contradiction, since, as we mentioned above, every positive c-vector of Q Λ is the dimension vector of an indecomposable KQ Λ -module. Three dimension vectors We have seen in Section 6 that there is a cluster-tilted algebra Λ associated to a quiver of tame representation type with the property that not every d-vector of A(Q Λ ) is the dimension vector of an indecomposable Λ-module. So we can ask if it is possible to express each such d-vector as a sum of a small number of such dimension vectors. Our final result shows that, for a cluster-tilted algebra Λ associated to a quiver of tame representation type, it is always possible to write a d-vector for A(Q Λ ) as the sum of at most three dimension vectors of indecomposable rigid Λ-modules. We do not know whether it is possible to write every d-vector for A(Q Λ ) as a sum of at most two dimension vectors of indecomposable rigid Λ-modules. It would also be interesting to know whether analogous results hold in the wild case. As before, we fix a quiver Q of tame representation type. We fix an arbitrary cluster-tilting object T in the corresponding cluster category, C. If M is (induced by) an indecomposable module in T , then there is a mesh M M in the AR-quiver of T corresponding to the almost split sequence with last term M . This is displayed in Figure 21, with the diagram on the left indicating the case when M is on the border of T . We denote the middle term whose quasilength is greater (respectively, smaller) than that of M by M U (respectively, M L ). Note that if M is on the border of T then M L does not exist. For objects X, Y of C we shall write We assume for the rest of this section that there is an indecomposable direct summand T 0 of T T with the property that every indecomposable direct summand of T T lies in the wing W T0 . (In the notation at the beginning of Section 2, we have s = 1). δ X,Y = 1, if X ∼ = Y ; 0, otherwise.d ′ V ( M ) = d ′ V ( M U ) + d ′ V ( M L ) − d ′ V (τ M ) + δ V, We assume further that the quasilength of T 0 (i.e. l 0 ) is equal to r − 1. We arrange the labelling, for simplicity, so that the quasisimple modules in W τ T are the Q i with i ∈ [0, r−2], so in the notation from Section 2, i 0 = 0. Let (7.1) D = {M i,l : 1 ≤ i ≤ r − 1, r − i ≤ l ≤ 2r − 2 − i}. Note that D can be formed from S 0 and its reflection in the line L through the modules of quasilength r − 1. It is a diamond-shaped region, with leftmost corner T 0 ∼ = M 1,r−1 and rightmost corner τ 2 T 0 ∼ = M r−1,r−1 . The lowest point is the unique quasisimple module Q r−1 not in W τ T0 and the highest point is the same as the highest point M 1,2r−3 of S 0 , immediately below Top 1 ; see Figure 22. Given an indecomposable module M = M i,l ∈ D, we define: (7.2) I M = {M j,r−j : 1 ≤ j ≤ i}, i.e. the set of indecomposable modules which are injective in W T0 and lie above or on the (lowest) intersection point, M i,r−i , between the ray through M and the coray through T 0 . We also set X M = M i,r−i−1 , Y M = M 0,i+l . Note that X M is the object in the part of the ray through M below M which is of maximal quasilength subject to not lying in D. Similarly, Y M is the nearest object to M in the part of the coray through M above M , which is of minimal quasilength subject to not lying in D. See Figure 22. d ′ V (M ) = d ′ V (X M ) + d ′ V (Y M ) + 1, V ∈ I M ; d ′ V (X M ) + d ′ V (Y M ), V ∈ I M . Proof. Suppose first that V is preprojective, i.e. V is an indecomposable direct summand of U . Since X M ∈ W τ T0 we have Hom C (V, X M ) = 0. Note that by Lemmas 3.2 and 3.5, dim Hom C (V, Q i ) = 0, 0 ≤ i ≤ r − 2; 1, i = r − 1. Figure 22. Here the rank of the tube is 11. The region D is indicated by the shaded area. The filled-in circles indicate the indecomposable direct summands of T , and the double circles indicate the elements of H 0 . The line L divides the region D into two and S 0 consists of the vertices in D on and above the line. The upper boxed region is I M (see (7.2)) and the lower boxed region is I ′ M (see (7.8)). ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ T0 T0 τ T0 τ T0 It follows from Lemma 3.2 that d ′ V (X) = 0 for any module X ∈ W τ T k . By Proposition 3.6, d ′ V (Q r−1 ) = 1, noting that Q r−1 is the unique quasisimple in T not in W τ T k . Using additivity as in the proof of Lemma 3.2, we see that d ′ V (X) = 1 if X ∈ D ∪ H 0 . Since X M ∈ W τ T0 , Y M ∈ H 0 and M ∈ D, we have d ′ V (M ) = 1, d ′ V (X M ) = 0 and d ′ V (Y M ) = 1, giving the result in this case. So we may assume that V is an indecomposable direct summand of T T . We prove the result in this case by induction on the minimal length of a path in T from T 0 to M . The base case is M ∼ = T 0 . Then I M = {T 0 }. Since d ′ V (τ M ) = 0, the result in this case follows directly from Lemma 7.1. We assume that M ∼ = T 0 and that the result is proved in the case where the minimal length of a path in T from T 0 to M is smaller. In particular, the result is assumed to be true for all modules in M M ∩ D other than M . Figure 23). By Lemma 7.1 and (7.3), we have: d ′ V (M ) = d ′ V (M U ) + d ′ V (M L ) − d ′ V (τ M ) + δ V,M = d ′ V (M U ) + d ′ V (X M ) − d ′ V (X MU ) + δ V,M = d ′ V (X MU ) + d ′ V (Y M ) + d ′ V (X M ) − d ′ V (X MU ) + δ V,M + 1, V ∈ I MU ; d ′ V (X MU ) + d ′ V (Y M ) + d ′ V (X M ) − d ′ V (X MU ) + δ V,M , V ∈ I MU ; = d ′ V (X M ) + d ′ V (Y M ) + 1, V ∈ I M ; d ′ V (X M ) + d ′ V (Y M ), V ∈ I M . Case Figure 23), we have: (7.4) d ′ V (M L ) = d ′ V (X M ) + d ′ V (Y ML ) + 1, V ∈ I ML ; d ′ V (X M ) + d ′ V (Y ML ), V ∈ I ML . Note that M U = Y M , τ M = Y ML , I M = I ML = {T 0 } and δ V,M = 0 (see Figure 23). By Lemma 7.1 and (7.4), we have: d ′ V (M ) = d ′ V (M U ) + d ′ V (M L ) − d ′ V (τ M ) + δ V,M = d ′ V (Y M ) + d ′ V (M L ) − d ′ V (Y ML ) + δ V,M = d ′ V (Y M ) + d ′ V (X M ) + d ′ V (Y ML ) − d ′ V (Y ML ) + 1, V ∈ I ML ; d ′ V (Y M ) + d ′ V (X M ) + d ′ V (Y ML ) − d ′ V (Y ML ), V ∈ I ML ; = d ′ V (X M ) + d ′ V (Y M ) + 1, V ∈ I M ; d ′ V (X M ) + d ′ V (Y M ), V ∈ I M . Case d ′ V (M U ) = d ′ V (X τ M ) + d ′ V (Y M ) + 1, V ∈ I τ M ; d ′ V (X τ M ) + d ′ V (Y M ), V ∈ I τ M ; (7.5) d ′ V (M L ) = d ′ V (X M ) + d ′ V (Y τ M ) + 1, V ∈ I M ; d ′ V (X M ) + d ′ V (Y τ M ), V ∈ I M ; (7.6) d ′ V (τ M ) = d ′ V (X τ M ) + d ′ V (Y τ M ) + 1, V ∈ I τ M ; d ′ V (X τ M ) + d ′ V (Y τ M ), V ∈ I τ M . (7.7) By Lemma 7.1 and (7.5)-(7.7), we obtain: d ′ V (M ) = d ′ V (M U ) + d ′ V (M L ) − d ′ V (τ M ) = d ′ V (X M ) + d ′ V (Y M ) + 1, V ∈ I M ; d ′ V (X M ) + d ′ V (Y M ), V ∈ I M . The result now follows by induction. Proof. If V is preprojective (i.e. an indecomposable direct summand of U ) then, since Z M ∈ W τ T0 , we have d ′ V (Z M ) = 0 by Lemma 3.2. Suppose that V is an indecomposable direct summand of T T . The quasisocle of Z M is Q 0 , which does not lie in W T0 . Since V ∈ W T0 , it follows from Corollary 2.3 that Hom(V, Z M ) = 0. Hence (using (1.6)), we have: Since V and X ′ M are both indecomposable direct summands of T , we have that Hom(V, τ X ′ M ) = 0. So, again using Lemma 2.2, V cannot lie in the rectangle with corners M 1,j−1 , M j−1,1 , M j−1,r−j and M 1,r−2 . Combining this fact with the statement in the previous paragraph, we see that Hom(Z M , τ 2 V ) = 0 if and only if V ∈ I, V ∼ = T 0 and V has quasilength greater than ql(X ′ M ) = r − j. coincides with the dimension vector of M . If not, then M must lie in the region D defined in (7.1) (after Lemma 7.1) (note that in addition it must have quasilength at most r − 1, but we don't need that here). By construction, the quasilengths of X M and Z M are both less than or equal to r − 1, so they are τ -rigid Λ-modules by Lemma 4.2. Since Y M lies in H 0 , it follows from Theorem 4.10 that Y M is a rigid Λ-module. By [5, Thm. A], the d-vector T0 . The result now follows from Proposition 7.5. (d V ) V ∈ind(T ) of x M satisfies d V (x M ) = d ′ V (M ) − δ V, Figure 2 . 2Part of the AR-quiver of KQ-mod, where Q is the quiver in (1.3). Figure 3 . 3The endomorphism algebra End C (T ) opp for the tilting module in Example 1.6. Proposition 1 . 13 . 113Let A, B, C be objects in D b (KQ). ( a ) aLet α : A → C and Hom(B, τ α) : Hom(B, τ A) → Hom(B, τ C) and Hom(α, B[1]) : Hom(C, B[1]) → Hom(A, B[1]) Then Hom(B, τ α) is nonzero (respectively, injective, surjective, or an isomorphism) if and only if Hom(α, B[1]) is nonzero (respectively, surjective, injective or an isomorphism). We illustrate the maps Hom(B, τ α) and Hom(α, B[1]) below for ease of reference. and consider the induced maps: Hom(β, τ A) : Hom(B, τ A) → Hom(C, τ A) and Hom(A, β[1]) : Hom(A, C[1]) → Hom(A, B[1]) Figure 4 . 4The 1 ≤ m ≤ l − 1 and j is congruent to an element of [i + l − m, i + l − 1] modulo r; K, if m ≥ l and j is congruent to an element of [i, i + l − 1] modulo r; 0, otherwise. 1 ≤ m ≤ l − 1 and j is congruent to an element of [i − m + 1, i] modulo r; K,if m ≥ l and j is congruent to an element of [i − m + 1, i − m + l] modulo r; 0, otherwise. Corollary 2. 3 . 3Let M, N, X be indecomposable objects in T , and suppose that M has quasilength at most r, and M ∈ W N . (a) If the quasisocle of X does not lie in W N then Hom(M, X) = 0. (b) If the quasitop of X does not lie in W N then Hom(X, M ) = 0. Lemma 2. 4 . 4Let M i,l be an indecomposable module in T . Then we have: Figure 5 . 5The left hand figure shows the modules X in T for which Hom(M i,l , X) = 0 (in the shaded region), for the case r = 5. The module M i,l is denoted by a filled-in circle. The right hand figure shows the modules X with Hom(X, M i,l ) = 0. Lemma 3 . 1 . 31The quasisocles of the indecomposable objects in Lemma 4. 1 . 1Let X = M i,l be an indecomposable module in T . Then: Lemma 4. 3 . 3Let A, B, C be indecomposable KQ-modules, and assume that Hom(A, B[1]) ∼ = K. Let ε ∈ Hom(A, B[1]) be nonzero. (a) If there is a map ϕ ∈ Hom(B, τ A) such that im(ϕ) has an indecomposable direct summand which does not lie in Figure 9 9Figure 9. The left hand diagram shows part of the AR-quiver of Λ-mod. The right hand diagram shows the same objects. The symbol for a module is circular if it is Schurian, filled-in with gray if it is rigid but not τ -rigid, and filled-in with black if it is τ -rigid. The symbol × represents a gap in the AR-quiver (corresponding to an indecomposable direct summand of τ T ) . . 1 . 1Let X be an indecomposable module in R. Then X is rigid if and only if it has quasilength less than or equal to 2.Proof. By[16, Thm. 2.6], every rigid module in a regular AR-component of a hereditary algebra has quasilength at most n − 2, where n is the number of simple modules. In this case, there are 5 simple modules, so no indecomposable module in R with quasilength at least 4 is rigid.Since KQ is hereditary and no module in R is projective, we have Ext(M, N ) ∼ = Ext(τ M, τ N ) for all M, N ∈ R. Hence an indecomposable module in R is rigid if and only if every module in its τ -orbit is rigid. Figure 12 . 12of quasilength 3, is not rigid. Hence every module in R of quasilength 1 or 2 is rigid, and no module in R of quasilength 3 is rigid, and we are done. Maps between indecomposable projective KQ-modules and the quiver with relations of End C (T ) opp . τ 2 T 3 ) = 0 and Hom(e, τ 2 T 3 ) = 0. Therefore, Hom(T 3 , τ −1 d[1]) = 0 and Hom(T 3 , τ −1 e[1]) = 0. Hence, Hom(T 3 , τ −1 d[1])(c) = (τ −1 d[1]) • c = 0, so dc = 0 in C. Since the domain of Hom(T 3 , τ −1 e[1]) is Hom F C (T 3 , P 4 ) = Hom(T 3 , τ −1 P 4 [1]), which is spanned by c, we have Hom(T 3 , τ −1 e[1])(c) = 0, so ec = 0 in C. Similarly, we can show that cb = 0 and dc = 0 and that the maps f b, ad, ae, be,bec, f bec and aec are all nonzero in C. Figure 13 . 13The projective cover of E Vertex i Action of ϕ basis for Figure 17 . 17Rigidity of X 2 and X 3 . Suppose M is an object in the cluster category C, with corresponding Λ-module M . The vertices of the quiver of Λ = End C (T ) are indexed by the indecomposable direct summands ind(T ) of T . The dimension vector of M is given by the tuple (d ′ V (M )) V , where V varies over the indecomposable direct summands of T . We have: d ′ V (M ) = dim Hom C (V, M ) = dim Hom(V, M ) + dim Hom(M, τ 2 V ). We shall also write d ′ V ( M ) for d ′ V (M ). Note that if M lies in add(τ T ) then M = 0 and d ′ V (M ) = 0 for all V ∈ ind(T ). Lemma 7. 1 . 1Let M be an indecomposable object in T with mesh M M as above. Then: Figure 21 . 21M , where the terms involving M L do not appear if M is on the border of T . The mesh ending at an indecomposable object in T . The diagram on the left indicates the case where M is on the border of T . Proof. If V ∼ = M then the mesh ending at M in Λ-mod is the image under Hom C (T, −) of the mesh ending at M in C (deleting zero modules corresponding to summands of τ T ). If V ∼ = M then M is an indecomposable projective module, so rad( M ) ∼ = M L ⊕ M R . Lemma 7. 2 . 2Let M ∈ D and let V be an indecomposable summand of T . Then we have Case I: If M = M i,r−i , with 1 ≤ i ≤ r − 1 lies on the lower left boundary of D then M M ∩D = {M U , M }. Applying the inductive hypothesis to M U and noting that Y MU = Y M , we have:(7.3) d ′ V (M U ) = d ′ V (X MU ) + d ′ V (Y M ) + 1, V ∈ I MU ; d ′ V (X MU ) + d ′ V (Y M ), V ∈ I MU ;Note that M L = X M , τ M = X MU , Y MU = Y M and I M = I MU ∪ {M } (see II: If M = M 1,l where r ≤ l ≤ 2r − 3 lies on the upper left boundary of D then M M ∩D = {M L , M }. Applying the inductive hypothesis to M L and noting that X ML = X M (see III: If M = M i,l with 1 ≤ i ≤ r − 1, r − i ≤ l ≤ 2r − 2 − i},but is not in one of the cases above, then M M ∩ D = {M L , M U , τ M, M }. Note that X MU = X τ M , Y MU = Y M , X ML = X M , Y ML = Y τ M and δ V,M = 0. We also have that I MU = I τ M and I ML = I M . Applying the inductive hypothesis to M L , M U and τ M , we have: Figure 23 . 23Proof of Lemma 7.2. The shaded region is the region D. Let I denote the set of all injective objects in W T0 and set (7.8) I ′ M = I \ I M , i.e. the set of objects in the coray through T 0 which are on or below the lowest intersection point with the ray through M . Suppose that there is an indecomposable direct summand of T in I ′ M . Let X ′ M ∼ = M j,r−j be such a summand with maximal quasilength and set Z M = M 0,j−2 . Otherwise, we set Z M = M 0,r−2 . Remark 7.3. In the first case above, the object Z M can be constructed geometrically as follows. Let Z ′ M = M j,r−2 be the unique object in the ray through X ′ M of quasilength r − 2. Then Z M = M 0,j−2 is the unique object in the coray through Z ′ M which is a projective in W τ T0 . See Figure 22. Lemma 7 . 4 . 74Let V be an indecomposable direct summand of T . Then d ′ V (Z M ) = 1, V ∈ I M \ T 0 ; 0, otherwise. d ′ V (Z M ) = dim Hom F C (V, Z M ) = dim Hom(Z M , τ 2 V ). Consider first the case where there is an indecomposable direct summand of T in I ′ M , so X ′ M is defined. We haveHom(Z M , τ 2 V ) = 0 if and only if Hom(τ −2 Z M , V ) = 0. By Lemma 2.2 and the fact that V ∈ W T0 , this holds if and only if V lies in the rectangle with corners τ −2 Z M = M 2,j−2 , M 2,r−2 , M j−1,1 and M j−1,r−j+1 In this case, dim Hom(Z M , τ 2 V ) = 1. Acknowledgements Both authors would like to thank the MSRI, Berkeley for kind hospitality during a semester on cluster algebras in Autumn 2012. RJM was Guest Professor at the Department of Mathematical Sciences, NTNU, Trondheim, Norway, during the autumn semester of 2014 and would like to thank the Department for their kind hospitality.Proof. This follows from Lemmas 7.2 and 7.4. Theorem 7.6. Let Q be a quiver of tame representation type and let C be the corresponding cluster category. Let T be a cluster-tilting object in C and Λ = End C (T ) opp the corresponding cluster-tilted algebra. Let Q Λ be the quiver of Λ and A(Q Λ ) the corresponding cluster algebra. Then any d-vector of A(Q Λ ) can be written as a sum of at most three dimension vectors of indecomposable rigid Λ-modules.Proof. Let M be a rigid indecomposable object in C which is not an indecomposable direct summand of τ T and x M the corresponding non-initial cluster variable of A(Q Λ ). By[5,Thm.A], if M is transjective or in a tube of rank r containing no indecomposable direct summand of T of quasilength r − 1 then the d-vector of x M coincides with the dimension vector of the Λ-module M .Suppose that M lies in a tube which contains an indecomposable direct summand T 0 of T of quasilength r − 1. If M is contained in the wing W τ T0 then the d-vector of x M again M0,1. 11M0,1 M1,1 . Mj−. 11Mj−1,1 Iyama and I. Reiten, τ -tilting theory. T Adachi, O , Compos. Math. 1503T. Adachi, O. Iyama and I. Reiten, τ -tilting theory. Compos. Math. 150 (2014), no. 3, 415-452. Modules over cluster-tilted algebras determined by their dimension vectors. I Assem, G Dupont, Comm. 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[ "UNIQUENESS IN INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING WITH PHASELESS NEAR-FIELD DATA AT A FIXED FREQUENCY", "UNIQUENESS IN INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING WITH PHASELESS NEAR-FIELD DATA AT A FIXED FREQUENCY" ]	[ "Xiaoxu Xu ", "B O Zhang ", "ANDHaiwen Zhang " ]	[]	[]	This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work (SIAM J. Appl. Math. 78 (2018), 1737-1753, where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, we use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called phaseless acoustic near-field data) on two spheres enclosing the scatterers generated by such incident fields on the two spheres. Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined from the phaseless acoustic near-field data at a fixed frequency. Moreover, the idea is also extended to the electromagnetic case, and it is proved that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless electric near-field data at a fixed frequency, that is, the modulus of the tangential component with the orientations e φ and e θ , respectively, of the electric total-field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarization vectors e φ and e θ , respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data.	10.3934/ipi.2020023	[ "https://arxiv.org/pdf/1906.05116v1.pdf" ]	186,206,725	1906.05116	0099c4bd71cdc7761054b4ae698727786f22a6a9	UNIQUENESS IN INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING WITH PHASELESS NEAR-FIELD DATA AT A FIXED FREQUENCY 3 Jun 2019 Xiaoxu Xu B O Zhang ANDHaiwen Zhang UNIQUENESS IN INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING WITH PHASELESS NEAR-FIELD DATA AT A FIXED FREQUENCY 3 Jun 2019Uniquenessinverse acoustic scatteringinverse electromagnetic scatteringphase- less near-fieldobstacleinhomogeneous medium AMS subject classifications 78A4635P25 This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work (SIAM J. Appl. Math. 78 (2018), 1737-1753, where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, we use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called phaseless acoustic near-field data) on two spheres enclosing the scatterers generated by such incident fields on the two spheres. Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined from the phaseless acoustic near-field data at a fixed frequency. Moreover, the idea is also extended to the electromagnetic case, and it is proved that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless electric near-field data at a fixed frequency, that is, the modulus of the tangential component with the orientations e φ and e θ , respectively, of the electric total-field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarization vectors e φ and e θ , respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data. 1. Introduction. Inverse scattering problems occur in many applications such as radar, remote sensing, geophysics, medical imaging and nondestructive testing. These problems aim at reconstructing the unknown scatterers from the measurement data of the scattered waves. In the past decades, inverse acoustic and electromagnetic scattering problems with phased data have been extensively studied mathematically and numerically. A comprehensive account of these studies can be found in the monographs [10,14]. In many practical applications, it is much harder to obtain data with accurate phase information compared with just measuring the intensity (or the modulus) of the data, and thus it is often desirable to study inverse scattering with phaseless data (see, e.g., [10,Chapter 8] and the references quoted there). In fact, inverse scattering problems with phaseless data have also been widely studied numerically over the past decades (see, e.g. [2, 3, 10-13, 18, 26, 32, 37, 38, 43-45] and the references quoted there). Recently, uniqueness and stability results have also been established for inverse scattering with phaseless data (see, e.g. [1, 19, 20, 24, 25, 27, 29-31, 35, 39-41, 46]). For example, for point source incidence uniqueness results have been established in [24,25] for inverse potential and acoustic medium scattering with the phaseless near-field data generated by point sources placed on a sphere enclosing the scatterer and measured in a small ball centered at each source position for an interval of frequencies, and in [30] for inverse acoustic medium scattering with the phaseless near-field data measured on an annulus surrounding the scatterer at fixed frequency. The purpose of this paper is to propose a new approach to establish uniqueness results for inverse acoustic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work [39], where uniqueness results have been proved for inverse acoustic scattering with phaseless far-field data corresponding to superpositions of two plane waves as the incident fields at a fixed frequency, we consider to utilize the superposition of two point sources at a fixed frequency as the incident field. However, the idea of proofs used in [39] can not be applied directly to the inverse scattering problem with phaseless near-field data. This is due to the fact that our proofs in [39] are based essentially on the limit of the normalized eigenvalues of the far-field operators. To overcome this difficulty, we consider to use two spheres, which enclose the scatterers, as the locations of such incident fields and the measurement surfaces of the modulus of the acoustic total-field (the sum of the incident field and the scattered field). In fact, many phase retrieval algorithms have been developed for inverse scattering problems with phaseless near-field data measured on two surfaces to ensure the reliability of the near-field phase reconstruction algorithms (see, e.g. [17,33,34]). Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of the inhomogeneous medium can be uniquely determined from the phaseless total-field data at a fixed frequency. Note that the superposition of two point sources was also used in [35] as the incident field to study uniqueness for phaseless inverse scattering problems. Some related uniqueness results can be found in [47,48]. The idea is also applied to phaseless inverse electromagnetic scattering which is more complicated than the acoustic case. In this case, the electric total field is a complex vector-valued function, so we need to define the phaseless data used in this paper. In many applications (see, e.g. [4,32,36]), the phaseless near-field data are based on the measurement of the modulus of the tangential component of the electric total field on the measurement surface. Further, it has been elaborated in [16] that the measurement data are based on two tangential components of the electric field on the measurement sphere (see [16, p.100]). Therefore, the phaseless near-field data used is the modulus of the tangential component in the orientations e φ and e θ , respectively, of the electric total field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarizations e φ and e θ , respectively. Following a similar idea as in the acoustic case, we prove that the impenetrable bounded obstacle or the refractive index of the inhomogeneous medium (under the condition that the magnetic permeability is a positive constant) can be uniquely determined by the phaseless total-field data at a fixed frequency. To the best of our knowledge, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data. It should be mentioned that our uniqueness results in this paper are based on parts of the PhD thesis [42]. The outline of this paper is as follows. The acoustic and electromagnetic scattering models considered are given in Section 2. Sections 3 and 4 are devoted to the uniqueness results for phaseless inverse acoustic and electromagnetic scattering problems, respectively. Conclusions are given in Section 5. 2. The direct scattering problems. We will introduce the acoustic and electromagnetic scattering models considered in this paper. To this end, assume that D is an open and bounded domain in R 3 with a C 2 −boundary ∂D such that the exterior R 3 \D is connected. Assume further that D ⊂ B R1 , where B R1 is a ball centered at the origin with radius R 1 > 0 large enough. 2.1. The acoustic case. In this paper, we consider the problem of acoustic scattering by an impenetrable obstacle or an inhomogeneous medium in R 3 . We need the following fundamental solution to the three-dimensional Helmholtz equation ∆w + k 2 w = 0 in R 3 with k > 0: Φ k (x, y) := e ik\|x−y\| 4π\|x − y\| , x, y ∈ R 3 , x = y. For arbitrarily fixed y ∈ R 3 \D consider the time-harmonic (e −iωt time dependence) point source w i := w i (x, y) = Φ k (x, y), x ∈ R 3 \D, which is incident on the obstacle D from the unbounded part R 3 \D, where k = ω/c > 0 is the wave number, ω and c are the wave frequency and speed in the homogeneous medium in the whole space. Then the problem of scattering of the point source w i by the impenetrable obstacle D is formulated as the exterior boundary value problem: ∆ x w s (x, y) + k 2 w s (x, y) = 0, x ∈ R 3 \D, (2.1) Bw = 0 on ∂D, (2.2) lim r→∞ r ∂w s ∂r − ikw s = 0, r = \|x\|, (2.3) where w s is the scattered field, w := w i + w s is the total field, and (2.3) is the Sommerfeld radiation condition imposed on the scattered field w s . The boundary condition B in (2.2) depends on the physical property of the obstacles D:    Bw := w on ∂D if D is a sound-soft obstacle, Bw := ∂w/∂ν + ηw on ∂D if D is an impedance obstacle, Bw := w on Γ D , Bw := ∂w/∂ν + ηw on Γ I if D is a partially coated obstacle, where ν is the unit outward normal to the boundary ∂D and η is the impedance function on ∂D satisfying that Im[η(x)] ≥ 0 for all x ∈ ∂D or x ∈ Γ I . We assume that η ∈ C(∂D) or η ∈ C(Γ I ), that is, η is continuous on ∂D or Γ I . When η = 0, the impedance boundary condition becomes the Neumann boundary condition (a soundhard obstacle). For a partially coated obstacle, we assume that the boundary ∂D has a Lipschitz dissection ∂D = Γ D ∪ Π ∪ Γ I , where Γ D and Γ I are disjoint, relatively open subsets of ∂D and having Π as their common boundary in ∂D (see, e.g., [8]). The problem of scattering of the point source w i by an inhomogeneous medium is modeled as follows: ∆ x w s (x, y) + k 2 n(x)w s (x, y) = k 2 (1 − n(x))w i (x, y), x ∈ R 3 , (2.4) lim r→∞ r ∂w s ∂r − ikw s = 0, r = \|x\|,(2.5) where w s is the scattered field and n in (2.4) is the refractive index characterizing the inhomogeneous medium. We assume that n − 1 has compact support D and n ∈ L ∞ (D) with Re[n(x)] > 0, Im[n(x)] ≥ 0 for all x ∈ D. The existence of a unique (variational) solution to the problems (2.1)-(2.3) and (2.4)-(2.5) has been proved in [7,14,21,22]. In particular, the scattered-field w s has the asymptotic behavior: w s (x, y) = e ik\|x\| \|x\| w ∞ (x, y) + 1 \|x\| , \|x\| → ∞ uniformly for all observation directionsx = x/\|x\| ∈ S 2 , where S 2 is the unit sphere in R 3 and w ∞ (x, y) is the far-field pattern of w s which is an analytic function ofx ∈ S 2 for each y ∈ R 3 \ D (see, e.g., [14, (2.13)]). In this paper, we also consider the superposition of two point sources w i = w i (x; y 1 , y 2 ) = w i (x, y 1 ) + w i (x, y 2 ) = Φ k (x, y 1 ) + Φ k (x, y 2 ) (2.6) as the incident field, where y 1 , y 2 ∈ R 3 \ D are the locations of the two point sources. It then follows by the linear superposition principle that the corresponding scattered field w s (x; y 1 , y 2 ) = w s (x, y 1 ) + w s (x, y 2 ) (2.7) and the corresponding total field w(x; y 1 , y 2 ) = w(x, y 1 ) + w(x, y 2 ), (2.8) where w s (x, y j ) and w(x, y j ) are the scattered field and the total field corresponding to the incident point source w i (x, y j ), respectively, j = 1, 2. The inverse acoustic obstacle (or medium) scattering problem we consider in this paper is to reconstruct the obstacle D and its physical property (or the index of refraction n of the inhomogeneous medium) from the phaseless total field \|w(x; y 1 , y 2 )\| for x, y 1 , y 2 on some spheres enclosing D and the inhomogeneous medium. 2.2. The electromagnetic case. In this paper, we consider two electromagnetic scattering models, that is, scattering by an impenetrable obstacle and scattering by an inhomogeneous medium. We will consider the time-harmonic (e −iωt time dependence) incident electric dipole located at y ∈ R 3 \ D and described by the matrices E i (x, y) and H i (x, y) defined by E i (x, y)p := i k curl x curl x [pΦ k (x, y)], H i (x, y)p := curl x [pΦ k (x, y)], x = y, for x ∈ R 3 \ D, where p ∈ R 3 is the polarization vector, k := ω/c > 0 is the wave number, ω and c := 1/ √ ε 0 µ 0 are the wave frequency and speed in the homogeneous medium in R 3 \ D, respectively, and ε 0 and µ 0 are the electric permittivity and the magnetic permeability of the homogeneous medium, respectively. A direct calculation shows that for x = y, E i (x, y) = ikΦ k (x, y)I + i k ∇ x ∇ x Φ k (x, y) (2.9) = i k k 2 + ik − 1 \|x − y\| 1 \|x − y\| I + x − y · x − y ⊤ f (\|x − y\|) Φ k (x, y), where I is a 3 × 3 identity matrix, ∇ x ∇ x := (∂ xi ∂ xj ) 3×3 , x − y = (x − y)/\|x − y\| and f (r) := 3/r 2 − 3ik/r − k 2 . Then the problem of scattering of the electric dipole E i and H i by the impenetrable obstacle D can be modeled as the exterior boundary value problem: curl x E s − ikH s = 0in R 3 \ D, (2.10) curl x H s + ikE s = 0in R 3 \ D, (2.11) BE = 0on ∂D, (2.12) lim r→∞ (H s × x − rE s ) = 0,r = \|x\|,(2.13) where (E s , H s ) is the scattered field, E := E i + E s and H := H i + H s are the electric total field and the magnetic total field, respectively, and (2.13) is the Silver-Müller radiation condition which holds uniformly for allx ∈ S 2 and ensures the uniqueness of the scattered field. The boundary condition B in (2.12) depends on the physical property of the obstacle D, that is, BE := ν × E on ∂D (called as the PEC condition) if D is a perfect conductor, where ν is the unit outward normal to the boundary ∂D, BE := ν × curlE − iλ(ν × E) × ν on ∂D if D is an impedance obstacle, where λ is the impedance function on ∂D, and BE := ν × E on Γ D , BE := ν × curlE − iλ(ν × E) × ν on Γ I if D is a partially coated obstacle, where ∂D has a Lipschitz dissection ∂D = Γ D ∪ Π ∪ Γ I with Γ D and Γ I being disjoint and relatively open subsets of ∂D and having Π as their common boundary in ∂D and λ is the impedance function on Γ I . We assume throughout this paper that λ ∈ C(∂D) with λ(x) ≥ 0 for all x ∈ ∂D or λ ∈ C(Γ I ) with λ(x) ≥ 0 for all x ∈ Γ I . The problem of scattering of an electric dipole by an inhomogeneous medium is modeled as the medium scattering problem: curl x E s − ikH s = 0 in R 3 (2.14) curl x H s + ikn(x)E s = ik(1 − n(x))E i in R 3 (2.15) lim r→∞ (H s × x − rE s ) = 0,r = \|x\|,(2.16) where (E s , H s ) is the scattered field and (E, H) := (E i , H i ) + (E s , H s ) is the total field. The refractive index n(x) in (2.15) is given by n(x) := 1 ε 0 ε(x) + i σ(x) ω . In this paper, we assume the magnetic permeability µ = µ 0 to be a positive constant in the whole space. We assume further that n − 1 has a compact support D and n ∈ C 2,γ (R 3 ) for 0 < γ < 1 with Re[n(x)] > 0 and Im[n(x)] ≥ 0 for all x ∈ D. The existence of a unique (variational) solution to the problems (2.10)-(2.13) and (2.14)-(2.16) has been established in [8,9,14]. In particular, it is well known that the electromagnetic scattered field E s has the asymptotic behavior: E s (x, y)p = e ik\|x\| \|x\| E ∞ (x, y)p + 1 \|x\| , \|x\| → ∞ uniformly for all observation directionsx = x/\|x\| ∈ S 2 , where E ∞ (x, y) is the electric far-field pattern of E s which is an analytic function ofx ∈ S 2 for each y ∈ R 3 \ D (see, e.g., [14, (6.23)]). Because of the linearity of the direct scattering problem with respect to the incident field, we can express the scattered waves by matrices E s (x, d) and H s (x, d), the total waves by matrices E(x, d) and H(x, d), and the far-field patterns by E ∞ (x, d) and H ∞ (x, d), respectively. We will also consider the following superposition of two electric dipoles as the incident field: E i = E i (x, y 1 , p 1 , τ 1 , y 2 , p 2 , τ 2 ) := τ 1 E i (x, y 1 )p 1 + τ 2 E i (x, y 2 )p 2 = i k curl x curl x [τ 1 p 1 Φ k (x, y 1 ) + τ 2 p 2 Φ k (x, y 2 )], H i = H i (x, y 1 , p 1 , τ 1 , y 2 , p 2 , τ 2 ) := τ 1 H i (x, y 1 )p 1 + τ 2 H i (x, y 2 )p 2 = curl x [τ 1 p 1 Φ k (x, y 1 ) + τ 2 p 2 Φ k (x, y 2 )], where x ∈ R 3 , y 1 , y 2 ∈ R 3 \ D, x = y 1 , x = y 2 , p 1 , p 2 ∈ R 3 and τ 1 , τ 2 ∈ {0, 1}. By the linear superposition principle, the electric total field and scattered field corresponding to the superposition of two electric dipoles as the incident field satisfy E s (x, y 1 , p 1 , τ 1 , y 2 , p 2 , τ 2 ) := τ 1 E s (x, y 1 )p 1 + τ 2 E s (x, y 2 )p 2 and E(x, y 1 , p 1 , τ 1 , y 2 , p 2 , τ 2 ) := τ 1 E(x, y 1 )p 1 + τ 2 E(x, y 2 )p 2 , (2.17) where E s (x, y j )p j and E(x, y j )p j are the electric scattered field and the electric total field corresponding to the incident field E i (x, y j )p j , respectively, j = 1, 2. Following [16,32,36], we measure the modulus of the tangential component of the electric total field on a sphere ∂B r centered at the origin with radius r > 0. To represent the tangential components, we introduce the following spherical coordinate      x 1 = r sin θ cos φ, x 2 = r sin θ sin φ, x 3 = r cos θ, with x := (x 1 , x 2 , x 3 ) ∈ R 3 and (r, θ, φ) ∈ [0, +∞) × [0, π] × [0, 2π). For any x ∈ ∂B r \ {N r , S r }, the spherical coordinate gives an one-to-one correspondence between x and (r, φ, θ). Here, N r := (0, 0, r) and S r := (0, 0, −r) denote the north and south poles of ∂B r , respectively. If we define e φ (x) := (− sin φ, cos φ, 0), e θ (x) := (cos θ cos φ, cos θ sin φ, − sin θ), then e φ (x) and e θ (x) are two orthonormal tangential vectors of ∂B r at x / ∈ {N r , S r }. Now, we can represent our phaseless measurement data by \|e m (x) · E(x, y 1 , e n (y 1 ), τ 1 , y 2 , e l (y 2 ), τ 2 )\| with x, y 1 , y 2 ∈ ∂B r \ {N r , S r }, x = y 1 , x = y 2 , m, n, l ∈ {φ, θ} and τ 1 , τ 2 ∈ {0, 1}. The inverse electromagnetic obstacle or medium scattering problem we consider in this paper is to reconstruct the obstacle D and its physical property or the index of refraction n of the inhomogeneous medium from the modulus of the tangential component of the electric total field, \|e m (x) · E(x, y 1 , e n (y 1 ), τ 1 , y 2 , e l (y 2 ), τ 2 )\|, for all x, y 1 , y 2 in some spheres enclosing D or the inhomogeneous medium, m, n, l ∈ {φ, θ} and τ 1 , τ 2 ∈ {0, 1}. The purpose of this paper is to prove uniqueness results for the above inverse acoustic and electromagnetic scattering problems. 3. Inverse acoustic scattering with phaseless total-field data. This section is devoted to the uniqueness results for inverse acoustic scattering with phaseless total-field data at a fixed frequency measured on two spheres enclosing the scatterers (see Denote by w s j and w j the scattered field and the total field, respectively, associated with the impenetrable obstacle D j (or the refractive index n j ) and corresponding to the incident field w i , j = 1, 2. Let B R2 denote the ball centered at the origin with radius R 2 > R 1 > 0 with ∂B R2 denoting the boundary of B R2 . By appropriately choosing Theorem 3.1. Let D 1 , D 2 be two bounded domains and let R 2 > R 1 > 0, it can be ensured that k 2 is not a Dirichlet eigenvalue of −∆ in B R2 \ B R1 . Here, k 2 is called a Dirichlet eigenvalue of −∆ in a bounded domain V ifR 2 > R 1 > 0 be large enough so that D 1 ∪ D 2 ⊂ B R1 . Assume that k 2 is not a Dirichlet eigenvalue of −∆ in B R2 \B R1 . (a) Assume that D 1 and D 2 are two impenetrable obstacles with boundary conditions B 1 and B 2 , respectively. If the corresponding total fields satisfy \|w 1 (x, y)\| = \|w 2 (x, y)\|, ∀(x, y) ∈ (∂B R1 × ∂B R1 ) ∪ (∂B R2 × ({y 0 } ∪ ∂B R2 )), x = y (3.1) and \|w 1 (x; y, y 0 )\| = \|w 2 (x; y, y 0 )\|, ∀(x, y) ∈ (∂B R1 × ∂B R1 ) ∪ (∂B R2 × ∂B R2 ), x = y, y 0 (3.2) for an arbitrarily fixed y 0 ∈ ∂B R1 , then D 1 = D 2 and B 1 = B 2 . (b) Assume that n 1 , n 2 ∈ L ∞ (R 3 ) are the indices of refraction of two inhomogeneous media with n j − 1 supported in D j , j = 1, 2. If the corresponding total fields satisfy (3.1) and (3.2), then n 1 = n 2 . To prove Theorem 3.1, we need the following lemmas on the property of the total field. Lemma 3.2. Let R 2 > R 1 > 0 and let D be a bounded domain such that D ⊂ B R1 . Suppose w(x, y) is the total field of the obstacle scattering problem (2.1)-(2.3) or the medium scattering problem (2.4)-(2.5) associated with the point source w i (x, y). Then, for any fixed y 0 ∈ ∂B R1 we have w(x, y 0 ) ≡ 0, x ∈ ∂B R1 , x = y 0 , (3.3) w(x, y 0 ) ≡ 0, x ∈ ∂B R2 , (3.4) w(x, y) ≡ 0, (x, y) ∈ ∂B R2 × ∂B R2 , x = y. (3.5) Proof. Since w(x, y) is singular at x = y 0 or y, we know that (3.3) and (3.5) are true. We now prove (3.4). Assume to the contrary that w( x, y 0 ) ≡ 0 for x ∈ ∂B R2 , that is, w s (x, y 0 ) = −Φ k (x, y 0 ) for x ∈ ∂B R2 . Then, by the uniqueness of the exterior Dirichlet problem it follows that w s (x, y 0 ) = −Φ k (x, y 0 ) for all x ∈ R 3 \ B R2 . Since the scattered field w s (x, y 0 ) is analytic for x ∈ R 3 \ D and Φ k (x, y 0 ) is analytic for x ∈ R 3 \ {y 0 }, we have w s (x, y 0 ) = −Φ k (x, y 0 ) for all x ∈ R 3 \ (D ∪ {y 0 }). This is a contradiction since Φ k (x, y 0 ) has a singularity at x = y 0 ∈ ∂B R1 and w s (x, y 0 ) is analytic when x is in a neighbourhood of y 0 . Thus, (3.4) is true. Lemma 3.3. Under the assumption of Lemma 3.2, we have the following results. (i) There exist two open sets U 1 , U 2 ⊂ ∂B R1 such that U 1 ∩U 2 = ∅ and w(x, y) = 0 for all (x, y) ∈ U 1 × U 2 . (ii) There exist two open sets U ′ 1 , U ′ 2 ⊂ ∂B R2 such that U ′ 1 ∩U ′ 2 = ∅ and w(x, y) = 0 for all (x, y) ∈ U ′ 1 × (U ′ 2 ∪ {y 0 }), where y 0 ∈ ∂B R1 . Proof. We only prove (ii). The proof of (i) is similar. By (3.4) we know that for y 0 ∈ ∂B R1 there exists x 0 ∈ ∂B R2 such that w(x 0 , y 0 ) = 0. Since w(x, y) is continuous for x, y ∈ R 3 \ D with x = y, there exists a neighbourhood U ′ ⊂ ∂B R2 of x 0 such that w(x, y 0 ) = 0 for all x ∈ U ′ . Further, since w(x, y) is analytic with respect to x ∈ ∂B R2 and y ∈ ∂B R2 , respectively, when x = y, then it follows from (3.5) that there exist two points x 1 ∈ U ′ and x 2 ∈ ∂B R2 such that w(x 1 , x 2 ) = 0 with x 1 = x 2 . Finally, again by the continuity of w(x, y) for x, y ∈ R 3 \ D with x = y, there exists a neighbourhood U ′ 1 ⊂ U ′ of x 1 and a neighbourhood U ′ 2 ⊂ ∂B R1 of x 2 such that U ′ 1 ∩ U ′ 2 = ∅ and w(x, y) = 0 for all (x, y) ∈ U ′ 1 × U ′ 2 . Thus, w(x, y) = 0 for all (x, y) ∈ U ′ 1 × (U ′ 2 ∪ {y 0 }). This completes the proof. Proof of Theorem 3.1. From (2.8) it is easy to see that (3.2) is equivalent to the equation \|w 1 (x, y) + w 1 (x, y 0 )\| = \|w 2 (x, y) + w 2 (x, y 0 )\| for all (x, y) ∈ (∂B R1 × ∂B R1 ) ∪ (∂B R2 × ∂B R2 ) with x = y, y 0 . This, together with (3.1), implies that Re{w 1 (x, y)w 1 (x, y 0 )} = Re{w 2 (x, y)w 2 (x, y 0 )} (3.6) for all (x, y) ∈ (∂B R1 × ∂B R1 ) ∪ (∂B R2 × ∂B R2 ) with x = y, y 0 . Define r j (x, y) := \|w j (x, y)\|, j = 1, 2. Then it follows from (3.1) that r 1 (x, y) = r 2 (x, y) =: r(x, y), for all x ∈ ∂B R1 , y ∈ ∂B R1 ∪ ∂B R2 with x = y, so we can write w j (x, y) = r(x, y)e iϑj (x,y) , ∀x, y ∈ ∂B R1 ∪ ∂B R2 , x = y, j = 1, 2 with real-valued functions ϑ j (x, y), j = 1, 2. Case 1. (3.6) holds with (x, y) ∈ ∂B R1 × ∂B R1 , x = y. Since w s j (x, y), j = 1, 2, are analytic functions of x ∈ ∂B R1 and y ∈ ∂B R1 , respectively, and Φ k (x, y) has a singularity at x = y, then, by Lemma 3.3 we can choose two open sets U 1 , U 2 ⊂ ∂B R1 small enough so that U 1 ∩ U 2 = ∅, r(x, y) = 0 for all (x, y) ∈ U 1 × (U 2 ∪ y 0 ), and ϑ j (x, y), j = 1, 2, are analytic with respect to x ∈ U 1 and y ∈ U 2 , respectively. Now, by (3.6) we have cos[ϑ 1 (x, y) − ϑ 1 (x, y 0 )] = cos[ϑ 2 (x, y) − ϑ 2 (x, y 0 )] (3.7) for all (x, y) ∈ U 1 × U 2 . Since ϑ j (x, y), j = 1, 2, are real-valued analytic functions of x ∈ U 1 and y ∈ U 2 , respectively, we have either ϑ 1 (x, y) − ϑ 1 (x, y 0 ) = ϑ 2 (x, y) − ϑ 2 (x, y 0 ) + 2pπ, ∀(x, y) ∈ U 1 × U 2 (3.8) or ϑ 1 (x, y) − ϑ 1 (x, y 0 ) = −[ϑ 2 (x, y) − ϑ 2 (x, y 0 )] + 2pπ, ∀(x, y) ∈ U 1 × U 2 , (3.9) where p ∈ Z. For the case when (3.8) holds, we have ϑ 1 (x, y) − ϑ 2 (x, y) = ϑ 1 (x, y 0 ) − ϑ 2 (x, y 0 ) + 2pπ, ∀(x, y) ∈ U 1 × U 2 . This implies that α(x) := ϑ 1 (x, y) − ϑ 2 (x, y) = ϑ 1 (x, y 0 ) − ϑ 2 (x, y 0 ) + 2pπ depends only on x ∈ U 1 . Then it follows that w 1 (x, y) = r(x, y)e iϑ1(x,y) = r(x, y)e iα(x)+iϑ2(x,y) = e iα(x) w 2 (x, y) for all x ∈ U 1 and y ∈ U 2 ∪ {y 0 }. By the analyticity of w 1 (x, y) − e iα(x) w 2 (x, y) in y ∈ ∂B R1 with y = x, we get w 1 (x, y) = e iα(x) w 2 (x, y), ∀x ∈ U 1 , y ∈ ∂B R1 , x = y.(3.10) Changing the variables x → y and y → x in (3.10) gives w 1 (y, x) = e iα(y) w 2 (y, x), ∀x ∈ ∂B R1 , y ∈ U 1 , x = y.(3.11) Use (3.10), (3.11) and the reciprocity relation that w s j (x, y) = w s j (y, x) for all x, y ∈ ∂B R1 , j = 1, 2 (see [14,Theorem 3.17]) to give e iα(x) w 2 (x, y) = e iα(y) w 2 (x, y), ∀x, y ∈ U 1 with x = y. (3.12) Since w j (x, y) has a singularity at x = y, and by (3.12) and the analyticity of w j (x, y) (j = 1, 2) with respect to x ∈ ∂B R1 and y ∈ ∂B R1 , respectively, with x = y, it follows that e iα(x) = e iα(y) for all x, y ∈ U 1 with x = y. This means that e iα(x) ≡ e iα for all x ∈ U 1 , where α is a real constant. Substituting this formula into (3.10) gives that w 1 (x, y) = e iα w 2 (x, y) for all x ∈ U 1 , y ∈ ∂B R1 with x = y. Again, by the analyticity of w j (x, y) (j = 1, 2) with respect to x ∈ ∂B R1 with x = y we have w 1 (x, y) = e iα w 2 (x, y) ∀x, y ∈ ∂B R1 with x = y,(3.13) which gives w s 1 (x, y) − e iα w s 2 (x, y) = (e iα − 1)Φ k (x, y), ∀x, y ∈ ∂B R1 with x = y. (3.14) Since w s j (x, y), j = 1, 2, are analytic for x ∈ G and y ∈ G, respectively, and Φ k (x, y) has a singularity at x = y, then passing the limit y → x in (3.14) gives that e iα = 1, so w s 1 (x, y) = w s 2 (x, y), ∀x, y ∈ ∂B R1 . (3.15) For the case when (3.9) holds, a similar argument as above gives w 1 (x, y) = e iβ w 2 (x, y), ∀x, y ∈ ∂B R1 with x = y (3.16) for a real constant β, that is, w s 1 (x, y) − e iβ w s 2 (x, y) = e iβ Φ k (x, y) − Φ k (x, y), ∀x, y ∈ ∂B R1 with x = y. Since w s j (x, y), j = 1, 2, are analytic for x ∈ G and y ∈ G, respectively, Re[Φ k (x, y)] has a singularity at x = y and Im[Φ k (x, y)] is analytic for all x, y ∈ R 3 , then e iβ = 1. Thus, it follows from (3.16) that w 1 (x, y) = w 2 (x, y), ∀x, y ∈ ∂B R1 with x = y. We now prove that both (3.17) and (3.19) can not hold simultaneously. Suppose this is not the case. Then define v(x) := w 1 (x, y 0 ) − w 2 (x, y 0 ) for x ∈ G with x = y 0 . Since Φ k (x, y) − Φ k (x, y) = i sin(k\|x − y\|)/(2π\|x − y\|) is analytic for all x, y ∈ R 3 , then, by the analyticity of w s j (x, y) (j = 1, 2) with respect to x ∈ G it follows that v can be extended as an analytic function of x ∈ G, denoted by v again. Further, since i sin(k\|x − y\|)/(2π\|x − y\|) and w s j (x, y) (j = 1, 2) as functions of x satisfy the Helmholtz equation ∆u + k 2 u = 0 in G, we have by (3.17) and (3.19) that v satisfies the Dirichlet boundary value problem: ∆v + k 2 v = 0 in B R2 \B R1 , v = 0 on ∂B R1 ∪ ∂B R2 . From the assumption that k 2 is not a Dirichlet eigenvalue of −∆ in B R2 \B R1 , it is known that v(x) = 0 for any x ∈ B R2 \B R1 . Thus w 1 (x, y 0 ) = w 2 (x, y 0 ) for all x ∈ B R2 \B R1 with x = y 0 . By the analyticity of w j (x, y 0 ) (j = 1, 2) with respect to x ∈ G with x = y 0 , we obtain w 1 (x, y 0 ) = w 2 (x, y 0 ), ∀x ∈ G, x = y 0 ,(3.20) which contradicts to the fact that w j (x, y 0 ) = Φ k (x, y 0 ) + w s j (x, y 0 ), j = 1, 2, satisfy the Sommerfeld radiation condition. We then conclude that both (3.17) and (3.19) can not hold simultaneously. This means that at least one of the formulas (3.17) and (3.19) does not hold. If (3.17) does not hold, then it follows that (3.15) holds. By the reciprocity relation, the well-posedness of the exterior Dirichlet problem and the analyticity of w s j (x, y) (j = 1, 2) with respect to x ∈ G and y ∈ G, respectively, it is easily derived from (3.15) that w s 1 (x, y) = w s 2 (x, y), ∀x, y ∈ G. (3.21) Then, by [14, Theorem 2.13] and the mixed reciprocity relation 4πw ∞ j (−d, z) = u s j (z, d) for all d ∈ S 2 and z ∈ G, j = 1, 2 (see [14,Theorem 3.16]) it is obtained on applying (3.21) that u ∞ 1 (x, d) = u ∞ 2 (x, d), ∀x, d ∈ S 2 ,(3.22) where u ∞ j is the far-field pattern associated with the obstacle D j (or the refractive index n j ) and corresponding to the incident field u i (x, d) = e ikx·d , j = 1, 2. Similarly, if (3.19) does not hold, then (3.18) holds and thus we can also show that (3.22) holds. Finally, for the case with two impenetrable obstacles D 1 and D 2 , by (3.22) and [14, Theorem 5.6] we have D 1 = D 2 and B 1 = B 2 , while for the case with two refractive indices n 1 and n 2 , we have by (3.22) and [21, Theorem 6.26] that n 1 = n 2 . Theorem 3.1 is thus proved. ✷ Remark 3.4. (i) Theorem 3.1 (a) remains true for the two-dimensional case, and the proof is similar. (ii) Theorem 3.1 (b) also holds in two dimensions if the assumption n 1 , n 2 ∈ L ∞ (R 3 ) is replaced by the condition that n j is piecewise in W 1,p (D j ) with p > 2, j = 1, 2. In this case, the proof is similar except that we need Bukhgeim's result in [6] (see also the theorem in Section 4.1 in [5]) instead of [21,Theorem 6.26] in the proof. (iii) Theorem 3.1 (b) generalizes the uniqueness results in [24,25,27,30] substantially in the sense that our uniqueness results only need the measurement data of the modulus of the total-field on two spheres enclosing the inhomogeneous medium at a fixed frequency, under no smoothness assumption on the refractive index, instead of the measurement data in a ball for each point source in a sphere for an interval of frequencies as used in [24,25,27] or in an open domain for each point source in another open domain at a fixed frequency as used in [30]. 4. Inverse electromagnetic scattering with phaseless electric total field data. In this section, we extend the uniqueness results in Section 3 for the acoustic case to the case of inverse electromagnetic scattering problems with phaseless electric total-field data at a fixed frequency. In this case, we consider the measurement of the modulus of the tangential component of the electric total-field on two spheres enclosing the scatterers, generated by superpositions of two electric dipoles located also on the two spheres. Denote by E j , E s j , H s j and H j the electric scattered-field, electric totalfield, magnetic scattered-field and magnetic total-field, respectively, associated with the obstacle D j (or the refractive index n j ) and corresponding to the incident electric field E i , j = 1, 2. Let B R2 denote the ball centered at the origin with radius R 2 > R 1 > 0 with ∂B R2 denoting the boundary of B R2 and let G denote the unbounded component of the complement of D 1 ∪ D 2 . Denote by N Rj and S Rj the north and south poles of ∂B Rj , respectively, j = 1, 2. See Figure 4.1 for the geometry of the electromagnetic scattering problem. j n (kR 1 ) y n (kR 1 ) j n (kR 2 ) y n (kR 2 ) = 0, j n (kR 1 ) + kR 1 j ′ n (kR 1 ) y n (kR 1 ) + kR 1 y ′ n (kR 1 ) j n (kR 2 ) + kR 2 j ′ n (kR 2 ) y n (kR 2 ) + kR 2 y ′ n (kR 2 ) = 0 (4.1) D E s ∂BR 1 ∂BR 2 E i (a) n E s ∂BR 1 ∂BR 2 E i (b) for all n = 1, 2, · · · , where j n and y n are the spherical Bessel functions and spherical Neumann functions of order n, respectively. Proof. Assume that (E, H) solves the interior Maxwell problem By the perfectly conducting boundary condition on ∂B R2 ∪ ∂B R1 we have      curlE − ikH = 0 in B R2 \ B R1 curlH + ikE = 0 in B R2 \ B R1 ν × E = 0 on ∂B R2 ∪ ∂B R1 .j n (kR 1 ) y n (kR 1 ) j n (kR 2 ) y n (kR 2 ) a m n c m n = 0 0 , (4.3) j n (kR 1 ) + kR 1 j ′ n (kR 1 ) y n (kR 1 ) + kR 1 y ′ n (kR 1 ) j n (kR 2 ) + kR 2 j ′ n (kR 2 ) y n (kR 2 ) + kR 2 y ′ n (kR 2 ) for all n = 1, 2, · · · , m = −n, · · · , n. By (4.1) we have a m n = b m n = c m n = d m n = 0 for all n = 1, 2, · · · , m = −n, · · · , n, and so k 2 is not a Maxwell eigenvalue in B R2 \ B R1 . On the other hand, if j n (kR 1 ) y n (kR 1 ) j n (kR 2 ) y n (kR 2 ) = 0 or j n (kR 1 ) + kR 1 j ′ n (kR 1 ) y n (kR 1 ) + kR 1 y ′ n (kR 1 ) j n (kR 2 ) + kR 2 j ′ n (kR 2 ) y n (kR 2 ) + kR 2 y ′ n (kR 2 ) = 0 for some n ∈ N * , then (4.3) or (4.4) has non-zero solutions. Thus there exists a nontrivial solution to the interior Maxwell problem (4.2), and so k 2 is a Maxwell eigenvalue in B R2 \ B R1 . The proof is thus complete. We have the following uniqueness results for the phaseless inverse electromagnetic scattering problems. Theorem 4.2. Let D 1 , D 2 be two bounded domains and let R 2 > R 1 > 0 be large enough such that D 1 ∪ D 2 ⊂ B R1 and k 2 is not a Maxwell eigenvalue in B R2 \ B R1 . (a) Assume that D 1 and D 2 are two impenetrable obstacles with boundary conditions B 1 and B 2 , respectively. If the corresponding electric total fields satisfy \|e m (x) · E 1 (x, y 1 , e φ (y 1 ), τ 1 , y 2 , e θ (y 2 ), τ 2 ) \| = \|e m (x) · E 2 (x, y 1 , e φ (y 1 ), τ 1 , y 2 , e θ (y 2 ), τ 2 ) \| (4.5) for all x, y 1 , y 2 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , y 2 , (τ 1 , τ 2 ) ∈ {(1, 0), (0, 1), (1, 1)}, m ∈ {φ, θ} and \|e m (x) · E 1 (x, y 1 , e n (y 1 ), τ 1 , y 2 , e l (y 2 ), τ 2 )\| = \|e m (x) · E 2 (x, y 1 , e n (y 1 ), τ 1 , y 2 , e l (y 2 ), τ 2 )\| (4.6) for all x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , y 2 ∈ ∂B R2 \ {N R2 , S R2 }, (τ 1 , τ 2 ) ∈ {(0, 1), (1, 1)}, m, n, l ∈ {φ, θ}, then D 1 = D 2 and B 1 = B 2 . (b) Assume that n 1 , n 2 ∈ C 2,γ (R 3 ) with γ > 0 are the refractive indices of two inhomogeneous media with n j − 1 supported in D j (j = 1, 2). If the corresponding electric total fields satisfy (4.5) and (4.6), then n 1 = n 2 . \|e m (x) · E 1 (x, y 1 , e φ (y 1 ), τ 1 , y 2 , e θ (y 2 ), τ 2 )\| = \|e m (x) · E 2 (x, y 1 , e φ (y 1 ), τ 1 , y 2 , e θ (y 2 ), τ 2 )\| (4.7) for all x, y 1 , y 2 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , y 2 , (τ 1 , τ 2 ) ∈ {(1, 0), (0, 1), (1, 1)}, then we have either e m (x) · E 1 (x, y 1 )e φ (y 1 ) = e m (x) · E 2 (x, y 1 )e φ (y 1 ), ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.8) e m (x) · E 1 (x, y 2 )e θ (y 2 ) = e m (x) · E 2 (x, y 2 )e θ (y 2 ), ∀x, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 (4.9) or e m (x) · E 1 (x, y 1 )e φ (y 1 ) = −e m (x) · E 2 (x, y 1 )e φ (y 1 ), ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.10) e m (x) · E 1 (x, y 2 )e θ (y 2 ) = −e m (x) · E 2 (x, y 2 )e θ (y 2 ), ∀x, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 . (4.11) Proof. We only consider the case m = φ since the case m = θ can be proved similarly. Using (2.17) and (4.7) and arguing similarly as in the proof of Theorem 3.1 give Re{[e φ (x) · E 1 (x, y 1 )e φ (y 1 )] × [e φ (x) · E 1 (x, y 2 )e θ (y 2 )]} = Re{[e φ (x) · E 2 (x, y 1 )e φ (y 1 )] × [e φ (x) · E 2 (x, y 2 )e θ (y 2 )]} (4.12) for all x, y 1 , y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , y 2 . For x, y ∈ ∂B R1 \ {N R1 , S R1 }, x = y, define r (φφ) j (x, y) := \|e φ (x) · E j (x, y)e φ (y)\|, r (φθ) j (x, y) := \|e φ (x) · E j (x, y)e θ (y)\|, j = 1, 2. It then follows from (4.7) with (τ 1 , τ 2 ) = (1, 0) and (τ 1 , τ 2 ) = (0, 1) that r (φφ) 1 (x, y) = r (φφ) 2 (x, y) =: r (φφ) (x, y), r (φθ) 1 (x, y) = r (φθ) 2 (x, y) =: r (φθ) (x, y) for all x, y ∈ ∂B R1 \ {N R1 , S R1 }, x = y. Therefore we can write e φ (x) · E j (x, y 1 )e φ (y 1 ) := r (φφ) (x, y 1 )e iϑ (φφ) j (x,y1) , ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , e φ (x) · E j (x, y 1 )e θ (y 2 ) := r (φθ) (x, y 2 )e iϑ (φθ) j (x,y2) , ∀x, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 , where ϑ (φφ) j and ϑ (φθ) j , j = 1, 2 are real-valued functions. We now prove that r (φφ) (x, y 1 ) ≡ 0, x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , and r (φθ) (x, y 2 ) ≡ 0, x, y 2 ∈ ∂B R1 \{N R1 , S R1 }, x = y 2 . In fact, fix y 1 ∈ ∂B R1 \{N R1 , S R1 } and define the circle C e φ (y1) := {x ∈ ∂B R1 : (x − y 1 ) · e φ (y 1 ) = 0}, which is the intersection of the sphere ∂B R1 with the plane whose normal vector is e φ (y 1 ) at y 1 . When x tend to y 1 along the circle C e φ (y1) , we have x − y 1 ⊤ e φ (y 1 ) → 0 and e φ (x) · e φ (y 1 ) → 1. Thus, by (2.9) it is known that e φ (x) · E i (x, y 1 )e φ (y 1 ) ∼ i k k 2 + ik − 1 \|x − y 1 \| 1 \|x − y 1 \| Φ k (x, y 1 ) (4.13) as x goes to y 1 along the circle C e φ (y1) . The singularity in (4.13) implies that r (φφ) (x, y 1 ) ≡ 0 for x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 since E s j (x, y 1 ) is analytic with respect to x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , respectively (j = 1, 2). Further, fix y 2 ∈ ∂B R1 \ {N R1 , S R1 } and define the circle C e φ (y2)+e θ (y2) := {x ∈ ∂B R1 : (x − y 2 ) · (e φ (y 2 ) + e θ (y 2 )) = 0}. Then, on letting x tend to y 2 along C e φ (y2)+e θ (y2) we have e φ (x) · e θ (y 2 ) → 0 and e φ (x) · x − y 2 · x − y 2 ⊤ e θ (y 2 ) → c 1 for a non-zero constant c 1 . Thus it follows from (2.9) that e φ (x) · E i (x, y 2 )e θ (y 2 ) = 1 \|x − y 2 \| 2 Φ k (x, y 2 ) [c 2 + o(1)] (4.14) as x → y 2 along C e φ (y2)+e θ (y2) , where c 2 is a non-zero constant. Therefore the singularity in (4.14) implies that r (φθ) (x, y 2 ) ≡ 0 for x, y 2 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 2 since E s j (x, y 2 ) is analytic with respect to x, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 , respectively (j = 1, 2). Then, similarly as in the proof of Theorem 3.1, we can show that there are three small enough open sets U, U 1 , U 2 ⊂ ∂B R1 \ {N R1 , S R1 } such that U, U 1 and U 2 are disjoint, r (φφ) (x, y 1 ) = 0 and r (φθ) (x, y 2 ) = 0 for all x ∈ U , y 1 ∈ U 1 and y 2 ∈ U 2 , and ϑ (φφ) j (x, y 1 ) and ϑ (φθ) j (x, y 2 ) are analytic with respect to x ∈ U , y 1 ∈ U 1 , y 2 ∈ U 2 , respectively, j = 1, 2. Now, by (4.12) we have cos[ϑ for all (x, y 1 , y 2 ) ∈ U × U 1 × U 2 . Since ϑ (φφ) j (x, y 1 ) and ϑ (φθ) j (x, y 2 ) are analytic functions of x ∈ U , y 1 ∈ U 1 and y 2 ∈ U 2 , respectively (j = 1, 2), we obtain that there holds either ϑ (φφ) 1 (x, y 1 ) − ϑ (φθ) 1 (x, y 2 ) = ϑ (φφ) 2 (x, y 1 ) − ϑ (φθ) 2 (x, y 2 ) + 2pπ (4.16) or ϑ (φφ) 1 (x, y 1 ) − ϑ (φθ) 1 (x, y 2 ) = −[ϑ (φφ) 2 (x, y 1 ) − ϑ (φθ) 2 (x, y 2 )] + 2pπ (4.17) for all (x, y 1 , y 2 ) ∈ U × U 1 × U 2 , where p ∈ Z. For the case when (4.16) holds, we have α(x) := ϑ (φφ) 1 (x, y 1 ) − ϑ (φφ) 2 (x, y 1 ) = ϑ (φθ) 1 (x, y 2 ) − ϑ (φθ) 2 (x, y 2 ) + 2pπ depends only on x, which is a real-valued analytic function in x ∈ U . Thus e φ (x) · E 1 (x, y 1 )e φ (y 1 ) = r (φφ) (x, y 1 )e iϑ (φφ) 1 (x,y1) = r (φφ) (x, y 1 )e iα(x)+iϑ (φφ) 2 (x,y1) = e iα(x) e φ (x) · E 2 (x, y 1 )e φ (y 1 ), e φ (x) · E 1 (x, y 2 )e θ (y 2 ) = r (φθ) (x, y 2 )e iϑ (φθ) 1 (x,y2) = r (φθ) (x, y 2 )e iα(x)+iϑ (φθ) 2 (x,y2) = e iα(x) e φ (x) · E 2 (x, y 2 )e θ (y 2 ) for all (x, y 1 , y 2 ) ∈ U × U 1 × U 2 . By the analyticity of E 1 (x, y) − e iα(x) E 2 (x, y) in y ∈ ∂B R1 for y = x, we obtain e φ (x) · E 1 (x, y 1 )e φ (y 1 ) = e iα(x) e φ · E 2 (x, y 1 )e φ (y 1 ), ∀x ∈ U, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.18) e φ (x) · E 1 (x, y 2 )e θ (y 2 ) = e iα(x) e φ · E 2 (x, y 2 )e θ (y 2 ), ∀x ∈ U, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 . (4.19) From (4.18) it follows that e φ (x) · [E s 1 (x, y 1 )e φ (y 1 ) − e iα(x) E s 2 (x, y 1 )e φ (y 1 )] = [e iα(x) − 1]e φ (x) · E i (x, y 1 )e φ (y 1 ) (4.20) for all x ∈ U and y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 . For arbitrarily fixed y 1 ∈ U , the left-hand side of (4.20) is analytic in x ∈ U , while, by (4.13) the right-hand side of (4.20) is singular when x is close to y 1 along the circle C e φ (y1) . Therefore, e iα(y1) = 1. Since y 1 ∈ U is arbitrary, we have e iα(x) = 1 for all x ∈ U , and so (4.18) and (4.19) become e φ (x) · E 1 (x, y 1 )e φ (y 1 ) = e φ · E 2 (x, y 1 )e φ (y 1 ), ∀x ∈ U, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.21) e φ (x) · E 1 (x, y 2 )e θ (y 2 ) = e φ · E 2 (x, y 2 )e θ (y 2 ), ∀x ∈ U, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 . (4.22) This, together with the analyticity of E j (x, y) (j = 1, 2) in x ∈ ∂B R1 with x = y, gives (4.8) and (4.9). Similarly, for the case when (4.17) holds, we can deduce e φ (x) · E 1 (x, y 1 )e φ (y 1 ) = e iβ(x) e φ (x) · E 2 (x, y 1 )e φ (y 1 ), ∀x ∈ U, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.23) e φ (x) · E 1 (x, y 2 )e θ (y 2 ) = e iβ(x) e φ (x) · E 2 (x, y 2 )e θ (y 2 ), ∀x ∈ U, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 , (4.24) where β is a real-valued analytic function of x ∈ U . From (4.23) it is easy to derive that e φ (x) · [E s 1 (x, y 1 ) − e iβ(x) E s 2 (x, y 1 )]e φ (y 1 ) = e φ (x) · [e iβ(x) E i (x, y 1 ) − E i (x, y 1 )]e φ (y 1 ) (4.25) for all x ∈ U , y 1 ∈ ∂B R1 , x = y 1 . For arbitrarily fixed y 1 ∈ U , the left-hand side of (4.25) is analytic in x ∈ U , but, by (2.9) and a direct calculation, the right-hand side of (4.25) has a singularity at x = y 1 unless e iβ(x) = −1 for x ∈ C e φ (y1) near y 1 . This means that e iβ(y1) = −1. By the arbitrariness of y 1 ∈ U , we have e iβ(x) = −1 for all x ∈ U , and so e iβ(x) E i (x, y) − E i (x, y) = −E i (x, y) − E i (x, y) = (k 2 I + ∇ x ∇ x ) i k Φ k (x, y) − Φ k (x, y) (4.26) is analytic in x ∈ R 3 and y ∈ R 3 , respectively, since Φ k (x, y) − Φ k (x, y) is analytic in x ∈ R 3 and y ∈ R 3 , respectively. Thus (4.23) and (4.24) are reduced to e φ (x) · E 1 (x, y 1 )e φ (y 1 ) = −e φ (x) · E 2 (x, y 1 )e φ (y 1 ), ∀x ∈ U, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.27) e φ (x) · E 1 (x, y 2 )e θ (y 2 ) = −e φ (x) · E 2 (x, y 2 )e θ (y 2 ), ∀x ∈ U, y 2 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 2 . (4.28) Both (4.10) and (4.11) then follow from the analyticity of E j (x, y) (j = 1, 2) in x ∈ ∂B R1 for x = y. The proof is thus complete. Lemma 4.5. Assume that the assumptions of Theorem 4.2 are satisfied. If for any fixed m, n, l ∈ {φ, θ} there holds \|e m (x) · E 1 (x, y 1 , e n (y 1 ), τ 1 , y 2 , e l (y 2 ), τ 2 )\| = \|e m (x) · E 2 (x, y 1 , e n (y 1 ), τ 1 , y 2 , e l (y 2 ), τ 2 )\| (4.29) for all x, 0), (0, 1), (1, 1)}, then we have either e m (x) · E 1 (x, y 1 )e n (y 1 ) = e m (x) · E 2 (x, y 1 )e n (y 1 ), ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , y 2 ∈ ∂B R2 \ {N R2 , S R2 }, (τ 1 , τ 2 ) ∈ {(1,y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 ,(4.30) e m (x) · E 1 (x, y 2 )e l (y 2 ) = e m (x) · E 2 (x, y 2 )e l (y 2 ), ∀x ∈ ∂B R1 \ {N R1 , S R1 }, y 2 ∈ ∂B R2 \ {N R2 , S R2 } (4.31) or e m (x) · E 1 (x, y 1 )e n (y 1 ) = −e m (x) · E 2 (x, y 1 )e n (y 1 ), ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , (4.32) e m (x) · E 1 (x, y 2 )e l (y 2 ) = −e m (x) · E 2 (x, y 2 )e l (y 2 ), ∀x ∈ ∂B R1 \ {N R1 , S R1 }, y 2 ∈ ∂B R2 \ {N R2 , S R2 }. (4.33) Proof. Since \|e m (x) · E 1 (x, y 2 )e l (y 2 )\| is analytic in x ∈ (∂B R1 \ {N R1 , S R1 }) and y 2 ∈ (∂B R2 \ {N R2 , S R2 }), respectively, we only need to distinguish between two cases: A) \|e m (x) · E 1 (x, y 2 )e l (y 2 )\| ≡ 0, ∀(x, y 2 ) ∈ (∂B R1 \ {N R1 , S R1 }) × (∂B R2 \ {N R2 , S R2 }) , B) \|e m (x) · E 1 (x, y 2 )e l (y 2 )\| ≡ 0, ∀(x, y 2 ) ∈ (∂B R1 \ {N R1 , S R1 }) × (∂B R2 \ {N R2 , S R2 }) . For the case when A) holds, by arguing similarly as in the proof of Lemma 4.4 it can be deduced from (4.29) that we have either e m (x) · E 1 (x, y 1 )e n (y 1 ) = e iα(x) e m (x) · E 2 (x, y 1 )e n (y 1 ), ∀x ∈ U, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.34) e m (x) · E 1 (x, y 2 )e l (y 2 ) = e iα(x) e m (x) · E 2 (x, y 2 )e l (y 2 ), ∀x ∈ U, y 2 ∈ ∂B R2 \ {N R2 , S R2 } (4.35) or e m (x) · E 1 (x, y 1 )e n (y 1 ) = e iβ(x) e m (x) · E 2 (x, y 1 )e n (y 1 ), ∀x ∈ U, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , (4.36) e m (x) · E 1 (x, y 2 )e l (y 2 ) = e iβ(x) e m (x) · E 2 (x, y 2 )e l (y 2 ), ∀x ∈ U, y 2 ∈ ∂B R2 \ {N R2 , S R2 }, (4.37) where U is some small open subset of ∂B R1 \ {N R1 , S R1 }, and α(x) and β(x) are real-valued functions of x. By (4.8) and (4.10) in Lemma 4.4 it follows easily that e iα(x) = 1 and e iβ(x) = −1. This, together with (4.34)-(4.37) and the analyticity of the total fields E j (x, y), j = 1, 2, in x for x = y, implies that either (4.30) and (4.31) hold or (4.32) and (4.33) hold. For the case when B) holds, it follows from (4.29) that \|e m (x) · E 2 (x, y 2 )e l (y 2 )\| ≡ 0, ∀(x, y 2 ) ∈ (∂B R1 \ {N R1 , S R1 }) × (∂B R2 \ {N R2 , S R2 }) .E 1 (x, y) = E 2 (x, y), ∀x, y ∈ G, x = y. (4.38) Proof. We first show that for any fixed m ∈ {φ, θ}, e m (x) · E 1 (x, y 1 )e n (y 1 ) = e m (x) · E 2 (x, y 1 )e n (y 1 ), ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , ∀n ∈ {φ, θ}. (4.39) To this end, for any fixed m ∈ {φ, θ} we need to distinguish between the following two cases. Case 1. Re[e m (x) · E 1 (x, y 1 )e l (y 1 )] = 0 for all x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 and for all l ∈ {φ, θ}. In this case, by Lemma 4.4 it follows that Re[e m (x) · E 2 (x, y 1 )e l (y 1 )] = 0 for all x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 and for all l ∈ {φ, θ}. By Lemma 4.4 again we have (4.39). Case 2. Re[e m (x) · E 1 (x, y 1 )e l (y 1 )] = 0 for some x, y 1 ∈ ∂B R1 \ {N R1 , S R1 } with x = y 1 , l ∈ {φ, θ}. Here, we only consider the case with l = φ. The case l = θ can be treated similarly. In this case, by Lemma 4.4 we have that either both (4.8) and (4.9) hold or both (4.10) and (4.11) hold. We can prove that both (4.10) and (4.11) can not hold simultaneously. Suppose this is not the case. Then we have e m (x) · [E 1 (x, y 1 )e n (y 1 )] = −e m (x) · [E 2 (x, y 1 )e n (y 1 )], ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , ∀n ∈ {φ, θ}.(4.40) This, together with Lemmas 4.4 and 4.5, implies that e m (x) · [E 1 (x, y 2 )e n (y 2 )] = −e m (x) · [E 2 (x, y 2 )e n (y 2 )], ∀n ∈ {φ, θ} (4.41) ∀x ∈ ∂B R1 \ {N R1 , S R1 }, y 2 ∈ ∂B R2 \ {N R2 , S R2 }. We now show that both (4.40) and (4.41) can not hold simultaneously. In fact, by the reciprocity relation E j (x, y) = [E j (y, x)] ⊤ for all x, y ∈ G (j = 1, 2), we deduce from (4.40) and (4.41) that This, together with the linear combination of e φ (y j ) and e θ (y j ) (j = 1, 2), gives that e n (y 1 ) · [E 1 (y 1 , x)e m (x)] = −e n (y 1 ) · [E 2 (y 1 , x)e m (x)], ∀n ∈ {φ, θ},ν(y 1 ) × [E 1 (y 1 , x)e m (x)] = −ν(y 1 ) × [E 2 (y 1 , x)e m (x)], (4.44) ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , ν(y 2 ) × [E 1 (y 2 , x)e m (x)] = −ν(y 2 ) × [E 2 (y 2 , x)e m (x)], (4.45) ∀x ∈ ∂B R1 \ {N R1 , S R1 }, y 2 ∈ ∂B R2 \ {N R2 , S R2 }. For any fixed x ∈ ∂B R1 \ {N R1 , S R1 } and m ∈ {φ, θ}, define E(y) := E 1 (y, x)e m (x) + E 2 (y, x)e m (x), y = x. Since 2Re[E i (y, x)] := E i (y, x) + E i (y, x) is analyticity for all x, y ∈ R 3 (see (4.26)), then, by the analyticity of E s j (y, x) with respect to y ∈ G (j = 1, 2), it follows that E can be extended as an analytic function of y ∈ G, which we denote by E again. Define H(y) := [1/(ik)]curl y E(y). Then (Re[E i (y, x)]e m (x), Im[H i (y, x)]e m (x)) and (E s j (y, x)e m (x), H s j (y, x)e m (x)) satisfy the Maxwell equations for x ∈ G, j = 1, 2. Thus it follows by (4.44), (4.45) and the analyticity of E j (y, x) in y ∈ G with y = x (j = 1, 2) that ( E, H) satisfies the interior Maxwell problem      curl E − ik H = 0 in B R2 \ B R1 , curl H + ik E = 0 in B R2 \ B R1 , ν × E = 0 on ∂B R1 ∪ ∂B R2 . Since k 2 is not a Maxwell eigenvalue in B R2 \B R1 , then E = 0 in B R2 \B R1 . Thus, and by the analyticity of E j (y, x) in y ∈ G with y = x (j = 1, 2), we have E 1 (y, x)e φ (x) = −E 2 (y, x)e φ (x) for all y ∈ G, y = x. This contradicts to the fact that E j (y, x)e m (x) = E i (y, x)e m (x) + E s j (y, x)e m (x), j = 1, 2, satisfy the Silver-Müller radiation condition. Therefore, (4.40) and (4.41) can not be true simultaneously, which means that both (4.10) and (4.11) can not hold simultaneously. This then implies that both (4.8) and (4.9) are true, and so (4.39) holds. Finally, by (4.39) and the linear combination of e φ and e θ we obtain that for arbitrarily fixed y 1 ∈ ∂B R1 \ {N R1 , S R1 } and n ∈ {φ, θ}, ν(x) × [E s 1 (x, y 1 )e n (y 1 )] = ν(x) × [E s 2 (x, y 1 )e n (y 1 )], ∀x ∈ ∂B R1 \ {N R1 , S R1 }. By the well-posedness of the exterior Maxwell problem in R 3 \ B R1 with the PEC condition on ∂B R1 it is deduced that for arbitrarily fixed y 1 ∈ ∂B R1 \ {N R1 , S R1 }, E s 1 (x, y 1 )e n (y 1 ) = E s 2 (x, y 1 )e n (y 1 ), ∀ x ∈ R 3 \ B R1 , ∀ n ∈ {φ, θ}. This, together with the reciprocity relation E s j (x, y) = [E s j (y, x)] ⊤ for all x, y ∈ G, j = 1, 2, implies that for any fixed x ∈ R 3 \ B R1 , ν(y) × E s 1 (y, x) = ν(y) × E s 2 (y, x), ∀y ∈ ∂B R1 . Again, by the well-posedness of the exterior Maxwell problem in R 3 \ B R1 with the PEC condition on ∂B R1 it is derived that for any fixed x ∈ R 3 \ B R1 , E s 1 (y, x) = E s 2 (y, x), ∀y ∈ R 3 \ B R1 . The required result (4.38) then follows from this, the reciprocity relation and the analyticity of E s j (x, y) (j = 1, 2) in x ∈ G and y ∈ G, respectively. Proof of Theorem 4.2. By Lemma 4.6 it follows from (4.5) and (4.6) that (4.38) holds. For j = 1, 2, denote by E ∞ j (x, y) the far-field pattern of E s j (x, y), x, y ∈ G, and by E s j (x, d) and E ∞ j (x, d) the electric scattered field and its far-field pattern associated with the obstacle D j (or the refractive index n j ) and corresponding to the incident electromagnetic plane waves described by the matrices E i (x, d), H i (x, d) defined by E i (x, d)p := i k curl curl pe ikx·d = ik(d × p) × de ikx·d , H i (x, d)p := curl pe ikx·d = ikd × pe ikx·d , where d ∈ S 2 and p ∈ R 3 denote the incident direction and polarization vector, respectively, and x ∈ R 3 . Then, by (4.38) in Lemma 4.6 and the mixed reciprocity relation that 4πE ∞ j (−d, x) = [E s j (x, d)] ⊤ for all x ∈ G, d ∈ S 2 and j = 1, 2 (see [14, Theorem 6.31]), we obtain that E s 1 (x, d) = E s 2 (x, d) for all x ∈ G and all d ∈ S 2 or E ∞ 1 (x, d) = E ∞ 2 (x, d) for allx, d ∈ S 2 . By the uniqueness result for inverse electromagnetic scattering with full far-field data (see [14,Theorem 7.1] for the obstacle case and [15,Theorem 4.9] for the inhomogeneous medium case) it follows easily that the uniqueness statements (a) and (b) of Theorem 4.2 are true. The theorem is thus proved. ✷ 5. Conclusions. This paper proposed a new approach to prove uniqueness results for inverse acoustic and electromagnetic scattering for obstacles and inhomogeneous media with phaseless near-field data at a fixed frequency. The idea is to use superpositions of two point sources at a fixed frequency as the incident fields and, as the phaseless near-field data, to measure the modulus of the acoustic total-field on two spheres enclosing the scatterers generated by such incident fields located on the two spheres, in the acoustic case. For the electromagnetic case, the idea is to utilize superpositions of two electric dipoles at a fixed frequency with the polarization vectors e φ and e θ , respectively, as the incident fields and, as the phaseless near-field data, to measure the modulus of the tangential component with the orientations e φ and e θ , respectively, of the electric total-field on a sphere enclosing the scatterers and generated by such incident fields located on the measurement sphere and another bigger sphere. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data. the the interior Dirichlet boundary value problem ∆u + k 2 u = 0 in V, u = 0 on ∂V has a nontrivial solution u. The above assumption on R 1 and R 2 can be easily satisfied since the Dirichlet eigenvalues of −∆ in a bounded domain are discrete and satisfy the strong monotonicity property [28, Theorem 4.7] (see also the arguments in the proof of [14, Theorem 5.2]). Let G denote the unbounded component of the complement of D 1 ∪ D 2 . Then we have the following global uniqueness results for the phaseless inverse scattering problems. Fig. 3 . 2 . 32Scattering by a medium. ( 3 . 17 ) 317Case 2. (3.6) holds with (x, y) ∈ ∂B R2 × ∂B R2 , x = y. By a similar argument as in Case 1, it can be obtained that there holds either w s 1 (x, y) = w s 2 (x, y), ∀x ∈ ∂B R2 , y ∈ ∂B R2 ∪ {y 0 } (3.18) or w 1 (x, y) = w 2 (x, y), ∀x ∈ ∂B R2 , y ∈ ∂B R2 ∪ {y 0 } with x = y.(3.19) Fig. 4 . 1 . 41Electromagnetic scattering by an obstacle (a) or a medium (b) By choosing appropriate R 1 and R 2 (see Lemma 4.1), it can be ensured that k 2 is not a Maxwell eigenvalue in B R2 \ B R1 . Here, k 2 is called a Maxwell eigenvalue in a bounded domain V if the interior Maxwell problem E − ikH = 0 in V curl H + ikE = 0 in V ν × E = 0 on ∂V has a nontrivial solution (E, H). Lemma 4.1. k 2 is not a Maxwell eigenvalue in B R2 \ B R1 if and only if argument as in the proof of [23, Theorems 2.48 and 2.50] gives the following expansion in the spherical vector harmonics of the electric fieldE in B R2 \ B R1 as a curl {xy n (k\|x\|)Y m n (x)} , x ∈ B R2 \ B R1 ,where Y m n , m = −n, . . . , n, n = 0, 1, 2, . . ., are the spherical harmonics. By[14, (6.71) and (6.72)], we have that for any x ∈ ∂B r with r ∈ [R 1 , R 2 ], {j n (kr) + krj ′ n (kr)}x × GradY m n {y n (kr) + kry ′ n (kr)}x × GradY m n (x). Remark 4 . 3 . 43Since E i j (x, y) = [E i j (y, x)] ⊤ , and by the reciprocity relation E s j (x, y) = [E s j (y, x)] ⊤ for all x, y ∈ G (see [14, Theorem 6.32]), j = 1, 2, we know that (4.5) with m = φ and (τ 1 , τ 2 ) = (0, 1) is equivalent to (4.5) with m = θ and (τ 1 , τ 2 ) = (1, 0). To prove Theorem 4.2, we need some results on the phaseless electric total-fields measured on S R1 . Lemma 4.4. Assume that the assumptions of Theorem 4.2 are satisfied. If for any fixed m ∈ {φ, θ} there holds (φφ) 1 (x, y 1 ) 11− ϑ (φθ) 1 (x, y 2 )] = cos[ϑ (φφ) 2 (x, y 1 ) − ϑ (φθ) 2 (x, y 2 )] (4.15) ( 4 . 442) ∀x, y 1 ∈ ∂B R1 \ {N R1 , S R1 }, x = y 1 , e n (y 2 ) · [E 1 (y 2 , x)e m (x)] = −e n (y 2 ) · [E 2 (y 2 , x)e m (x)], ∀n ∈ {φ, θ} (4.43) ∀x ∈ ∂B R1 \ {N R1 , S R1 }, y 2 ∈ ∂B R2 \ {N R2 , S R2 }. 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[ "Anomalous couplings in associated V H production with Higgs decay to massive b quarks at NNLO in QCD", "Anomalous couplings in associated V H production with Higgs decay to massive b quarks at NNLO in QCD" ]	[ "Wojciech Bizoń *[email protected]†[email protected]‡[email protected]§[email protected] \nInstitute for Theoretical Particle Physics\nKIT\nKarlsruheGermany\n\nInstitute for Astroparticle Physics\nKIT\nKarlsruheGermany\n", "Fabrizio Caola \nRudolf Peierls Centre for Theoretical Physics\nClarendon Laboratory\nUK and Wadham College\nParks RoadOX1 3PU, OX1 3PNOxford, OxfordUK\n", "Kirill Melnikov \nInstitute for Theoretical Particle Physics\nKIT\nKarlsruheGermany\n", "Raoul Röntsch \nTheoretical Physics Department\nCERN\n1211Geneva 23Switzerland\n" ]	[ "Institute for Theoretical Particle Physics\nKIT\nKarlsruheGermany", "Institute for Astroparticle Physics\nKIT\nKarlsruheGermany", "Rudolf Peierls Centre for Theoretical Physics\nClarendon Laboratory\nUK and Wadham College\nParks RoadOX1 3PU, OX1 3PNOxford, OxfordUK", "Institute for Theoretical Particle Physics\nKIT\nKarlsruheGermany", "Theoretical Physics Department\nCERN\n1211Geneva 23Switzerland" ]	[]	We combine the NNLO QCD description of Higgs boson production in association with an electroweak vector boson V = W or Z with a similarly-precise description of Higgs boson decays into a pair of massive b quarks and with the anomalous couplings that modify interactions of the Higgs and electroweak vector bosons. The resulting numerical code provides the most advanced theoretical tool to investigate such anomalous couplings in the associated Higgs boson production process.We study the impact of anomalous couplings on fiducial cross sections and differential distributions and argue that, with increased QCD precision, smaller anomalous couplings become accessible in kinematic regions where the effects of higher-dimensional operators in the Standard Model Effective Field Theory remain small and the EFT expansion is under control.	10.1103/physrevd.105.014023	[ "https://arxiv.org/pdf/2106.06328v1.pdf" ]	235,417,262	2106.06328	717b8417abd080bcddb1afcb881ffcf4ca889e63	Anomalous couplings in associated V H production with Higgs decay to massive b quarks at NNLO in QCD 11 Jun 2021 Wojciech Bizoń [email protected]†[email protected]‡[email protected]§[email protected] Institute for Theoretical Particle Physics KIT KarlsruheGermany Institute for Astroparticle Physics KIT KarlsruheGermany Fabrizio Caola Rudolf Peierls Centre for Theoretical Physics Clarendon Laboratory UK and Wadham College Parks RoadOX1 3PU, OX1 3PNOxford, OxfordUK Kirill Melnikov Institute for Theoretical Particle Physics KIT KarlsruheGermany Raoul Röntsch Theoretical Physics Department CERN 1211Geneva 23Switzerland Anomalous couplings in associated V H production with Higgs decay to massive b quarks at NNLO in QCD 11 Jun 20211 We combine the NNLO QCD description of Higgs boson production in association with an electroweak vector boson V = W or Z with a similarly-precise description of Higgs boson decays into a pair of massive b quarks and with the anomalous couplings that modify interactions of the Higgs and electroweak vector bosons. The resulting numerical code provides the most advanced theoretical tool to investigate such anomalous couplings in the associated Higgs boson production process.We study the impact of anomalous couplings on fiducial cross sections and differential distributions and argue that, with increased QCD precision, smaller anomalous couplings become accessible in kinematic regions where the effects of higher-dimensional operators in the Standard Model Effective Field Theory remain small and the EFT expansion is under control. I. INTRODUCTION Studies of Higgs boson properties in experiments at the Large Hadron Collider (LHC) have converged to the conclusion [1] that the Standard Model of particle physics describes Higgs couplings to gauge bosons and to (some) matter fields with a precision between 10 and 30 percent. To reach a higher precision, new experimental measurements as well as refined theoretical descriptions of major Higgs production processes are needed. The forthcoming Run III of the LHC, as well as its high-luminosity phase, will play an important role in achieving these goals. From a theoretical perspective, some of the simplest but also most interesting Higgs boson production processes are those where Higgs bosons are produced in association with vector bosons, i.e. pp → W H and pp → ZH. Indeed, at lowest order in perturbative QCD, both of these processes are of the Drell-Yan type pp → V → V H, so that their description through next-to-next-to-leading order (NNLO) in perturbative QCD is quite straightforward. These production processes allow for a study of Higgs-gauge interactions. Associated production processes also provide an environment in which the decay of the Higgs to a bb pair can be observed [2][3][4][5][6], allowing the study of the Higgs coupling to b quarks. Therefore, describing both the production pp → V H and the decay H → bb processes with the same precision is important. Moreover, it is important to consider b quarks as massive since in this case one can apply conventional jet algorithms to identify b jets and reconstruct Higgs boson kinematics. Indeed, it was shown recently in Ref. [7] that working with massless b quarks may lead to sizeable differences in theoretical predictions for the associated production process pp → W H. The status of theoretical predictions for V H processes in the Standard Model (SM) is quite advanced. Refined predictions that include both QCD [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and electroweak [24][25][26] radiative corrections are available. Recently, NNLO QCD corrections for W H production in association with a hard jet were computed [27]. A major difference between the W H and ZH final states is that, starting from NNLO QCD, the latter receives large contributions from the gg → ZH process. The O(α 2 s ) contribution of this process has been known for a long time [28,29]. Approximate results for the O(α 3 s ) contributions due to gg → ZH suggest that these can be quite large [30,31] and should be included for reliable predictions despite being formally subleading. In the recent past, significant effort went into their computation [32][33][34][35], using either numerical methods or phenomenologically-motivated analytic approximations. Strategies to extract the gg → ZH contribution from experimental data have been investigated in Ref. [36]. Both the W H [37,38] and ZH [38][39][40] processes have been matched to parton shower Monte Carlo tools retaining NNLO QCD accuracy. Ref. [40] also includes NNLO QCD corrections to the H → bb decay. Dedicated parton shower Monte Carlo tools including both next-to-leading order (NLO) QCD and electroweak corrections [41] and a refined treatment of the gg → ZH contribution [42] also exist. Similarly, advanced theoretical predictions for the H → bb decay are available. The total rate is known up to N 4 LO QCD in the limit of massless bottom quarks [43][44][45][46][47][48][49][50][51]. Electroweak corrections are also known [52][53][54][55]. A comprehensive review of computations of the H → bb inclusive branching ratio and its uncertainties can be found in Refs. [56,57]. At the differential level, QCD corrections for massless b quarks are known at NNLO [22,[58][59][60] and N 3 LO [61,62]. NNLO QCD results retaining the full b-quark mass dependence have been presented in Refs. [7,[63][64][65]. Top-quark effects have been studied in Refs. [66,67]. Given this level of sophistication, it is interesting to extend the precise modeling of associated Higgs boson production and H → bb decay to cases where Higgs couplings to gauge and matter fields differ from the ones in the Standard Model Lagrangian. A convenient way to do this is provided by the Standard Model Effective Field Theory (SMEFT), see e.g. Ref. [68] for a review. In principle, one may aim at the complete description of the processes pp → V H(bb) in the SMEFT, taking into account contributions of all dimension-six operators present in the SMEFT Lagrangian. However, since in this case the number of operators that one has to consider becomes quite large, it makes sense to first restrict oneself to a subset of operators. A particularly interesting choice is those that modify the couplings of the Higgs boson to electroweak gauge bosons. Indeed, this approach has been adopted in Ref. [69] where this process was studied with NLO QCD accuracy. The main goal of this work is to promote this analysis to full NNLO QCD. The paper is organized as follows. In Section II we briefly describe the computation of NNLO QCD corrections to the processes pp → V H(bb), with V = W, Z, in the Standard Model. Such a computation for the W H final state was discussed earlier in Ref. [7]; the results for ZH(bb) with massive b quarks are new. In Section III we describe calculations that include both anomalous couplings and NNLO QCD corrections to Higgs boson associated production and its decay into a bb pair. We consider scenarios where anomalous couplings modify fiducial cross sections only slightly so that the availability of highly precise theoretical predictions for fiducial cross sections and kinematic distributions becomes important. We conclude in Section IV. II. ASSOCIATED PRODUCTION pp → V H(bb) IN THE STANDARD MODEL In this section, we briefly describe the computation of NNLO QCD radiative corrections to the associated production process pp → V H(bb) in the Standard Model, keeping the b-quark masses nonzero. We note that the W H(bb) final state was discussed recently in Ref. [7]. The results for the ZH(bb) final state, which we mostly focus on in this section, are new. As we already mentioned in the introduction, the computation of NNLO QCD radiative corrections to pp → V H(bb) involves two major ingredients: QCD corrections to the production process pp → V H and QCD corrections to the decay process H → bb. An earlier computation of NNLO QCD corrections to pp → W H with the decay H → bb for massless b quarks was described in Ref. [22]. This computation was based on the nested soft-collinear subtraction scheme introduced in Ref. [70], and employed simple analytic formulas derived for the production and decay processes of color-singlet states in Refs [60,71]. This earlier computation was recently extended by including Higgs boson decays to massive b quarks [7], using predictions for H → bb decay from Ref. [64] and modifying the calculation of NNLO QCD corrections to the production process in Ref. [22] to exclude b quarks from being active partons in a proton. As we already mentioned in the introduction, working with massive quarks allows us to employ conventional jet algorithms to describe b-flavored jets. In this paper, we have extended the above computations to the ZH final state. From the point of view of soft and collinear subtractions, such an extension is straightforward since the analytic formulas derived in Ref. [71] there are other classes of contributions proportional to the top Yukawa coupling for which the exact two-loop amplitudes are unknown. Here we followed the approach of Ref. [20], which is based on the analysis of Ref. [15]. In the notation of Ref. [15], we have included the so-called V I,II contributions keeping only the leading term in the m top → ∞ expansion. We have also included exact R I contributions, but discarded R II terms since they have been shown to be very small for phenomenologically relevant setups [15]. Similarly, we have neglected effects of top quark loops in Drell-Yan type diagrams pp → V * , V * → V H as they too have negligible phenomenological impact [15]. As the first step in our discussion, we present cross sections and differential distributions for the two associated production processes pp → W + H → (ν e e + )(bb) ,(1)pp → ZH → (e − e + )(bb) ,(2) at the 13 TeV LHC. We treat both decay processes V → ¯ and H → bb in the narrow-width approximation. Following Ref. [7], we write differential cross sections as Finally, we take the CKM matrix to be diagonal. We set the factorization and renormalization scales equal to each other, µ r = µ f = µ. We use We define W + H and ZH final states using kinematic selection criteria for charged leptons and b-flavored jets. To this end, we require that an event contains at least two b jets that are defined with the anti-k t jet algorithm [75,76] and we choose the jet radius R = 0.4. dσ pp→V H(bb) ∝ Br(H → bb) × dσ pp→V H × dΓ H→bb Γ H→bb ,(3) Pseudo-rapidities and transverse momenta of charged leptons and b jets should satisfy the following constraints \|η l \| < 2.5 , p t,l > 25 GeV , \|η j b \| < 2.5 , p t,j b > 20 GeV .(4) Finally, following experimental analyses, we may additionally require that the vector boson has large transverse momentum, p t,V > 150 GeV. We always state explicitly when this cut is applied. and factorization scales in the production process is set to one half of the invariant mass of the V H system, i.e. µ r = µ f = µ = 1 2 (p V + p H ) 2 . The renormalization scale for the decay process is set to the Higgs boson mass, µ r,dec = M H ; it is kept fixed for all results reported in this paper. The uncertainty of the cross sections is obtained by varying the scale in the production process by a factor of two around the central value. We present fiducial cross sections for the above set of cuts at leading order (LO), nextto-leading order (NLO) and NNLO in QCD for the W + H and ZH production processes in Tables I and II. The contribution of the gluon-initiated process gg → ZH is reported separately. As we already mentioned, this contribution is rather large. Indeed, it follows from Tables I and II that We also see from Tables I and II that uncertainties of quark-initiated contributions at NNLO are less than a percent if no p t,V cut is applied, and around two percent in the presence of this cut. The inclusion of the gg → ZH contribution increases the uncertainty significantly, to about four percent without the additional p t,V cut and to about seven percent if we require p t,V > 150 GeV. To reduce this uncertainty, the gg → ZH contribution has to be computed at NLO in perturbative QCD and, as we already mentioned in the introduction, a significant 1 We note that in the gg → ZH channel there is a strong cancellation between the box and triangle diagrams, which however is only active for SM couplings. If e.g. the top Yukawa coupling were to be different, a very strong enhancement of this contribution could be expected [77]. (p V + p H ) 2 . The lower panes show the ratio of the NNLO results to the NLO ones. See text for details. effort in this direction is currently underway [32][33][34][35]. Before we turn to the discussion of anomalous couplings, we show a few kinematic distributions in the Standard Model. Since we have discussed the W + H process in some detail earlier [7], we focus exclusively on the distributions for the pp → ZH process. In Fig. 1 we display the invariant mass of the Higgs boson and Z boson system. We note that the invariant mass is reconstructed from the "true" Higgs boson momentum p H and the Z-boson momentum p Z , i.e. M 2 ZH = (p H + p Z ) 2 ; however, to be included in the plot, an event is required to pass the kinematic cuts described above. We observe large changes in this distribution starting at M ZH ∼ 350 GeV where the gg → ZH contribution becomes significant. However, the NNLO QCD corrections to the quark-initiated processes change the NLO QCD distribution only slightly; they are about −5% at low invariant masses and become slightly positive at larger M ZH values. The transverse momentum and rapidity distributions of the Higgs boson are shown in Figs. 2 and 3, respectively. In these plots, the Higgs momentum is reconstructed from two b jets as described earlier. We note that if more than two b jets appear in the final state, we choose the pair whose invariant mass is closest to the Higgs boson mass M H = 125 GeV. The corrections to the Higgs transverse momentum distribution show a pattern that is similar to Kinematic distributions that are integrated over the Higgs boson transverse momentum and ZH invariant masses do not exhibit local enhancements due to the gg → ZH process but, rather, show an overall increase similar to fiducial cross sections. This is the case for e.g. the (reconstructed) Higgs rapidity distribution shown in Fig. 3; we observe there that with or without gg → ZH contributions, the rapidity distribution is modified by an almost constant K-factor. Finally, we give an example of a kinematic distribution that exhibits large NNLO QCD corrections that are not related to the gg → ZH process. In Fig. 4 In this paper, we will follow Ref. [69] and only consider operators that modify interactions of the Higgs boson to electroweak gauge bosons. The part of the SMEFT Lagrangian that is relevant to us reads L anom = − 1 4Λ g (1) hzz Z µν Z µν h − 1 2Λ g (1) hww W µν W † µν h − 1 Λ g (2) hzz Z ν ∂ µ Z µν h − 1 Λ g (2) hww W ν ∂ µ W † µν h + h.c. − 1 4Λg hzz Z µνZ µν h − 1 2Λg hww W µνW † µν h .(5) The energy scale associated with this Lagrangian is denoted by Λ; in what follows we will set Λ to 1 TeV for definiteness. Parametrically, modifications of the Standard Model predictions due to operators in Eq. (5) are controlled by the quantities g (i) hV V v/Λ where v = 246 GeV is the Higgs field vacuum expectation value. In what follows, we will consider values of the couplings that lead to relatively small deviations from Standard Model predictions and discuss to what extent better quality theoretical predictions for the associated production processes pp → V H may help with detecting and analyzing such scenarios. It is straightforward to incorporate effects of the anomalous couplings into theoretical predictions for cross sections and kinematic distributions. To this end, we note that the above Lagrangian leads to the following HV (q 1 )V (q 2 ) interaction vertex − g µν c 1 + c 2 (q µ 1 q ν 2 + q ν 1 q µ 2 ) + c 3 µναβ q 1,α q 2,β + c 4 (q µ 1 q ν 1 + q µ 2 q ν 2 ) .(6) In Eq. (6) the coefficient c 1 is a first-degree polynomial in q 2 1 , q 2 2 and q 1 q 2 , whereas coefficients c 2,3,4 are independent of the external momenta. Also, the coefficients c 1,..,4 are functions of the various couplings g hV V in the Lagrangian Eq. (5); the exact relations between the coefficients c 1..4 and the various g hV V couplings can be found in Fig. 1 of Ref. [69] and we do not reproduce them here. As we already mentioned in Section II, analytic formulas for the integrated subtraction terms required for the NNLO QCD description of color-singlet production [71] are generic. For this reason all that we need to do in order to incorporate effects of the anomalous couplings into a NNLO QCD description of the pp → V H process is to provide scattering amplitudes for hard processes qq → Hl 1l2 , qq → Hl 1l2 + g, qq → Hl 1l2 + gg etc. that include the anomalous couplings. Once these amplitudes are available -and it is quite straightforward to calculate them -they can be immediately included in a numerical code for computing cross sections and kinematic distributions for the associated production processes through NNLO in perturbative QCD. We have seen in the previous section that existing predictions for the gg → ZH partonic process are insufficiently precise, leaving up to six percent uncertainty in predictions for fiducial cross sections. For this reason, it is desirable to reduce the impact of this contribution. Since the relevance of the gg → ZH channel grows at high invariant masses of the ZH system, putting an upper kinematic cut on M ZH is useful to increase the quality of theoretical predictions without reducing fiducial cross sections. At the same time, an upper kinematic cut on M ZH has the additional benefit of removing contributions of high-energy tails of distributions where an EFT expansion may become unreliable. We note that since it is not possible to impose such a cut on the W + H system, we also restrict the transverse momentum of the vector boson following the experimental analysis [80]. Hence, we will study fiducial cross sections and kinematic distributions of the associated production processes pp → V H including the anomalous couplings by imposing the kinematic cuts of Eq. (4) as well as the following constraints: ZH : 75 GeV < p t,Z < 250 GeV, M e + e − bb < 320 GeV, W + H : 150 GeV < p t,W < 250 GeV. The notation M e + e − bb emphasizes the fact that the invariant mass of the ZH system is calculated using the four-momenta of the two charged leptons and the two b jets used for the Higgs boson reconstruction. A. ZH process We begin with the discussion of the pp → ZH process and consider the following scenarios: Setup 1: g (1) hzz = + 2.80 , g(2) hzz = − 0.60 ,g hzz = + 0.00 , hzz = − 1.00 ,g hzz = + 2.00 . Although these choices look quite random, the corresponding scenarios were chosen to provide almost identical cross sections both at leading and, especially, at next-to-leading order in QCD, subject to the kinematic constraints shown in Eq. (4) and Eq. (7). This can be clearly seen from the results for fiducial cross sections summarized in Table III. We observe that NNLO QCD corrections in these cases are not insignificant; they lead to important shifts compared to next-to-leading order predictions. Also, the very strong degeneracy of the four scenarios at NLO is lifted at NNLO. However, the differences between predictions for the different scenarios remain within a few percent of each other, making NNLO QCD precision for these cases essential. It also follows from Table III fiducial cross sections of processes with anomalous couplings may help with analyzing cases where differences between various scenarios are marginal. Another way to lift the degeneracies of different scenarios is to explore kinematic distributions. In many cases kinematic distributions offer more opportunities to detect the anomalous couplings since their effects can be quite profound even if they are small in fiducial cross sections. However, for the four scenarios that we considered, the situation is slightly more subtle, as we will illustrate now. In Fig. 5 we show distributions of the Higgs boson transverse momentum, the transverse momentum of the hardest b jet, the angular separation between the hardest b jet and the hardest lepton ∆R b , and the transverse momentum distribution of the hardest lepton. The quantity ∆R b is defined as ∆R b = (y b − y ) 2 + (ϕ b − ϕ ) 2 ,(9) where y b (y ) and ϕ b (ϕ ) are the rapidity and the azimuthal angle of the hardest b jet (the hardest lepton), respectively. We consider four different scenarios of the anomalous couplings and we choose them in a way that makes the differences between fiducial cross sections marginal. The four scenarios are: Setup 1: g (1) hww = − 1.20 , g(2) hww = − 0.25 ,g hww = + 0.00 , Setup 2: g (1) hww = + 1.00 , g hww = + 0.00 ,g hww = + 0.80 , Setup 3: g (1) hww = + 0.00 , g(2) hww = − 0.10 ,g hww = − 1.10 , Setup 4: g (1) hww = + 0.70 , g(2) hww = − 0.05 ,g hww = − 1.05 . The fiducial cross sections at various orders of perturbation theory are reported in Table IV. We observe that the NLO QCD predictions for cross sections for the four scenarios agree to within a few percent. At variance with ZH case, however, adding NNLO QCD corrections does not change the situation in a significant way except that the uncertainty on the theoretical predictions is reduced compared to the NLO QCD case. However, we again observe that the NNLO QCD corrections are not constant across the four scenarios. In Fig. 6 we show kinematic distributions for the pp → W + H process for the four scenarios Before concluding we would like to illustrate the potential impact of the calculations described in this paper on bounds on the anomalous couplings that can be obtained from measurements of fiducial cross sections of pp → V H processes. We consider a hypothetical measurement of a fiducial W + H cross section and find the allowed values for various combinations of the anomalous couplings. We use the same setup as described earlier in this section to define the fiducial W + H cross section. We assume that it has been measured and the value σ W + H fid,exp = 3.40 (14) fb was obtained. We assigned a four percent uncertainty to the measured cross section; this corresponds to projections for the high-luminosity LHC that can be found in Ref. [81]. We would like to understand how regions of allowed anomalous couplings change when we increase the accuracy of theoretical computations. We note that the fiducial cross section σ W H fid = f (g (1) hww , g(2) hww ,g hww ) is a polynomial in the couplings. For this reason, it is enough to sample it for ten different points to determine the full function. We then use these results Our results are presented in Fig. 7 where we show two-dimensional projections of the g (i) hww parameter space. Shaded areas mark couplings that are compatible with the results of the measurement at the 68% confidence level. We note that NLO QCD corrections change the LO predictions significantly; for this reason, we do not display the latter. Changes are smaller when moving from NLO to NNLO predictions; nevertheless, we observe some distortion of shapes of the allowed region. This effect is a consequence of the fact that corrections do, in fact, depend on the anomalous couplings, a feature that we have already discussed when talking about Table IV. The thickness of the bands representing the allowed regions is only marginally reduced when NNLO predictions are used instead of NLO predictions. This is a consequence of the fact that the experimental uncertainty is fixed at 4 percent which is comparable to the scale uncertainty of the theoretical prediction at NLO and is larger than that of the NNLO prediction. In order to highlight potential benefits of using NNLO theory predictions, we display similar exclusion limits in Fig. 8 but assume that the experimental uncertainty is significantly reduced. For the sake of argument, we take it to be zero. We can now clearly see the thick- ness of the bands decreasing as we move from NLO to NNLO, as a result of the decreased theoretical error. Of course, it is unrealistic to assume no experimental uncertainty, but using this assumption does allow us to highlight the benefits of NNLO-accurate theoretical predictions. We note, in this regard, that projections in Ref. [81] are only estimates and that it is quite possible that the actual results will outperform these projections. If this happens, we anticipate that fully-differential NNLO theoretical predictions will become not only preferable but perhaps even necessary for studies of the anomalous couplings in the pp → V H process. IV. CONCLUSIONS In this paper, we presented computations of NNLO QCD corrections to Higgs boson production in association with a W or Z boson. We included NNLO QCD corrections to H → bb decays, retaining full b-quark mass dependence. This allowed us to present our results in a setup which is close to the actual experimental analyses. In addition to NNLO QCD corrections and the effects of the b-quark masses, our computation also includes anomalous couplings in the V V H interaction vertex. We have shown that QCD corrections to fiducial cross sections depend non-trivially on the anomalous couplings since they change the relative importance of various kinematic regions that contribute to the fiducial cross sections. We have argued that the availability of NNLO QCD predictions may allow one to search for the anomalous couplings in kinematic regions where the EFT framework based on the momentum expansion is more trustworthy than in the high-energy tails of distributions. We have also shown how the NNLO QCD theory predictions may be used to improve exclusion limits for the anomalous couplings; this becomes especially relevant if experimental uncertainties on fiducial cross sections of pp → V H production measured at the high-luminosity LHC reach the few percent level. The computation reported in this paper describes the most advanced and realistic way to simulate the associated production process pp → V H at the LHC, but further improvements are possible. On the SM side, it is definitely important to refine the calculation of the gg → ZH subprocess and to update the contributions to V H production that depend on top loops since many of them are still only known as expansions in the inverse mass of the top quark. On the EFT side, one can gradually include contributions of other dimension-six operators, gradually moving towards a full EFT analysis. Although such extensions of the current computation are not trivial, they are clearly possible given recent developments in both methods for loop computations and subtraction technology. We look forward to providing such refined predictions for V H associated production in the future. and we do not perform an expansion of Br(H → bb) in a series of α s , treating it as an input parameter. For numerical computations we take Br(H → bb) = 0.5824 as recommended by the Higgs Cross Section Working Group [72]. We set the Higgs boson mass to M H = 125 GeV, the vector boson masses to M W = 80.399 GeV and M Z = 91.1876 GeV, respectively, the on-shell b-quark mass to m b = 4.78 GeV, and the top quark mass to m t = 173.2 GeV. We use the Fermi constant G F = 1.16639×10 −5 GeV −2 and the weak mixing angle sin 2 θ W = 0.2226459. The widths of vector bosons are taken to be Γ W = 2.1054 GeV and Γ Z = 2.4952 GeV. V + p H ) 2 for the central value and the uncertainties are calculated by varying the scale µ by a factor of two in both directions. See text for details. We note that the b-quark Yukawa coupling that enters the H → bb decay rate is computed using the MS b-quark mass calculated at µ = M H , m b (µ = M H ) = 2.81 GeV [73, 74]. However, since the physical cross sections in Eq. (3) are proportional to the ratio dΓ H→bb /Γ H→bb , the dependence on the Yukawa coupling to a large extent cancels out in the results that are presented below. (ν e e + )(bb) and pp → ZH → (e − e + )(bb) at the 13 TeV LHC at various orders of QCD perturbation theory calculated with massive b quarks. We set the factorization and renormalization scales equal to each other, µ r = µ f = µ. We use µ = 1 2 (p V + p H ) 2 for the central value and the uncertainties are calculated by varying the scale µ by a factor of two in both directions. See text for details. it increases the fiducial cross section by 15 percent if no cut on the Z transverse momentum is applied, and by about 25 percent if the Z boson is required to have a transverse momentum in excess of 150 GeV. 1 Figure 2 . 2The transverse momentum distribution of the reconstructed Higgs boson at NLO (blue) and NNLO (red) at the 13 TeV LHC with the fiducial cuts discussed in the text. We present the NNLO results without (left) and with (right) the gg → ZH contribution. We display results forthe central scale µ = 1 2 (p V + p H ) 2 .The lower panes show the ratio of the NNLO results to the NLO ones. See text for details.what is seen in the ZH invariant mass distribution. Indeed, in the case of quark-initiated ZH production, the NNLO QCD corrections are negative and decrease NLO distributions by no more than five percent, whereas if gg → ZH is included in the theoretical prediction, there are large modifications of the p t,H distribution starting at p t,H ∼ 150 GeV. Figure 3 . 3we display the Higgs boson invariant mass distribution where the Higgs boson is reconstructed from two b jets whose invariant mass is the closest to Higgs boson mass. The gg → ZH process contributes only to the M H(bb) = M H bin since it has at most two b jets in the final state and the invariant mass of these b jets is equal to M H . However, similar to the case of the W + H The rapidity distribution of the reconstructed Higgs boson at NLO (blue) and NNLO (red) at the 13 TeV LHC with the fiducial cuts discussed in the text. We present the NNLO results without (left) and with (right) the gg → ZH contribution. We display results for the central scale µ = 1 2 (p V + p H ) 2 . The lower panes show the ratio of the NNLO results to the NLO ones. See text for details.final state discussed in Ref.[7], we observe very large NNLO QCD effects in the M H(bb) distributions away from the peak at the true mass of the Higgs boson due to initial-and final-state radiation.III. ASSOCIATED V H PRODUCTION WITH ANOMALOUS COUPLINGSTheoretical predictions for the associated production processes can be modified by both higher-order QCD effects and by contributions of physics beyond the Standard Model (BSM).Under certain circumstances, the latter can be described by an effective Lagrangian that parameterizes possible deviations from the Standard Model in terms of operators with increasing mass dimensions. A convenient description is provided by the so-called Standard Model Effective Field Theory (SMEFT), see Ref.[68] for a review. Figure 4 . 4The invariant mass distribution of the reconstructed Higgs boson at NLO (blue) and NNLO (red) at the 13 TeV LHC. We present the NNLO results including the gg → ZH contribution.The left plot includes the standard fiducial cuts described in the text, the right plot includes the additional p t,V > 150 GeV cut. We display results for the central scale µ = 1 2 (p V + p H ) 2 . The lower panes show the ratio of the NNLO results to the NLO ones. See text for details. It is to be expected that generic choices of anomalous couplings would lead to significant changes in cross sections and kinematic distributions. For such cases an extraction of the values of the anomalous couplings from data, rather than the detection of anomalies, would benefit from precise predictions for observables that include the anomalous couplings. On the other hand, there are also cases where, even with anomalous couplings, changes in cross sections are marginal. In this situation, studies of kinematic distributions and precise theoretical predictions may be needed to both detect the presence of anomalies and distinguish between different scenarios. In what follows we present a few examples. = − 1.00 ,g hzz = − 3.30 , Figure 5 . 5It follows fromFig. 5that there are kinematic regions where the differences between the four scenarios are more pronounced than in fiducial cross sections. For example, if we look at the p t,H(bb) distribution, there are noticeable differences at low transverse momenta, whereas in the peak region all four distributions are similar. The same applies to the other three distributions. For example, the ∆R bl distributions peak at ∆R bl ∼ 3; in that region the four Kinematic distributions in the process pp → Z(e + e − )H(bb) at the 13 TeV LHC for various SMEFT scenarios. In the lower panes, ratios of SMEFT to SM distributions are shown. We set the factorization and the renormalization scales in the production process to half the invariant mass of the ZH system. See text for details. scenarios provide very similar results. The differences become noticeable at ∆R bl ∼ 1 but the number of events for such values of ∆R bl is reduced by an order of magnitude. Given the fact that we deal here with O(1 fb) cross sections, losing an order of magnitude in the number of events is not optimal. However, the availability of highly accurate NNLO QCD predictions in peak regions of kinematic distributions and identifiable differences between various scenarios in distribution tails should allow one to optimize analysis strategies and benefit from measurements across accessible kinematic regions. IV. Fiducial cross sections for pp → W + H → (ν e e + )(bb) at the 13 TeV LHC at various orders of QCD perturbation theory calculated with massive b quarks. We show the results for various scenarios including anomalous couplings. We set the factorization and renormalization scales equal to each other, µ r = µ f = µ. We use µ = 1 2 (p V + p H ) 2 for the central value and the uncertainties are calculated by varying the scale µ by a factor of two in both directions. See main text for details.B. W + H processWe repeat the analysis of the previous subsection for W + H production. We focus exclusively on the fiducial region defined in Eq. (4) with additional restrictions on the W -boson transverse momentum, shown in Eq.(8). Figure 6 . 6Kinematic distributions in the process pp → W + (e + ν)H(bb) at the 13 TeV LHC for various SMEFT scenarios. In lower panes ratios of SMEFT to SM distributions are shown. We set the factorization and the renormalization scales in the production process to half the invariant mass of the W H system. See text for details. considered above. Overall, the situation is similar to what has been already discussed in case of pp → ZH: in peak regions of all distributions the different scenarios provide very similar predictions; away from peak regions clear differences are seen in some of them. These differences, as well as reduced theoretical uncertainties in peak regions, should eventually enable improved studies of the anomalous couplings in W + H production. to check which combinations of anomalous couplings are compatible with the result of the hypothetical measurement. The uncertainty in the experimental cross section is fixed to four percent and the uncertainties in the theoretical predictions is determined by varying the scale by a factor of two around the central scale µ = M W H /2. We note that the factorization and the renormalization scales are chosen to be equal. Figure 7 . 7Examples of contours (68% confidence level) of allowed combinations of anomalous couplings based on a hypothetical measurement of the fiducial cross section of W + H production at the 13 TeV LHC. Color coding describes NLO (lighter blue) and NNLO (darker blue) calculations. The SM result is shown as an orange cross. Upper row: full contours of allowed couplings; lower row: contours close to the SM configuration, i.e. for small anomalous couplings. A 4% experimental uncertainty was assumed. See text for details. Figure 8 . 8As for Fig. 7 but with the experimental uncertainty removed. See text for details. are applicable to all color-singlet final states. For this reason, a transition from the W H final state to the ZH final state only requires us to change the relevant matrix elements and adjust flavors of colliding partons. However, an important difference between ZH and W H final states is the contribution of the gg → ZH partonic process which only exists in the former case. Thanks to a large gluon flux, this contribution is significant; accounting for it in the theoretical prediction for the ZH final state is important, especially for large values of ZH invariant masses. In our calculation, we have included the exact O(α 2 s ) contributions to the gg → ZH channel but we have neglected higher-order terms which so far are not available. Apart from corrections to the gg channel, Table II. Fiducial cross sections in the boosted region (p t,V > 150 GeV) for pp → W + H →The fiducial cross sections are calculated with the four-flavor parton distribution function (PDF) set NNPDF31_nnlo_as_0118_nf_4. We emphasize that we employ NNLO PDFs to compute LO, NLO and NNLO cross sections in what follows. Moreover, we use α s (M Z ) = 0.118 and perform the running of the strong coupling at three loops with five active flavors. For all numerical results presented in this paper, the central value of the renormalization Order σ W + H fid [fb] σ ZH fid [fb] σ ZH+ggZH fid [fb] LO 3.89 0.97 − NLO 4.79 +0.13 −0.10 1.20 +0.03 −0.03 − NNLO 4.79 +0.02 −0.06 1.22 +0.03 −0.03 1.52 +0.11 −0.09 For example, if the top Yukawa coupling had the opposite sign[78,79], the NNLO cross section inTable Iwould increase to about 10 fb.RatioFigure 1. The invariant mass M ZH distribution at NLO (blue) and NNLO (red) at the 13 TeV LHC with the fiducial cuts discussed in the text. We present the NNLO results without (left) and with (right) the gg → ZH contribution. We display results for the central scale µ = 110 −4 10 −3 10 −2 dσ/dM ZH [ fb/GeV ] without ggZH NLO NNLO 200 300 400 500 600 700 800 900 1000 M ZH [ GeV ] 1.0 1.2 1.4 Ratio 10 −4 10 −3 10 −2 dσ/dM ZH [ fb/GeV ] with ggZH NLO NNLO 200 300 400 500 600 700 800 900 1000 M ZH [ GeV ] 1.0 1.2 1.4 2 that for such situations it is important to include higher-order QCD corrections to the description of processes with anomalous couplings since simply re-weighting the leading order predictions with Standard Model K-factors may be insufficient. At any rate, having high-precision predictions forTable III. Fiducial cross sections for pp → ZH → (e − e + )(bb) at the 13 TeV LHC at various orders of QCD perturbation theory calculated with massive b quarks. We show the results for various are calculated by varying the scale µ by a factor of two in both directions. See text for details.σ ZH fid [fb] SM Setup 1 Setup 2 Setup 3 Setup 4 LO 0.894 +0.032 −0.041 0.782 +0.028 −0.034 0.854 +0.031 −0.038 0.786 +0.027 −0.034 0.780 +0.027 −0.034 NLO 1.289 +0.025 −0.017 1.266 +0.012 −0.007 1.273 +0.018 −0.010 1.276 +0.009 −0.004 1.269 +0.008 −0.003 NNLO 1.356 +0.009 −0.011 1.423 +0.003 −0.006 1.379 +0.014 −0.004 1.454 +0.003 −0.006 1.445 +0.004 −0.003 NNLO(+ggZH) 1.419 +0.024 −0.023 1.476 +0.015 −0.015 1.443 +0.028 −0.015 1.499 +0.014 −0.015 1.490 +0.014 −0.011 scenarios including anomalous couplings. We set the factorization and renormalization scales equal to each other, µ r = µ f = µ. We use µ = 1 2 (p V + p H ) 2 for the central value and the uncertainties Table . Tev, 10.1007/JHEP08(2016)045arXiv:1606.02266JHEP. 0845hep-exTeV," JHEP 08 (2016) 045, arXiv:1606.02266 [hep-ex]. Jet substructure as a new Higgs search channel at the LHC. J M Butterworth, A R Davison, M Rubin, G P Salam, 10.1103/PhysRevLett.100.242001arXiv:0802.2470Phys. Rev. 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[ "Risk-sensitive Markov Control Processes with Strictly Convex Risk Maps ", "Risk-sensitive Markov Control Processes with Strictly Convex Risk Maps ", "Risk-sensitive Markov Control Processes with Strictly Convex Risk Maps ", "Risk-sensitive Markov Control Processes with Strictly Convex Risk Maps " ]	[ "Yun Shen ", "Wilhelm Stannat ", "Klaus Obermayer ", "Yun Shen ", "Wilhelm Stannat ", "Klaus Obermayer " ]	[]	[]	We fully develop the Lyapunov approach to optimal control problems of Markov control processes on general Borel spaces equipped with risk maps, especially, with strictly convex risk maps including the entropic map. To ensure the existence and uniqueness of a solution to the associated nonlinear Poisson equation with possibly unbounded costs, we propose a new set of conditions: 1) Lyapunov-type conditions on both risk maps and cost functions that control the growth speed of iterations, and 2) Doeblin's conditions that generalize the known conditions for Markov chains. In the special case of the entropic map, we show that the above conditions can be replaced by the existence of a Lyapunov function, a local Doeblin's condition for the underlying Markov chain, and a growth condition for cost functions.	null	[ "https://arxiv.org/pdf/1403.3321v4.pdf" ]	119,613,482	1403.3321	39d66b9f9c49cce31b2192f9fd9de9f0e3a30a60	Risk-sensitive Markov Control Processes with Strictly Convex Risk Maps * 19 Mar 2014 Yun Shen Wilhelm Stannat Klaus Obermayer Risk-sensitive Markov Control Processes with Strictly Convex Risk Maps * 19 Mar 2014arXiv:1403.3321v2 [math.OC]Markov control processesPoisson equationrisk-sensitive controlrisk measuresstability of nonlinear operatorsDoeblin's conditionLyapunov Stability AMS subject classifications 60J0593E2093C5547H0791B06 We fully develop the Lyapunov approach to optimal control problems of Markov control processes on general Borel spaces equipped with risk maps, especially, with strictly convex risk maps including the entropic map. To ensure the existence and uniqueness of a solution to the associated nonlinear Poisson equation with possibly unbounded costs, we propose a new set of conditions: 1) Lyapunov-type conditions on both risk maps and cost functions that control the growth speed of iterations, and 2) Doeblin's conditions that generalize the known conditions for Markov chains. In the special case of the entropic map, we show that the above conditions can be replaced by the existence of a Lyapunov function, a local Doeblin's condition for the underlying Markov chain, and a growth condition for cost functions. Introduction In recent years, many authors [29,3,33] have developed a general framework of risk-sensitive sequential decision-making problems on Borel spaces by applying coherent/convex risk measures [1,14], which were originally employed in mathematical finance, to the classical discrete-time risk-neutral Markov control processes (MCPs, see, e.g., [20,21], and [28] under the name Markov decision processes). Within the framework, two infinite-horizon risk-sensitive criteria, discounted total risk and average risk, are optimized under various settings. Among them, we applied in our previous work [33] weighted norm spaces to incorporate possibly unbounded costs and stated Lyapunov-type stability conditions that generalized known conditions for Markov chains to ensure the existence of solutions to the optimality equation for the average-risk criterion. More specifically, we introduced the concept of upper modules [33, Subsection 3.1] for risk maps and assumed [33, Assumption 3.1] that 1) a Lyapunov function exists to the upper module, and 2) the Doeblin's condition holds over a compact subspace. However, given a strictly convex (i.e., convex but not coherent) risk map, the upper module may be infinite for unbounded cost functions. Hence, a (realvalued) Lyapunov function need not exist and the theory developed in our previous work remains valid only if 1) costs are bounded and 2) the Doeblin's condition holds over the whole space. This excludes the applicability of our framework to Markov models with non-compact state spaces, where Doeblin's condition does not hold over the whole space (see, e.g., [26,Chapter 5]). As an example, consider the entropic map R x (c) := 1 λ log X e λc(y) P x (dy) , x ∈ X, λ > 0, which is a widely applied risk map in the literature of risk sensitive MCPs (see, e.g., [22,7,2,4,11,8,13,18,5]). Here, c : X → R is a cost function, P denotes a transition kernel of a time-homogeneous Markov chain on (X, B(X)) and the positive constant λ is used to control the risk sensitivity. Then, the upper module of R,R, is defined asR x (c) := sup γ∈R,γ =0 1 γ R x (γc) = P x -esssup c. Its value becomes infinite if c is unbounded from above and the probability measure P x (·) has full support on X (e.g., the autoregressive model of order one with Gaussian noise [ where ρ ∈ R and h : X → R are unknown, plays a central role in solving the optimal control problem associated to risk-sensitive MCPs with the average criterion. In our previous work [33], the Poisson equation is solved by an iterative technique, and the sequence of iterations {T n , n = 1, 2, . . .} is required to be uniformly bounded under a weighted norm determined by a Lyapunov function w satisfyingR x (w) ≤ γw(x) + K, ∀x ∈ X, with some constants γ ∈ (0, 1) and K > 0. However, as discussed above, a real-valued w need not exist for the entropic map. Hence, the theory developed in our previous work can only allow for bounded costs with compact state spaces, when applying strictly convex risk maps like the entropic map. However, in the last four decades, the entropic map is the most widely applied risk map for risk-sensitive control in the framework of MCPs (see, e.g., [22,7,2,4,11,8,13,18,5]). It is, therefore, of great importance to cover this special type of risk maps in our framework of risk-sensitive MCPs on general Borel spaces with possibly unbounded costs. The purpose of this paper is to solve the above mentioned problems within the same framework by introducing 1) constraints on the cost functions to control the growth of iterations (Assumption 3.1(i)), and 2) additional minorization properties on small sets (Assumption 3.1(ii)) that generalize the original Doeblin's condition. Under these conditions, we show the existence of a bounded forward invariant subset that covers the whole iterations. Restricted to the bounded subset, we assume the existence of the Lyapunov function (Assumption 3.7) for the weaker type of upper module, which is called upper envelope in this paper, to ensure the existence of a unique solution to the optimality equation for the average-risk criterion. As a special case, we show in Section 4 that, when applying the entropic map, the above conditions are satisfied, if 1) a Lyapunov function exists for the entropic map, 2) the local Doeblin's condition holds for the underlying Markov chain, and 3) a growth condition for cost functions. Most of the existing literature on risk-sensitive MCPs, especially that applies the entropic map, considers finite or countable state spaces (see, e.g., [2,4,8,13,18,5]), or bounded cost functions (see, e.g., [3]). Comparing with the few literature [10,11] (for detailed comparisons, see Remark 4.16) of the same general settings, i.e., Borel spaces and unbounded cost functions, we provide in this paper a more general framework which can be applied to all types of risk maps, and more importantly, with a conceptually simpler proof, whereas the methods developed in [10,11] can be only applied to the entropic map. Moreover, the conditions we stated in Section 4 for the entropic map are easier to verify than the conditions stated in [10,11]. The paper is organized as follows. In Section 2, we briefly review the definitions and basic properties of the weighted norm space, risk measures and risk maps. In Section 3, we develop a general theory of nonlinear Poisson equation for risk maps. As a special case, we show in Section 4 that under proper assumptions, the entropic map fits the theoretical framework developed in the previous section. In Section 5, the theory is applied to solve the optimal control problems within the framework of risk-sensitive MCPs, and an example with the entropic map applied to discretized ergodic diffusions is presented in Subsection 5.3. Preliminaries Let X be a Borel space, which is a Borel subset of a complete separable metric space, and its Borel σ-algebra is denoted by B(X). Weighted norm Let w : X → [1, ∞) be a given real-valued B(X)-measurable function. Consider the w-norm u w := sup x∈X \|u(x)\| w(x) . Let B w be the space of real-valued B(X) measurable functions with bounded w-norm. It is obvious that B ⊂ B w , where B denotes the space of bounded B(X)-measurable functions. Let µ be a signed measure on B(X). Define µ w := sup u w ≤1 \| X udµ\| = X wd\|µ\| ≥ µ T V , where · T V denotes the total variation norm of probability measures. The following w-seminorm is used throughout this paper: v s,w := sup x =y \|v(x) − v(y)\| d w (x, y) , where d w (x, y) := 0 x = y w(x) + w(y) x = y . This seminorm is originally used by Hairer and Mattingly (2011) [17] to study the ergodicity of Markov chains. In particular, when restricting to the space B, i.e., setting w ≡ 1, the seminorm is called span-norm in [19] or Hilbert seminorm in [16]. In the following, we restate the Lemma 2.1 in [17]. Lemma 2.1 (see Lemma 2.1, [17]). v s,w = min c∈R v + c w , ∀v ∈ B w . Risk maps without control The partial ordering ≤ between elements in B w is defined as v ≤ u if v(x) ≤ u(x) ∀x ∈ X. A real number u ∈ R can be viewed as a constant-valued function which belongs also to B w . We now define risk measures on B w . A mapping ν : B w → R is said to be a risk measure (cf. [1,14] ) if (I) (Monotonicity) ν(v) ≤ ν(u), whenever v ≤ u ∈ B w ; (II) (Translation invariance) ν(v + u) = ν(v) + u, for any u ∈ R; (III) (Centralization) ν(0) = 0. Risk measures can be categorized as follows: a risk measure ν is said to be convex, if for all α ∈ [0, 1], v, u ∈ B w , ν(αv + (1 − α)u) ≤ αν(v) + (1 − α)ν(u); concave, if ν(·) := −ν(−·) is a convex risk measure; homogeneous, if for all λ ∈ R + and v ∈ B w , ν(λv) = λν(v); coherent, if ν is convex and homogeneous; and strictly convex, if it is convex but not homogeneous. Proposition 2.2. A convex risk measure ν satisfies \|ν(v)\| ≤ ν(\|v\|), ∀v ∈ B w . Proof. By monotonicity, ν(v) ≤ ν(\|v\|). Next we show −ν(v) ≤ ν(−v) ≤ ν(\|v\|). Indeed, due to the convexity, we obtain that 1 2 (ν(v) + ν(−v)) ≥ ν( 1 2 (v − v)) = 0. Let {X t , t = 0, 1, 2, . . .} be a time-homogeneous Markov chain on (X, B(X)) with P as its transition kernel. We generalize the idea of risk measures to risk maps equipped with a Markov chain. A mapping R(x, v) : X × B w → R is said to be a risk map (cf. [33]) on the Markov chain {X t } with transition kernel P , if (i) for each x ∈ X, R x (·) := R(x, ·) is a risk measure; and (ii) R(·, v) ∈ B w for each v ∈ B w . Analogously, a risk map R is said to be convex (resp. concave, homogenous, coherent, strictly convex ) if R x is convex (resp. concave, homogenous, coherent, strictly convex ), for all x ∈ X. Remark 2.3. Comparing with the standard literature [1,14,30], we define risk measures/maps on the weighted space B w , rather than the space of bounded random variables, L ∞ , since the weighted space is more suitable for investigating the stability properties of the underlying Markov chain (see, e.g., [26,17,33]) and is also more general than L ∞ . We will specify later in Section 3 and 4 the choice of w, depending on the form of risk maps and the properties of the underlying Markov chain as well. Examples The standard conditional expectation R x (v) = E P x [v] := vdP x with aR x (v) = 1 λ log e λv dP x , λ > 0, where the parameter λ controls risk-sensitivity. It is easy to check that R is strictly convex. This risk map is intensively studied in the field of optimal control (see, e.g., [22,7,2,4,11,8,13,18,5]). Mean-semideviation trade-off [27,31] consider the trade-off between the one-step conditional mean and semideviation, R x (v) := E P x [v] + λ E P x (v − E P x [v]) r + 1/r(1) where r ≥ 1 and λ ∈ [0, 1] denotes the risk-sensitivity parameter. Moreover, this map is coherent (for the proof of convexity, see, e.g., [31]). Nonlinear Poisson Equations Let c ∈ B w be a fixed cost function. In this section, we shall prove under some sufficient conditions the existence of a unique solution to the following (nonlinear) Poisson equation c + R(h) = ρ + h,(2) and the existence of an invariant risk measure, ν, satisfying ν(c + R(v)) = ν(v) + ρ, ∀v ∈ B w : v s,w ≤ C.(3) with properly chosen weight function w and constant C > 0. As in the theory of MCPs, both Poisson equation and invariant risk measures play important roles in studying the stability properties of risk maps and the optimization of the average risk (see Section 5). Bounded forward invariant subset Define an operator T : B w → B w , T (v) := c + R(v). and its nth iteration, T n (v) := T T n−1 (v) , n = 2, 3, . . .. In this subsection, we state a set of sufficient conditions that guarantee the existence of a bounded (under w-seminorm) forward invariant subset covering the whole sequence of {T n (v)}. More specifically, we consider subspaces of the following form B (C) w := {v ∈ B w \| v s,w ≤ C} .(4) Here, we choose the w-seminorm, since we shall prove the contraction property under the w-seminorm as in the standard literature of Markov chain and MCPs (see, e.g., [17]). Assumption 3.1. There exist a B(X)-measurable function w 0 : X → [0, ∞) and constants γ 0 ∈ (0, 1), K 0 > 0 andK 0 > K 0 such that (i) (c(x) + R x (w 0 )) ∨ (−c(x) − R x (−w 0 )) ≤ γ 0 w 0 (x) + K 0 , ∀x ∈ X,(5) and (ii) for all x, y ∈ B := {x ∈ X\|w 0 (x) ≤ R 0 := 2K0 1−γ0 }, the following inequality R x (v) − R y (v) ≤ 2(K 0 − K 0 ) + R x (w 0 ) − R y (−w 0 )(6) holds for all v ∈ B 1+w0 satisfying \|v\| ≤ w 0 +K 0 . Remark 3.2. (a) If R is a convex risk map, a sufficient condition to guarantee the assumption (i) is \|c\| + R(w 0 ) ≤ γ 0 w 0 + K 0 ,(7) since by Proposition 2.2, −R(−w 0 ) ≤ R(w 0 ). (b) The assumption (i) can be replaced by two conditions for the cost function c and risk map R separately, 1) w 0 is a Lyapunov function satisfying R(w 0 ) ∨ (−R(−w 0 )) ≤γ 0 w 0 +K 0 , and 2) \|c\| ≤γ 0 w 0 + C 0 with some constantsγ 0 ∈ (0, 1 −γ 0 ) and C 0 > 0. Hence, comparing with our previous work [33], the assumption on the cost function c is more restrictive in the present work. (c) The assumption (ii) is more general than the Doeblin's condition assumed in [33, Assumption 3.1(ii)]. Indeed, given a convex risk map R, the Doeblin's condition implies that for allK 0 > 0, there exists a constant α ∈ (0, 1) such that R (6). x (v) − R y (v) ≤ 2(1 − α)K 0 + R x (w 0 ) − R y (−w 0 ), which implies (d) Applying the entropic map, we will show in Section 4 some sufficient conditions ensuring (ii) based on properties of the underlying Markov chain. Theorem 3.3. Suppose Assumption 3.1 holds. Then c + R(v) s,1+K −1 0 w0 ≤K 0 , whenever v s,1+K −1 0 w0 ≤K 0 . Proof. Let β 0 :=K −1 0 . Note that adding a constant to v will not change the required inequality. Due to Lemma 2.1, we assume that \|v\| ≤ β −1 0 + w 0 . By the definition of w-seminorm, the task is to prove \|c(x) + R x (v) − c(y) − R y (v)\| ≤ 2β −1 0 + w 0 (x) + w 0 (y), ∀x = y ∈ X. Note that since switching x and y will not change the right-hand side of the inequality, it is sufficient to show that c(x) + R x (v) − c(y) − R y (v) ≤ 2β −1 0 + w 0 (x) + w 0 (y), ∀x = y ∈ X.(8) We consider the following two cases. Case I: (8) holds for this case. w 0 (x) + w 0 (y) ≥ R 0 . By (5), we have for all β 0 > 0, c(x) + R x (v) ≤ c(x) + β −1 0 + R x (w 0 ) ≤β −1 0 + γ 0 w 0 (x) + K 0 , and −c(y) − R x (v) ≤ −c(y) + β −1 0 − R y (−w 0 ) ≤β −1 0 + γ 0 w 0 (y) + K 0 . By the choice of R 0 , 2β −1 0 + γ 0 (w 0 (x) + w 0 (y)) + 2K 0 ≤ 2β −1 0 + w 0 (x) + w 0 (y) holds. Hence, Case II: w 0 (x) + w 0 (y) ≤ R 0 . Then both x and y are in the subset B. By (6), c(x) + R x (v) − c(y) − R y (v) ≤c(x) − c(y) + 2(K 0 − K 0 ) + R x (w 0 ) − R y (−w 0 ) ≤2(K 0 − K 0 ) + γ 0 w 0 (x) + γ 0 w 0 (y) + 2K 0 ≤2β −1 0 + w 0 (x) + w 0 (y). Combining I and II, we obtain the required inequality. As a special case, if R is a (strictly) convex risk map, by Proposition 2.2 and repeating the above proof, we immediately obtain the following corollary. Corollary 3.4. Suppose R is a convex risk map satisfying that there exist a B(X)measurable function w 0 : X → [0, ∞) and constants γ 0 ∈ (0, 1), K 0 > 0 andK 0 > K 0 such that (i) \|c(x)\| + R(w 0 ) ≤ γ 0 w 0 (x) + K 0 , ∀x ∈ X, and (ii) for all x, y ∈ B := {x ∈ X\|w 0 (x) ≤ R 0 := 2K0 1−γ0 }, the following inequality R x (v) − R y (v) ≤ 2(K 0 − K 0 ) + R x (w 0 ) + R y (w 0 ) holds for all v ∈ B 1+w0 satisfying \|v\| ≤ w 0 +K 0 . Then c + R(v) s,1+K −1 0 w0 ≤K 0 , whenever v s,1+K −1 0 w0 ≤K 0 . Theorem 3.3 shows that (together with Lemma 2.1) starting from some v satisfying \|v\| ≤ w 0 +K 0 , for each n ∈ N, there exists a real number A ∈ R such that \|T n (v) + A\| ≤ w 0 +K 0 . Hence, we may set w = 1 +K −1 0 w 0 and the corresponding forward invariant subset is B (K0) w . Geometric contraction Given a risk map satisfying Assumption 3.1, we can then restrict ourselves to the invariant subset B (C) w (with C =K 0 ) rather than the whole set B w . We introduce in the following the concept of upper envelope, which weakens the sub-and uppermodules employed in our previous work [33]. Definition 3.5. A coherent risk measureν (w,C) is said to be an upper envelope of a risk measure ν given a bound C ∈ R + , if the following inequality holds ν(v) − ν(u) ≤ν (w,C) (v − u), ∀v, u ∈ B (C) w . (9) Analogously, a coherent risk mapR (w,C) is said to be an upper envelope of a risk map R given a bound C ∈ R + , if for all v, u ∈ B (C) w , R x (v) − R x (u) ≤R (w,C) x (v − u), ∀x ∈ X. Remark 3.6. Apparently, if ν (resp. R) is coherent, then ν (resp. R) is an upper envelope of itself for all bounds C > 0, due to its sublinearity (for proof see, e.g., [9]). We now prove the contraction property based on the following assumption, which is similar to Assumption 3.1 in [33]. Assumption 3.7. There exist two real-valued B(X)-measurable functions, w 0 : X → [0, ∞) and w : X → [1, ∞) satisfying that (i) B 1+w0 = B w ; (ii) there exist constants γ ∈ (0, 1), K > 0 and an upper envelopeR (w,C) such that R (w,C) (w 0 ) ≤ γw 0 + K; and (iii) for all v ≥ u ∈ B 1+w0 , there exist a constant α ∈ (0, 1) and a probability measure µ on (X, B(X)) such that R (w,C) x (v) −R (w,C) x (u) ≥ α (v(x) − u(x)) µ(dx), ∀x ∈ B, where B := {x ∈ X\|w 0 (x) ≤ R} for some R > 2K 1−γ . Theorem 3.8. Suppose Assumption 3.7 holds. Then there exist constantsᾱ ∈ (0, 1) and β > 0 such that R(v) − R(u) s,1+βw0 ≤ᾱ v − u s,1+βw0 , ∀v, u ∈ B (C) w . Proof. Define w ′ := 1 + βw 0 for some β ∈ R + , whose value will be specified later. Suppose v − u s,w ′ = A ∈ R + . Due to Lemma 2.1 and the fact that adding any constant to v and u will not change the values of both sides of the required inequality, we may assume that v − u w ′ = A. By the definition of upper envelope, we then have R x (v) − R x (u) ≤R (w,C) x (v − u) ≤R (w,C) x (\|v − u\|), ∀x ∈ X, where the last inequality is due to Proposition 2.2. Switching v and u, we obtain \|R x (v) − R x (u)\| ≤R (w,C) x (\|v − u\|) ≤ v − u w ′R (w,C) x (w ′ ), ∀x ∈ X.(10) Case I: w 0 (x) + w 0 (y) ≥ R and set γ 0 := γ + 2K R < 1 and γ 1 := 2+βRγ0 2+βR for some β > 0. It is easy to verify that γ 1 ∈ (0, 1). Then (10) yields \|R x (v) − R x (u) − R y (v) + R y (u)\| ≤\|R x (v) − R x (u)\| + \|R y (v) − R y (u)\| ≤A(2 + βR C x (w 0 ) + βR C y (w 0 )) ≤ A(2 + βγw 0 (x) + βγw 0 (y) + 2βK) ≤A(2 + βγ 0 w 0 (x) + βγ 0 w 0 (y)) ≤ Aγ 1 (w ′ (x) + w ′ (y)).(11) Case II: w 0 (x) + w 0 (y) ≤ R. Hence both x and y are in the subset B. We define for all x ∈ B,R x (v) := 1 1 − α R x (v) − α 1 − α µ(v), and R (w,C) x (v) := 1 1 − αR (w,C) x (v) − α 1 − α µ(v). It is easy to verify thatR (w,C) x is a valid coherent risk measure on B 1+βw0 = B 1+w0 = B w for all x ∈ B. Indeed, the monotonicity is satisfied due to Assumption 3.7(iii). Hence,R x (v) −R x (u) ≤R (w,C) x (v − u), which indicates thatR (w,C) x is an upper envelope ofR x for all x ∈ B. Hence, \|R x (v) − R x (u) − R y (v) + R y (u)\| =(1 − α)\|R x (v) −R x (u) −R y (v) +R y (u)\| ≤(1 − α)\|R x (v) −R x (u)\| + (1 − α)\|R y (v) −R y (u)\| ≤(1 − α)R (w,C) x (\|v − u\|) + (1 − α)R (w,C) x (\|v − u\|) ≤2A(1 − α) + A(1 − α)β R(w,C) x (w 0 ) +R (w,C) y (w 0 ) . Note that since (1 − α)R (w,C) x (w 0 ) ≤R (w,C) x (w 0 ) holds for all x ∈ B, we obtain \|R x (v) − R x (u) − R y (v) + R y (u)\| ≤2A(1 − α) + Aβ R (w,C) x (w 0 ) +R (w,C) y (w 0 ) (12) ≤2A(1 − α) + Aβ(γw 0 (x) + γw 0 (y) + 2K). We select β := α0 K for some α 0 ∈ (0, α). Setting γ 2 : = (1 − α + α 0 ) ∨ γ ∈ (0, 1) yields for all x = y \|R x (v) − R x (u) − R y (v) + R y (u)\| ≤2A(1 − α + α 0 ) + Aγβ(w 0 (x) + w 0 (y)) ≤ Aγ 2 (w ′ (x) + w ′ (y)).(13) Hence, settingᾱ := γ 1 ∨ γ 2 < 1, (11) and (13) imply for all x = y \|R x (v) − R x (u) − R y (v) + R y (u)\| ≤ v − u s,w ′ᾱ(w ′ (x) + w ′ (y)), the required inequality. Poisson equation We set w ′ = 1 + βw 0 as in Theorem 3.8, w = 1 +K −1 0 w 0 and C =K 0 as in Theorem 3.3. Hence, apparently B w ′ = B w . v, u ∈ B (C) w , lim n→∞ 1 n T n (v) − T n (u) w = 0. Proof. It is sufficient to show that T n (v) − T n (u) w is uniformly bounded, which is equivalent to requiring that T n (v) − T n (u) w ′ is uniformly bounded. Indeed, by Assumption 3.7(ii), setting K ′ := βK + 1 − γ, we have \|T (v) − T (u)\| ≤R (w,C) (\|v − u\|) ≤ v − u w ′ (γw ′ + K ′ ) where the first inequality is due to Proposition 2.2. On the other hand, by Theorem 3.3, T n (v) s,w ≤ C holds for all n ∈ N + . Hence, by induction w.r.t. n, we have for n = 2, 3, . . . \|T n (v) − T n (u)\| ≤R (w,C) (\|T n−1 (v) − T n−1 (u)\|) ≤ v − u wR (w,C) γ n−1 w ′ + K ′ n−2 k=0 γ k ≤ v − u w γ n w ′ + K ′ n−1 k=0 γ k , which implies that T n (v) − T n (u) w ′ ≤ K ′ 1−γ . LetB w ′ = B w ′ / ∼ be the quotient space, which is induced by the equivalence relation ∼ on B w ′ defined by v ∼ u if and only if there exists some constant A ∈ R such that v(x) − u(x) = A ∀x ∈ X,ν(c + R(v)) = ν(v) + ρ, ∀v ∈ B (C) w . Proof. (i) Starting from any v satisfying v s,w ≤ C, {v n := T n (v)} is a Cauchy sequence inB w ′ under the w ′ -seminorm due to Theorem 3.3 and Theorem 3.8. Then by the fixed point argument w.r.t. the w ′ -seminorm applied in the proof of [33, Theorem 3.14], there exists a fixed point h ∈ B w ′ (= B w ) such that T (h) − h s,w ′ = 0. Hence, by Lemma 2.1, there exists a constant ρ ∈ R such that T (h) = c + R(h) = h + ρ. Uniqueness of ρ. Suppose there are two solutions (ρ, h) and (ρ ′ , h ′ ) in R × B w . Then, T n (h) = h + nρ and T n (h ′ ) = h ′ + nρ ′ . By Lemma 3.9, 1 n T n (h) − T n (h ′ ) w = 1 n h − h ′ + n(ρ − ρ ′ ) w → 0 as n → ∞ implies that ρ = ρ ′ . (ii) Let µ 0 ∈ M w ′ be a probability measure and h be one solution in B w ′ . We show first that lim m→∞ sup n≥m \|µ 0 [T n (v) − T n (h)] − µ 0 [T m (v) − T m (h)]\| = 0, ∀v ∈ B (C) w .(14) Indeed, define v n := T n (v) and h n := T n c (h), n = 1, 2, . . . . and we have sup v−h s,w ′ ≤A \|µ 0 [v n − h n ] − µ 0 [v m − h m ]\| ≤ sup v1−h1 s,w ′ ≤ᾱA \|µ 0 T n−1 (v 1 ) − T n−1 (h 1 ) − µ 0 T m−1 (v 1 ) − T m−1 (h 1 ) \| ≤ sup vm−hm s,w ′ ≤ᾱ m A \|µ 0 T n−m (v m ) − T n−m (h m ) − µ 0 [v m − h m ]\|. by which (14) follows immediately. Define D(·) := T (·) − ρ and µ n (·) := µ 0 (D n (·)). (14) is equivalent to lim m→∞ sup n≥m \|µ 0 [D n (v) − D m (v)]\| = lim m→∞ sup n≥m \|µ n (v) − µ m (v)\| = 0, ∀v ∈ B (C) w . Hence, µ n converges to a mapping µ ∞ : B w ′ → R satisfying µ ∞ (D(v)) = µ ∞ (v), ∀v ∈ B (C) w . On the other hand, for each n, µ n satisfies the axioms of risk measures except the axiom of centralization. Hence, µ ∞ preserves two axioms of risk measures and by setting ν(·) := µ ∞ (·) − µ ∞ (0) we obtain the required risk measure. Remark 3.11. If R is coherent, its upper envelopeR (C) w becomes R itself. In this case, Assumption 3.1 is no longer needed to determine a priori the size of the bounded forward invariant subset, C. Moreover, 1) Assumption 3.7(iii) implies Assumption 3.1(ii) due to (12), and 2) Theorem 3.10(ii) holds for all v ∈ B w . For instance, the mean-semideviation map defined in (1) is coherent. It is shown in [33,Section 6] that the mean-semideviation map with r = 2 in a 1-dimensional linear model satisfy Entropic Map Recall that, given a Markov transition kernel P , the entropic map is defined as R x (v) := 1 λ log e λv dP x , λ > 0.(15) Without loss of generality, in the remaining part of this paper, we set λ = 1. Upper envelope We now derive the upper envelope for entropic measures. Suppose that for all v ∈ B w , e \|v\| dµ < ∞ holds. Then (i) ν(v) ≤ µ(e v v) µ(e v ) , and (ii) ν (w,C) (u) := sup v∈B (C) w e v udµ e v dµ is an upper envelope for ν given C. Proof. Given any two u, v ∈ B w , we obtain ν(v) − ν(u) = log µ(e v ) µ(e u ) = log µ(e u e v−u ) µ(e u ) ≥ µ(e u (v − u)) µ(e u ) ,(16) where the last inequality is due to Jensen's inequality. Hence, log [µ(e v )] ≥ µ(e u v) µ(e u ) − µ e u µ(e u ) (u − log [µ(e u )]) , ∀u, v ∈ B w . Define ξ u := e u µ(e u ) . Restricting u and v to be in the subset B (C) w , the above inequality yields log[µ(e v )] ≥ sup ξ= e u µ(e u ) ,u∈B (C) w µ(ξv) − µ(ξ log(ξ)). Since the equality holds by taking ξ * := e v µ(e v ) , we obtain log[µ(e v )] = sup ξ= e u µ(e u ) ,u∈B (C) w µ(ξv) − µ(ξ log(ξ)).(17) The second term µ(ξ log(ξ)) on the right-hand side of the above equation is the relative entropy and is always nonnegative (for proof see, e.g., [25, Section 5.1]). Hence, we obtain (i). Finally, (ii) is followed by log[µ(e v )] − log[µ(e u )] ≤ sup ξ= e f µ(e f ) ,f ∈B (C) w µ(ξ(v − u)) = sup f ∈B (C) w e f (v − u)dµ e f dµ . and it is easy to verify thatν (w,C) (u) = sup f ∈B (C) w e f udµ e f dµ is a valid coherent risk measure. Remark 4.2. The inequality in (17) is similar to the dual representation of convex risk measures on L ∞ [14,15] or on more general spaces such as Orlicz hearts [6]. However, since we consider a different functional space, namely, the weighted norm space B w , the existing result cannot be directly applied here. On the other hand, for other types of convex risk measures, their dual representation provide us with a generic approach to calculate their upper envelopes, as shown in the above proposition. By Proposition 4.1, we obtain one upper envelope for the entropic map: R (C) x (u) = sup f ∈Bw: f s,w ≤C e f udP x e f dP x ,(18) provided that P x (e f ) < ∞ holds for all f ∈ B w and x ∈ X. Lyapunov functions Now we investigate properties of Lyapunov functions w.r.t. the entropic map. Definition 4.3. A function w is said to be a Lyapunov function w.r.t. a risk map R, if (i) w : X → [0, ∞) is B(X)-measurable and unbounded from above, and (ii) there exist constants γ ∈ (0, 1) and K > 0 satisfying R x (w) ≤ γw(x) + K, ∀x ∈ X. We also introduce the following notation of level-sets. For any unbounded nonnegative B(X)-measurable function w and any real number R ∈ R, we define B w (R) := {x ∈ X\|w(x) ≤ R} and B c w (R) its complementary set. We then make the following assumption. Assumption 4.4. There exists a Lyapunov function w 1 ≥ 1 w.r.t. the entropic map R, with constants γ 1 ∈ (0, 1) and K 1 > 0. If the above assumption holds and setting w 0 := w p 1 with any p ∈ (0, 1), then for all f ∈ B w0 , there exists a constant K f (depending on p and f w0 ) satisfying \|f (x)\| ≤ f w0 w 0 (x) ≤ w 1 (x) + K f , ∀x ∈ X. We immediately have P x (e f ) ≤ P x (e w1+K f ) ≤ e K f e γ1w1+K1 < ∞, ∀x ∈ X and therefore, the upper envelope for the entropic map in (18) is well defined. In the following theorem, we show that if w 1 is a Lyapunov function w.r.t. R, then w 0 = w p 1 with any p ∈ (0, 1) is a Lyapunov function w.r.t. the upper envelope of R. Theorem 4.5. Suppose that Assumption 4.4 holds. Let w 0 := w p 1 with p ∈ (0, 1). Then, for any constant C > 0, there exist constants γ 2 ∈ (0, 1) (depending only on p and λ 1 ) and K 2 > 0 (depending on p, C, λ 1 and K 1 ) such that sup f :f ∈Bw 0 ,\|f \|≤w0+C P x (e f w 0 ) P x (e f ) ≤ γ 2 w 0 (x) + K 2 . Proof. Due to Assumption 4.4, for any λ ∈ (γ 1 , 1), we have R x (w 1 ) ≤ λw 1 (x), ∀x ∈ B c w1 (A), A := K 1 λ − γ 1 . It implies that for all x ∈ B c w1 (A), B c w 1 (λw1(x)) P x (dy) e w1(y)−λw1(x) − 1 ≤ Bw 1 (λw1(x)) P x (dy) 1 − e w1(y)−λw1(x) .(19) Taking some γ 2 ∈ (λ p , 1), by the definition of w 0 , we then have B c w0 (γ 2 w 0 (x)) ⊂ B c w1 (λw 1 (x)), ∀x ∈ X.(20) Indeed, for any y ∈ B c w0 (γ 2 w 0 (x)), it satisfies w 0 (y) > γ 2 w 0 (x), which is equivalent to w(y) > (γ 2 ) 1/p w 1 (x) > λw 1 (x). Hence, y ∈ B c w1 (λw 1 (x)) as well. We will need the following two lemmas (Lemma 4.6 and 4.7) to complete the proof of Theorem 4.5. Lemma 4.6. For any η ∈ (0, 1 − λ), p ∈ (0, 1) and γ 2 ∈ ((λ + η) p , 1), there exists a constant R 1 > 0 such that for all y ∈ B c w0 (γ 2 w 0 (x)), x ∈ B c w1 (R) and R ≥ R 1 , e w0(y)+C+ηw1(x) (w 0 (y) − γ 2 w 0 (x)) ≤ e w1(y)−λw1(x) − 1. Proof. It is sufficient to show that there exists a constant R 1 > 0 satisfying w 0 (y) + log w 0 (y) + C + log 2 + ηw 1 (x) ≤ w 1 (y) − λw 1 (x) (21) for all y ∈ B c w0 (γ 2 w 0 (x)), x ∈ B c w1 (R) and R ≥ R 1 . Note that for any p ∈ (0, 1) and ǫ ∈ (0, 1), there exists a constant D (depending on p and ǫ) satisfying x p + p log x ≤ ǫx + D, ∀x ≥ 1, which implies that w 0 (x) + log w 0 (x) ≤ ǫw 1 (x) + D, ∀x ∈ X. Hence, for all y ∈ B c w0 (γ 2 w 0 (x)), we have w 1 (y) − w 0 (y) − log w 0 (y) − (λ + η)w 1 (x) ≥(1 − ǫ)w 1 (y) − (λ + η)w 1 (x) − D ≥ (1 − ǫ)γ 1/p 2 − λ − η w 1 (x) − D. Choosing γ 2 ∈ ((λ + η) p , 1), ǫ < 1 − λ+η γ 1/p 2 and R 1 := D+C+log 2 (1−ǫ)γ 1/p 2 −λ−η , (21) holds for all y ∈ B c w0 (γ 2 w 0 (x)), x ∈ B c w (R) and R ≥ R 1 . Lemma 4.7. For any η > 0, p ∈ (0, 1) and C ≥ 0, there exists a constant R 2 such that for all y ∈ B w1 (λw 1 (x)), x ∈ B c w1 (R) and R ≥ R 2 , e −w0(y)+ηw1(x)−C (γ 2 w 0 (x) − w 0 (y)) ≥ 1 − e w1(y)−λw1(x) . Proof. It is sufficient to show that e −w0(y)+ηw1(x)−C (γ 2 w 0 (x) − w 0 (y)) ≥ 1 under the same condition. Note that there exists a constant D > 0 such that γ 2 η x p ≤ x + D, ∀x ≥ 1, which yields −w 0 (y) + ηw 1 (x) − C ≥ −w 0 (y) + γ 2 w 0 (x) − C − D and hence, e −w0(y)+ηw1(x)−C (γ 2 w 0 (x) − w 0 (y)) ≥ e γ2w0(x)−w0(y)−C−D (γ 2 w 0 (x) − w 0 (y)) . For all y ∈ B w1 (λw 1 (x)), we have γ 2 w 0 (x) − w 0 (y) ≥ (γ 2 − λ p )w 0 (x). Hence, e γ2w0(x)−w0(y)−C−D (γ 2 w 0 (x) − w 0 (y)) ≥ e (γ2−λ p )w0(x)−C−D (γ 2 − λ p )w 0 (x). Due to the fact that g(x) = e x ·x is an increasing function on R + , we can chooseR 2 > 0 such that eR 2 ·R 2 = e C+D . Hence, we have for all y ∈ B w1 (λw 1 (x)), x ∈ B c w0 (R) and R ≥R 2 , e −w0(y)+ηw1(x)−C (γ 2 w 0 (x) − w 0 (y)) ≥ 1 holds. Finally, setting R 2 =R 1/p 2 , the assertion is obtained. Proof of Theorem 4.5 continued: hence, by Lemma 4.6 and 4.7, for all x ∈ B c w1 (R 1 ∨ R 2 ∨ A), B c w 0 (γ2w0(x)) P x (dy)e w0(y)+C+ηw1(x) (w 0 (y) − γ 2 w 0 (x)) (Lemma 4.6) ≤ B c w 0 (γ2w0(x)) P x (dy) e w1(y)−λw1(x) − 1 (20) ≤ B c w 1 (λw1(x)) P x (dy) e w1(y)−λw1(x) − 1 (19) ≤ Bw 1 (λw1(x)) P x (dy) 1 − e w1(y)−λw1(x) (Lemma 4.7) ≤ Bw 1 (λw1(x)) P x (dy)e −w0(y)+ηw1(x)−C (γ 2 w 0 (x) − w 0 (y)) (20) ≤ Bw 0 (γ2w0(x)) P x (dy)e −w0(y)+ηw1(x)−C (γ 2 w 0 (x) − w 0 (y)) , which implies that for all f ∈ B w0 satisfying \|f \| ≤ w 0 + C, P x (dy)e f (y) (w 0 (y) − γ 2 w 0 (x)) ≤ 0, ∀x ∈ B c w1 (R 1 ∨ R 2 ∨ A). (23) Finally, for all x ∈ B w1 (R 1 ∨ R 2 ∨ A) and f ∈ B w0 satisfying \|f \| ≤ w 0 + C, P x (e f w 0 ) P x (e f ) ≤ P x (e w0+C w 0 ) P x (e −w0−C ) ≤ e 2C P x (e w0 w 0 ) · P x (e w0 ) Using again the fact that there exists some constant D > 0 satisfying x p + p log x ≤ x + D, ∀x ≥ 1, we obtain that P x (e w0 w 0 ) ≤ e D P x (e w1 ) which is upper bounded on B w1 (R 1 ∨R 2 ∨A). Hence, there exists a K 2 > 0 such that for all f ∈ B w0 satisfying \|f \| ≤ w 0 + C, P x (e f w 0 ) P x (e f ) ≤ K 2 , ∀x ∈ B w1 (R 1 ∨ R 2 ∨ A), which together with (23) implies the required inequality. Remark 4.8. The statement of Theorem 4.5 can be easily generalized as follows: for any positive C and A, there exist constants γ 2 ∈ (0, 1) and K 2 ∈ R + such that sup f :f ∈Bw 0 ,\|f \|≤Aw0+C P x (e f w 0 ) P x (e f ) ≤ γ 2 w 0 (x) + K 2 . Corollary 4.9. Suppose that Assumption 4.4 holds. Then, for any p ∈ (0, 1), w 0 := w p 1 , there exist constantsγ 0 ∈ (0, 1) (depending on p and γ 1 ) andK 0 (depending on p, γ 1 and K 1 ) satisfying R x (w 0 ) ≤γ 0 w 0 (x) +K 0 , ∀x ∈ X. Proof. By Proposition 4.1(i), R x (w 0 ) = log P x (e w0 ) ≤ Px(e w 0 w0) Px(e w 0 ) , ∀x ∈ X. Then, by Theorem 4.5, there exist constantsγ 0 ∈ (0, 1) (depending on p and γ 1 ) andK 0 > 0 (depending on p, γ 1 and K 1 ) such that P x (e w0 w 0 ) P x (e w0 ) ≤ sup f ∈Bw 0 :\|f \|≤w0 P x (e f w 0 ) P x (e f ) ≤γ 0 w 0 +K 0 , which yields the required inequality. In summary, if Assumption 4.4 holds, then (a) by Corollary 4.9, w 0 is a Lyapunov function w.r.t. the entropic map with constantŝ γ 0 andK 0 ; (b) by Theorem 4.5, the same w 0 is also a Lyapunov function with constants γ 2 and K 2 , which satisfies satisfying Assumption 3.7(i) (see Remark 3.2(a) and (b) ) if the cost function c satisfies \|c\| ≤γ 0 w 0 + C 0 with someγ 0 ∈ (0, 1 − γ 2 ) and C 0 > 0; (c) combining (a) and (b), Assumption 3.7(i) also holds. Minorization properties We investigate now the properties of the entropic map restricted to bounded levelsets. We introduce first the local Doeblin's condition (see [12] and references therein) as follows. λ − C µ C (A ∩ C) ≤ P x (A ∩ C) ≤ λ + C µ C (A ∩ C), ∀x ∈ C, A ∈ B(X). The following proposition indicates the connection to the standard Doeblin's condition. Proposition 4.11. The following two conditions are equivalent: (i) there exist a measure µ C and a constant λ − C > 0 such that µ C (C) > 0 and P x (A ∩ C) ≥ λ − C µ C (A ∩ C), ∀x ∈ C, A ∈ B(X).(24) (ii) there exist a probability measure µ and a constant α > 0 such that P x (A) ≥ αµ(A), ∀x ∈ C, A ∈ B(X).(25) Proof. First, it is clear that (25) implies (24). Conversely, assume that (24) holds. Then, µ(·) := µ C (C∩·) µ C (C) , and α := λ − C µ C (C) satisfy (25). ∈ (0, 1). Then for any positive constant K 0 > 0, there exists a positive constantK 0 > K 0 such that for all v ∈ B w0 satisfying \|v\| ≤ w 0 + K 0 , the following inequality holds R x (v) − R y (v) ≤ 2(K 0 − K 0 ) + R x (w 0 ) − R y (−w 0 ), ∀x, y ∈ B. Proof. Let C := B w0 (R) ⊃ B = B w0 (R 0 ) with R > R 0 . Then P x (e v ) P y (e v ) = P x (e v 1 C ) + P x (e v 1 C c ) P y (e v 1 C ) + P y (e v 1 C c ) ≤ P x (e v 1 C ) + P x (e v 1 C c ) P y (e v 1 C ) .(26) We first consider the quotient Px(e v 1 C c ) Py (e v 1 C ) . By \|v\| ≤K 0 + w 0 , we obtain P x (e v 1 C c ) P y (e v 1 C ) ≤ e 2K0 P x (e w0 1 C c ) P y (e −w0 1 C ) = e 2K0 θ(x, C)P x (e w0 ) θ ′ (y, C)P y (e −w0 ) where we define θ(x, C) := Px(e w 0 1 C c ) Px(e w 0 ) and θ ′ (y, C) := Py(e −w 0 1 C ) Py(e −w 0 ) . By Theorem 4.5, there exist some constants γ 2 ∈ (0, 1) and K 2 > 0 such that θ(x, C) ≤ 1 C c w0 P x (e w0 w 0 ) P x e w0 ≤ 1 C c w0 sup \|v\|≤w0 P x (e v w 0 ) P x e v ≤ 1 C c w0 (γ 2 w 0 (x) + K 2 ). Hence, θ(x, C) ≤ 1 C c w0 sup x∈B (γ 2 w 0 (x) + K 2 ) ≤ γ2R0+K2 R . Similarly, we have θ ′ (y, C) = 1 − P y (e −w0 1 C c ) P y (e −w0 ) ≥ 1 − γ 2 R 0 + K 2 R Hence, sup x,y∈B θ(x,C) θ ′ (y,C) → 0 as R → ∞, which implies that for any K 0 > 0, we can select sufficiently large R such that log θ(x, C) θ ′ (y, C) ≤ −2K 0 − log 2, ∀x, y ∈ B.(27) Thus for anyK 0 > K 0 > 0, there exists a sufficiently large R (depending on K 0 ) such that P x (e v 1 C c ) P y (e v 1 C ) ≤ e 2(K0−K0)+Rx(w0)−Ry(−w0)−log 2 .(28) Now we consider the first quotient in (26). By Assumption 4.10, we immediately have Px(e v 1 C ) Py(e v 1 C ) ≤ λ + C λ − C . Hence, setting K 0 := K 0 + 1 2 log 2 + log( λ + C λ − C ),(29) we obtain Px(e v 1 C ) Py(e v 1 C ) ≤ e 2(K0−K0)+Rx(w0)−Ry(−w0)−log 2 . Together with (28), it yields the required inequality: Px(e v ) Py(e v ) ≤ e 2(K0−K0)+Rx(w0)−Ry(−w0) , whereK 0 is chosen according to (29), while R is determined by (27). We investigate now the minorization property of the upper envelopeR (w,C) of the entropic map R, which is required in Assumption 3.7(iii). (w 0 ) < ∞ for all x ∈ B. Then there exist a constant α ∈ (0, 1) and a probability measure µ on (X, B(X)) satisfyinḡ R (w,C) x (v) −R (w,C) x (u) ≥ αµ(v − u), ∀x ∈ B, v ≥ u ∈ B 1+w0 . Proof. Note that sinceR (w,C) x (w 0 ) < ∞, we have for all v ∈ B 1+w0 and x ∈ B, \|R (w,C) x (v)\| ≤R (w,C) x (\|v\|) ≤ v 1+w0R (w,C) x (1 + w 0 ) < ∞. By (18), we have for all x ∈ B and v ≥ u ∈ B 1+w0 , R (w,C) x (v) −R (w,C) x (u) = sup h∈B (C) w P x (e h v) P x (e h ) − sup h ′ ∈B (C) w P x (e h ′ u) P x (e h ′ ) = inf h ′ ∈B (C) w sup h∈B (C) w P x (e h v) P x (e h ) − P x (e h ′ u) P x (e h ′ ) ≥ inf h ′ ∈B (C) w P x e h ′ (v − u) P x (e h ′ ) . By Proposition 4.11, Assumption 4.10 implies that there exist a probability measure µ B and α B such that P x (v) ≥ α B µ B (v) for all nonnegative measurable function v. Hence, for all x ∈ B and h ′ ∈ B (C) w , we have P x e h ′ (v − u) P x (e h ′ ) ≥ α B µ B e h ′ (v − u) P x (e Cw ) ≥ α B µ B e −Cw (v − u) max x∈B P x (e Cw ) = α B µ B (e −Cw ) max x∈B P x (e Cw ) µ B e −Cw (v − u) µ B (e −Cw ) Hence, α := α B µ B (e −Cw ) max x∈B Px(e Cw ) and the probability measure dµ := e −Cw dµ B e −Cw dµ B are the required constant and probability measure respectively. The following theorem shows that applying the entropic map, together with an additional growth condition for cost functions (see (30) below), Assumption 4.4 and 4.10 are sufficient for Assumption 3.1 and 3.7. Theorem 4.14. Let R be the entropic map with λ = 1. Assume that Assumption 4.4 and 4.10 hold with a Lyapunov function w 1 , and that the cost function c satisfies c ∈ B w q 1 with some q ∈ (0, 1). Then Assumption 3.1 holds with w 0 = w p 1 for any p ∈ (q, 1), and someK 0 > 0, and Assumption 3.7 holds with w 0 and w = 1 +K −1 0 w 0 . Proof. Fix one p ∈ (q, 1) and let w 0 = w p 1 . Then by Corollary 4.9, there existŝ γ 0 ∈ (0, 1) andK 0 > 0 satisfying R x (w 0 ) ≤γ 0 w 0 (x) +K 0 . By assumption, there exists some C > 0 and q ∈ (0, 1) such that \|c\| ≤ Cw q 1 . Choosing one γ (c) 0 ∈ (0, 1 −γ 0 ), there exists a constant K (c) 0 > 0 satisfying Cw q 1 (x) ≤ γ (c) 0 w 0 (x) + K (c) 0 . Hence, Assumption 3.1(i) holds with γ 0 :=γ 0 + γ (c) 0 ∈ (0, 1) and K 0 :=K 0 + K (c) 0 . Due to Proposition 4.11, Assumption 3.1(ii) holds with some constantK 0 > 0. Next, by Theorem 3.3 and 4.5, Assumption 3.7(i) and (ii) hold with w := 1 +K −1 0 w 0 and C :=K −1 0 . Assumption 3.7(iii) holds due to Proposition 4.13. By Theorem 3.8 and 3.10, the above theorem implies the following corollary. (30). We compare our results with two mostly related results in the literature. Comparison with [11]. (a) The assumption (A4) in [11,Section 4] requires a positive continuous density, i.e., there exists a positive function q satisfying Q(dy\|x, a) = q(x, a, y)µ(dy) for some reference probability measure µ, which implies the local Doeblin's condition in Assumption 4.10. Hence, our assumption is more general than its counterpart in [11]. (b) The assumption (A3) set in [11,Section 3] for the cost function c is implicit and difficult to be verified. On the contrary, the sufficient growth condition for c, (30), is explicit in form of the Lyapunov function w 1 w.r.t. the entropic map. Note that, in the example provided by [11], the assumption (A3) is also verified with the help of a Lyapunov function. (c) As an advantage, in comparison with [11], the convergence rate of iterations towards the solution to the Poisson equation is explicitly specified byᾱ in Theorem 3.8 under the chosen seminorm. Comparison with [24]. Among others, Kontoyiannis and Meyn (2005) developed in [24] (see also their earlier work on the same topic [23]) a spectral theory of multiplicative Markov processes, where the Poisson equation w.r.t. the entropic map (called multiplicative Poisson equation in [24]) plays the central role. Though our assumptions are less general than the assumptions stated in [24,23], our proof that generalizes the Hairer-Mattingly approach [17] is conceptually simpler than the one provided in [24,23], and can also be applied to other types of risk maps. Note again that, in our approach, the convergence rate of iterations towards the solution to the Poisson equation is explicitly specified byᾱ in Theorem 3.8 under the chosen seminorm. Optimal Risk-sensitive Control Markov control processes In this subsection, we introduce the framework of Markov control processes, where we mostly follow the notations of Hernández-Lerma & Lasserre (1999) [21]. A Markov control process, (X, A, {A(x)\|x ∈ X}, Q, c), consists of the following components: state space X and action space A, which are Borel spaces; the feasible action set A(x), which is a nonempty Borel space of A, for a given state x ∈ X; the transition model Q (B\|x, a), B ∈ B(X), (x, a) ∈ K: a stochastic kernel on X given K, where K denotes the set of feasible state-action pairs K := {(x, a)\|x ∈ X, a ∈ A(x)}, which is a Borel subset of X × A; and the cost function c: K → R, B(K)-measurable. Random variables are denoted by capital letters, e.g. X t and A t , whereas realizations of the random variables are denoted by lowercase letters, e.g., x t and a t . We consider in this paper Markov policies, π = [π 0 , π 1 , π 2 , . . .], where each singlestep policy π t (·\|x t ), which denotes the probability of choosing action a t at x t , (x t , a t ) ∈ K, is Markov (independent of the states and actions before t) and, therefore, a stochastic kernel on A given X. We use the boldface to represent a sequence of policies while using the normal typeface for a single-step policy. Let ∆ denote the set of all stochastic kernels on A given X, µ, such that µ(A(x)\|x) = 1 and Π M = ∆ ∞ denotes the set of all Markov policies. A policy f ∈ ∆ is deterministic if for each x ∈ X, there exists some a ∈ A(x) such that f ({a}\|x) = 1. Let ∆ D ⊂ ∆ denote the set of all deterministic single-step policies. A policy π is said to be stationary, if π = π ∞ for some π ∈ ∆. For each x ∈ X and single-step policy π ∈ ∆, define The following average cost is used as an objective: (32) S := lim sup T →∞ 1 T S T , where S T := T t=0 c(X t , A t ). The optimization problem is then to minimize the expected objective (33) inf π∈ΠM E π [S\|X 0 = x] by selecting a policy π. We notice that the finite-stage objective function can be decomposed as follows, E π X0 [S T ] = c π0 (X 0 ) + E π0 X0 c π1 (X 1 ) + E π1 X1 c π2 (X 2 ) + . . . + E πT −1 XT −1 [c πT (X T )] . . . ,(34) where E πt Xt [v(X t+1 )] := v(X t+1 )P πt (dX t+1 \|X t ) denotes the conditional expectation of the function v of the successive state X t+1 given current state X t . Average risk-sensitive MCPs To incorporate risk as in our previous work [33], we directly replace the conditional expectation E πt Xt with a risk map R πt Xt , which is similar to the risk mapping defined in [30] and is formally defined as follows. A mapping R(v\|x, a) : K× B w → R (simply written as R) is said to be a risk map on an MCP (X, A, {A(x)\|x ∈ X}, Q), if (i) for each (x, a) ∈ K, R(·\|x, a) : B w → R is a risk measure; and (ii) for each v ∈ B w , R(v\|·) is a real-valued B(K)-measurable function. Furthermore, we define for any π ∈ ∆, R π (v\|x) := A(x) π(da\|x)R(v\|x, a). For convenience, we sometimes write R x,a (v) := R(v\|x, a) and R π x (v) := R π (v\|x). Replacing the conditional expectation in (34) with a risk map R, we obtain J T (x, π) := c π0 (x) + R π0 x (c π1 + R π1 (c π2 + . . . + R πT −1 (c πT ) . . .)) , π ∈ Π M , x ∈ X, and the risk-sensitive objective considered in this paper is the average risk (AR): J(x, π) := lim sup T →∞ 1 T J T (x, π).(35) Remark 5.1. Applying the same constructive approach as above, other two widely used objectives in the literature of MCPs, the finite-stage total cost and the discounted cost, can be analogously extended to risk-sensitive objectives [33], the finite-stage total risk and the discounted risk, respectively. Among them, the finite-stage risk can be optimized by dynamic programming [29]. For the discounted case, we refer to our previous work [33, Subsection 5.1], where the same problem for strictly convex risk maps, i.e., a Lyapunov function w.r.t. the upper module of the risk map need not exist, can be easily solved by replacing the upper module by the upper envelope defined in this paper. In the rest of this section, for convenience, the AR objective can be considered as functions on X within the space B w by using the notation J(π), as well as J * . Analogous to classical MCPs, we need further assumptions to guarantee the existence of the "selector" in the optimization problem. Define the following operators F π (v) := c π + R π (v), F (v) := inf π∈ΠM F π (v), v ∈ B w .(36) Proposition 5.3 (see Proposition 5.1 in [33]). Suppose R is a risk map satisfying Assumption 5.2. Then, for all v ∈ B w and x ∈ X, there exists a deterministic policy f ∈ ∆ D , such that c f (x) + R f (v\|x) = F (v\|x) = inf π∈∆ {c π (x) + R π (v\|x)} . We now extend Assumption 3.1 and 3.7 to the MCP-framework. Assumption 5.4. There exist a B(X)-measurable function w 0 : X → [0, ∞), constants γ 0 , γ ∈ (0, 1),K 0 > K 0 > 0, K > 0 and α ∈ (0, 1) such that (i) (c(x, a) + R x,a (w 0 )) ∨ (−c(x, a) − R x,a (−w 0 )) ≤ γ 0 w 0 (x) + K 0 , ∀(x, a) ∈ K; (ii) for all x, x ′ ∈ B 0 := {x ∈ X\|w 0 (x) ≤ R 0 := 2K0 1−γ0 }, a ∈ A(x), a ′ ∈ A(x ′ ), the inequality R x,a (v) − R x ′ ,a ′ (v) ≤ 2(K 0 − K 0 ) + R x,a (w 0 ) − R x ′ ,a ′ (−w 0 ) holds for all real-valued B(X)-measurable function v satisfying \|v\| ≤ w 0 +K 0 ; (iii) letting w := 1 +K −1 0 w 0 , there exists an upper envelopeR (w,K0) such that R (w,K0) x,a (w 0 ) ≤ γw 0 (x) + K, ∀(x, a) ∈ K; and (iv) for all x, x ′ ∈ B := {x ∈ X\|w 0 (x) ≤ R, R > 2K 1−γ }, a ∈ A(x), a ′ ∈ A(x ′ ), yield F x (v) − F x (u) + F y (u) − F y (v) ≤ R fu x (v) − R fu x (u) + R fv y (u) − R fv y (v) for all x, y ∈ X. By Assumption 5.4 and repeating the proof in Theorem 3.8, we obtain the required inequality. and furthermore, ρ * = J * (x) = J(x, f ∞ ) for all x ∈ X, where f denotes the optimal selector in the right hand side of the AROE (37). R (w,K0) x,a (v) −R (w,K0) x,a (u) ≥ α (v(x) − u(x)) µ(dx), ∀v ≥ u ∈ B 1+w0 . Proof. The existence of a unique solution to the AROE is simply due to Lemma 5.5 and Theorem 3.10(i). For the proof of the remaining part, we refer to the proof of [33, Theorem 5.10]. Remark 5.7. When applying the entropic map with λ = 1, analogous to the withoutcontrol case stated in Corollary 4.15, Assumption 5.4 can be replaced by the following conditions: (i) there exist a function w 1 : X ∈ [1, ∞), constants γ 1 ∈ (0, 1) and K 1 > 0 such that R x,a (w 1 ) ≤ γ 1 w 1 (x) + K 1 , ∀(x, a) ∈ K, (ii) for all p ∈ (0, 1) and all level-sets C := B w p 1 (R), R > 0, there exist a measure µ C and constants λ + C > λ − C > 0 such that µ C (C) > 0 and λ − C µ C (A ∩ C) ≤ Q x,a (A ∩ C) ≤ λ + C µ C (A ∩ C), ∀x ∈ C, a ∈ A(x) and ∀A ∈ B(X), and (iii) the cost function c satisfies thatc(x) := sup a∈A(x) \|c(x, a)\| belongs to B w q 1 for some q ∈ (0, 1). In the following example, we present one MCP that satisfies the above conditions (i) and (ii) with the entropic map. Entropic map with discretized ergodic diffusions Let X = R d . Consider the following discretized ergodic diffusion {x n ∈ R d } (cf. the example in [11,Section 6]): x n+1 = Ax n + b(x n , a n ) + D(x n , a n )w n , where {w n ∈ R d } is a sequence of i.i.d. standard white noise, D : K → R d×d is a continuous bounded matrix-valued function which is uniformly elliptic, i.e., there exists a constant L > 0 such that L −1 ξ 2 ≤ ξ ⊤ D(x, a)D ⊤ (x, a)ξ ≤ L ξ 2 , ∀(x, a) ∈ K, ξ ∈ R d ,(38) and b : K → R d is a continuous bounded vector function, and A is a matrix satisfying that there exists a constantγ ∈ (0, 1) such that ξ ⊤ A ⊤ Aξ ≤γ ξ 2 , ∀ξ ∈ R d . Then the transition kernel Q(dy\|x, a) has the following density w.r.t. the Lebesgue measure, where Σ = (DD ⊤ ) −1 . Take one γ ∈ (γ, 1) and consider the following weight function w 1 (x) = ǫ 2 x 2 , with some positive ǫ ≤ γ −γ γ L −1 < L −1 .(40) Hence, Σ(x, a) − ǫI is positive definite for all (x, a) ∈ K. We show thatŵ 1 is a Lyapunov function w.r.t. the entropic map satisfying the condition (i) in Remark 5.7 as follows. By settingx := Ax + b, we obtain Q(dy\|x, a)eŵ 1(y) =(2π) −d/2 \|Σ\| 1/2 e − 1 2 (y ⊤ (Σ−ǫI)y−2y ⊤ Σx+x ⊤ Σx) dy = \|Σ\| 1/2 \|Σ − ǫI\| 1/2 e By (38) and the choice of ǫ in (40), we have 1 2 x ⊤ A ⊤ Σ (Σ − ǫI) −1 − Σ −1 ΣAx ≤ γǫ 2 x 2 = γŵ 1 (x), ∀(x, a) ∈ K. Finally, due to the uniform boundedness of b and log \|Σ\| 1/2 \|Σ−ǫI\| 1/2 , we can always select a γ 1 ∈ (γ, 1) andK 1 > 0 such that log Q(dy\|x, a)eŵ 1 (y) ≤ γ 1ŵ1 (x) +K 1 , ∀(x, a) ∈ K, which confirms thatŵ 1 ≥ 0 is a Lyapunov function w.r.t. the entropic map. Hence, the condition (i) in Remark 5.7 holds with w 1 :=ŵ 1 + 1, γ 1 and K 1 :=K 1 + 1 − γ 1 . Next, since the transition kernel Q has a positive continuous density function, q (see (39)), w.r.t. the Lebesgue measure, the local Doeblin's condition (ii) in Remark 5.7 is obviously satisfied. 26 , 26Section 2.1.1]). Given a cost function c and a risk map R, the nonlinear Poisson equation T (h) := c + R(h) = ρ + h, Lemma 3 . 9 . 39Suppose Assumption 3.1 and 3.7 hold. Then for any endowed with the quotient norm induced by the weighted seminorm. Theorem 3 . 10 . 310Suppose Assumption 3.1 and 3.7 hold. Then there exist (i) a solution (ρ, h) ∈ R×B w to the Poisson equation (2), where ρ is unique and (ii) a risk measure ν satisfying Assumption 3.7(ii) and (iii), based on which the existence and uniqueness of a solution to the Poisson equation is guaranteed. Proposition 4. 1 . 1Let ν(v) := log e v dµ with a probability measure µ on (X, B(X)). Assumption 4 . 10 . 410Let w 0 : X → [0, ∞) be a B(X)-measurable function. For any level-set C := B w0 (R), R > 0, there exist a measure µ C and constants λ + C , λ − C > 0 such that µ C (C) > 0 and Theorem 4 . 12 . 412Suppose Assumption 4.10 and Assumption 4.4 hold. Let w 1 be the Lyapunov function and B = B w0 (R 0 ) be a bounded levels-set with some R 0 > 0, where w 0 := w p 1 , p Proposition 4 . 13 . 413Let w : X → [1, ∞) be a B(X)-measurable function and B := B w (R) with some R > 0. Suppose Assumption 4.10 holds. Assume further that R (w,C) x Corollary 4 . 15 . 415Let R be the entropic map with λ = 1. Suppose Assumption 4.4 and 4.10 hold and w 1 is the Lyapunov function. Assume further that the cost function c satisfies(30). Then for any p ∈ (q, 1), there exist (i) constantsᾱ ∈ (0, 1) and β > 0such that R(v) − R(u) s,w ≤ᾱ v − u s,w with w := 1 + βw p 1 , and (ii) a solution (ρ, h) ∈ R × B w to the Poisson equation c + R(h) = ρ + h, where ρ is unique.Remark 4.16. Hence, for the entropic map, the required sufficient conditions in Assumption 3.1 and 3.7 can replaced by the existence of Lyapunov function in Assumption 4.4, the local Doeblin's condition in Assumption 4.10 and the growth condition for the cost function in B\|x, a)π(da\|x), B ∈ B(X). Assumption 5.2. For each x ∈ X, (i) the cost function c(x, a) is lower semi-continuous on A(x), (ii) the action space A(x) is compact, and (iii) the function u ′ (x, a) := R x,a (u) is continuous in a ∈ A(x) for any u ∈ B w . Lemma 5. 5 . 5Suppose Assumption 5.2 and 5.4 hold. Then there existsᾱ ∈ (0, 1) andβ > 0 such that F (v) − F (u) s,1+βw0 ≤ᾱ v − u s,1+βw0 , for all v, u ∈ B (K0) w . Proof. By Proposition 5.3, there exist deterministic policies f v , f u ∈ ∆ D such that F (v) = F fv (v) and F (u) = F fu (u). Thus F (v) − F (u) ≤ F fu (v) − F fu (u) = R fu (v) − R fu (u) and F (u) − F (v) ≤ F fv (u) − F fv (v) = R fv (u) − R fv (v) Theorem 5 . 6 . 56Suppose Assumption 5.2 and 5.4 hold. 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[ "A GENERALIZATION OF STANLEY'S MONSTER RECIPROCITY THEOREM", "A GENERALIZATION OF STANLEY'S MONSTER RECIPROCITY THEOREM" ]	[ "Guoce Xin " ]	[]	[]	By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the "error term" is given in the case when the system does not have the reciprocity property. We also give a short proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.	10.1016/j.jcta.2007.03.007	[ "https://export.arxiv.org/pdf/math/0504425v1.pdf" ]	15,587,785	math/0504425	c45efa751fd34627e76ac3346d874bb767ce7df2	A GENERALIZATION OF STANLEY'S MONSTER RECIPROCITY THEOREM 21 Apr 2005 Guoce Xin A GENERALIZATION OF STANLEY'S MONSTER RECIPROCITY THEOREM 21 Apr 2005arXiv:math/0504425v1 [math.CO]Reciprocity propertylinear Diophantine systemLaurent seriesMalcev-Neumann series By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the "error term" is given in the case when the system does not have the reciprocity property. We also give a short proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems. Introduction Let A be an r by n matrix with integer entries, and let b be an r-vector in Z r . Many combinatorial problems turn out to be equivalent to finding all nonnegative integral (column) vectors α ∈ N n satisfying (1.1) Aα = b, especially in the homogeneous case when b equals 0, of which the solution space is a rational cone. Such problems are also known as solving a linear Diophantine system. There are two closely related generating functions associated to (1.1): E(x; b) = (α 1 ,...,αn)∈N n x α 1 1 · · · x αn n ,Ē(x; b) = (α 1 ,...,αn)∈P n x α 1 1 · · · x αn n , where the first sum ranges over all α = (α 1 , . . . , α n ) such that Aα = b, and the second sum ranges over all positive integral α such that Aα = −b. We omit b in the homogeneous case. The following well-known reciprocity theorem for homogeneous linear diophantine equations was given by Stanley as [3, Theorem 4.1]. Theorem 1.1 (Reciprocity Theorem). Let A be an r by n integral matrix of full rank r. If there is at least one α ∈ P n such that Aα = 0, then we have as rational functions E(x 1 , . . . , x n ) = (−1) n−rĒ (x −1 1 , . . . , x −1 n ). Previous proofs of this theorem used decompositions into simplicial cones or lattice cones, or complicated algebraic technique. See [4, p. 214] and [5] for further information. We will give a short proof using a signed cone decomposition and induction. In the general situation, the best known result (up to now) is the monster reciprocity theorem, which was given by Stanley [4] in 1974. The theorem will be stated later after new notation is introduced. It includes as special cases many combinatorial reciprocity theorems, such as the reciprocity theorem for homogeneous linear Diophantine system, that for Ehrhart polynomials, and that for P-partitions, etc. We will give a simple approach to this theorem. As applications, we give detailed, and short, implication of the reciprocal domain theorem [4, Proposition 8.3]. The new approach uses the idea of Malcev-Neumann series [2; 7; 9], which defines a total ordering on the group of monomials to clarify the series expansion of rational functions. We study the reciprocity property of an object that is more general, but less combinatorial, than that was studied in [4]. The new objects we are going to study are Elliott-rational functions, while the previous objects are Elliott-rational functions with a monomial numerator. By an Elliott-rational function, we mean the one that can be written as F (λ 1 , . . . , λ r , x) = p(λ 1 , . . . , λ r , x) m i=1 (y i − z i ) , where p is a polynomial and y i and z i are monomials. In this larger set of objects, it is much easier to build up the reduction steps. Theorem 3.8, a general result that gives a reciprocity formula for Elliott-rational functions, turns out to be easy to prove. We shall use this result to formulate the monster reciprocity theorem (Theorem 4.2). In Section 2, we introduce the basic idea of Malcev-Neumann series and reformulate the reciprocity of linear Diophantine system in terms of constant terms. In Section 3, we develop the reciprocity theorem for Elliott-rational functions. We apply our result in Section 4 to give the generalized monster reciprocity theorem. In section 5, we illustrate the monster reciprocity theorem by examples, and as an application, we give a simple derivation of Theorem 1.1. Section 6 includes an inductive (combinatorial) proof of Theorem 1.1. Reciprocity in Terms of Constant Terms Solving a linear Diophantine system (LD-system for short) means finding all vectors α ∈ N n that satisfy Aα = b, where A is an r by n matrix with integral entries. More precisely, we want to solve the following system of equations: a 1,1 α 1 + a 1,2 α 2 + · · · + a 1,n α n = b 1 a 2,1 α 1 + a 2,2 α 2 + · · · + a 2,n α n = b 2 · · · · · · = · · · (2.1) a r,1 α 1 + a r,2 α 2 + · · · + a r,n α n = b r . We assume the rank of A\|b equals the rank of A, for otherwise, the LD-system has no solution even in Q. Let C i be the ith column vector of A. Then the above system is the same as C 1 α 1 + C 2 α 2 + · · · + C n α n = b. Now let E(b) andĒ(b) be the sets of all such solutions in N n and P n respectively. It is interesting to study the following two associated generating functions of (2.1): E(x; b) = α∈E(b) x α ,Ē(x; b) = α∈Ē(−b) x α (2.2) where x = (x 1 , . . . , x n ) and if α = (α 1 , . . . , α n ), then x α := x α 1 1 · · · x αn n . The above equation defines two rational functions in x. If as rational functions E(x; b) = (−1) n−rĒ (x −1 ; b), then we say that the system (2.1) has the R-property (short for reciprocity property). We can compute E(x; b) by replacing the r linear constraints with r new variables λ 1 , λ 2 , . . . , λ r and then take the constant terms. Let Λ be (λ 1 , . . . , λ r ), and let CT Λ F be the constant term of F in Λ. We have E(x; b) = α∈N n CT Λ λ a 1,1 α 1 +···+a 1,n αn−b 1 1 · · · λ a r,1 α 1 +···+ar,nαn−br r x α = CT Λ λ −b 1 1 · · · λ −br r n i=1 (1 − λ a 1,i 1 λ a 2,i 2 · · · λ a r,i r x i ) = CT Λ Λ −b n i=1 (1 − Λ C i x i ) , (2.3) with the working ring C[Λ, Λ −1 ][[x]], where Λ −1 means (λ −1 1 , . . . , λ −1 r ). The above conversion can be trait back to MacMahon [1]. Similarly we get E(x; b) = CT Λ Λ b n i=1 Λ C i x i n i=1 (1 − Λ C i x i ) . (2.4) We define E(Λ, x; b) andĒ(Λ, x; b) to be the crude generating functions of E(x, b) andĒ(x; b) as E(Λ, x; b) = Λ −b n i=1 (1 − Λ C i x i ) ,Ē(Λ, x; b) = Λ b n i=1 Λ C i x i n i=1 (1 − Λ C i x i ) , (2.5) and observe that as rational functions E(Λ −1 , x −1 ; b) = (−1) n E(Λ, x; b). However, the series expansion of the two sides of the above equation are different. The change of variables by Λ → Λ −1 , which corresponds to multiplying each row of (2.1) by −1, will not make a difference when taking constant terms. Therefore, the system has the R-property if and only if as rational functions CT Λ E(Λ, x) = (−1) r CT Λ ′ E(Λ, x), where we expand E(Λ, x) on the LHS at x = 0, while on the RHS at x = ∞. As we shall see later, the different expansions appearing in the above equation is easily explained in the context of Malcev-Neumann series. The group of monomials in Λ and x can be given a total ordering " ρ " that is compatible with its group structure; i.e., for any monomials A, B and C, A B implies AC ρ BC. This is equivalent to a total ordering ≤ ρ on the additive group Z n+r . An important such ordering ≤ is the reverse lexicographical ordering on Z n+r . Then a Malcev-Neumann series (or MN-series for short) with respect to ρ is a formal series on Λ and x with a well-ordered support: the set of monomials corresponds to the nonzero terms. Recall that a well-ordered set is a totally ordered set such that every nonempty subset has a minimum. For our purpose, ρ will denote an injective endomorphism of Z n+r (a nonsingular integral matrix), and ≤ ρ will be the induced total ordering defined by a ≤ ρ b if and only if ρ(a) ≤ ρ(b). We denote by C ρ Λ, x the corresponding field of MN-series with respect to ρ. The field of iterated Laurent series C Λ, x , where ρ is the identity map and is omitted, has been studied in [8; 9]. For a more general setting of MN-series, the readers are referred to [7; 9] or [2,Chapter 13]. The series expansion of MN-series will be explained in more details in the next section. Let us review some properties of MN-series [7] to see that such fields are suitable for dealing with different kinds of series expansions of rational functions. For any total ordering ≤ ρ , C ρ Λ, x is a field. In particular, C Λ, x is the field of iterated Laurent series [8]. The field C(Λ, x) of rational functions is naturally embedded into C ρ Λ, x for any ρ. This follows from the field structure of C ρ Λ, x and the fact that every polynomial has a finite support. Every rational function F (Λ, x) has a unique expansion in C ρ Λ, x . The expansions of F for different ρ are usually different. For instance, the expansion of 1/(x − y) in K x, y is 1 x − y = 1 x · 1 1 − y/x = 1 x k≥0 y k /x k , but the expansion in K y, x is 1 x − y = 1 −y · 1 1 − x/y = 1 −y k≥0 x k /y k . Note that we can write K y, x as K ρ x, y where ρ is defined by the matrix 0 1 1 0 , or by abuse of notation, ρ(x) = y and ρ(y) = x. Recall also that every subset of a well-ordered set is well-ordered. Thus the following operators CT λ , PT λ , and NT λ are well-defined for MN-series. CT λ k∈Z b k λ k = b 0 , PT λ k∈Z b k λ k = k≥0 b k λ k , and NT λ k∈Z b k λ k = k<0 b k λ k . Obviously, for an MN-series F (λ), CT λ F (λ) = PT λ F (λ)\| λ=0 . The constant term operators are commutative so that taking the constant term in a set of variables is defined by iteration. Now it is easy to see that Theorem 1.1 is a consequence of the following proposition. Proposition 2.1. Suppose thatĒ is nonempty. Then CT Λ E(x; 0) = (−1) rank(A) CT Λ ρ E(x; 0), (2.6) where ρ is the endomorphism defined by ρ(x i ) = x −1 i and ρ(λ i ) = λ i . On the other hand, it is easy to deal with the case of rank(A) < r. So Theorem 1.1 is equivalent to Proposition 2.1, whose proof will be given in section 6. The following lemma asserts that elementary row operation will not change the solution space of an LD-system. We give it here to show that all the work can be done algebraically. If Φ ∈ K[x, x −1 ] y , then for f i = x b i1 1 · · · x b in n with det(b ij ) 1≤i,j≤n = 0, CT x Φ(f 1 , . . . , f n ) = CT x Φ(x 1 , . . . , x n ). Reciprocity of Elliott-Rational Functions It is convenient for our purpose to denote by K the field C(x). The field of rational functions C(Λ, x) can be identified with K(Λ). Usually we are taking constant terms in the ring C[Λ, Λ −1 ][[x] ]. This ring can be embedded into C ρ Λ, x , as long as ρ is compatible with the relation x i ≻ ≻ ρ λ j for all i and j, where A ≻ ≻ ρ B means that A ≻ ρ B k for any positive integer k. The case r = 1 is illustrative for our understanding of the series expansion for MNseries, and in this particular case, we need not restrict ourselves to Elliott-rational functions. Let us consider the following problem. Problem: Given a rational function Q(λ) (short for Q(λ, x)) of λ and x, compute PT ρ λ Q(λ, x), where the notation PT ρ λ indicates that Q(λ) is treated as an element of C ρ λ, x , and we use similar notations for the CT and NT operators. To deal with this problem, we shall understand that Q(λ) is not only an element of K(λ), but also an element of C ρ λ, x . As an element of K(λ), Q(λ) can be written as p(λ)/q(λ), where p(λ) and q(λ) are both in K[λ]. As an element of C ρ λ, x , the denominator q(λ) plays an important role. Recall that C ρ λ, x is equipped with a total ordering ρ on its group of monomials and that its elements have well-ordered supports. Thus for a nonzero element η, we can define its order ord η to be min supp(η), and its initial term to be the term with the least order. The order of 0 is treated as ∞. Let us write q(λ) = d i=0 a i λ i , with a i ∈ C(x) and a d = 0. To expand Q(λ) into a series in C ρ λ, x , we need to find the λ-initial term a j λ j , i.e., the j such that ord(a j λ j ) ≺ ρ ord(a i λ i ) for all i = j. This can be achieved because of the different powers in λ. Then 1 q(λ) = 1 a j λ j 1 1 + i =j a i /a j λ i−j = 1 a j λ j k≥0 (−1) k i =j a i /a j λ i−j k . This expansion is justified by the composition law [9, Theorem 2.2]. It is now clear that we have the following three situations. (1) If j equals 0, then for any polynomial p(λ), p(λ)/q(λ) contains only nonnegative powers in λ. In this case, we say that 1/q(λ) is PT ρ in λ. (2) If j equals d, then for any polynomial p(λ) of degree in λ less than d, p(λ)/q(λ) contains only negative powers in λ. In this case, we say that 1/q(λ) is NT ρ in λ. (3) If j equals neither 0, nor d, then 1/q(λ) contains both positive and negative powers in λ. Thus 1/q(λ) is neither PT ρ nor NT ρ in λ. Lemma 3.1. Let q 1 and q 2 be polynomials in λ. Then for any total ordering ≤ ρ • Both 1/q 1 (λ) and 1/q 2 (λ) are PT ρ in λ if and only if 1/(q 1 q 2 ) is. • Both 1/q 1 (λ) and 1/q 2 (λ) are NT ρ in λ if and only if 1/(q 1 q 2 ) is. • For all the other cases, 1/(q 1 q 2 ) is neither PT ρ in λ nor NT ρ in λ. Proof. We prove the first case for PT as follows. The other cases are similar. Write q 1 = d 1 i=0 a i λ i , q 2 = d 2 i=0 b i λ i , and q 1 q 2 = d 1 +d 2 i=0 c i λ i . Suppose that a j 1 λ j 1 and b j 2 λ j 2 are the λ-initial term of q 1 and q 2 respectively. Now if we expand the product q 1 q 2 but do not collect terms, then a j 1 b j 2 λ j 1 +j 2 is the unique term with the least order. So the order of c j 1 +j 2 λ j 1 +j 2 has to equal the order of a j 1 b j 2 λ j 1 +j 2 . This implies that the λ-initial term of q 1 q 2 is c j 1 +j 2 λ j 1 +j 2 . The assertion for PT in the lemma hence follows from the fact that j 1 + j 2 = 0 ⇔ j 1 = 0 and j 2 = 0. (Remember that j 1 , j 2 ≥ 0). A direct consequence of the above lemma is the following corollary. Corollary 3.2. If 1/q 1 (λ) is PT ρ in λ and 1/q 2 (λ) is NT ρ in λ, then q 1 (λ) and q 2 (λ) cannot have a nontrivial common divisor in K[λ] , i.e., they are relatively prime. Definition 3.3. If q(λ) can be factored as q 1 (λ)q 2 (λ) such that 1/q 1 (λ) is PT ρ in λ and 1/q 2 (λ) is NT ρ in λ, then we say that q(λ) is ρ-factorable, and q(λ) = q 1 (λ)q 2 (λ) is a ρ-factorization. Such factorization is unique (if it exists) up to a constant in K. Theorem 3.4. Let p(λ), q(λ) ∈ K[λ]. If q(λ) is ρ-factorable, then CT ρ λ p(λ)/q(λ) is in K, i.e., is rational. Proof. Suppose q(λ) = q 1 (λ)q 2 (λ) is such a ρ-factorization. Since 1/q 1 (λ) is PT ρ in λ and 1/q 2 (λ) is NT ρ in λ, q 1 (λ) and q 2 (λ) are relatively prime in K[λ] by Corollary 3.2. Thus we have the unique partial fraction expansion in K(λ): p(λ) q(λ) = p 0 (λ) + p 1 (λ) q 1 (λ) + p 2 (λ) q 2 (λ) , (3.1) where p i are polynomials in λ for i = 0, 1, 2 and deg p i (λ) < deg q i (λ) for i = 1, 2. Since when expanded as series in C ρ λ, x , p 0 (λ) and p 1 (λ)/q 1 (λ) contains only nonnegative powers in λ, and p 2 (λ)/q 2 (λ) contains only negative powers in λ, we have PT λ ρ p(λ) q(λ) = p 0 (λ) + p 1 (λ) q 1 (λ) . Thus CT ρ λ = p 0 (0) + p 1 (0)/q 1 (0) is in C(x). This theorem generalizes a result of Hadamard [6, Proposition 4.2.5 ], which says that the Hadamard product of two rational power series is rational. This statement can be easily seen from the following observation: Let f = k≥0 f k x k and g = k≥0 g k x k . Then the Hadamard product of f and g is k≥0 f k g k x k = CT λ f (λ)g(x/λ), where we are taking the constant term in C λ, x for the RHS of the above equation. For any total ordering ρ on the monomials of K(λ), we let ρ be the total ordering such that m 1 ≺ρ m 2 if and only if m 1 ≻ ρ m 2 for all monomials m 1 and m 2 . Then we have a sort of reciprocity invariant, for which we need three more notations. We use the notation CT λ=0 F (λ) to indicate that F (λ) is treated as an element in K((λ)) and CT λ=∞ F (λ) to indicate that F (λ) is treated as an element in K((λ −1 )). We define I λ F (λ) = CT λ=0 F (λ) + CT λ=∞ F (λ). (3.2) Theorem 3.5. Suppose that p(λ), q(λ) ∈ K[λ] , with q(λ) being ρ-factorable. Then the following is always true as rational functions in K: CT λ ρ p(λ) q(λ) + CT λρ p(λ) q(λ) = I λ p(λ) q(λ) . (3.3) Theorem 3.5 gives an invariant of a rational function when taking the constant term in λ. This invariant is independent of the choice of the total ordering ≤ ρ when applicable. This fact is the key in our new approach to the monster reciprocity theorem in Section 4. Proof of Theorem 3.5. Write q(λ) as q 1 (λ)q 2 (λ)λ s , such that 1/q 1 (λ) is PT ρ in λ, 1/q 2 (λ) is NT ρ , and q 1 (0)q 2 (0) = 0. Clearly s ≥ 0 and the partial fraction decomposition of p(λ)/q(λ) can be written as p(λ) q(λ) = p −1 (λ) λ s + p 0 (λ) + p 1 (λ) q 1 (λ) + p 2 (λ) q 2 (λ) , where deg p −1 < s, deg p 1 < deg q 1 , deg p 2 < deg q 2 , and p 0 is a polynomial. Now we are going to apply different operators on this partial fraction decomposition. Applying CT ρ λ to p(λ)/q(λ) gives us p 0 (0)+p 1 (0)/q 1 (0), and applying CTρ λ to p(λ)/q(λ) gives us p 0 (0) + p 2 (0)/q 2 (0). Therefore CT λ ρ p(λ) q(λ) + CT λρ p(λ) q(λ) = 2p 0 (0) + p 1 (0) q 1 (0) + p 2 (0) q 2 (0) . Applying CT λ=0 to p(λ)/q(λ) gives us p 0 (0)+p 1 (0)/q 1 (0)+p 2 (0)/q 2 (0), and applying CT λ=∞ to p(λ)/q(λ) gives us p 0 (0). Thus the theorem follows. Remark 3.6. In the proof of Theorem 3.5, we see thatρ can be replaced with σ if σ switches the PT and NT properties of 1/q 1 (λ) and 1/q 2 (λ) with respect to ρ. As an element of K[λ], q(λ) can be factored into the product of irreducible polynomials. Let q(λ) = q 1 (λ) · · · q k (λ) be such a factorization. By Lemma 3.1 q(λ) is ρ-factorable if and only if every 1/q i is either PT ρ or NT ρ . When this is the case, the ρ-factorization can be obtained by collecting similar terms. Elliott-rational functions are ρ-factorable for any ρ. Such a function F can be written as follows: (3.4) F = p(λ) (λ j 1 − a 1 ) · · · (λ jn − a n )(λ k 1 − b 1 ) · · · (λ km − b m ) , where p(λ) is a polynomial in λ, j i and k i are positive integers, m and n are nonnegative integers, and a i and b l are monomials independent of λ. For a particular ρ, we require that 1/(λ j i − a i ) is NT ρ in λ, and 1/(λ k i − b i ) is PT ρ in λ. Note that a 1 can be 0. "The method of Elliott" [1, p. 111-114] shows that CT ρ λ F is always Elliott-rational. A rational function of λ is proper in λ if the degree in λ of its numerator is less than that of its denominator. Corollary 3.7. Let F (λ) be of the form (3.4). If F (0) = 0, and F (λ) is proper in λ, then for any ρ, we have a reciprocity formula CT λ ρ F (λ) = − CT λρ F (λ), where both sides are regarded as elements in K. More generally, a rational function F is said to have the R-property with respect to ρ if CT Λ ρ F = (−1) d CT Λρ F. (3.5) for some integer d. Here we restrict our interest to the case when d equals r, the number of λ's. We have the following reciprocity formula for Elliott-rational functions. Theorem 3.8. Let F (Λ, x) be an Elliott-rational function and let ≤ ρ be a total ordering on Z n+r that is compatible with its additive group structure. Then CT Λρ F = (−1) r CT Λ ρ F + r−1 i=0 (−1) i CT λr,...,λ i+2 ρ I λ i+1 CT λ i ,...,λ 1 ρ F, (3.6) where CT ρ λ i ,...,λ 1 is the identity operator for i = 0 and similar for CTρ λr,...,λ i+2 when i = r − 1. Proof. Since we are always taking constant terms in λ i , we omit the λ for convenience. We compute the following in two different ways. r−1 i=0 (−1) i CT r,...,i+1 ρ CT i,...,1 ρ F + (−1) i CT r,...,i+2 ρ CT i+1,...,1 ρ F. (3.7) Using Theorem 3.5, we can rewrite (3.7) as r−1 i=0 (−1) i CT r,...,i+2 ρ I i+1 CT i,...,1 ρ F. On the other hand, most of the terms in (3.7) cancel with each other. The only terms left are given by CT r,...,2,1ρ F + (−1) r−1 CT r,...,2,1 ρ F. The proposition then follows. Theorem 3.8 gives the error term of the reciprocity formula. A different error term representation was given in [5] in terms of cohomology. However, the computation of this error term saved only a little work for general r. Our formula for the error term is true for any fixed order of λ 1 , . . . , λ r , and any fixed order of x 1 , . . . , x n . This suggests that some simplifications might exist and a better formula is possible. We have not succeeded in finding a such formula. Since simple equivalent condition for F to have the R-property is unlikely, we search for a sufficient condition. Corollary 3.9 and Proposition 3.10 below play important roles in our formulating the monster theorem. A rational function F is said to have the I-property with respect to ρ if for i = 1, 2, . . . , r, we have This result is a direct consequence of Theorem 3.8. The special case of this corollary when the Elliott-rational function has a monomial numerator was shown by a complicated computation in [4, Lemma 9.2]. I λ i CT λ i−1 ρ · · · CT λ 1 ρ F = 0. (3.8) In the case r = 1, Theorem 3.8 gives the equivalence between the I-property and the R-property. Moreover, we have, as shown below, a nice equivalent condition [4, Proposition 10.3] for the R-property that contains no algebraic expression. . Let E(x; b) be the crude generating function associated to an LD-system consisting of a single equation Aα = a 1 α 1 + · · · + a n α n = b: E(x; b) = λ −b (1 − λ a 1 x 1 ) · · · (1 − λ an x n ) . Then the following four conditions are equivalent for any ρ: (1) E(x; b) has the R-property. The proof of this proposition, which is not given in full detail here, proceeds by showing that CT λ=0 E(x; b) and CT λ=∞ E(x; b) have no common terms when expanded as Laurent series. The reader is referred to [4, Proposition 10.3] for details. The Monster Reciprocity Theorem Consider an LD-system Aα = b as in (2.1). The crude generating function E(Λ, x; b) is an Elliott-rational function with a monomial numerator. We say such a function has the matrix form since we are going to represent it by a matrix. The problem is to find a simple sufficient condition for E(Λ, x; b) to have the R-property. A homology version solution can be found in [5]. The best known result was Stanley's monster reciprocity theorem [4, Theorem 10.2], which says that the LD-system has the R-property if certain linear combinations of its equations have the R-property. We present here a simple approach to this problem. The central idea of our approach to this problem, as in [4], is to apply Corollary 3.9 and Proposition 3.10. If the following checking procedure returns a true, then E(Λ, x; b) has the I-property and hence the R-property with respect to ρ. Note that the converse of this statement is false. The checking procedure for E(Λ, x; b): (1) Let T 1 = E(Λ, x; b). If I λ 1 T 1 (Λ) = 0 then return false. (2) Write CT λ 1 ρ T 1 (Λ) as a sum of matrix forms in an efficient way. For every matrix form T 2 , if I λ 2 T 2 = 0, then return false. (3) Repeat the above step for every matrix form T 2 with respect to λ 2 , and then for every T 3 with respect to λ 3 , . . . , until we have checked if I λr T r (λ r ) = 0. If no false is returned, then return true. The basic tool in finding these T i 's is partial fraction decomposition of rational functions. Using the constant term operators seems neater than using residue operators as in [4]. Our task is to find a simple equivalent condition for the checking procedure to return a true. Such a condition will be our monster reciprocity theorem. In order to do so, we represent a matrix form T as an augmented matrix. In fact, we can keep track everything by adding a row of monomials in the x's on the top and a column of monomials in the λ's to the left of an LD-system. Therefore, the checking procedure will be done by matrix operations. Note that using matrix operations is one important aspect of the monster reciprocity theorem. We use the following identification: T = y n+1 Λ −b n i=1 (1 − Λ C i y i ) ≡ y 1 · · · y n y n+1 C 1 · · · C n b , where y i are monomials in x, and C i are column vectors. It would be clearer if we add λ i to the left of the ith row, but this is unnecessary after applying Lemma 2.2 and requiring that the ith row (with i ≥ 2) of T s is indexed by λ s+i−1 . The row operations we are going to perform will never involve the top row. The column operations, when acting on the first row, are treated as multiplications instead of additions for the obvious reason. We alow fractional entries and fractional powers. Roots of unity might appear, but will not be a trouble. Three special matrix operations will be useful. We define T ← C i to be the matrix obtained from T by adding −a 1,j /a 1,i times the ith column to the jth column for all j = i. This operation is exactly Gaussian column elimination by taking the (2, i)th entry of T as the pivot. Similarly we define the Gaussian row elimination T ← R i . The third operation T ← D i is defined to be the matrix obtained from T by deleting the second row and the i-th column. Combination of the operations will also be used from left to right. For instance, T ← CR i := T ← C i ← R i . Since row operations commute with column operations, we have T ← CR i = T ← RC i . It is easy to verify the following. T ← CRD i = T ← RCD i = T ← CD i . For example, if T is given by T = λ −b 1 λ −c 2 (1 − λ 3 1 x 1 /λ 2 )(1 − λ 2 x 2 /λ 1 )(1 − x 3 /λ 2 1 λ 2 ) ≡    x 1 x 2 x 3 1 3 −1 −2 b −1 1 −1 c    , (4.1) then T ← C 1 =    x 1 x 2 x 1 3 1 x 3 x 2 3 1 x − b 3 1 3 0 0 0 −1 2 3 − 5 3 c + b 3    , T ← R 1    x 1 x 2 x 3 1 3 −1 −2 b 0 2 3 − 5 3 c + b 3    , and T ← CD 1 = T ← CRD 1 = x 2 x 1 3 1 x 3 x 2 3 1 x − b 3 1 2 3 − 5 3 c + b 3 . The above three operations are generalized to sequences of integers. For instance, T ← R i 1 , . . . , i p is the matrix obtained from T by applying Gaussian row elimination by first taking the (2, i 1 )th entry of T as the pivot, then taking the (3, i 2 )th entry as the pivot, and so on. However, the elimination cannot go backwards. For instance, we are not allowed to eliminate the nonzero entries in the second row when taking the (3, i 2 )th entry as the pivot. More precisely, pick out the i 1 , . . . , i p th columns of T 1 , and rearrange them as follows: T 1 (i 1 , . . . , i p ) :=         x i 1 x i 2 · · · x ip a 1,i 1 a 1,i 2 · · · a 1,ip a 2,i 1 a 2,i 2 · · · a 2,ip . . . . . . . . . . . . a r,i 1 a r,i 2 · · · a r,ip         . If all the pivots encountering are nonzero, then when ignoring the top row T 1 (i 1 , . . . , i p ) ← R i 1 , . . . , i p =           x i 1 x i 2 · · · x ip a 1,i 1 a 1,i 2 · · · a 1,ip 0 a ′ 2,i 2 · · · a ′ 2,ip . . . . . . . . . . . . 0 0 · · · a ′ p,ip 0 0 · · · 0           (4.2) will be an upper triangular square matrix followed by a zero matrix, and T 1 (i 1 , . . . , i p ) ← RC i 1 , . . . , i p =           x i 1 y i 2 · · · y ip a 1,i 1 0 · · · 0 0 a ′ 2,i 2 · · · 0 . . . . . . . . . . . . 0 0 · · · a ′ p,ip 0 0 · · · 0           (4.3) will be a diagonal square matrix followed by a zero matrix. Since the matrix operations we have performed do not change the determinants, the a ′ s,is can be inductively computed by the formulas a ′ 1,i 1 = a 1,i 1 , and s j=1 a ′ j,i j = det(a k,i l ) 1≤k,l≤s . We denote by M(z 1 , . . . , z k ) is a generic monomial in z 1 , . . . , z k whose exact expression is not needed. Though we can formulate the monster reciprocity theorem for any ρ, the result seems nicer if we assume that ρ satisfies the following condition: ∀y = M(x), y ≻ ρ λ s ⇒ yM(λ s+1 , λ s+2 , . . . ) ≻ ≻ ρ λ s ,( ). For example, condition ( ) holds for any injective ρ such that ρ(x i ) is a monomial in x and ρ(λ i ) is a monomial in Λ. We will explain two such ρ in detail in the next section. Definition 4.1. With notation as in (4.3), we define (i 1 , . . . , i p ) of distinct entries ranging from 1 to n to be a contribution sequence of length p with respect to ρ if y sign(−a ′ s,is ) is ≺ ρ λ s for all s. The empty sequence is a contribution sequence of length 0. The name contribution sequence is in correspondence with the "pole sequence" in [4]. The condition in this definition will be replaced with simple ones for two special ρ in the next section. Theorem 4.2 (Generalized Monster Reciprocity Theorem). Let T be a matrix form corresponding to an r by n matrix of full rank, and let ρ be a total ordering on the group of monomials in Λ and x satisfying condition ( ). If for every contribution sequence (i 1 , . . . , i p ) of T with p < r, the second row of T ← RD i 1 , . . . , i p has the R-property, then T has the R-property with respect to ρ. Proof of Theorem 4.2. For given T and ρ . The checking procedure will return a true if and only if every T p encountered has the property that I λp T p = 0, which is the same as the condition that the second row of T p has the R-property by Proposition 3.10. We claim that T p must be similar to the following form for some contribution sequence (i 1 , . . . , i p−1 ): T 1 ← CD i 1 , . . . , i p−1 = T 1 ← RCD i 1 , . . . , i p−1 . (4.4) The term similar will be explained later. Assuming T p be given by (4.4), we can complete the proof of the theorem as follows. We observe that the C operations after the R operations do not affect the (p + 1)st row (and below) of T 1 . See (4.2). Therefore the second row of T p is the same as the second row of T 1 ← RD i 1 , . . . , i p−1 . We prove the claim by induction on p. The claim is trivial for p = 1. Now assume the claim is true for p = s and we need to show that the claim is true for p = s + 1. By choosing appropriate positive integer N and letting λ = λ 1/N s , (note that T s+1 will be independt of the choice of N), we can assume that T s (λ s ) = T ′ s (λ) = λ −bỹ m+1 m i=1 (1 − λ a iỹ i ) =    y 1 · · · y m y m+1 a 1 · · · a m b * · · · * *    , where a k and b are integers,ỹ k = y k M(λ s+1 , λ s+2 , . . . ), and the * 's are column of integers that we do not care. Dividing the second row of T ′ s by N will give us the the second row of T s . Since we have deleted s − 1 columns, m equals n − s + 1. We observe that y k = x k ′ M(x i 1 , . . . , x i s−1 ) for k ≤ m, where k ′ −k equals the number of j's such that k ′ > i j . Therefore y 1 , . . . , y m are independent of each other. It is now straightforward to check that the partial fraction decomposition of T ′ s (λ) with respect to λ is: T ′ s (λ) = L(λ) + m k=1 \|a k \| j=1 −ζ k,jỹ −1/a k k λ − ζ k,jỹ −1/a k k · (ζ k,jỹ −1/a k k ) −bỹ m+1 a k m i=1,i =k (1 − (ζ k,jỹ −1/a k k ) a iỹ i ) , where L(λ) is a Laurent polynomial in λ, and ζ k,j ranges over all a k th roots of unity. By Proposition 3.10, I λs T s = 0 implies that CT λs=∞ T s = 0 and hence CT ρ λs L(λ) = 0. Together with the fact that for any u independent of λ, CT λs ρ 1 λ − u = 0, if u ≻ ρ λ, (−u) −1 , if u ≺ ρ λ, we have CT λs ρ T s = k \|a k \| j=1 (ζ k,jỹ −1/a k k ) −bỹ m+1 a k m i=1,i =k (1 − (ζ k,jỹ −1/a k k ) a iỹ i ) , where the sum ranges over all k such that ζ k,jỹ −1/a k k ≺ ρ λ = λ 1/N s , which, by condition ( ), is equivalent to y sign(−a i ) i ≺ ρ λ s . For such k, we can check that (i 1 , . . . , i s−1 , k ′ ) is a contribution sequence. Let T k (x −1/a k k ′ ) = (ỹ −1/a k k ) −bỹ m+1 m i=1,i =k (1 − (ỹ −1/a k k ) a iỹ i ) = T s ← CD k , where we emphasize T k as a function of x −1/a k k ′ . By delaying the deletion procedure, we can check that T k (x −1/a k k ′ ) = T 1 ← CD i 1 , . . . , i s−1 , k ′ . Then CT λs T s is a sum of T s+1 's, each have the form T (j) k = 1 a k T k (ζ k,j x −1/a k k ′ ) for some j with 1 ≤ j ≤ \|a k \| and k with (i 1 , . . . , i s−1 , k ′ ) being a contribution sequence. We say that T (j) k is similar to T k and it is clear that T If T satisfies the condition in Theorem 4.2, then for p ≤ r, it follows from the proof that CT ρ λ 1 ,...,λp T can be expressed as a sum of group terms indexed by contribution sequences (i 1 , . . . , i p ) of T , with the corresponding group being a sum of terms similar to T 1 ← i 1 , . . . , i p . In particular, CT ρ Λ can be expressed as at most a sum of n(n − 1) · · · (n − r + 1) groups, since there are at most n(n − 1) · · · (n − r + 1) contribution sequences. A fast way to compute the sum for each group and an effective way to reduce the number of contribution sequences will result in an efficient algorithm for computing E(x, b). Examples and Applications To apply Theorem 4.2 to a particular LD-system, we need to choose a working field C ρ Λ, x to work with. The choice of ρ is not unique, but we will concentrate on two special cases that always work. One is the case that ρ is the identity; the other is equivalent to that in [4]. In both cases, we can simplify the condition in finding the contribution sequence. Case 1: Let C Λ, x be the working field. The condition y sign(−a ′ s,is ) is ≺ λ s in Definition 4.1 can be replaced with sign(a ′ s,is k l ) > 0, where if we write y is = x k 1 1 · · · x kn n , then l is the largest such that k l = 0. In practice, we put the sign of k l at the upper front of y j . Example 5.1. Let (E, (b, c)) the following LD-system: 3α 1 − α 2 − 2α 3 = b, −α 1 + α 2 − α 3 = c. Then the crude generating function T = T 1 of this LD-system is given by (4.1). Using Maple, we find that (E, (b, c)) has the R-property for all (b, c) plotted by •, and has I-property for all (b, c) plotted by • in the following Figure 1, where we tested all −12 ≤ b, c ≤ 12. Thus the R-property does not implies the I-property. Let C Λ, x be the working field. We want to apply Theorem 4.2 to find such pairs. Since the second row of T 1 has only one positive entries, only (1) is a contribution sequence of length 1. So after eliminating λ 1 , we get a sum of three terms similar to T 2 given by T 2 = T 1 ← CD 1 = ⊘ + x 2 x 1 3 1 + x 3 x 2 3 1 x − b 3 1 ⊘ 2 3 − 5 3 c + b 3 , where we kept the first column to keep track the original column numbers. Now it is easy to see that the only contribution sequence of length 2 is (1, 2), though we do not need it. Therefore, Theorem 4.2 tells us that the LD-system has the R-property if the following two equations have the R-property: 3α 1 −α 2 −2α 3 = b, 2α 2 −5α 3 = 3c + b. Using Maple, we find all such (b, c) as plotted by • in Figure 1: Example 5.2. Consider the equivalent LD-system (E ′ (−c, b)): α 1 − α 2 + α 3 = −c, 3α 1 − α 2 − 2α 3 = b, where we multiplied both sides of the second equation by −1 and switched the two equations. We need to find (b, c) for S to have the R-property, where S =    x 1 x 2 x 3 1 1 −1 1 −c 3 −1 −2 b    . This time we have two contribution sequences of length 1: (1) and (3). Therefore, Theorem 4.2 tells us that the LD-system has the R-property if the following three equations have the R-property, where the second and third equation are from S ← 1 and S ← 3 : α 1 −α 2 +α 3 = −c, 2α 2 −5α 3 = b + 3c, 5α 1 −3α 2 = b − 2c. Using Maple, we obtain the same pairs (b, c) as in the previous one, i.e., those plotted by • in Figure 1. All these three equations are needed to apply Theorem 4.2. The following coincidence is worth mentioning: if we only consider the second and the third equation, we will get all (b, c) plotted by • in Figure 1, i.e., those (b, c) such that (E ′ , (b, c)) has the R-property. Since the first equation comes from the empty contribution sequence, we come back to check the previous example, which is obviously not the case. Case 2: Let ρ be the injective homomorphism into C x, Λ, t by ρ(x i ) = x i t and ρ(λ i ) = λ r−i+1 . Then the condition in Definition 4.1 can be replaced with sign(da ′ s,is ) > 0, where d is the total degree of y is in the x's. Since we only need to keep track of the total degree of the x's, the x i in the top row of T can be replaced with 1. The monster reciprocity theorem obtained this way is similar to that of [4], in which the computation used integration along the circles \|λ i \| = 1 − ǫ i with 1 > > ǫ 1 > > ǫ 2 > > · · · , where > > means "much greater", and the x i is taken to satisfy \|x i \| = δ < 1 for some positive real number δ. In fact, the condition as in Definition 4.1 was completely written in terms of determinants. Detailed example for this case, which will not be given here, can be found in [4, p. 245]. Now let us consider Linear homogeneous Diophantine system (LHD-system for short). We shall use Theorem 4.2 to derive the following theorem, which implies the reciprocal domain theorem [4, Proposition 8.3] including Theorem 1.1. If we let ρ be the identity map, then we get Theorem 1.1. If we let ρ(x i ) = x i for i = 1, . . . , p and ρ(x i ) = x −1 i for i = p + 1, . . . n for p with 1 < p < n, and ρ(λ i ) = λ i for all i, then we will get the reciprocal domain theorem [4, Proposition 8.3]. Proof of Theorem 5. 3. If T has the R-property, then CT Λ ρ T = (−1) r CT Λρ T. The implication thus follows. Now we show the converse is true. Obviously we can suppose r > 0, CT ρ Λ T = 0 and CTρ Λ T = 0. We first show that the second row of T has the R-property. Since T corresponds to an LHD-system, we can write T = 1 n i=1 (1 −ỹ i λ a i 1 ) , whereỹ i is a monomial independent of λ 1 . If some of the a i are positive and some of the a i are negative, then Corollary 3.7 applies and the second row of T has the R-property. Otherwise, one of CT λ 1 =0 T and CT λ 1 =∞ T will be 0 and the other will be nonzero. (Note that since the LHD-system has full rank, the case that a i = 0 for all i will not happen.) The statement then follows by Proposition 3.10. Now by Lemma 2.2, if T ′ is obtained from T by elementary row operations, then CT ρ Λ T ′ = 0 and CTρ Λ T ′ = 0. Therefore, the second row of T ′ has the R-property. This means every linear combination of the equations of T has the R-property. Thus the theorem follows from Theorem 4.2. Remark 5.4. The proof of the theorem, together with Remark 4.3, in fact shows the following statement: If CT ρ Λ T = 0 and CTρ Λ T = 0, then CT ρ λ 1 ,...,λp T is proper in all λ i for i > p. On the other hand, a simple proof of the statement will lead to a simple proof of Theorem 1.1. If we restrict ourself in C[Λ, Λ −1 ][[x]], the best known proof of Theorem 1.1 should be that given by the author in [9], which is included in the next section. The above remark suggest a way to reduce the number of contribution sequences of an LHD-system: Following the notation as in Remark 4.3, since every T p has the R-property for λ p , we have a choice to choose all those terms with contribution or all those terms (with a minus sign) without contribution. The author is managing to develop a computer program implementing these techniques. Linear Homogeneous Diophantine Systems We are concentrating on linear homogeneous Diophantine systems (LHD-systems for short), i.e., Aα = 0. Recall that C i is the ith column vector of A. We omit the 0 so that E andĒ are the sets of all solutions of Aα = 0 in N n and P n respectively, and similar for other notations. Since the proof closely related the linear system and its associate generating functions, we restate them as follows. E(x) = CT Λ E(x),Ē(x) = (−1) n CT Λ E(x −1 ). (6.1) We are going to prove Proposition 2.1, i.e., to show that ifĒ is nonempty, then CT Λ E(x) = (−1) rank(A) CT Λ ρ E(x), (6.2) where we are taking constant term of MN-series and ρ(x i ) = x −1 i for all i. We shall see that all of the work is done algebraically. First, let us see some facts. Exchanging column i and j corresponds to exchanging x i and x j . Row operations, which will not change the solutions of Aα = 0, are equivalent to multiplying A on the left by an invertible matrix. This fact can be obtained by applying Lemma 2.2. Let us see the simple case of r = 1. In this case, E(x) has the form: E(x) = n i=1 1 1 − λ a i x i . The condition thatĒ is nonempty is equivalent to saying that some of a i have to be positive and some of a i have to be negative. Thus when written in the normal form of a rational function in λ, E(x) is proper and its numerator divides λ. So Proposition 2.1 follows from Corollary 3.7. The general case does not seem to work along this line because of two problems. One is how to use the conditions thatĒ is nonempty, and the other is how to connect to the rank of A. The proof we are going to give uses induction and Elliott's reduction identity [1, p. 111-114], which is easy to check and is not given here. Clearly if a 11 , . . . , a 1,n are all positive or are all negative, thenĒ is empty. So we can assume that a 11 > 0 and a 12 < 0. Applying Elliott's reduction identity on λ 1 , we get: E(x) = 1 1 − Λ C 1 +C 2 x 1 x 2 1 1 − Λ C 1 x 1 + 1 1 − Λ C 2 x 2 − 1 i≥3 1 1 − Λ C i x i Now expand E(x) according to the middle term, and denote the resulting three summans by E 1 , E 2 , and E 3 respectively. We have E(x) = E 1 (x 1 , x 1 x 2 , x 3 , . . . ) + E 2 (x 1 x 2 , x 2 , x 3 , . . . ) − E 3 (x 1 x 2 , x 3 , . . . ). (6.3) Then these E i are very similar to E. Correspondingly, they are associated to matrices, and hence solution spaces that lie in N n and P n . More precisely, E i , i = 1, 2, 3, are associated to A 1 = (C 1 , C 1 + C 2 , C 3 , . . . , C n ), A 2 = (C 1 + C 2 , C 2 , C 3 , . . . , C n ), and A 3 = (C 1 + C 2 , C 3 , . . . , C n ) respectively. Thus E i , E i (x) andĒ i ,Ē i (x) are defined correspondingly. Now the matrix A 1 is obtained from A by adding the second column to the first; the matrix A 2 is obtained from A by adding the first column to the second. They are obtained from A through a column operation. So the rank of A 1 and A 2 are both equal to that of A. The rank of A 3 might not equal the rank of A. Applying CT Λ and (−1) n CT ρ Λ to (6.3) respectively, we get our key induction equations. E(x) = E 1 (x 1 , x 1 x 2 , x 3 , . . . ) + E 2 (x 1 x 2 , x 2 , x 3 , . . . ) − E 3 (x 1 x 2 , x 3 , . . . ), (6.4)Ē (x) =Ē 1 (x 1 , x 1 x 2 , x 3 , . . . ) +Ē 2 (x 1 x 2 , x 2 , x 3 , . . . ) + (−1) rank(A)−rank(A 3 )Ē 3 (x 1 x 2 , x 3 , . . . ). (6.5) Looking more closely at these E i , we can see that up to isomorphism, E 1 , E 2 , and E 3 are obtained from E by intersecting the half spaces α 1 ≥ α 2 , α 1 ≤ α 2 , and the hyperplane α 1 = α 2 respectively. For instance, (α 1 , α 2 , . . . , ) belongs to E with α 1 ≥ α 2 if and only if (α 1 − α 2 , α 2 , . . . ) belongs to E 1 . Thus Elliott's reduction identity in fact corresponds to a signed decomposition of E. Equation (6.4) and (6.5) could be explained directly from geometry. We need two more lemmas to give our proof of Proposition 2.1. We shall see that the condition onĒ plays an important role. IfĒ is nonempty, then dim E = dimĒ = n − rank(A). Clearly, the dimension of the solution space of Aα = 0 is n − rank(A). Let γ ∈Ē, and let Υ 1 , . . . , Υ n−rank(A) be a Z-basis of the solution space in Z n with Υ 1 = γ. Then for sufficiently large m, mγ + Υ 1 , . . . , mγ + Υ n−rank(A) will be a linearly independent set inĒ. Lemma 6.1. Suppose thatĒ is nonempty, and thatĒ i is defined as above for i = 1, 2, 3. Then any two of theĒ i being nonempty implies that they are all nonempty. Proof. Suppose thatĒ 1 andĒ 2 are nonempty. Then we have elements β and γ inĒ such that β = (β 1 , β 2 , . . . ) with β 1 > β 2 and γ = (γ 1 , γ 2 , . . . ) with γ 1 < γ 2 . Then (γ 2 − γ 1 )β + (β 1 − β 2 )γ is inĒ with the first two entries being equal. This meansĒ 3 is nonempty. Suppose thatĒ 1 andĒ 3 are nonempty. Then we have elements β and δ inĒ such that β = (β 1 , β 2 , . . . ) with β 1 > β 2 and δ = (δ 1 , δ 2 , . . . ) with δ 1 = δ 2 . Then for sufficiently large m, mδ − β is inĒ with the first entry being smaller than the second. This meansĒ 2 is nonempty. The case thatĒ 2 andĒ 3 are nonempty is similar to the previous case. Lemma 6.2. If all of theĒ i are nonempty, then rank(A 3 ) = rank(A). Proof. By hypothesis, it is clear that E is not contained in the hyperplane α 1 = α 2 . Thus the intersection of E with the hyperplane has dimension dim E − 1. So dim E 3 is also dim E − 1 and the rank of A 3 equals n − 1 − dim E 3 = rank(A). Proof of Proposition 2.1. The base case, when A is the zero matrix, is trivial. By exchanging rows, we can assume that not all of the entries in the first row of A are zero. Moreover, since the entries can not be all positive or negative, we can assume the first entry is positive and the second is negative by exchanging columns. We use induction on S 1 (A), which is defined to be the sum of the absolute values of all the entries in the first row. Now the above argument applies, and it is easy to see that S 1 (A i ) < S 1 (A) for i = 1, 2, 3. Applying Lemma 6.1, we can reduce the seven cases of E i being nonempty or not into the following four cases: Case 1: onlyĒ 1 is nonempty. Let β inĒ be such that β 1 > β 2 . We claim that all α with Aα = 0 satisfy the condition α 1 > α 2 , so that E 2 (x 1 x 2 , x 2 , x 3 , . . . ) equals E 3 (x 1 x 2 , x 3 , . . . ), and hence by induction we have E(x) = E 1 (x 1 , x 1 x 2 , x 3 , . . . ) = (−1) rank(n−A 1 )Ē 1 (x −1 1 , x −1 1 x −1 2 , x −1 3 , . . . ) = (−1) n−rank(A)Ē (x −1 ). If the claim does not hold, then α 1 ≤ α 2 . But for sufficiently large m, mβ − α will produce an element inĒ 2 orĒ 3 , a contradiction. Case 2: onlyĒ 2 is nonempty. This is similar to case 1. Case 3: onlyĒ 3 is nonempty. This means that E is contained in the hyperplane α 1 = α 2 . Thus for i = 1, 2, and that E 3 (x 2 , x 3 , . . . ) = (−1) n−1−rank A 3Ē (x −1 2 , x −1 3 , . . . ). Lemma 2.2 ([7], Corollary 3.18). Suppose y is another set of variables. Corollary 3. 9 . 9If an Elliott-rational function has the I-property, then it has the Rproperty. ( 2 ) 2E(x; b) has the I-property. (3) CT λ=0 E(x; b) = 0 and CT λ=∞ E(x; b) = 0. (4) The following two conditions are both satisfied: (a) There does not exist a β ∈ Z with Aβ = b such that β e < 0 if a e > 0 and β e ≥ 0 if a e < 0. (b) There does not exist a γ ∈ Z with Aγ = b such that γ e ≥ 0 if a e > 0 and γ e < 0 if a e < 0. I-property if and only if T k has. This completes the proof of the claim. Figure 1 . 1The R-property and I-property for (E, (b, c)). Theorem 5 . 3 . 53Suppose T is a matrix form corresponding to an LHD-system of full rank. Then for any ρ satisfying ( ), T has the R-property if and only if either CT ρ Λ T = CTρ Λ T = 0 or CT ρ Λ T = 0 and CTρ Λ T = 0. E 1 1(x 1 , x 1 x 2 , x 3 , . . . ) = E 2 (x 1 x 2 , x 2 , x 3 , . . . ) = E 3 (x 1 x 2 , x 3 , . . . ), and we have rank(A 3 ) = n − 1 − dim(E 3 ) = n − dim(E) − 1 = rank(A) − 1. So E(x) = E 3 (x 1 x 2 , x 3 , . . . ) = (−1) n−1−rank(A 3 )Ē 3 (x −1 1 x −1 2 , x −1 3 , . . . ) = (−1) n−rank(A)Ē (x −1 ). Case 4: all ofĒ i are nonempty. By induction, we see that E i (x) = (−1) n−rank(A i )Ē i (x −1 ) From Lemma 6.2, rank(A 3 ) = rank(A). Thus together with our key induction equations (6.4) and (6.5), we getAcknowledgement: I am very grateful to Richard Stanley for introducing me to his inspiring work. . P A Macmahon, Combinatory Analysis. 2ChelseaReprintedP. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, Cam- bridge, 1915-1916, Reprinted: Chelsea, New York, 1960. The Algebraic Structure of Group Rings. D S Passmann, Wiley-InterscienceNew YorkD. S. Passmann, The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1985. Linear homogeneous Diophantine equations and magic labelings of graphs. R P Stanley, Enumerative Combinatorics. Cambridge University Press40Invent. Math.. 2nd ed.R. P. Stanley, Linear homogeneous Diophantine equations and magic labelings of graphs, Duke Math. J. 40 (1973), 607-632. 4. , Combinatorial reciprocity theorems, Adv. in Math. 14 (1974), 194-253. 5. , Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), 175-193. 6. , Enumerative Combinatorics, 2nd ed., vol. 1, Cambridge University Press, 1997. The Ring of Malcev-Neumann Series and the Residue Theorem. G Xin, arXiv:math.CO/0409190.8arXiv:math.CO/0405133R58, 20 pp. 9. 11Brandeis University ; Department of Mathematics, Brandeis UniversityA fast algorithm for MacMahon's partition analysis. Ph.D. thesis. Waltham MA 02454-9110 E-mail address: [email protected]. Xin, A residue theorem for Malcev-Neumann series, Adv. Appl. Math., to appear, arXiv:math.CO/0409190. 8. , A fast algorithm for MacMahon's partition analysis, Electron. J. Combin. 11 (2004), R58, 20 pp. 9. , The Ring of Malcev-Neumann Series and the Residue Theorem, Ph.D. thesis, Brandeis University, 2004, arXiv:math.CO/0405133. Department of Mathematics, Brandeis University, Waltham MA 02454-9110 E-mail address: [email protected]	[]
[ "Projected free energies for polydisperse phase equilibria", "Projected free energies for polydisperse phase equilibria" ]	[ "Peter Sollich \nDepartment of Physics and Astronomy\nUniversity of Edinburgh\nEH9 3JZEdinburghU.K\n", "Michael E Cates \nDepartment of Physics and Astronomy\nUniversity of Edinburgh\nEH9 3JZEdinburghU.K\n" ]	[ "Department of Physics and Astronomy\nUniversity of Edinburgh\nEH9 3JZEdinburghU.K", "Department of Physics and Astronomy\nUniversity of Edinburgh\nEH9 3JZEdinburghU.K" ]	[ "Physical Review Letters" ]	A 'polydisperse' system has an infinite number of conserved densities. We give a rational procedure for projecting its infinite-dimensional free energy surface onto a subspace comprising a finite number of linear combinations of densities ('moments'), in which the phase behavior is then found as usual. If the excess free energy of the system depends only on the moments used, exact cloud, shadow and spinodal curves result; two-and multi-phase regions are approximate, but refinable indefinitely by adding extra moments. The approach is computationally robust and gives new geometrical insights into the thermodynamics of polydispersity.	10.1103/physrevlett.80.1365	[ "https://arxiv.org/pdf/cond-mat/9711312v1.pdf" ]	14,180,634	cond-mat/9711312	b9e4452a177cba8b17f92ec039d02a4401906725	Projected free energies for polydisperse phase equilibria 28 Nov 1997 Peter Sollich Department of Physics and Astronomy University of Edinburgh EH9 3JZEdinburghU.K Michael E Cates Department of Physics and Astronomy University of Edinburgh EH9 3JZEdinburghU.K Projected free energies for polydisperse phase equilibria Physical Review Letters 28 Nov 1997 A 'polydisperse' system has an infinite number of conserved densities. We give a rational procedure for projecting its infinite-dimensional free energy surface onto a subspace comprising a finite number of linear combinations of densities ('moments'), in which the phase behavior is then found as usual. If the excess free energy of the system depends only on the moments used, exact cloud, shadow and spinodal curves result; two-and multi-phase regions are approximate, but refinable indefinitely by adding extra moments. The approach is computationally robust and gives new geometrical insights into the thermodynamics of polydispersity. struct from f [ρ(σ)] an (optimally) projected free energy surface in a reduced subspace of density variables. For these variables, we choose linear combinations of densities, the 'generalized moments' m i = dσ w i (σ)ρ(σ) of ρ(σ), defined by certain weight functions w i (σ); these are ordinary (non-normalized) moments if w i (σ) = σ i . The simplest imaginable case is where the free energy f depends only on a finite set of K such moments: f = f (m i ), i = 1 . . . K(1) In coexisting phases one demands equality of particle chemical potentials, defined as µ(σ) = δf /δρ(σ) = i (∂f /∂m i )w i (σ) = i µ i w i (σ), for all σ. But this implies that all 'moment' chemical potentials, µ i ≡ ∂f /∂m i , are likewise equal among phases. The osmotic pressures Π of all phases also must be equal; simple algebra establishes that −Π = f − i µ i m i which involves only the moments m i and their chemical potentials µ i . Finally, if the overall σ-distribution is ρ (0) (σ), and there are p coexisting phases with σ-distributions ρ (α) (σ), each occupying a fraction φ (α) of the total volume (α = 1 . . . p), then conservation of particles implies the usual 'lever rule' (or material balance) among species: α φ (α) ρ (α) (σ) = ρ (0) (σ), ∀σ. Multiplying this by a weight function w i (σ) and integrating over σ shows that the lever rule also holds for the moments: p α=1 φ (α) m (α) i = m (0) i(2) These results express the fact that any linear combination of conserved densities (a generalized moment) is itself a conserved density in thermodynamics. Therefore, if the free energy of the system depends only on K moments m i . . . m K we can view these as the densities of K 'quasispecies' of particles, and construct the phase diagram via the usual construction of tangencies and the lever rule. Formally this has reduced the problem to finite dimensionality by a projection, although this is trivial here because f , by construction, has no dependence on any variables other than the m i (i = 1 . . . K). Of course, it is uncommon for the free energy f to obey (1). In particular, the 'ideal gas' (or, for polymers, Flory-Huggins) entropy term, in mixtures of many species, is definitely not of this form. On the other hand, in very many thermodynamic (especially mean field) models the free energy takes the form (k B = 1) f =f (m i ) + T dσ ρ(σ) [ln (ρ(σ)/R(σ)) − 1] ,(3) in which the excess free energyf does depend only on K moments. Examples include polydisperse hard spheres [8], polydisperse homo-and copolymers [2][3][4][5]9], and van der Waals fluids with factorized interaction parameters [10]. Note that, in the ideal gas term of (3), we have included a dimensional factor R(σ) inside the logarithm: since the resulting contribution is linear in densities, this has no effect in rigorous thermodynamics. However, it will play a central role in our approach. In principle, the phase equilibria stemming from (3) can be computed exactly by a finite algorithm. Specifically, the spinodal stability criterion involves a Kdimensional square matrix [11][12][13][14] whereas calculation of p-phase equilibrium involves solution of (p − 1)(K + 1) strongly coupled nonlinear equations. This method has certainly proved useful [2][3][4]9,10,14], but is cumbersome, particularly if one is interested mainly in cloud-and shadow-curves, rather than coexisting compositions deep within multi-phase regions [2,4,7,9]. Various ways of simplifying the procedure exist [6,11,[15][16][17], but there has been, up to now, no systematic alternative to the full computation. Note also that the nonlinear phase equilibrium equations permit no simple geometrical interpretation or qualitative insight akin to the familar rules for constructing phase diagrams from the free energy surface of a finite mixture. Our method instead proceeds by deriving from (3) a 'projected' free energy that depends only on a finite set of moments. We argue that the most important moments to treat correctly are those that actually appear in the excess free energyf (m i ). Accordingly we divide the infinite-dimensional space of σ-distributions into two orthogonal subspaces: a 'moment subspace', which contains all the degrees of freedom of ρ(σ) that contribute to the moments m i (this subspace is spanned by the weight functions w i (σ)), and a 'transverse subspace' which contains all remaining degrees of freedom (as can be varied without affecting the chosen moments m i ). Physically, it is reasonable to expect that these 'leftover' degrees of freedom play a relatively minor role in the phase equilibria of the system, a view justified a posteriori below. Accordingly, we now allow violations of the lever rule, so long as these occur solely in the transverse space. The 'transverse' degrees of freedom, instead of obeying the strict particle conservation laws, are chosen so as to minimize the free energy: they are treated as 'annealed'. If, as assumed above,f =f (m i ) only depends on the moments retained, this amounts to maximizing the entropy in (3), while holding fixed the values of the moments m i . At this point, the factor R(σ) in (3), which is immaterial if all conservation laws are strictly obeyed, becomes central. Indeed, maximizing the entropy over all distributions ρ(σ) at fixed moments m i yields ρ(σ) = R(σ) exp i λ i w i (σ)(4) where the Lagrange multipliers λ i are chosen to give m i = dσ w i (σ) R(σ) exp i λ i w i (σ)(5) The corresponding minimum value of f then defines our projected (i.e., annealed) free energy f pr (m i ) =f + T i λ i m i − m 0(6) In the last term, m 0 = dσ ρ(σ) is the 'zeroth moment' which is identical to the overall particle density ρ defined previously. If this is among the moments used for the projection, the resulting linear term can be dropped; otherwise it must be retained (with m 0 now expressed as a function of the m i , via the λ i ). Our maximum entropy method yields a free energy f pr (m i ) which only depends on the chosen set of moments: i.e., (6) is of the form (1) [18]. A finite dimensional phase diagram can now be constructed from it according to the usual rules. Obviously, though, the results now depend on R(σ) which is formally a 'prior distribution' for the entropy maximization. To understand its thermodynamic role, we recall that our projected free energy f pr (m i ) was constructed as the minimum of f [ρ(σ)] at fixed m i ; that is, f pr is the lower envelope of the projection of f onto the moment subspace. Crucially, the shape of this envelope depends on how, by choosing a particular prior distribution R(σ), we 'tilt' the infinitedimensional free energy surface before projecting it. To find the optimum choice of prior, we note that R(σ) serves physically to determine which distributions ρ(σ) lie within the maximum-entropy family (4) that the annealed system can have. Typically, one is interested in a system where a fixed overall 'parent' (or 'feed') distribution ρ (0) (σ) becomes subject to separation into various phases. In such circumstances, we should generally choose this parent distribution as our prior, R(σ) = ρ (0) (σ), thereby guaranteeing that it is contained within the family (4). Having done this, we note that the annealing procedure will be exactly valid, to whatever extent the σ-distributions actually arising in the various coexisting phases of the system under study are members of the family (4). (This statement of exactness, and similar ones below, of course hold only if (3) is valid.) In fact, the condition just described does hold whenever all but one of a set of coexisting phases are of infinitesimal volume compared to the majority phase. This is because the σ-distribution, ρ (0) (σ), of the majority phase is negligibly perturbed, whereas that in each minority phase differs from this by an exponential Gibbs-Boltzmann factor, of exactly the form required for (4). Accordingly, our projection method yields exact cloudcurves and shadow-curves. By the same argument, critical points (which in fact lie at the intersection of these two curves) are exactly determined. Moreover, all spinodals are also found exactly by our annealing method. For, at a spinodal, there exists an instability direction (in the full space) along which the curvature of the free energy vanishes; in all other directions f has positive curvature. One can show that such an instability direction always connects neighboring distributions within the same maximum entropy family (4), and hence that only the free energy of such distributions (i.e., the projected free energy with the parental R(σ)) is needed to calculate spinodals. The geometrical interpretation of this result, and also proofs of it and the others stated above, will be given elsewhere [19]. The method does, however, give only approximate results for coexistences involving finite amounts of different phases. This is because linear combinations of different σ-distributions obeying (4), corresponding to two (or more) phases arising from the same parent (ρ (0) (σ) = R(σ)) do not necessarily add to recover the parent distribution itself. Moreover, according to Gibbs' phase rule, a projected free energy depending on n moments will not normally predict more than n + 1 coexisting phases, whereas a polydisperse system can in principle separate into an arbitrary number of phases. Both of these shortcomings can be overcome by systematically including additional moments within the annealing procedure. (The above exact results are unaffected, because these do not exclude a null dependence off on certain of the m i .) Indeed, by adding further moments one can indefinitely expand the maximum-entropy family (4) of σ-distributions, thereby approaching with increasing precision the actual distributions in all phases present; this yields phase diagrams of ever-refined accuracy. How quickly convergence to the exact results occurs depends on the choice of weights functions for the additional moments; this will be quantified elsewhere [19]. To demonstrate the power of our approach, we consider a specific example. This is a simplified model of chemical fractionation, in which one considers species of continuously variable chemical character (such as aromaticity) governed by a parameter σ between 0 and 1. We suppose that the interaction energy between species varies as (σ − σ ′ ) 2 , so that the most different species repel each other most strongly. For simplicity we take a molten system, choosing volume units so that the overall density is constrained as dσ ρ(σ) = 1. Within a mean-field treat- ment, the system is then described by a free energy of the form (3), with an excess free energy (in units of k B T ) off = −χm 2 1 , m 1 = dσ σρ(σ) (up to irrelevant terms linear in ρ(σ)). This model differs only by a rescaling of parameters (with powers of polymer molecular weight) from the Flory-Huggins treatment of random AB copolymers, in which χ is the usual interaction parameter and σ is the proportion of A monomers in a chain [2][3][4]. The model should show fractionation into an ever-increasing number of phases as χ is increased. It is therefore an interesting test case for our projection approach (and the method of adding further moments), yet simple enough for exact phase equilibrium calculations to remain feasible, allowing detailed comparisons to be made. We consider phase separation from parent phases with σ distributions of the form ρ(σ) ∝ exp(λσ) (for 0 ≤ σ ≤ 1); λ is thereby fixed in terms of the parental m 1 = m 1 . Even for the minimal set of moments (n = 1) the point where phase separation first occurs on increasing χ is predicted correctly (this is a cloud point for the given parent). As more moments are added, the annealed coexistence curves approach the exact one to higher and higher precision [20]. As expected, the precision decreases at high χ, where fractionated phases proliferate; in this region, the number of coexisting phases predicted by the projection method increases with n. However, it is not always equal to n + 1, as one might expect from a naive use of Gibbs' phase rule; three-phase coexistence, for example, is first predicted for n = 4 [21]. Note that the stability of the results to the addition of extra moments provides, in this example, a good test of convergence on the coexistence curves. For the computational implementation of both the annealing and the exact method we used a Newton-Raphson nonlinear equation solver. The annealed calculation turned out to be significantly more robust with respect to the choice of initial values, the size of χ increments etc. due to an effective decoupling of the equations: Equality of chemical potentials is achieved using the moments contained in the excess free energy, while the lever rule is satisfied (increasingly accurately) using the remaining moments [19]. This advantage should be much more pronounced in more complex cases, as should savings in computer time (which are modest in our simple example). With exact results for cloud-and shadow-curves, critical points and spinodals, as well as refinably accurate coexistence curves and multi-phase regions, our annealing method allows rapid and accurate computation of the phase behavior of many polydisperse systems. Moreover, by establishing the link to a projected free energy f pr (m i ) as a function of a finite set of conserved densities m i , it restores to the problem much of the geometrical interpretation and insight (as well as the computational methodology) associated with phase diagrams for finite mixtures. This contrasts with procedures commonly used for systems in which the excess free energy involves a finite set of moments (3) [2][3][4][5]9,10]. Some previous approximations to that problem have used (generalized) moments as coordinates; see e.g. [11,14,15,17,22]. Our annealing method provides a rational basis for these methods and, by a careful choice of prior, guarantees that many properties of interest are found exactly. Finally, our method may extend to models for which the excess free energy cannot be written directly in terms of a finite number of moments as in (3). For example, many mean-field theories correspond to a variational minimization of the free energy: F ≤ E 0 − T S 0 , where subscript 0 refers to a trial Hamiltonian [23]. In such a case, one might choose to first make a physically motivated decision about which (and how many) moments m i to keep, and then include among the variational parameters the annealed "transverse" degrees of freedom. This would lead directly to a mean-field estimate of the projected free energy without assuming Eq. (3). Note that a good choice of prior R(σ) will again be important. Although no exact results can be guaranteed, this approach may form a promising basis for future developments. Acknowledgements: After this work was substantially complete, we learned from P. B. Warren [24] that he has independently developed an approach which, though based on distinctly different principles, yields a formalism broadly equivalent to our own [19]. We thank him, and also N. Clarke, R. M. L. Evans, T. McLeish, P. Olmsted, and W. C. K. Poon, for helpful discussions. * are the values of m1 of the coexisting phases; horizontal lines guide the eye where new phases appear. Curves are labeled by n, the number of moments retained in the projected free energy. Predictions for n = 10 are indistinguishable from an exact calculation (in bold). the predictions from our projected free energy with n moments (m i = dσ σ i ρ(σ), i = 1 . . . n) retained. Comparable results are found for other m (0) . Royal Society Dorothy Hodgkin Research Fellow. Electronic address: [email protected]. Royal Society Dorothy Hodgkin Research Fellow. Elec- tronic address: [email protected]. R T Dehoff, Thermodynamics in Material Science. New YorkMcGraw-HillR. T. DeHoff, Thermodynamics in Material Science (McGraw-Hill, New York, 1992). . B J Bauer, Polymer Eng. Sci. 251081B. J. Bauer, Polymer Eng. Sci. 25, 1081 (1985). . M T Rätzsch, C Wohlfarth, Adv. Polymer Sci. 9849M. T. Rätzsch and C. Wohlfarth, Adv. 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London A. 375397P. Irvine and M. Gordon, Proc. R. Soc. London A 375, 397 (1981). . S Beerbaum, J Bergmann, H Kehlen, M T Rätzsch, Proc. R. Soc. London A. 406103S. Beerbaum, J. Bergmann, H. Kehlen, and M. T. Rätzsch, Proc. R. Soc. London A 406, 63 (1986) and 414, 103 (1987). . E M Hendriks, Ind. Eng. Chem. Res. 271728E. M. Hendriks, Ind. Eng. Chem. Res. 27, 1728 (1988). . E M Hendriks, A R D Vanbergen, Fluid Phase Eq. 7417E. M. Hendriks and A. R. D. Vanbergen, Fluid Phase Eq. 74, 17 (1992). . R L Cotterman, J M Prausnitz, Ind. Eng. Chem. Proc. Des. Dev. 24434R. L. Cotterman and J. M. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev. 24, 434 (1985). . S K Shibata, S I Sandler, R A Behrens, Chem. Eng. Sci. 421977S. K. Shibata, S. I. Sandler, and R. A. Behrens, Chem. Eng. Sci. 42, 1977 (1987). . P Bartlett, J. Chem. Phys. 107188P. Bartlett, J. Chem. Phys. 107, 188 (1997). Numerically, it is easier to view the annealed free energy as a function of the Lagrange multipliers λi rather than the moments mi, to avoid having to invert eqs. Numerically, it is easier to view the annealed free energy as a function of the Lagrange multipliers λi rather than the moments mi, to avoid having to invert eqs. (5). . M E Cates, P Sollich, P B Warren, in preparationM. E. Cates, P. Sollich, and P. B. Warren, in preparation. The precision of the annealed coexistence curves should depend on the total number of moments n, rather than the number of moments in the excess free energy, K. Hence the minimal annealed theory (n = K), poor for our K = 1 case, should perform better for more realistic (larger K) models. Computational effort, on the other hand. is mainly sensitive to K (the dimension of the space to be searched for new phases), rather than nThe precision of the annealed coexistence curves should depend on the total number of moments n, rather than the number of moments in the excess free energy, K. Hence the minimal annealed theory (n = K), poor for our K = 1 case, should perform better for more realistic (larger K) models. Computational effort, on the other hand, is mainly sensitive to K (the dimension of the space to be searched for new phases), rather than n. For fractionation problems such as this (but not more generally), the low temperature limit suggests that to obtain n + 1 phases, one may have to include up to 2n moments. Using localized weight functions (rather than powers of σ) for the extra moments can reduce this number. but gives less accurate coexistence curves [19For fractionation problems such as this (but not more generally), the low temperature limit suggests that to obtain n + 1 phases, one may have to include up to 2n moments. Using localized weight functions (rather than powers of σ) for the extra moments can reduce this num- ber, but gives less accurate coexistence curves [19]. Note also that the commonplace method of 'binning' the σ-distribution into discrete 'pseudo-components' is a special case of our annealing method in which each weight function is zero outside the corresponding bin. Note also that the commonplace method of 'binning' the σ-distribution into discrete 'pseudo-components' is a spe- cial case of our annealing method in which each weight function is zero outside the corresponding bin. R P Feynman, Statistical Mechanics. Benjamin, Reading, MAR. P. Feynman, Statistical Mechanics (Benjamin, Read- ing, MA, 1972). P B Warren, ??, ???? (1998). P. B. Warren, Phys. Rev. Lett. ??, ???? (1998).	[]
[ "Some Spacetimes with Higher Rank Killing-Stäckel Tensors", "Some Spacetimes with Higher Rank Killing-Stäckel Tensors" ]	[ "G W Gibbons \nWilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K\n", "T Houri \nOsaka City University Advanced Mathematical Institute (OCAMI)\n3-3-138 Sugimoto558-8585SumiyoshiOsakaJapan\n", "D Kubizňák \nWilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K\n", "C M Warnick \nWilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K\n\nQueens' College\nCB3 9ETCambridgeU.K\n" ]	[ "Wilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K", "Osaka City University Advanced Mathematical Institute (OCAMI)\n3-3-138 Sugimoto558-8585SumiyoshiOsakaJapan", "Wilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K", "Wilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K", "Queens' College\nCB3 9ETCambridgeU.K" ]	[]	By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four-and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson-Schouten-Nijenhuis algebra. We discuss the extension to the quantum regime.Emails:	10.1016/j.physletb.2011.04.047	[ "https://arxiv.org/pdf/1103.5366v1.pdf" ]	119,153,812	1103.5366	39978082ca6aaf4837ee76bc992fbeef7398b9eb	Some Spacetimes with Higher Rank Killing-Stäckel Tensors 28 Mar 2011 January 18, 2013 G W Gibbons Wilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K T Houri Osaka City University Advanced Mathematical Institute (OCAMI) 3-3-138 Sugimoto558-8585SumiyoshiOsakaJapan D Kubizňák Wilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K C M Warnick Wilberforce RoadD.A.M.T.PCB3 0WACambridge, CambridgeU.K Queens' College CB3 9ETCambridgeU.K Some Spacetimes with Higher Rank Killing-Stäckel Tensors 28 Mar 2011 January 18, 2013 By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four-and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson-Schouten-Nijenhuis algebra. We discuss the extension to the quantum regime.Emails: Introduction Since Carter's tour de force in separating variables for the Hamilton-Jacobi and Klein-Gordon equations in the Kerr metric [1] there has been a great deal of work on spacetimes {M, g ab } admitting a second rank Killing-Stäckel tensor K ab = K ba which is responsible for the additive separability of the Hamilton-Jacobi equation. Almost nothing is known about higher rank totally symmetric tensors K a1a2...ap satisfying the condition that ∇ (a1 K a2a3...ap+1) = 0 . (1) While it is known that any such tensor gives rise to a homogeneous function on the cotangent bundle T ⋆ M, K p = K a1...ap p a1 . . . p ap of degree p in momenta, which Poisson commutes with the Hamiltonian H = 1 2 g ab p a p b generating the geodesic flow, no non-trivial (i.e. irreducible) examples appear to be known. Given any two such Killing-Stäckel tensors of rank p and q respectively their Schouten-Nijenhuis bracket [K p , K q ] a1a2...ap+q−1 is defined in terms of the standard Poisson bracket {K p , K q } as follows K p , K q = ∂K p ∂q i ∂K q ∂p i − ∂K q ∂q i ∂K p ∂p i ≡ [K p , K q ] a1a2. ..ap+q−1 p a1 p a2 . . . p ap+q−1 . While examples of spacetimes admitting more than one quadratic Killing tensor satisfying a non-trivial Poisson or Schouten-Nijenhuis bracket algebra exist [2], no such higher rank examples appear to be known. This may well be because the quickest route for finding quadratic Killing tensors is to follow Carter's original path [1] and seek to separate variables in the Hamilton-Jacobi and Klein-Gordon equations. This route is not available for higher rank Killing-Stäckel tensors since there is no obvious connection between their existence and separability. By theorems in [3,4] only rank two Killing tensors apply to separability of the Hamilton-Jacobi equation. In some cases it is possible to go further and "quantize" the system. In the case of quadratic Killing-Stäckel tensors it is known that subject to certain conditions on the K ab and the Ricci tensor R ab , the second order differential operator −∇ a K ab ∇ b commutes with the wave operator −∇ a g ab ∇ b and this is related to the multiplicative separability of the Klein-Gordon equation [5]. A recent survey of quantum integrability of quadratic Killing-Stäckel tensors may be found in [6]. To our knowledge, there are few if any results to date on the higher rank case. The paper is organized as follows. In section 2 we give details of the lightlike Eisenhart lift and in particular how constants of the motion are lifted. In section 3 we give examples of spacetimes generated from the classical examples of Liouville integrable dynamical systems describing heavy tops. In section 4 we discuss how to obtain a supersymmetric spacetime by lifting dynamical systems in E 2 and give a superintegrable example. We conclude in section 5 and include a brief summary of conventions in the appendix. The Eisenhart Lift Our examples are all obtained by taking the Eisenhart lift or oxidation [2,7,8,9] of a dynamical system with an n-dimensional configuration space {Q n , g ij , V, A i } with Lagrangian L = 1 2 g ij (q k , t)q iqj − V (q k , t) + A i (q k , t)q i ,(3) to give a system of geodesics in an (n + 2)-dimensional Bargmann spacetime {M, g ab , ∂ s }, which admits a covariantly constant null Killing vector field ∂ s . The original dynamical trajectories are obtained by a null reduction along the orbits of ∂ s . Since all Bargmann metrics admit a covariantly constant null vector field, it follows that the holonomy is contained within E(2) ⊂ SO(3, 1), the two-dimensional Euclidean group which stabilizes a null vector. Thus the null congruence is geodesic, expansion, shear and vorticity free. Thus it is also contained within the class of Kundt spacetimes. It is simplest to work with the Hamiltonian formulation in order to see how the lift affects constants of the motion. We consider dynamics on the cotangent bundle, T * M , of some manifold M which is equipped with a natural symplectic form given in local coordinates by ω = dq i ∧ dp i , with associated Poisson bracket {, }. We assume that the Hamiltonian is a polynomial of degree two in momenta: H = H (2) + H (1) + H (0) ,(4) where H (i) has degree i in momenta. We do not need to assume that H is independent of t. We lift H to a Hamiltonian on T * (M ×R 2 ) by promoting t to a configuration space coordinate and introducing a new coordinate s. The conjugate momenta are denoted p t , p s and the new symplectic form is ω ′ = ω + dt ∧ dp t + ds ∧ dp s , with associated Poisson bracket {, } ′ . The Hamiltonian on this enlarged phase space is H = H (2) + p s H (1) + p 2 s H (0) + p s p t .(5) Projecting the integral curves of this system onto the T * M × R t factor of the phase space gives integral curves of the original Hamiltonian. Suppose now that the system (H, T * M ) has a constant of the motion which is a polynomial in momenta: K = k i=0 K (i) .(6) We calculate the variation of K along an integral curve of (H, T * M ) and find after collecting terms according to their degree in momenta that 0 = dK dt = {K, H} + ∂K ∂t = k i=0 {K (i−1) , H (2) } + {K (i) , H (1) } + {K (i+1) , H (0) } + ∂K (i) ∂t ,(7) Since K should be constant along any integral curve, the terms in the sum should vanish independently for each i. We lift K to the extended phase space as K = k i=0 p k−i s K (i) .(8) Now, along an integral curve of (H, T * (M × R 2 )) we have dK dλ = {K, H} ′ = k i=0 p k−i+1 s {K (i−1) , H (2) } + {K (i) , H (1) } + {K (i+1) , H (0) } + ∂K (i) ∂t ,(9) Clearly this vanishes iff K is a constant of the motion for the original system. Furthermore, since H is a homogeneous polynomial of degree two in momenta we may interpret it as generating the geodesic flow of a (pseudo-)Riemannian metric. K is a constant along geodesics which is a homogeneous polynomial in momenta and so corresponds to a Killing tensor of this metric. A similar calculation shows that for constants of the motion for the original system K 1 , K 2 , K 3 which lift to K 1 , K 2 , K 3 we have {K 1 , K 2 } = K 3 , ⇔ {K 1 , K 2 } ′ = K 3 .(10) As a result, the Shouten-Nijenhuis algebra of the Killing tensors in the lifted spacetime will be the same as the Poisson algebra of the constants of the motion for the original dynamical system. We also note that whilst we have increased the dimension of the configuration space by two, we have also gained 1 two new constants of the motion: p s and p t . Thus the degree of integrability of the system is unchanged by the lift-if the original system is Liouville integrable (i.e. admits n functionally independent constants of the motion in involution) or super-integrable (admits further constants of the motion) then so will the lifted system be. Applying this method to the system {Q n , g ij , V, A i } defined above, we find that the lifted system is equivalent to geodesic motion on the spacetime with metric ds 2 = g ij (q k , t)dq i dq j − 2V (q k , t)dt 2 + 2A i (q k , t)dq i dt + 2dtds .(11) Eisenhart lift of Goryachev-Chaplygin and Kovalevskaya's Tops Eisenhart lift of the Goryachev-Chaplygin Top In this section we shall illustrate our general procedure by starting with the well-known Liouville integrable system known as the Goryachev-Chaplygin top [10,11]. After introducing the Goryachev-Chaplygin Hamiltonian and the corresponding constant of motion, we proceed to their Eisenhart lift. We demonstrate that the obtained four-dimensional Lorentzian spacetime, which we call the Goryachev-Chaplygin spacetime, admits a rank-3 irreducible Killing tensor. We conclude by making several comments on the quantization of the Goryachev-Chaplygin top and the corresponding results in the Goryachev-Chaplygin spacetime. Goryachev-Chaplygin Top Following Whittaker [10] we consider the motion of Goryachev-Chaplygin top as a constrained motion of a heavy top with principle moments of inertia A = B = 4C and whose centre of gravity lies in the plane determined by the two equal moments of inertia, so we start with: L top = 1 2 (θ 2 + sin 2 θφ 2 ) + 1 8 (ψ + cos θφ) 2 − α 2 sin θ sin ψ.(12) Proceeding to the Hamiltonian formulation, we find p φ = sin 2 θφ + 1 4 cos θ(ψ + cos θφ) , p θ =θ , p ψ = 1 4 (ψ + cos θφ) ,(13) and hence the Hamiltonian is H top = 1 2 p 2 θ + 2p 2 ψ + 1 2 ( p φ sin θ − cot θp ψ ) 2 + α 2 sin θ sin ψ = 1 2 M 2 1 + M 2 2 + 4M 2 3 + α 2 x 2 ,(14) which, in notations of the appendix, is the Hamiltonian (1) considered by Komarov [11]. It is obvious that coordinate φ is cyclic and hence p φ equals constant. The Hamiltonian of Goryachev-Chaplygin top is obtained if one sets p φ = 0 , H GC = 1 2 cot 2 θ + 4 p 2 ψ + 1 2 p 2 θ + α 2 sin θ sin ψ .(15) The Hamiltonian (14) has a remarkable property such that the function K top = M 3 (M 2 1 + M 2 2 ) − α 2 M 2 x 3(16) obeys {H top , K top } = α 2 p φ M 1 .(17) Hence, for p φ = 0, i.e. for Goryachev-Chaplygin top, (16) is a constant of motion and reads K GC = p ψ p 2 θ + cot 2 θp 3 ψ + α 2 cos θ sin ψ cot θp ψ − cos ψp θ .(18) Introducing the following functions (projections of standard functions M i ): m 1 = − sin ψp θ − cos ψ cot θp ψ , m 2 = cos ψp θ − sin ψ cot θp ψ , m 3 = p ψ ,(19) we may write the Goryachev-Chaplygin top Hamiltonian and the corresponding constant of motion as H GC = 1 2 m 2 1 + m 2 2 + 4m 2 3 + α 2 x 2 , K GC = m 3 (m 2 1 + m 2 2 ) − α 2 m 2 x 3 .(20) Eisenhart lift: Goryachev-Chaplygin spacetime Using the results of section 2 the Hamiltonian (20) lifts to the four-dimensional Hamiltonian H = m 2 1 + m 2 2 + 4m 2 3 + 2α 2 p 2 s x 2 + 2p s p t .(21) This generates the geodesic flow of the four-dimensional Lorentzian 4-metric with Killing vector fields k = ∂ t and l = ∂ s , the latter of which is lightlike and covariantly constant, g = −2α 2 sin θ sin ψdt 2 + 2dtds + dθ 2 + dψ 2 cot 2 θ + 4 .(22) The constant of motion (20) now reads K = m 3 (m 2 1 + m 2 2 ) − α 2 p 2 s m 2 x 3(23) and defines a rank-3 Killing tensor K, K = K abc p a p b p c , with non-zero contravariant components K θθψ = 1 3 , K θss = − α 2 3 cos ψ cos θ , K ψψψ = cot 2 θ , K ψss = α 2 3 cos 2 θ sin ψ sin θ ,(24) together with the other components related by symmetry. One may verify directly that K satisfies the Killing equation, ∇ (a K bcd) = 0, however, it is not covariantly constant. We can see in an elementary way that K is not decomposable into lower rank Killing tensors. This follows from the fact that k and l are the only Killing vectors of the spacetime (22). Suppose K were decomposable, then it would be the sum of terms of the form K (a (1) K bc) (2) , or K (a (3) K b (4) K c) (5) ,(25) where the K (i) are Killing tensors. Since a rank 1 Killing tensor is a Killing vector, by our assumption at least one of the factors in each term must be either k or l. Such terms will only have non-zero components when at least one of a, b, c is either t or s. Since K has a non-zero ψψψ-component, K cannot be decomposed into a sum of lower rank Killing tensors. One may verify that the following holds: [k, l] = 0 , L k K = 0 , L l K = 0 ,(26) which implies that the associated constants of the geodesic motion are in involution; the motion is Liouville integrable. Let us finally mention some properties of the Goryachev-Chaplygin spacetime. The spacetime is not Ricci flat, nor does the Ricci scalar vanish. This means that it does not admit a Killing spinor, e.g., [12]. We also note that R ab l b = 0,(27) however R ab clearly has rank 3 (for typical values of the coordinates) and so R ab = Am a m b for any vector m a . The Einstein tensor has non-zero components G tt = −12α 2 (3 cos 4 θ − 10 cos 2 θ + 6) sin θ sin φ (3 cos 2 θ − 4) 2 , G ts = − 2(3 cos 2 θ + 2) (3 cos 2 θ − 4) 2 ,(28) and obeys G ab l a l b = 0 , which is, of course, obvious from the equivalent result for the Ricci tensor, together with the fact that l is null. Quantum mechanics of Goryachev-Chaplygin Top The quantum mechanics of the Goryachev-Chaplygin top was studied by Komarov [11]. Specifically, it was shown that (17) admits a quantum analogue [Ĥ top ,K top ] = −α 2 J 1 ∂ φ ,(29) where operatorsĤ top andK top are given bŷ H top = 1 2 J 2 1 + J 2 2 + 4J 2 3 + α 2 x 2 ,K top = J 3 (J 2 1 + J 2 2 ) − 1 4 J 3 − 1 2 α 2 (J 2 x 3 + x 3 J 2 ) ,(30) and J i are defined in (70). This means that acting on a wave function independent of φ, the operators (30) commute. By employing the Eisenhart lift on these operators one finds that the operatorŝ H top = J 2 1 + J 2 2 + 4J 2 3 + 2α 2 x 2 ∂ 2 s + 2∂ s ∂ t ,K top = J 3 (J 2 1 + J 2 2 ) − 1 4 J 3 − 1 2 α 2 (J 2 x 3 + x 3 J 2 )∂ 2 s ,(31)obey [Ĥ top ,K top ] = −2α 2 J 1 ∂ 2 s ∂ φ , and hence commute on φ-independent wave function. The former operator is precisely the standard wave operator on the Lorentzian 5-space with the metric g top , obtained by the Eisenhart lift of H top . So we have, top ≡ g ab top ∇ a ∇ b =Ĥ top , where g top = 2dsdt − 2α 2 x 2 dt 2 + (σ 1 ) 2 + (σ 2 ) 2 + 1 4 (σ 3 ) 2 ,(32) and σ i are the left invariant forms on SU (2) defined in (68). Moreover, the latter operator can be written aŝ K top = K abc (top) ∇ a ∇ b ∇ c + 3 2 (∇ a K abc (top) )∇ b ∇ c − 1 2 K (top)a ab ∇ b ,(33) where K (top) is a symmetric rank-3 tensor. Introducing the basis L s = ∂ s , L t = ∂ t , L i = J i ,(34) one finds that non-vanishing contravariant components of K (top) are K ss2 (top) = −2α 2 x 3 /3 , K 113 (top) = K 223 (top) = 2/3 ,(35) and that the tensor satisfies ∇ (a K (top) bcd) = −α 2 L (a s L b s (∂ φ ) c L d) 1 . Hence, if we restrict to geodesic motion on 5-space with metric g top such that p φ vanishes, K abc (top) p a p b p c defines a constant of motion. One might wonder whether it is possible to directly carry over the quantization to the Goryachev-Chaplygin four-dimensional spacetime discussed in the previous subsection. The 'naive quantization' of (20) giveŝ H GC = 1 2 j 2 1 + j 2 2 + 4j 2 3 + α 2 x 2 ,K GC = j 3 (j 2 1 + j 2 2 ) − 1 4 j 3 − 1 2 α 2 (j 2 x 3 + x 3 j 2 ) ,(36) where we have defined the operators (projections of J i ) j 1 = − sin ψ∂ θ − cos ψ cot θ∂ ψ , j 2 = cos ψ∂ θ − sin ψ cot θ∂ ψ , j 3 = ∂ ψ .(37) By lifting the operators (36), one findŝ H = j 2 1 + j 2 2 + 4j 2 3 + 2α 2 x 2 ∂ 2 s + 2∂ s ∂ t ,K = j 3 (j 2 1 + j 2 2 ) − 1 4 j 3 − 1 2 α 2 (j 2 x 3 + x 3 j 2 )∂ 2 s .(38) It is easy to verify that [Ĥ,K] = 0. However, the operatorĤ is not a standard (geometrical) wave operator on the Goryachev-Chaplygin spacetime. In fact, one finds ≡ g ab ∇ a ∇ b =Ĥ − 3 cot θ 4 + cot 2 θ ∂ θ .(39) It is an interesting question whether the operators (36) provide the 'correct quantization' of the Goryachev-Chaplygin top, in which case the operators (38) are 'preferred operators' in the Goryachev-Chaplygin spacetime, or whether some alternative quantization is more appropriate. We leave this problem for the future. We also remark that we were not able to find an operator linear in the Killing tensor K, (24), which commutes with the wave operator associated with the Goryachev-Chaplygin metric (22). Kovalevskaya's Spacetime: Quartic Killing Tensor In this case one considers a heavy top with principle moments of inertia A = B = 2C whose centre of gravity lies in the plane determined by the two equal moments of inertia. The Lagrangian is L K = 1 2 (θ 2 + sin 2 θφ 2 ) + 1 4 (ψ + cos θφ) 2 − α 2 sin θ cos ψ .(40) Clearly φ is ignorable and the Hamiltonian H K = 1 2 p 2 θ + ( p φ sin θ − cot θp ψ ) 2 + 2p 2 ψ + α 2 sin θ cos ψ = 1 2 M 2 1 + M 2 2 + 2M 2 3 + α 2 x 1(41) is constant. Kovalevskaya found another constant [10,13] which reads K K = p 2 θ + ( p φ sin θ − cot θp ψ ) 2 2 + 4α 4 sin 2 θ − 2α 2 sin θ e iψ ( p φ sin θ − cot θp ψ + ip θ ) 2 + c.c. = (M 2 1 + M 2 2 ) 2 + 4α 4 (x 2 1 + x 2 2 ) − 4α 2 x 1 (M 2 1 − M 2 2 ) + 2x 2 M 1 M 2 .(42) This will lift to give a quartic Killing tensor. In order to get a four-dimensional spacetime we perform again the reduction along the φ-direction. So we consider H = 1 2 m 2 1 + m 2 2 + 2m 2 3 + α 2 x 1 , K = (m 2 1 + m 2 2 ) 2 + 4α 4 (x 2 1 + x 2 2 ) − 4α 2 x 1 (m 2 1 − m 2 2 ) + 2x 2 m 1 m 2 .(43) The Hamiltonian lifts to H = m 2 1 + m 2 2 + 2m 2 3 + 2α 2 p 2 s x 1 + 2p s p t ,(44) which generates geodesic flow of the Lorenzian 4-metric g = −2α 2 sin θ cos ψdt 2 + 2dsdt + dθ 2 + dψ 2 cot 2 θ + 2 ,(45) admitting the rank-4 irreducible tensor K, given by K θθθθ = 1 , K θθψψ = 1 3 cot 2 θ , K ssθθ = 2 3 α 2 sin θ cos ψ , K ψψψψ = cot 4 θ , K ssθψ = − 2 3 α 2 cos θ sin ψ , K ssψψ = − 2 3 α 2 cos ψ cos θ cot θ , K ssss = 4α 4 sin θ 2 . Properties of the Kovalevskaya spacetime are very similar to properties of the Goryachev-Chaplygin spacetime. In particular, the spacetime admits a covariantly constant null Killing vector l = ∂ s , it is not Ricci flat, and does not admit a Killing spinor. We also have that =Ĥ, with the latter obtained by a naive quantization described in previous section. One can again consider a 5D spacetime instead, g K = −2α 2 sin θ cos ψdt 2 + 2dsdt + (σ 1 ) 2 + (σ 2 ) 2 + 1 2 (σ 3 ) 2 ,(47) where one has [14] K = g ab K ∇ a ∇ b =Ĥ K = J 2 1 + J 2 2 + 2J 2 3 + 2α 2 x 1 ∂ 2 s + 2∂ s ∂ t , K K = 1 2 (K + K − + K − K + ) − 2(J + J − + J − J + ) ,(48) where J ± = J 1 ± iJ 2 , K ± = J 2 ± − 2α 2 x ± ∂ 2 s and x ± = x 1 ± ix 2 . In this caseK K is a real symmetry of the wave operator, [ K ,K K ] = 0. It is related to the five-dimensional rank-4 irreducible Killing tensor K (K) aŝ K K = K abcd (K) ∇ a ∇ b ∇ c ∇ d + 2(∇ a K abcd (K) )∇ b ∇ c ∇ d + 3(∇ a ∇ b K abcd (K) )∇ c ∇ d −2K abc (K) c ∇ a ∇ b − 3 4 K ab (K) ab L c 3 L d 3 ∇ c ∇ d ,(49) where in the basis (34) the components of the Killing tensor K (K) are written as K ssss (K) = 4α 4 (x 2 1 + x 2 2 ) , K ss11 (K) = −K ss22 (K) = −2α 2 x 1 /3 , K ss12 (K) = −2α 2 x 2 /3 , K 1111 (K) = 3K 1122 (K) = K 2222 (K) = 1 .(50) 4 Superintegrable systems in E 2 : SUSY plane waves In this section we consider Hamiltonians of the form H = 1 2 (p 2 x + p 2 y ) + V (x, y) .(51) For some choices of the potential V this Hamiltonian is superintegrable, e.g., [15] and references therein. The Hamiltonian (51) lifts to H = p 2 x + p 2 y + 2V (x, y)p 2 s + 2p s p t ,(52) which generates geodesic flows of Lorentzian 4-metric g = dx 2 + dy 2 − 2V (x, y)dt 2 + 2dtds .(53) In quantum mechanics, one has the quantized Hamiltonian H = ∂ 2 x + ∂ 2 y + 2V (x, y)∂ 2 s + 2∂ s ∂ t(54) and this coincides with the Laplacian of the metric (53), i.e., one has ≡ ∇ a g ab ∇ b =Ĥ . Let us mention some basic properties of the spacetime (53). The Ricci curvature has only tt-component, R tt = (∂ 2 x + ∂ 2 y )V ,(55) and the scalar curvature vanishes, R = 0. Hence G ab = R ab and R ab l b = 0 ,(56) where l ≡ ∂/∂s is a covariantly constant null Killing vector. Since the "transverse" x-y space is flat the metric (53) admits a covariantly constant spinor field ǫ such that ǫγ a ǫ = l a = (∂ s ) a(57) and hence a covariantly constant null 2-form ℓ ab =ǭγ [ab] ǫ(58) such that ℓ ab l b = 0 . There are many examples of interesting (superintegrable) systems of the type (51) which give rise to higherrank Killing tensors and non-trivial Schouten-Nijenhuis brackets. We refer the reader to recent paper by Kalnins et al. [15] and references therein as well as to Chapter 4.4 in [16]. To illustrate the theory we give the following recent example: Post-Winternitz example In [17], Post and Winternitz give a (Hamilton-Jacobi non-separable) classical super-integrable example of the form (51) with the potential V = αy x 2 3 ,(59) such that X = 3p 2 x p y + 2p 3 y + 9αx 1 3 p x + 6αyp y x 2 3 ,(60)Y = p x 4 + 4αyp 2 x x 2 3 − 12αx 1 3 p x p y − 2α 2 (9x 2 − 2y 2 ) x 4 3 ,(61) both Poisson commute with H and satisfy the Heisenberg algebra {X, Y } = 108α 3 .(62) The spacetime reads g = 2dsdt − 2y x 2 3 dt 2 + dx 2 + dy 2 .(63) The constants X, Y are lifted and give {X , Y} = 108α 3 p 6 s .(64) Thus, consistent with previous cases ( [2] and references therein), the central element in the Heisenberg algebra (62) may be interpreted as the (sixth power of) a null translation. The spacetime admits rank-3 and rank-4 Killing tensors. Their components X abc and Y abcd can be read of from X = X abc p a p b p c = 3p 2 x p y + 2p 3 y + 9αx 1 3 p x p 2 s + 6αyp y p 2 s x 2 3 ,(65)Y = Y abcd p a p b p c p d = p x 4 + 4αyp 2 x p 2 s x 2 3 − 12αx 1 3 p x p y p 2 s − 2α 2 (9x 2 − 2y 2 ) x 4 3 p 4 s .(66) Since l a dx a = dt, we have l a X abc = 0 = l a Y abcd . Conclusions In this paper we have shown that by applying Eisenhart's lightlike lift to dynamical systems admitting constants of the motion of degree greater than two in momenta, one may obtain spacetimes admitting Killing tensors of higher rank than two. Our examples by no means exhaust the possibilities. In [13,14,15,18,19,20,21,22] more complicated examples are given, but our examples illustrate the point we wish to make. In some cases we find the Poisson-Schouten-Nijenhuis algebra to be non-trivial. We have also constructed differential operators which realize the classical algebra as → 0. In some, but not all, cases the Hamiltonian corresponds to the Laplace or wave operator. In general the wave operator must be augmented by quantum corrections which are not always expressible in purely geometric terms. The higher rank conserved quantities also receive quantum corrections not expressible solely in terms of the Killing tensor. In some ways this is one of the most interesting of our findings and is certainly worthy of further study. Defining the functions x 1 = sin θ cos ψ , x 2 = sin θ sin ψ , x 3 = cos θ , we have the additional relations [J i , x j ] = −ǫ ijk x k ,(73) where we interpret the functions x i as operators on functions, acting by multiplication. Both the Goryachev-Chaplygin and the Kovalevskaya tops discussed in the main text are examples of tops whose centre of gravity does not coincide with the pivot point. They admit a description in terms of the Lie algebra of the Euclidean group E(3) and since this is used in some of the literature, e.g. [11,13,14,18,19,23], we give it here. If M is the angular momentum of the top one has, in the rotating framė M + ω × M = −mgx 0 × x , k + ωx = 0 ,(74) where x is unit vector which is constant in the inertial frame (the constancy of \|x\| is a consequence of these equations of motion) and points in the opposition direction to the local direction of gravity and x 0 is a constant vector in the rotating from which gives the centre of gravity. An alternative interpretation, used in analyzing the Stark effect, is that x 0 is the electric dipole moment and and mgx is in the direction of the applied electric field. The system of equations admits three constants of the motion x · x , x · M , 1 2 ω · M + mgx 0 · x .(75) Choosing coordinates such that the centre of mass relative to the pivot (normalized to unit length) are given by (72), we find that the potential energy of the top is given by V = mg(x 0 sin θ cos ψ + y 0 sin θ sin ψ + z 0 cos θ) , The Poisson algebra of M and x then turns out to be that of the Euclidean group e(3). Thus the system of equations (74) may also be interpreted as a Hamiltonian system moving on e ⋆ (3) the dual of the Lie algebra e(3). As a consequence one has an isomorphism with the problem of a rigid body moving in a fluid. However it should be noted that the latter has phase space T ⋆ E(3) which is 12-dimensional while the top has phase space has phase space T ⋆ (SO (3)) which is 6-dimensional. As pointed out in [23] if one imposes the constraints x · x = 1 , M · x = 0, one gets the standard symplectic structure on T ⋆ S 2 . Post and Winternitz have provided a quantization of their model. Thus if [x, p x ] = i etc, then all products are replaced by half their anti-commutator and in addition one must subtract 5 2 72x 2 from the expression for H and add 25 4 1296x 4 to the expression for Y . and one may construct a Lagrangian on T SO(3) and a Hamiltonian on T ⋆ SO(3) which depend on the principle moments of inertia (A, B, C). For the Goryachev-Chaplygin top we have A = B = 4C, and the centre of gravity lies in the plane defined by the two principal axes with equal moments of inertia. The moment maps for left actions of rotations M 1 = − sin ψp θ + cos ψ sin θ p φ − cos ψ cot θp ψ , M 2 = cos ψp θ + sin ψ sin θ p φ − sin ψ cot θp ψ , M 3 = p ψ . The equations of motion derived from H imply that pt = const − E(t)/ps, where E(t) is the energy of the original system, thus when E is constant, we do not lose this constant of the motion by lifting. AcknowledgmentsWe would like to thank C. Duval, P. Horvathy and G. 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[ "Condensed Fraction of an Atomic Bose Gas Induced by Critical Correlations", "Condensed Fraction of an Atomic Bose Gas Induced by Critical Correlations" ]	[ "Robert P Smith \nCavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom\n", "Naaman Tammuz \nCavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom\n", "Robert L D Campbell \nCavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom\n", "Markus Holzmann \nLPTMC\nUMR 7600\nCNRS\nUniversité P. et M. Curie\n75752ParisFrance\n\nLPMMC\nUMR 5493\nCNRS\nUniversité J. Fourier\n38042GrenobleFrance\n", "Zoran Hadzibabic \nCavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom\n" ]	[ "Cavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom", "Cavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom", "Cavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom", "LPTMC\nUMR 7600\nCNRS\nUniversité P. et M. Curie\n75752ParisFrance", "LPMMC\nUMR 5493\nCNRS\nUniversité J. Fourier\n38042GrenobleFrance", "Cavendish Laboratory\nUniversity of Cambridge\nJ. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom" ]	[]	We study the condensed fraction of a harmonically-trapped atomic Bose gas at the critical point predicted by mean-field (MF) theory. The non-zero condensed fraction f0 is induced by critical correlations which increase the transition temperature Tc above T MF c . Unlike the Tc shift in a trapped gas, f0 is sensitive only to the critical behaviour in the quasi-uniform part of the cloud near the trap centre. To leading order in the interaction parameter a/λ0, where a is the s-wave scattering length and λ0 the thermal wavelength, we expect a universal scaling f0 ∝ (a/λ0) 4 . We experimentally verify this scaling using a Feshbach resonance to tune a/λ0. Further, using the local density approximation, we compare our measurements with the universal result obtained from Monte-Carlo simulations for a uniform system, and find excellent quantitative agreement. PACS numbers: 03.75.Hh, 67.85.-d c −Nc ∝ (a/λ0) 2 , is dominated by the density shift outside the central critical region, and is not directly related to the nc shift. (b) If N is increased to N MF c > Nc, a small condensate induced by critical correlations forms within the critical region of size ∝ a/λ0. The condensed atom number N0 ∝ (a/λ0) 4 directly relates to the critical density shift ∆nc ∝ a/λ0. arXiv:1106.6295v1 [cond-mat.quant-gas]	10.1103/physrevlett.107.190403	[ "https://arxiv.org/pdf/1106.6295v1.pdf" ]	24,834,160	1106.6295	42e51494aa7a295b11979158e4fe704fe2804e06	Condensed Fraction of an Atomic Bose Gas Induced by Critical Correlations 30 Jun 2011 Robert P Smith Cavendish Laboratory University of Cambridge J. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom Naaman Tammuz Cavendish Laboratory University of Cambridge J. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom Robert L D Campbell Cavendish Laboratory University of Cambridge J. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom Markus Holzmann LPTMC UMR 7600 CNRS Université P. et M. Curie 75752ParisFrance LPMMC UMR 5493 CNRS Université J. Fourier 38042GrenobleFrance Zoran Hadzibabic Cavendish Laboratory University of Cambridge J. J. Thomson AvenueCB3 0HECambridgeUnited Kingdom Condensed Fraction of an Atomic Bose Gas Induced by Critical Correlations 30 Jun 2011(Dated: July 1, 2011) We study the condensed fraction of a harmonically-trapped atomic Bose gas at the critical point predicted by mean-field (MF) theory. The non-zero condensed fraction f0 is induced by critical correlations which increase the transition temperature Tc above T MF c . Unlike the Tc shift in a trapped gas, f0 is sensitive only to the critical behaviour in the quasi-uniform part of the cloud near the trap centre. To leading order in the interaction parameter a/λ0, where a is the s-wave scattering length and λ0 the thermal wavelength, we expect a universal scaling f0 ∝ (a/λ0) 4 . We experimentally verify this scaling using a Feshbach resonance to tune a/λ0. Further, using the local density approximation, we compare our measurements with the universal result obtained from Monte-Carlo simulations for a uniform system, and find excellent quantitative agreement. PACS numbers: 03.75.Hh, 67.85.-d c −Nc ∝ (a/λ0) 2 , is dominated by the density shift outside the central critical region, and is not directly related to the nc shift. (b) If N is increased to N MF c > Nc, a small condensate induced by critical correlations forms within the critical region of size ∝ a/λ0. The condensed atom number N0 ∝ (a/λ0) 4 directly relates to the critical density shift ∆nc ∝ a/λ0. arXiv:1106.6295v1 [cond-mat.quant-gas] Some of the most interesting fundamental problems of many-body physics involve strong inter-particle correlations, and cannot be addressed by mean-field (MF) theories. Harmonically trapped ultracold atomic gases are promising candidates for highly controllable "quantum simulation" of such intricate many-body scenarios [1]. However, for testing the existing theories of spatially uniform systems, it is often important to experimentally extract information on local properties of a non-uniform trapped gas (see e.g. [2][3][4]). The effect of interactions on Bose-Einstein condensation of a dilute gas is a classic example of a difficult beyond-MF problem, which has challenged theorists for decades [5][6][7][8][9][10][11][12][13][14][15][16][17]. It is also an example of a situation where harmonic confinement both quantitatively and qualitatively alters the physics [18][19][20][21][22][23][24][25][26][27]. For a uniform gas the interaction shift of the critical temperature T c cannot be calculated to any order in the interaction strength using perturbation theory, owing to strong correlations that develop near the critical point. On the other hand, non-uniformity of a trapped atomic gas results in a significant MF shift of T c [18]. More importantly, it diminishes the more interesting beyond-MF effects, in essence because near T c only a small fraction of the cloud is actually in the critical regime (see Fig. 1). Only recently have the beyond-MF effects on condensation of an atomic gas become experimentally accessible [27]. Many questions remain open since beyond-MF effects in a uniform and a trapped gas have different dependence on the strength of interactions, and quantitative connections between the two are highly non-trivial. In this Letter, we study the condensed fraction (f 0 ) of an atomic Bose gas at the critical point predicted by MF theory. By definition f 0 vanishes within MF theory, and directly measures the effect of critical correlations which shift T c above T MF c . Moreover, while the T c shift itself strongly depends on the global properties of a non-uniform gas, f 0 measurements directly probe the quasi-uniform critical region near the centre of the trap. To leading order in the strength of interactions we predict a universal scaling f 0 ∝ (a/λ 0 ) 4 , where a > 0 is the s-wave scattering length and λ 0 the thermal wavelength at the ideal gas critical temperature T 0 c . Using a Feshbach resonance in a 39 K gas to tune a/λ 0 , and accurately measuring condensed fractions in the range 0.1 − 1%, we experimentally verify this prediction. Further, we directly relate our measurements to the universal critical behaviour seen in the classicalfield Monte-Carlo simulations of a uniform system [28], and find excellent quantitative agreement. In Fig. 1(a) we illustrate the difference between the beyond-MF shifts of the critical point in a uniform and a trapped system, and in Fig. 1(b) the expected scaling of the condensed fraction at the MF critical point. For visual clarity, here we fix the temperature of the gas and consider the interaction shift of the critical density n c (in the centre of the trap) and the critical atom number N c . Surprisingly, the beyond-MF shifts of n c and N c are not directly related to each other. The quadratic beyond-MF N c shift is of direct relevance to the experimentally pertinent case of a trapped gas, but from the point of view ∆n c~ a / λ0 n n 2 . 6 1 2 (solid blue line) is compared with the mean-field prediction (dashed red line). In the trap centre we expect n MF c −nc ∝ a/λ0, characteristic of the critical behaviour in a uniform system. However the experimentally measured Nc shift, N MF of the theory of critical behaviour the linear n c shift is actually more interesting. Here we will show how to use a trapped atomic cloud to experimentally obtain information about the critical behaviour in a uniform system. λ 3 N M F c N c ( a ) N M F c N 0~ ( a / λ0 ) 4 ( b ) ∆N c~ ( a / λ0 ) 2~ a / λ0~ a / λ0 We first outline some general scaling arguments, then present our experimental results, and finally return to a quantitative comparison of our measurements with the theory based on the classical-field Monte-Carlo simulations of Ref. [28] for a uniform system. In a uniform system, ideal-gas condensation occurs at a chemical potential µ 0 c = 0, and a critical phase space density nλ 3 = ζ(3/2) ≈ 2.612, where ζ is the Riemann function. In an interacting gas there is no T c shift at MF level, i.e. T MF c = T 0 c . To leading order in a/λ 0 1 the expected beyond-MF T c shift is given by [7][8][9][10][11][12][13][14][15][16][17] ∆T c T 0 c ≈ c a λ 0 ,(1) where ∆T c = T c − T 0 c , and c ≈ 1.8 [11,12]. Equivalently, the n c shift at constant T is ∆n c /n 0 c ≈ −(3/2)∆T c /T 0 c . An important point is that, at both MF and beyond-MF level, the interactions differently affect T c (or equivalently n c ) and the critical chemical potential µ c . The simple MF shift βµ MF c = 4 ζ(3/2)a/λ 0 , where β = 1/k B T , has no effect on condensation, and to lowest beyond-MF order [29]: βµ c ≈ βµ MF c + B 2 a λ 0 2 .(2) The qualitative difference between Eqs. (1) and (2) highlights the fact that the problem of the T c shift is non-perturbative and near criticality the equation of state does not have a regular expansion in µ (otherwise one would get ∆n c ∝ µ M F c − µ c ). In a harmonically trapped gas T c is defined for a given atom number N , rather than for a given density n. For an ideal gas k B T 0 c =hω [N/ζ(3)] 1/3 , where ζ(3) ≈ 1.202. Within the local density approximation (LDA) one expects the uniformsystem results for n c and µ c to apply in the centre of the trap, r = 0. Elsewhere in the trap the local chemical potential is µ(r) = µ(0)−mω 2 r 2 /2, where m is the atom mass and ω the trapping frequency. The result for the T c shift however does not carry over so easily to the trapped case; the basic reason for this is that at T c only a small fraction of the non-uniform cloud is actually in the critical regime. The size of the central critical region is r c ∼ (a/λ 0 )R T , where R T = k B T /mω 2 is the thermal radius [22]. Combining this with ∆n c ∼ a/λ 0 implied by Eq. (1), we obtain a very small beyond-MF shift of the critical number of atoms within the critical region, of the order (a/λ 0 ) 4 . However the interaction shift of µ c affects the density everywhere in the trap. One famous consequence of this is the negative MF shift of T c in a harmonically trapped gas [18]: while n MF c = n 0 c , repulsive interactions broaden the density distribution so that N MF c > N 0 c . More generally, the experimentally observed T c shift in a trapped gas [27] qualitatively mirrors Eq. (2): ∆T c T 0 c ≈ b 1 a λ 0 + b 2 a λ 0 2 .(3) Here b 1 ≈ −3.426 is an analytical, strictly MF result [18], and b 2 = 46 ± 5 was measured in [27]. The quadratic T c shift depends on the beyond-MF correlations, but does not directly correspond to the lowest-order (beyond-MF) T c shift in a uniform system [Eq. (1)], which should enter the in-trap T c only at the (a/λ 0 ) 4 level. We can qualitatively understand the similarity of Eqs. (2) and (3) by noting that: (i) Away from the critical point the equation of state is regular in µ and the local density shift is simply proportional to µ c , at both MF and beyond-MF level [30], and (ii) The contribution to N c from the non-critical region outweighs the contribution from within the critical region by a large factor ∼ (λ 0 /a) 3 . To summarize this analysis: On the one hand we expect n MF c − n c ∝ a/λ 0 , characteristic of the critical behaviour in a uniform system. On the other hand N MF c − N c ∝ (a/λ 0 ) 2 is dominated by the effect of the µ c shift on the density outside the critical region. The latter result was observed in [27]; the former cannot be experimentally verified without a direct probe of the local density in a 3D cloud. By studying the condensed fraction f 0 at the MF-predicted critical point we overcome the problem of the absence of the local density probe, and gain more direct insight into the critical behaviour in the centre of the trap. Simply put, instead of asking how N c is reduced with respect to N MF c by critical correlations, we ask how many atoms pile up in the condensate if (at constant T ) we increase the total atom number to N MF c > N c . Experimentally, the obvious advantage is that the condensed and thermal component can be clearly distinguished in standard time-of-flight (TOF) expansion, thus allowing us to use a "global" measurement technique to access the local behaviour of the gas within the critical region. Theoretically, the analogous quantity for a uniform gas, n 0 /n (where n 0 is the condensate density), was first considered by Holzmann and Baym [31]. Although the formal proof is rather involved, the main scaling result is intuitive, n 0 /n ∝ ∆n c ∝ a/λ 0 [32]. From this result we immediately obtain f 0 ∝ (a/λ 0 ) 4 , as illustrated in Fig. 1(b). The presence of the harmonic trapping potential still affects the scaling of f 0 with a/λ 0 , but in this case the results for a harmonic and a uniform system are trivially related by the volume of the critical region, ∝ (a/λ 0 ) 3 . To experimentally measure f 0 we use an optically trapped cloud of 39 K atoms in the \|F, m F = \|1, 1 hyperfine state, in which the strength of interactions can be tuned via a Feshbach resonance centred at 402.5 G [33]. Our experimental system and the procedure for making precise and accurate measurements close to the critical point are described in detail in [27,34,35]. Briefly, we prepare partially condensed clouds at various values of the scattering length a, and then let the number of atoms in the trap gradually decay through inelastic processes, while finite trap depth and sufficiently high rate of elastic collisions ensure that the sample remains in equilibrium at an approximately constant temperature. For the measurements presented here, N ≈ (4 − 5) × 10 5 , the geometric mean of the trapping frequencies in our nearly isotropic trap isω/2π ≈ 80 Hz, and T ≈ 250 nK, corresponding to λ 0 ≈ 10 4 a 0 , where a 0 is the Bohr radius. To discern condensed fractions as low as ∼ 0.1 % we turn off the interactions (switch a close to zero) during TOF, thus minimizing the condensate expansion [27]. We numerically calculate N MF c using standard MF theory (see also [30]) and measure the condensed atom number N 0 at the point where the total atom number is N = N MF c . To eliminate various sources of a-independent systematic errors (including absolute N andω calibration) we perform "reference" measurements in a weakly interacting gas with a/λ 0 ≈ 0.005 [27]. At this reference point the expected value of f 0 is < 10 −5 (see below), and we neglect it in our analysis. Our experimental results are summarized in Fig. 2. Starting at zero for small a (in agreement with MF theory), the condensed fraction f 0 grows to ∼ 1% at a ≈ 350 a 0 . The use of a Feshbach resonance in principle allows us to increase a further, but in the more strongly interacting gases the unfavorable ratio of the three-body loss rate to the two-body elastic collision rate precludes reliable equilibrium measurements [27]. We fit our f 0 data with a function (a/λ 0 ) x where x is a free parameter. The fit yields x = 3.9 ± 0.4, in agreement with the predicted x = 4. This confirmation of the expected scaling of f 0 with a/λ 0 is the first main result of this paper. We now quantitatively relate our measurements to Monte-Carlo (MC) calculations for a uniform gas. Following [28] we first define the reduced chemical potential X = µ − µ c 32π 3 (a/λ 0 ) 2 k B T .(4) Next, following [11] we calculate X 0 , the value of X in the centre of the trap for N = N MF c (due to logarithmic corrections this is a slightly different condition from µ(0) = µ MF c , but this distinction is not experimentally observable). We use the experimental value b 2 = 42 ± 2, and b MF 2 = 11.7 ± 0.1 [27,30] to get X 0 ≈ 3 ζ(3) 32π 3 ζ(2) b 2 − b MF 2 = 0.067 ± 0.005 .(5) For a uniform system the reduced condensate densityf (X), defined by n 0 λ 3 0 = 16π 3 (a/λ 0 )f (X), was tabulated in [28] using MC simulations. Invoking LDA, for a harmonically trapped gas we get N 0 N 0 c = √ 2(4π) 7 4ζ(3) a λ 0 4 X0 0f (X) X 0 − X dX .(6) Writing (N 0 /N 0 c ) 1/4 = α(a/λ 0 ) and numerically evaluating the integral in Eq. (6), using the results of [28], we get the numerical coefficient α MC = 10.4 ± 0.4. In Eq. (6) N 0 is calculated at N = N MF c but normalised to N 0 c . This expression therefore differs from f 0 by a factor N MF c /N 0 c . This difference does not affect the leading (a/λ 0 ) 4 term and is relevant only at the (a/λ 0 ) 5 level. Nevertheless, for a direct quantitative comparison, in Fig. 3 we normalise the measured N 0 values to N 0 c , and assume the quartic dependence on a/λ 0 . The linear fit to (N 0 /N 0 c ) 1/4 yields the experimental value α exp = 10.3 ± 0.3, in excellent agreement with the Monte-Carlo result. For another comparison, it is interesting to convert X 0 into N 0 using the standard Thomas-Fermi (TF) law. This MF law is valid well below T c , where N 0 ≈ N , but should not hold close to the critical point. For a given X 0 , the TF law also predicts N 0 ∝ (a/λ 0 ) 4 . However it corresponds tof (X) = X and gives α TF = 8.2 ± 0.4. This result underestimates the condensed fraction f 0 by a factor (α MC /α TF ) 4 ≈ 2.6, and we experimentally exclude it at about 4 sigma level. This confirms that near T c mean-field theory fails on both sides of the critical point. In conclusion, we have studied the condensed fraction of an atomic gas induced by inter-particle correlations at a point where no condensate is predicted by mean-field theory. Building on the recent observation of correlation effects on the condensation temperature of a trapped gas, this work makes a more direct connection with the critical behaviour in a homogeneous system. We experimentally confirm the predicted scaling f 0 ∝ (a/λ 0 ) 4 , which highlights the conceptual difference between the interaction shifts of the critical density (characteristic of a uniform system) and the critical atom number in a harmonically confined cloud. Moreover, we demonstrate excellent quantitative agreement between our experiments and Monte-Carlo simulations for a homogeneous gas. In a more general context, this provides an example of the potential of ultracold atomic gases for quantitative quantum simulation of intricate beyond-mean-field phenomena in uniform many-body systems. We thank J. Dalibard for comments on the manuscript. This work was supported by EPSRC (Grant No. EP/1010580/1). R.P.S. acknowledges support from the Newton Trust. . 1. (color online) Beyond-mean-field effects near the critical point in a harmonically trapped Bose gas. (a) For a fixed temperature, the density distribution at the critical point N = Nc < N MF c FIG. 2 . 2(color online) Condensed fraction of an atomic gas induced by critical correlations. The condensed fraction f0 is measured at the point where the total atom number is N = N MF c > Nc. A fit to the data (solid line) with the function f0 ∝ (a/λ0) x gives an exponent x = 3.9 ± 0.4, in agreement with the predicted x = 4. Vertical error bars are statistical and horizontal error bars reflect the 0.1 G uncertainty in the position of the Feshbach resonance. The insets show representative column density distributions after 19 ms TOF. FIG. 3 . 3(color online) Comparison with Monte-Carlo calculations for a uniform system. To quantitatively compare our data with the MC simulations we plot (N0/N 0 c ) 1/4 versus a/λ0 (see text). 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[ "Accurate Ground-based Astrometry of Naked-eye Stars: The United States Naval Observatory Bright-Star Astrometric Database", "Accurate Ground-based Astrometry of Naked-eye Stars: The United States Naval Observatory Bright-Star Astrometric Database" ]	[ "Jeffrey A Munn \nU. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA\n", "John P Subasavage \nU. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA\n\nThe Aerospace Corporation\n2310 E. El Segundo Boulevard90245El SegundoCAUSA\n", "Hugh C Harris \nU. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA\n", "Trudy M Tilleman \nU. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA\n" ]	[ "U. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA", "U. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA", "The Aerospace Corporation\n2310 E. El Segundo Boulevard90245El SegundoCAUSA", "U. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA", "U. S. Naval Observatory\n10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA" ]	[]	We present the United States Naval Observatory (USNO) Bright-Star Astrometric Database (UBAD), a current-epoch high-accuracy astrometric catalog. The catalog consists of 364 bright northern hemisphere stars, including all but five such stars with either V < 3.5 or with I < 3.2 and V < 6, as well as a large fraction of slightly fainter stars; 36 of the brightest catalog stars are not included in Gaia Early Data Release 3 (EDR3). Observations were conducted with the USNO, Flagstaff Station, Kaj Strand 61-inch Astrometric Reflector. Target stars were imaged through a small 12.5-magnitude neutral-density spot, while the remainder of the stars in the field of view were unattenuated. This allowed for unsaturated images of the bright target stars to be calibrated directly against much fainter reference stars from Gaia EDR3. The median position errors are 1.9 mas in both right ascension and declination at the catalog epoch of 2017.0, with 90% of catalog stars having errors less than 2.6 mas; systematic errors are 1 -3 mas. Combining UBAD observations with Hipparcos-2 positions yields proper motions with median errors of 0.045 and 0.049 mas year −1 in right ascension and declination, respectively, with 90% of stars having errors less than 0.1 mas year −1 ; systematic errors are about 0.1 mas year −1 . Single-frame accuracy for positions of the target stars is typically 5 -6 mas. Gaia EDR3 astrometry for these bright stars, which are heavily saturated in the Gaia observations, is validated over the magnitude range 2 G 6.	10.3847/1538-3881/ac41d2	[ "https://arxiv.org/pdf/2202.08369v1.pdf" ]	246,900,906	2202.08369	625a86ef016e422665f4b3d89feffd0adcc8adb4	Accurate Ground-based Astrometry of Naked-eye Stars: The United States Naval Observatory Bright-Star Astrometric Database February 18, 2022 Jeffrey A Munn U. S. Naval Observatory 10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA John P Subasavage U. S. Naval Observatory 10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA The Aerospace Corporation 2310 E. El Segundo Boulevard90245El SegundoCAUSA Hugh C Harris U. S. Naval Observatory 10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA Trudy M Tilleman U. S. Naval Observatory 10391 W. Naval Observatory Road86005-8521Flagstaff Station, FlagstaffAZUSA Accurate Ground-based Astrometry of Naked-eye Stars: The United States Naval Observatory Bright-Star Astrometric Database February 18, 2022Draft version Typeset using L A T E X preprint2 style in AASTeX631 We present the United States Naval Observatory (USNO) Bright-Star Astrometric Database (UBAD), a current-epoch high-accuracy astrometric catalog. The catalog consists of 364 bright northern hemisphere stars, including all but five such stars with either V < 3.5 or with I < 3.2 and V < 6, as well as a large fraction of slightly fainter stars; 36 of the brightest catalog stars are not included in Gaia Early Data Release 3 (EDR3). Observations were conducted with the USNO, Flagstaff Station, Kaj Strand 61-inch Astrometric Reflector. Target stars were imaged through a small 12.5-magnitude neutral-density spot, while the remainder of the stars in the field of view were unattenuated. This allowed for unsaturated images of the bright target stars to be calibrated directly against much fainter reference stars from Gaia EDR3. The median position errors are 1.9 mas in both right ascension and declination at the catalog epoch of 2017.0, with 90% of catalog stars having errors less than 2.6 mas; systematic errors are 1 -3 mas. Combining UBAD observations with Hipparcos-2 positions yields proper motions with median errors of 0.045 and 0.049 mas year −1 in right ascension and declination, respectively, with 90% of stars having errors less than 0.1 mas year −1 ; systematic errors are about 0.1 mas year −1 . Single-frame accuracy for positions of the target stars is typically 5 -6 mas. Gaia EDR3 astrometry for these bright stars, which are heavily saturated in the Gaia observations, is validated over the magnitude range 2 G 6. INTRODUCTION The brightest stars in the sky, including those visible to the naked eye, are used as reference sources in many government and commercial applications, including star trackers, navigation systems, and weapon systems. Before the European Space Agency's (ESA) Gaia mission (Gaia Collaboration et al. 2016), the most accurate as-errors increase linearly going forward in time. Further, the Hipparcos satellite took observations for only 3.5 years and thus was not sensitive to detecting multiple systems with periods appreciably longer than its mission lifetime. During the building stage of the Gaia mission, the bright magnitude limit was defined to be G ∼ 5.7. Given the importance of accurate bright-star astrometry, in particular for United States Department of Defense applications, the United States Naval Observatory (USNO), Flagstaff Station, commenced a program in 2012 to obtain accurate currentepoch ground-based astrometry of bright northern hemisphere stars, those that Gaia would not observe, including all such stars in the magnitude range 3 < V < 6. The basic technique was to observe the bright target star through a small neutral-density spot located in front of the detector, such that only the light from the target star and its immediate vicinity is attenuated, while the rest of the field of view remains unattenuated. The first observations were conducted with a 9-magnitude neutral-density spot, allowing the target star to be directly imaged with, and thus calibrated against, reference stars more than nine magnitudes fainter. These reference stars would be unsaturated on the Gaia instrument and thus would eventually have accurate Gaia astrometry once Gaia released its first data. This technique, using the same 9-magnitude neutraldensity spot in a different camera, was first developed at USNO to validate Hipparcos astrometry (Harris et al. 1997) and later to validate a program of wide-angle absolute astrometry with the Navy Precision Optical Interferometer (Stone 2004;Zavala 2011). In 2014 the Gaia Collaboration announced that they had modified the object detection and collection algorithms on the spacecraft to allow the collection of data for the brightest stars, though it was unclear whether they would be able to achieve the same astrometric accuracy for these stars given that their images would be heavily saturated (Martín-Fleitas et al. 2014;Sahlmann et al. 2016). In light of these developments, we decided to change the focus of our work to the very brightest stars, those that would be most challenging for Gaia. It was also realized that while Gaia was likely to provide more accurate astrometry on even these bright stars than we could achieve from the ground, our survey could provide one of the few external validations of Gaia astrometry for bright stars. The 9-magnitude neutral-density spot was replaced with a 12.5-magnitude spot and a much brighter sample of stars was targeted. This paper presents the results of the latter survey and compares it with Gaia Early Data Release 3 (EDR3, Gaia Collaboration et al. 2021;Lindegren et al. 2021;Rowell et al. 2021;Fabricius et al. 2021). Section 2 describes the instrumentation and observations. Section 3 describes the processing of the individual observations and characterizes the quality of those observations. Section 4 describes the production of the catalog from the individual observations and compares it with Gaia and Hipparcos-2. Section 5 presents the catalog, and Section 6 summarizes our results. OBSERVATIONS The USNO Bright-Star Astrometric Database (UBAD) targets all northern hemisphere (δ > 0 • ) stars in the original Hipparcos catalog with V < 3.5, or with I < 3.5 and V < 6, with the following exceptions: 1. Polaris, at a declination δ = 89 • 16 , lies above the useful declination limit of the telescope used for the survey, and thus was excluded; 2. 46 stars that were previously observed in the first survey with the 9-magnitude neutral-density spot were excluded -all have V > 3.5, and only four have I < 3.2; 3. an additional five stars with 6 < V < 7 were included, to extend the magnitude range of overlap with Gaia. The final sample numbers 364 stars. Observations were conducted with the USNO, Flagstaff Station, Kaj Strand 61-inch Astrometric Reflector (Strand 1964). The telescope, commissioned in 1963, was designed specifically for precise narrow-field differential astrometry and has a long history determining stellar parallaxes in both the optical (Dahn et al. 2017, and references therein) and infrared (Vrba et al. 2004). The camera was built in-house using a 2048 × 4102 e2v CCD with 15-micron pixels, which in the 61-inch focal plane yields a pixel scale of 0.2025 arcsec pixel −1 . A clear filter covers the detector with a 5-mm diameter 12.5-magnitude neutral-density spot deposited in the center on the sky-facing surface of the filter. The target star is observed through the spot, which has measured attenuations of 12.48 magnitudes in V and 12.89 magnitudes in I. A total of 4807 observations were obtained between March 2016 and October 2020. All observations were taken using either an SDSS i or z filter, mostly within an hour of the meridian, to minimize differential chromatic refraction (DCR). At a minimum, each star was targeted with two visits on different nights, with two observations per visit. After the initial set of data was processed and the final catalog generated, we decided to observe a set of fainter stars to further extend the magnitude range of overlap with Gaia and thus better constrain any dependence on magnitude in Gaia's bright-star astrometry. Thus, an additional 151 observations were obtained in i between March and May 2021, targeting stars with 4.0 < G < 6.5 (the survey was cut short due to a detector failure). These observations are not used in the UBAD catalog; they are only used in Figure 9 below. IMAGE PROCESSING Object Detection and Characterization Figure 1 displays a typical survey image, using a non-linear scaling to reveal various features. The dark spot in the center of the frame is the area of the CCD covered by the neutral-density filter. The target bright star is visible in the center of the spot, its flux attenuated by over 12 magnitudes. Scattered light from this bright star is evident around the spot and extending at a much lower level over about half the chip. Two out-of-focus images of the target star from multiple reflections between surfaces in the camera dominate the central half of the chip. Fringing is also evident. A close-up of the portion of the image under the neutral-density spot is shown in Figure 2. The ring around the target star is present in all images, though its brightness is only a few percent of the peak brightness. The background is considerably less uniform than a typical astronomical image. The images are flat-fielded using a median of three dome flats. As it is impossible to get adequate counts under the neutral-density spot without saturating the rest of the CCD, an archival flat produced before the spot was installed is used for the region under the spot; there is no way to track changing dust spots in that portion of the image. SExtractor (Bertin & Arnouts 1996) is used to detect and measure stars on the images. A smaller bin size for the sky is used than usual, to better track the varying background. SExtractor is run separately on the portion of the image under the neutraldensity spot, with tweaked deblending parameters so as not to deblend the ring from the target star. Centers are measured using SExtractor's windowed first-order moments (XWIN IMAGE and YWIN IMAGE). Each survey image was astrometrically calibrated against reference stars from Gaia EDR3 using custom software, built using several invaluable community software packages high- lighted below. The following cuts were applied to the Gaia catalog to yield reliable astrometric calibrators: Astrometric Calibration 1. astrometric params solved = 31 or 95 (5-or 6-parameter solution), 2. ruwe < 1.4, 3. ipd gof harmonic amplitude < 0.15, 4. ipd frac multi peak <= 3, 5. ipd frac odd win < 10. No stars were used within a large circle on the images encompassing the brightest portion of the out-of-focus images of the target star. The initial match of detected objects on the image against Gaia was performed using astroalign (Beroiz et al. 2020), or in those cases where astroalign fails, with Astrometry.net . A simple affine transformation was fit for each image; no additional optics terms were found necessary. Fits were performed after first transforming Gaia coordinates to observed place, using place conversion routines provided by Astropy (Astropy Collaboration et al. 2013, 2018, including its interface to Essential Routines for Fundamental Astronomy (PyERFA, Kerkwijk et al. 2020), and corrected for polar motion using the Positional Astronomy Library (PALpy, Jenness & Berry 2013). Separate residual maps in i and z were created from the initial fits by calculating the clipped mean residual for the calibration stars, separately in right ascension and declination, binned in 128 × 128 pixel regions of the images. The set of images used to generate the residual maps were restricted to those where 1) the rms of the residuals in both right ascension and declination were less than 12 mas, 2) the calibration used at least 30/25 stars for observations using the i/z filter, and 3) the seeing was less than 1.5/1.7 arcsec for observations using the i/z filter; looser constraints were used in z due to the smaller number of observations taken in that filter. Figures 3 and 4 display the residual maps in i and z, respectively. Residuals in i have peak-to-peak systematics of about -20 to 25 mas, while in z they are about -13 to 18 mas. The z residual map is considerably noisier as there were far fewer observations in z than i. Separate residual maps in both bands were generated spanning periods of several months, rather than over the entire survey, to look for changes in the residual maps with time; no significant changes were seen. Thus, a single residual map was used in each band for the entire survey. The pipeline was then rerun after correcting the on-chip positions, using corrections derived by interpolating the residual maps with bivariate splines. Next, corrections for DCR were derived. For each image, a line was fit to the residuals along the parallactic angle versus the colors of the reference stars, using Pan-STARRS (Chambers et al. 2016;Magnier et al. 2020) r − i for images taken in i, and i − z for images taken in z. As this is a noisy measurement for individual frames, the ensembles for the slope and intercept values, limited to the same set of good images used to generate the residual maps, were then fit with a linear function against refraction, separately in i and z, with each fit forced to go through 0 at a refraction of 0. The results are shown in Figures 5 and 6. The fits yield the following corrections for DCR in i and z as a function of refraction and star color: DCR i (mas) = 0.3057 θ (r − i − 0.3893), (1) DCR z (mas) = 0.2541 θ (i − z − 0.2220), (2) where θ is the refraction in arcseconds. The pipeline is then run a final time, now applying corrections for both the residual maps and DCR. Figure 7 shows the distributions of the rms residuals for the astrometric solutions for all survey images. The distributions peak near 4 mas, with 85% of the solutions having rms values less than 10 mas. Figure 8 displays the distribution of the number of calibrating stars used for each image. The sharp drop after 30 stars is because the pipeline only uses stars with better than 0.5% photometry, but will use less well-exposed stars if necessary to get at least 30 stars. Not all fields have 30 stars available, well-exposed or not. The astrometric solutions were then used to derive ICRS astrometric-place coordinates for the target star on each image, at the epoch of the observation. DCR corrections to the target stars were applied using synthesized Pan-STARRS r − i and i − z colors from the ATLAS All-Sky Stellar Reference Catalog (Tonry et al. 2018). Note that the median parallax (as measured by Hipparcos-2) for stars in the survey is 10 mas, with nearly a third of the stars having parallaxes greater than 20 mas, easily detectable at the level of astrometry achieved by the survey. . The left and right panels display the maps for right ascension and declination, respectively. The CCD manufacturing process leads to discontinuities between regions on the CCD 1024 × 512 pixels in size, evident in the maps. The large hole in the center of each map corresponds to the portion of the CCD not used in the astrometric calibrations due to the bright out-of-focus images of the target star. Comparison with Gaia for Single Observations Of the 4958 observations which comprise the UBAD survey, 4607 target a star which is in Gaia EDR3. Figures 9 and 10 plot the differences between the target star ICRS coordinates measured on each UBAD image and the matching Gaia EDR3 catalog positions, propagated to astrometric-place coordinates at the epoch of the UBAD observations using Gaia's proper motions and parallaxes, for the 1998 i and 371 z observations which meet the following criteria: 1. seeing is less than 2 arcsec; 2. the astrometric calibration uses at least 15 reference Gaia stars (relaxed for a few sparse fields) and yields rms residuals of less than 20 mas in both right ascension and declination; 3. visual inspection of the target star on the UBAD image indicates no issues (blends, close neighbors, etc); 4. Hipparcos-2 flags the star as a single star with a satisfactory 5-parameter astrometric fit; 5. the matching Gaia star has a 5-or 6parameter solution and ruwe < 3. Black and gray points are for stars in the main survey and the faint extension, respectively (no z observations were taken for the faint extension). The blue lines are the median differences for stars in the main survey. The solid and dotted orange lines indicate the medians and interquartile ranges, respectively, of the differences in bins of 100 stars in G. The comparison is excellent, with no magnitude-dependent systematic trends in either the fully-expected median differences (discussed below) or the dispersions over the four magnitudes of overlap in G between UBAD and Gaia. Figures 11 and 12 display histograms of the differences between the UBAD and Gaia EDR3 positions in i and z shown in Figures 9 and 10, limited to stars in the main survey. The median differences (Gaia − UBAD) in i are 8.5 and -6.6 mas in right ascension and declination, respectively. In z the median differences are 11.1 and -9.3 mas in right ascension and declination, respectively. Systematic offsets of such size are expected as the residual map was not generated under the spot, and so these off- sets can just be thought of as part of the residual map. These median offsets are added to all UBAD individual observation positions and are applied for all subsequent analyses. Note that since we use the offsets to calibrate UBAD, we cannot validate that Gaia EDR3 itself does not have a magnitude-independent systematic bias over the magnitude range covered by UBAD, however, given that there is no magnitudedependent systematic offset over that magnitude range, Gaia is likely free of any such over- all systematic error. The rms differences in i are 4.5 and 5.0 mas in right ascension and declination, respectively. In z the rms differences are 6.5 and 6.1 mas in right ascension and declination, respectively, giving a good indication of the single-frame astrometric accuracy of the UBAD observations (typical errors in the Gaia positions at the UBAD observation epochs are less than a mas). Different quality cuts to determine which stars are included in the analysis change the offsets by of order 0.5 mas in i and 1 -2 mas in z, which is one indication of the size of systematic errors in the UBAD astrometry. After correcting for the median offsets, there remain systematic differences between Gaia EDR3 and UBAD declinations as a function of declination. Figure 13 plots these differences. The orange solid and dashed lines indicate the median and interquartile range offsets, respectively, in bins of 200 stars. No similar systematic differences are seen in right ascension. We don't understand the cause of the systematic offsets. One possible source is the ring of reflected light around the target star seen in Figure 2. The star is offset along the x-axis with respect to that ring; the x-axis is parallel to the declination axis. It's possible that as the telescope flexes at different declinations (it is on an equatorial mount), the offset between the star and the center of the ring changes, affecting the measured center of the star. The brightness of the ring is only a few percent that of the peak of the star, however, the systematic offset is small, varying roughly linearly by about 3 mas over the entire declination range. Another possible source is DCR perhaps not being fully corrected by the adopted correction model; most observations were taken within an hour of the meridian, so refraction is predominantly along declination. It is far less likely that the systematics are in the Gaia EDR3 positions; Gaia's scanning law is roughly symmetric in ecliptic coordinates, and without concerns regarding atmospheric refraction or gravitational flexure, there is unlikely to be any systematic with declination. As the systematic is poorly understood and given that the comparison is with heavily-saturated Gaia stars, we choose not to correct it but note it as another potential source of systematic error. Estimating Astrometric Errors for Single Observations Characterizing the single-frame astrometric errors is complicated as the target star is observed in an environment (through 12.5 magnitudes of attenuation and on a background containing reflected and scattered light from the bright target star) very different from the remaining stars on the image. The errors are characterized by comparison with Gaia as the Gaia errors (less than a mas) are much smaller than the UBAD single-frame errors. Based on that comparison, it is clear that simply combining the center errors measured by SExtractor in quadrature with the calibration errors underes-timates the true errors. A floor of 3.5 mas is imposed on the center error of the target star on individual frames, which is then added to the center error measured by SExtractor (typically about one mas). This is then added in quadrature with the rms of the residuals of the astrometric calibration divided by the square root of the number of calibrating stars. This yields typical errors of around five mas separately in right ascension and declination, consistent with the histograms of differences between Gaia and UBAD positions seen in Figures 11 and 12. Figure 14 plots the differences between UBAD and Gaia EDR3 positions, normalized by the expected errors in the differences based on the modified UBAD error estimates. The mean and rms differences are indicated in G magnitude bins along with the individual star differences. These normalized differences should follow a Gaussian distribution with an rms of 1 if the UBAD error estimates are valid (since the UBAD errors are significantly larger than the Gaia errors, the UBAD errors dominate this analysis); that is the case at all magnitudes. CATALOG ASTROMETRY Positiona and Proper Motion Fits For each survey star, position and proper motion are fit separately in right ascension and declination using the individual UBAD observations combined with the Hipparcos-2 catalog position (with some exceptions, described below). As we lack enough UBAD observations to fit a parallax, UBAD positions are corrected for parallax using the Hipparcos-2 parallax. Only UBAD observations taken in less than 2 arcsec seeing and with rms residuals in both right ascension and declination of less than 20 mas are used. All stars have at least four UBAD observations. Individual-frame positions are weighted by their inverse variance, and a linear fit is performed of position versus epoch. Since the Hipparcos-2 position errors are less than a mas while the UBAD errors for the restricted set of observations are all around 5 mas, this is essentially an unweighted fit to the UBAD positions with the Hipparcos-2 position as a fixed point in the fit. The fits are displayed in Figure 15. Fig. Set 15. Position and Proper Motion Fits Sixty two of the stars have fits in right ascension and/or declination with χ 2 ν values greater than 2. Thirty three of these stars were refit without the Hipparcos-2 position, leading to much better fits; most are known or suspected multiple systems. These include the following stars: The remaining 29 stars with χ 2 ν values greater than 2 were processed including the Hipparcos-2 position, as excluding the Hipparcos-2 position did not significantly decrease the fit χ 2 ν values. There are an additional 6 and 10 unplotted stars with χ 2 ν > 3 in right ascension and declination, respectively. Figure 17 plots the distributions of the differences between Gaia EDR3 and UBAD catalog positions, after propagating the Gaia positions to the UBAD catalog epoch. The sample is limited to stars with good fits in both UBAD (χ 2 ν < 2 in both right ascension and declination) and Gaia (5-or 6-parameter fits and ruwe < 3), and that were fit as single stars in Hipparcos-2. The median offsets are -0.05 and 0.17 mas in right ascension and declination, respectively, with rms scatters of 1.8 and 2.3 mas. The median error for the Gaia positions for the stars in Figure 17, propagated to the UBAD catalog epoch, is around 0.2 mas, with 90% of the stars having errors less than around 0.35 mas. The typical errors for the UBAD stars are around 2 mas, consistent with the scatter in Figure 17. Thus, with position errors approximately 10 times as large as those of Gaia, UBAD can't validate Gaia bright-star astrometry to the stated accuracy in the Gaia catalog. However, UBAD does confirm Gaia's brightstar astrometry at approximately the 2 mas level, providing external validation of Gaia's astrometry for these heavily-saturated stars in their survey. The formal errors in the UBAD positions appear to underestimate the true errors. Figure 18 plots the differences between the Gaia and UBAD catalog positions, normalized by the expected errors in the differences. The rms differences in equal-sized bins of the formal UBAD position error are indicated by the orange points. If the error estimates are correct, the rms values should be near one. Clearly, either the UBAD or Gaia position errors are underestimated. The Gaia errors would have to be underestimated by about a factor of eight to explain the discrepancy, which is unlikely. Thus, the final UBAD catalog position errors are corrected by adding a floor error of 1.5 mas in quadrature to the formal errors. Figure 19 repeats Figure 18, but now using the inflated error estimates; the rms of the position differ- ences now match the expected value of one for all bins in formal error. Figures 20 and 21 display the distributions of the differences between the UBAD proper motions and those of Hipparcos-2 and Gaia EDR3, respectively, limited to the same clean sample used in Figure 17. There are no significant systematic offsets between the catalogs. The formal proper motion errors for UBAD are smaller than those for Gaia by typical factors of two to six and considerably smaller than those for Hipparcos-2, benefiting from the roughly 26year epoch difference between Hipparcos and UBAD (clearly one could combine Gaia EDR3 and Hipparcos-2 positions to derive proper mo- tions more accurate than UBAD's motions). Given the underestimated UBAD position errors, it's likely that the UBAD proper motions errors are also underestimated. THE CATALOG The UBAD catalog is presented in Table 1 (10) and (11) are the number of UBAD i and z observations used in the fits, respectively. Column (12) indicates whether the Hipparcos-2 coordinate was used in the fits (1=yes, 2=no). Columns (13) -(16) are the rms of the residuals for the UBAD observations only and the reduced χ 2 of the fits. Column (17) is the visual classification of the target star on the UBAD images (s=single, n=resolved neighbor, r=resolved blend, u=unresolved blend). Table 1 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. used in the fits, position errors (at the catalog epoch of 2017.0), and proper motion errors for those stars which included and excluded the Hipparcos-2 position, respectively. The median position errors are 1.9 mas in both right ascension and declination; 90% of catalog stars have position errors of less than 2.6 mas in either coordinate. We estimate systematic errors in the positions of order 1 -3 mas. For the 331 catalog stars whose fits include the Hipparcos-2 position, the median proper motion errors are 0.044 and 0.047 mas year −1 in right as-cension and declination, respectively, with 90% of such stars having proper motion errors less than 0.075 mas year −1 . Those stars whose fits excluded the Hipparcos-2 position have proper motion errors of order a few mas year −1 , depending on the time span of the UBAD observations for each star. Systematic errors in the proper motions are harder to estimate, as no bright-star catalog in this magnitude range has at least formally better errors in proper motions than UBAD, due to the long time span by including Hipparcos-2 positions in the proper mo- (12) and (13) are the mean FWHM and elongation of stars on the images, respectively. Table 2 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. tion calculations (though combining Hipparcos-2 with Gaia EDR3 would yield such a catalog). The systematic errors in the positions alone would imply systematic errors in proper motions of order 0.1 mas year −1 . CONCLUSION Gaia has revolutionized astrometry. While not designed for bright-star astrometry, Gaia EDR3 includes all but a few dozen of the brightest stars. Stars brighter than G ∼ 5 are saturated on the Gaia detectors, and the brightest stars are heavily saturated. Given the importance of such stars to government and commercial interests, it is important to provide some external validation for these stars. UBAD provides such validation at the few milliarcsecond level, as well as the most accurate current-epoch astrometry for those bright northern hemisphere stars excluded from the current Gaia catalog. UBAD gives precise positions and proper motions for 364 bright northern-hemisphere stars, including all but five such stars with either V < 3.5 or with I < 3.2 and V < 6; 36 of the stars are not included in Gaia EDR3. The bright target stars are exposed through a small 12.5 neutral-density spot, allowing for unsaturated images of the target stars to be calibrated directly against a dense set of stars with accurate Gaia astrometry more than 12 magnitudes fainter on the same image frame. Combining multiple UBAD observations for each star with Hipparcos-2, the median position error for UBAD stars is 1.9 mas in both right ascension and declination at the catalog epoch of 2017, with 90% of the stars having errors less than 2.6 mas; systematic errors are estimated at 1 - Figure 25, but for the 33 stars whose fits exclude the Hipparcos-2 position. There are an additional two and one unplotted stars with errors greater than 10 mas year −1 in right ascension and declination, respectively. 3 mas. For those stars that include Hipparcos-2 in their solution, the median error in proper motion is 0.045 mas year −1 in right ascension and 0.049 mas year −1 in declination, with 90% of stars having errors less than 0.1 mas year −1 . Coupled with a faint extension that is not included in the UBAD catalog, UBAD observations overlap Gaia in the magnitude range 2 G 6, extending from nearly the brightest stars to the non-saturated regime for Gaia. It is not possible to calibrate any optical distortion under the neutral-density spot relative to the rest of the field of view, thus we can not constrain the possibility of an overall systematic error in Gaia in this magnitude range. However, no magnitude-dependent systematic offset between Gaia and UBAD is evident at the milliarcsec level; as the comparison extends into the non-saturated regime for Gaia, an overall systematic error in Gaia is unlikely. After correction for the median offset between UBAD and Gaia, UBAD positions are consistent with Gaia to approximately 2 mas, providing external validation of Gaia's bright-star astrometry to that level. Gaia fundamentally changes the nature of ground-based astrometry. By providing a dense set of calibrating stars with sub-milliarcsecond positions, it changes the field from one dominated by errors in reference catalogs to one dominated by instrumentation and atmospheric limitations. There remains a role for ground-based astrometry with small telescopes in the Gaia era. A telescope such as the 61-inch, which has been producing sub-milliarcsecond differential astrometry for decades, can now produce milliarcsecond or better absolute astrometry, with the potential for significant impacts in the study of solar system bodies, resident space objects, and similar fields with astrometric requirements beyond what Gaia provides. Figure 1 . 1Typical survey image, taken through the i filter. The color bar indicates the non-linear scaling used to highlight various features in the image. Figure 2 . 2Close-up under the neutral-density spot of the image displayed inFigure 1, using a log scaling to highlight low-level features in the image. Figure 3 . 3Residual maps for observations taken in i (catalog position minus observed position) Figure 4 . 4Same asFigure 3, but for observations taken in z. Figure 5 .Figure 6 .Figure 7 .Figure 8 . 5678The slope and intercept values for the DCR fits for the ensemble of i images, plotted against refraction. The black lines are the fits to the ensemble, used to apply DCR corrections in i. Blue circles were included in the fits, orange squares were iteratively rejected as greater than 3 Same asFigure 5, but for the z images. Distribution of the rms residuals for the astrometric solutions for all images in the survey, separately in right ascension and declination. Distribution of the number of calibration stars used on each image in the survey. An additional 18 observations have more than 150 calibration stars. Figure 9 . 9Differences between Gaia EDR3 and UBAD individual-frame positions versus Gaia G magnitude for a clean set of i observations (right ascension in the top panel, declination in the bottom panel). The black points are observations for the main survey while the gray points are for the faint survey extension, used only in this figure. The blue line is the median difference for stars in the main survey. The solid and dotted orange lines indicate the median and interquartile range, respectively, of the differences in bins of 100 stars in G. Figure 10 .Figure 11 .Figure 12 . 101112Same asFigure 9, but for z observations. Histograms of the differences between Gaia and UBAD positions for a clean set of i observations. The differences in right ascension and declination are shown in the top and bottom panels, respectively. Same asFigure 11, but for z observations. Figure 13 . 13Offsets in declination between Gaia and UBAD positions as a function of declination, after correction for the median offsets. The orange solid and dashed lines are the median and interquartile range of the offsets, respectively, in bins of 200 stars. Figure 14 . 14Differences between Gaia and UBAD positions (Gaia -UBAD, right ascension in the top panel, declination in the bottom panel), normalized by the expected error in the differences. The black points are the individual stars. The blue and orange points are the mean and rms differences, respectively, in equal-sized bins in G. The rms values should be near one (indicated by the dotted cyan line). The dashed cyan line is a reference line at zero normalized error. Figure 15 . 15Fits to position and proper motion in the tangent plane, ξ in the top panel, η in the bottom panel, for HIP 000677. The circles with error bars are the individual observations and the lines are the fits. The Hipparcos-2 position (left panels) is taken as the tangent point and is plotted separate from the UBAD observations (right panels) due to the large epoch difference. The complete figure set (364 images) is available in the online journal. 8. HIP 003179, HIP 049637, HIP 077070, HIP 084379, HIP 087833, HIP 097278, HIP 112724: These are in the WDS but not marked as suspect in Hipparcos-2 or listed in the ∆µ catalog.9. HIP 054539, HIP 057399, HIP 083000, HIP 085693, HIP 086742, HIP 117863: These six stars are not known or suspected doubles. Their fits have χ 2 ν values in right ascension/declination of 2.6/1.0, 1.0/2.5, 3.6/1.7, 0.5/3.4, 2.2/5.9, and 2.1/0.9, respectively. The fits all look reasonable, and may just represent the tail of the χ 2 ν distributions. Figure 16 Figure 16 . 1616displays the distribution of χ 2 ν values from the fits. These fits yield the astrometric parameters given in the UBAD catalog.All UBAD catalog positions are for the epoch 2017.0, roughly the median epoch of the UBAD observations.4.2. Comparison of Catalog Astrometry withGaia and Hipparcos-χ 2 ν distributions for the position and proper motion fits in right ascension (top panel) and declination (bottom panel). Figure 17 . 17Distributions of differences in the catalog positions between Gaia and UBAD in right ascension (top panel) and declination (bottom panel). Figure 18 . 18Differences between Gaia and UBAD catalog positions (Gaia -UBAD, right ascension in the top panel, declination in the bottom panel), normalized by the expected error in the differences, plotted against the formal UBAD position error. The black points are the differences for individual stars. The blue and orange points are the mean and rms differences, respectively, in equal-sized bins. If the error estimates are correct, the rms values should be near one (indicated by the dotted cyan line). The dashed cyan line is a reference line at zero normalized error. Figure 19 . 19Same asFigure 18, except that the UBAD position errors used to normalize the differences now include a floor error of 1.5 mas added in quadrature with the formal error. The x-axis remains the formal error, without the addition of the floor error. . The catalog contains 364 stars, 36 of which are not included in Gaia EDR3. All catalog positions are given for the epoch 2017.0, roughly the median epoch of the observations. Table 2 lists the positions measured on each of the individual images comprising the survey. Positions are given as ICRS astrometric-place coordinates at the epoch of the observation (uncorrected for parallax and proper motion). Not all positions in this Figure 20 .Figure 21 .Figure 22 .Figure 23 . 20212223Distribution of the differences between Hipparcos-2 and UBAD proper motions for a clean sample of stars. There are an additional nine and six stars outside the histogram limits in right ascension and declination, respectively. Same asFigure 20, but comparing UBAD and Gaia proper motions. There are an additional six and one stars outside the histogram limits in right ascension and declination, respectively. Stacked histograms of catalog stars in Hipparcos-2 H p magnitude. The hatched blue and unhatched orange histograms are for stars with and without an entry in Gaia EDR3, respectively. Distribution of the number of UBAD observations used in the astrometry fits. Figure 24 .Figure 25 .Figure 26 . 242526Distributions of catalog position errors (right ascension in top panel, declination in bottom panel). There are an additional two and four unplotted stars with errors greater than 4.5 mas in right ascension and declination, respectively. Distributions of catalog proper motion errors for the 331 stars whose fits include the Hipparcos-2 position (right ascension in top panel, declination in bottom panel). Same as Table were wereused in compil- ing the final catalog (see Section 4.1). Figures 22, 23, 24, 25, and 26 display the dis- tributions of the catalog stars in Hipparcos-2 H p magnitude, number of UBAD observations Table 1. UBAD Catalog Hip α σ α cos δ δ σ δ µ α cos δ σ µα cos δ (deg) (mas) (deg) (mas) (mas year −1 ) (mas year −1 ) (1) (2) (3) (4) (5) (6) (7) 677 2.0976353 1.78 29.0896637 1.70 134.564 0.0369 746 2.2993446 1.74 59.1489328 1.68 523.668 0.0339 1067 3.3089709 1.98 15.1835486 1.83 1.681 0.0499 1168 3.6511466 1.65 20.2067096 2.45 91.341 0.0271 2219 7.0127137 2.12 17.8932221 2.42 114.618 0.0583 µ δ σ µ δ N i N z Hip used RMS α χ 2 ν,α RMS δ χ 2 ν,δ Class (mas year −1 ) (mas year −1 ) (mas) (mas) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) -162.824 0.0314 8 0 1 2.60 0.378 2.18 0.282 s -179.656 0.0293 12 0 1 2.95 0.481 2.58 0.365 s -9.443 0.0410 11 0 1 4.14 0.756 3.77 0.571 s 1.746 0.0754 9 0 1 1.89 0.165 5.79 1.440 s 20.559 0.0740 12 0 1 5.29 1.139 6.85 1.931 s Note-Column (1) is the Hipparcos identifier. Columns (2) -(5) are the ICRS catalog mean place coordi- nates and their errors at the catalog epoch of J2017. Columns (6) -(9) are the proper motions and their errors. Columns Table 2 . 2Single-epoch Positions Note-Column (1) is the observation epoch. Column (2) is the Hipparcos identifier. Columns (3) -(6) are the ICRS astrometric place coordinates and their errors at the observation epoch. Columns (7) -(8) are the observation exposure time and weighted mean airmass, respectively. Column (9) is the number of Gaia EDR3 reference stars used in the plate solution. Columns (10) -(11) are the rms residuals of the plate solution. ColumnsEpoch Hip α σ α cos δ δ σ δ (year) (deg) (mas) (deg) (mas) (1) (2) (3) (4) (5) (6) 2016.190580 26727 85.1897079 6.75 -1.9425936 5.35 2016.190593 26727 85.1897094 5.69 -1.9425971 5.35 2016.190680 34912 108.3475446 4.62 51.4288135 4.62 2016.190704 34912 108.3475466 4.62 51.4288122 4.59 2016.190761 37946 116.6638232 4.83 37.5174545 4.75 T X N s RMS α RMS δ Seeing Elong (s) (mas) (mas) (arcsec) (7) (8) (9) (10) (11) (12) (13) 180.0 1.279 27 23.35 9.87 1.457 1.057 360.0 1.288 27 16.50 13.13 1.604 1.035 750.0 1.042 35 3.37 3.31 1.259 1.015 600.0 1.043 35 2.52 3.00 1.177 1.039 360.0 1.002 30 3.72 3.54 1.155 1.074 We gratefully acknowledge the work of the outstanding engineering group at USNO, Flagstaff Station (some now retired), including Mike Divittorio, Fred Harris, Mike Schultheis, Al Rhodes, and Andrew Cenko. We also thank Bob Zavala for helpful comments on the text. 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[ "Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator", "Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator" ]	[ "W Arendt ", "A F M Ter Elst ", "M Warma " ]	[]	[]	In the very influential paper[4]Caffarelli and Silvestre studied regularity of (−∆) s , 0 < s < 1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea[16]and Galé, Miana and Stinga [7] gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.2010 Mathematics Subject Classification. 35R11, 35B65, 47A07.	10.1080/03605302.2017.1363229	[ "https://arxiv.org/pdf/1608.05707v1.pdf" ]	119,304,851	1608.05707	d2830089d2228fab141764faae718d7cd522decf	Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator 19 Aug 2016 W Arendt A F M Ter Elst M Warma Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator 19 Aug 2016 In the very influential paper[4]Caffarelli and Silvestre studied regularity of (−∆) s , 0 < s < 1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea[16]and Galé, Miana and Stinga [7] gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.2010 Mathematics Subject Classification. 35R11, 35B65, 47A07. Introduction In the very influential article [4] Caffarelli and Silvestre study the fractional powers (−∆) s , 0 < s < 1, on R N of the operator −∆ by identifying the operator (−∆) s with a Dirichlet-to-Neumann operator with respect to an extension to the upper half-plane. Subsequently, such extensions have been studied in more abstract settings by Stinga and Torrea [16] as well as by Galé, Miana and Stinga [7]. They obtain in particular a representation formula for the associated Dirichlet problem analogous to the Poisson formula. We also refer to [5,14] and their references for the case of symmetric second-order elliptic operators in divergence form with smooth coefficients on bounded open sets in R N subject to zero Dirichlet and Neumann boundary conditions on ∂Ω. Our contribution goes in the same direction. Instead of Banach spaces and generators of semigroups as in the papers [7,16] mentioned above, we concentrate on Hilbert spaces and sectorial operators. This allows us to obtain precise regularity results and well-posedness of the Dirichlet and the Neumann problem. The Dirichlet-to-Neumann operator will be shown to be an isomorphism between two interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power of the given sectorial operator. In this way, we prove, may be for the first time, uniqueness of the extensions. To be more specific, we consider a Hilbert space H and a sectorial operator A on H which is defined by a continuous, coercive form E : V × V → C, where V is a Hilbert space continuously and densely embedded in H, that is, V [3]. We also use the same representation formula used in [7,16]. Our proofs, however, are self-contained, using merely a few results of interpolation theory. The rest of the paper is structured as follows. We start with a short motivation for the result and the methods, by considering the square root of a bounded operator (Section 2). In Section 3 we put together some properties of the mixed Sobolev spaces related to fractions. The Dirichlet and Neumann problem is studied in Section 4. The main result on the identification of the Dirichlet-to-Neumann map with the fractional power in the coercive case is obtained in Section 5. In Section 6 we drop the condition that E is coercive and assume merely that E is sectorial with vertex zero. for all x ∈ H. Then there exists a unique accretive operator, denoted by A 1 2 , such that (A 1 2 ) 2 = A, see e.g. [9,Theorem V.3.35]. This operator A 1 2 can be realized as a Dirichlet-to-Neumann operator in the following way. We consider the Sobolev space W 1,2 ((0, ∞); H) := {u ∈ L 2 ((0, ∞); H); u ′ ∈ L 2 ((0, ∞); H)} and we recall that We have the following result. Proposition 2.1. For each x ∈ H there exists a unique u ∈ W 2,2 ((0, ∞); H) such that    −u ′′ (t) + Au(t) = 0, t ∈ (0, ∞), u(0) = x. (2.2) Proof. Define the sesquilinear form b : W 1,2 ((0, ∞); H) × W 1,2 ((0, ∞); H) → C by b(u, v) = ∞ 0 u ′ (t), v ′ (t) H + Au(t), v(t) H dt. Then b is continuous and from (2.1) we have that Re b(u, u) ≥ ∞ 0 u ′ (t) 2 H + α u(t) 2 H dt ≥ α u 2 W 1,2 ((0,∞);H) . (2.3) So b is coercive. Next we show existence. Let x ∈ H. There exists a function φ ∈ W 2,2 ((0, ∞); H) such that φ(0) = x. By the Lax-Milgram Lemma there exists a unique w ∈ W 1,2 0 ((0, ∞); H) such that b(w, v) = b(φ, v) for all v ∈ W 1,2 0 ((0, ∞); H), where W 1,2 0 ((0, ∞); H) := {v ∈ W 1,2 ((0, ∞); H) : v(0) = 0}. Let u := φ − w. Then u ∈ W 1,2 ((0, ∞); H) and ∞ 0 u ′ (t), v ′ (t) H + Au(t), v(t) H dt = 0 for all v ∈ W 1,2 0 ((0, ∞); H). This implies that u ′′ = Au weakly. Since Au ∈ L 2 ((0, ∞); H), one has that u ∈ W 2,2 ((0, ∞); H) and so u is a solution of (2.2). To show uniqueness, let u ∈ W 2,2 ((0, ∞); H) be a solution of (2.2) such that u(0) = 0. Then (2.3) gives 0 = ∞ 0 −u ′′ (t), u(t) H + Au(t), u(t) H dt ≥ α u 2 W 1,2 ((0,∞);H) . Hence u = 0. Now we define the Dirichlet-to-Neumann operator D : H → H as follows. Let x ∈ H. Let u ∈ W 2,2 ((0, ∞); H) be the unique solution of (2.2). Define Dx := −u ′ (0). Then the following result holds. Proof. We first show that D 2 = A. Let x ∈ H and u ∈ W 2,2 ((0, ∞); H) be such that u(0) = x and −u ′′ (t) + Au(t) = 0 for all t ∈ (0, ∞). Let w := u ′ . Then −w ′′ (t)+Aw(t) = 0 weakly for all t ∈ (0, ∞). This shows that w ∈ W 2,2 ((0, ∞); H) and w is a solution of (2.2) with w(0) = u ′ (0). Then −w ′ (0) = Du ′ (0) = D(−Dx). Moreover, u ∈ W 3,2 ((0, ∞); H) and Ax = Au(0) = u ′′ (0) = w ′ (0) = D 2 x. Next we show that D is accretive. Let x ∈ H and let u be the unique solution of (2.2). Then Re Dx, x H = Re −u ′ (0), u(0) H = Re ∞ 0 d dt u ′ (t), u(t) H dt = Re ∞ 0 u ′ (t), u ′ (t) H + u ′′ (t), u(t) H dt = ∞ 0 u ′ (t) 2 H + Re Au(t), u(t) H dt ≥ 0. Hence D is accretive. The crucial argument in the proof above is to differentiate the differential equation −u ′′ +Au = 0. This is possible since the operator A is bounded. For unbounded operators different arguments are needed. For fractional powers other than squares, weighted Sobolev spaces are needed. They are introduced in the next section. Sobolev spaces The Dirichlet and the Neumann problems we have in mind are well posed in mixed Sobolev spaces which are known from interpolation theory. We give the definition, cite results we shall need and prove an integration by parts formula. Let X be a Hilbert space. We will consider spaces of integrable functions on (0, ∞) with values in X. Derivatives will be taken in the distributional sense; i.e. using the elements of the scalar space C ∞ c ((0, ∞)) of all infinitely differentiable C-valued functions with compact support as test functions. Here is the precise definition. Definition 3.1. (a) Let u, v ∈ L 1 loc (X) := L 1 loc ((0, ∞); X). We say that v is the weak deriva- tive of u if − ∞ 0 ϕ ′ (t)u(t) dt = ∞ 0 ϕ(t)v(t) dt for all ϕ ∈ C ∞ c ((0, ∞)). In that case we write u ′ := v. (b) Let E be a subspace of L 1 loc (X) and let u ∈ L 1 loc (X). We say that u ′ ∈ E if there exists a v ∈ E such that v is the weak derivative of u. The weak derivative is unique if it exists and for all u ∈ C 1 ((0, ∞); X) the weak and classical derivatives coincide. Let X, Y be Hilbert spaces such that Y d ֒→ X. This means that Y is a dense subspace of X and the injection of Y into X is continuous. Fix 0 < s < 1. We define the space W s (X, Y ) := {u ∈ L 1 loc (Y ) : u ′ ∈ L 1 loc (X), t → t s u(t) ∈ L ⋆ 2 (Y ) and t → t s u ′ (t) ∈ L ⋆ 2 (X)}, where for Z = X or Z = Y , L ⋆ 2 (Z) := L 2 Z, dt t = L 2 (0, ∞); Z, dt t . In order to avoid clutter we write t s for the function t → t s . It is clear that W s (X, Y ) endowed with the norm u Ws(X,Y ) = t s u 2 L ⋆ 2 (Y ) + t s u ′ 2 L ⋆ 2 (X) 1 2 = ∞ 0 u(t) 2 Y + u ′ (t) 2 X t 2s−1 dt 1 2 is a Banach space and it is even a Hilbert space. We quote the following result from [11, Proposition 1.2.10]. Proposition 3.2. Let u ∈ W s (X, Y ). Then u(0) := lim t↓0 u(t) exists in the norm on X. Moreover, u(0) ∈ [X, Y ] 1−s . The map u → u(0) from W s (X, Y ) into [X, Y ] 1−s is continuous and surjective. Recall that [X, Y ] θ is the complex interpolation space between X and Y for all 0 < θ < 1. Note that the complex interpolation space [X, Y ] θ coincides with the trace-method real interpolation space (X, Y ) θ,2 since we restrict ourselves to Hilbert spaces. Let C ∞ c ([0, ∞); Y ) := {u : [0, ∞) → Y : u is infinitely differentiable and supp u is compact in [0, ∞)}. Clearly C ∞ c ([0, ∞); Y ) is a subspace of W s (X, Y ) . We need the following density result. ∞ c ([0, ∞); Y ) is dense in W s (X, Y ). (b) If s < 1 2 , then the space {u ∈ W s (X, Y ) ∩ C ∞ ((0, ∞); Y ) : supp u is a bounded set in (0, ∞)} is dense in W s (X, Y ). The proof of Proposition 3.3 requires quite some preparation. Let Z be a Hilbert space and θ ∈ (0, 1). Define the space W θ (Z) = {u ∈ L 1 loc (Z) : t θ u ∈ L * 2 (Z)}, with the norm u W θ (Z) = t θ u L * 2 (Z) . Note that W θ (Z) = L 2 ((0, ∞); Z, t 2θ−1 dt). Lemma 3.4. Let θ ∈ (0, 1). Let u ∈ W θ (Z) and r ∈ (0, ∞). Define L r u : (0, ∞) → Z by (L r u)(t) = u(r −1 t). Then L r u ∈ W θ (Z) and L r u W θ (Z) = r θ u W θ (Z) . Moreover, lim r→1 L r u = u in W θ (Z) for all u ∈ W θ (Z). Proof. Let u ∈ W θ (Z) and r ∈ (0, ∞). Clearly L r u ∈ L 1 loc (Z). Moreover, t θ L r u 2 L * 2 (Z) = ∞ 0 t θ u(r −1 t) 2 Z dt t = r 2θ ∞ 0 t θ u(t) 2 Z dt t = r 2θ u 2 W θ (Z) . This proves the first two claims. If u is a step function, then it is easy to see that lim r→1 L r u = u in W θ (Z). Since the step functions are dense in L 2 ((0, ∞); Z, t 2θ−1 dt) by [1, Lemma 3.26(1)], the lemma follows. Remark 3.5. The space (0, ∞) with multiplication is a one-dimensional Lie group. The Haar measure is dt t and the corresponding L 2 -space is L * 2 . Lemma 3.4 states that the vector valued left representation in W θ (Z) is well-defined and is a continuous representation of the group (0, ∞) in W θ (Z). For the remaining of this section fix for all n ∈ N a function ρ n ∈ C ∞ c ((0, ∞)) such that ρ n ≥ 0, supp ρ n ⊂ (1 − 1 2n , 1 + 1 n ) and lim n→∞ ∞ 0 ρ n (t) dt t = 1. For all χ ∈ C ∞ c ((0, ∞)), θ ∈ (0, 1) and u ∈ W θ (Z) define χ * u : (0, ∞) → Z by (χ * u)(t) = ∞ 0 χ(r)u(r −1 t) dr r . Clearly χ * u ∈ C ∞ ((0, ∞); Z). Lemma 3.6. Let θ ∈ (0, 1) and u ∈ W θ (Z). Then the following assertions hold. (a) If χ ∈ C ∞ c ((0, ∞)), then χ * u ∈ W θ (Z) ∩ C ∞ ((0, ∞); Z). (b) lim n→∞ ρ n * u = u in W θ (Z). Proof. (a). Let t ∈ (0, ∞). Then t θ (χ * u)(t) Z ≤ ∞ 0 r θ \|χ(r)\| (r −1 t) θ u(r −1 t) Z dr r . So ∞ 0 t θ (χ * u)(t) 2 Z dt t 1 2 ≤ ∞ 0 t θ \|χ(t)\| dt t · ∞ 0 t θ u(t) 2 Z dt t 1 2 . Therefore χ * u ∈ W θ (Z). (b). Let n ∈ N. Set λ n = ∞ 0 ρ n (r) dr r . Then ρ n * u − λ n u = ∞ 0 ρ n (r)(L r u − u) dr r . So ρ n * u − u W θ (Z) ≤ \|1 − λ n \| u W θ (Z) + ∞ 0 ρ n (r) L r u − u W θ (Z) dr r . Then the statement follows from Lemma 3.4 together with the condition that lim n→∞ λ n = 1. As an immediately consequence we obtain the next proposition. Proposition 3.7. Let θ ∈ (0, 1). Then the space W θ (Z) ∩ C ∞ ((0, ∞); Z) is dense in W θ (Z). Now we are able to prove Proposition 3.3. Proof of Proposition 3.3. The proof is in several steps. Step 1. Let s ∈ (0, 1). We claim that the space W s (X, Y ) ∩ C ∞ ((0, ∞); Y ) is dense in W s (X, Y ). Indeed, let u ∈ W s (X, Y ). Then ρ n * u ∈ C ∞ ((0, ∞); Y ) for all n ∈ N and lim n→∞ ρ n * u = u in W s (Y ) by Lemma 3.6(b). For all n ∈ N define ψ n ∈ C ∞ c ((0, ∞)) by ψ n (r) = 1 r ρ n (r) . Then ψ n ≥ 0 and supp ψ n ⊂ (1 − 1 2n , 1 + 1 n ). Moreover, lim n→∞ ∞ 0 ψ n (r) dr r = 1. Hence lim n→∞ (ρ n * u) ′ = lim n→∞ ψ n * (u ′ ) = u ′ in W s (X) by Lemma 3.6(b), this time applied with ρ n replaced by ψ n . So lim n→∞ ρ n * u = u in W s (X, Y ) and the claim is proved. Step 2. Let s ∈ (0, 1). We show that the space {u ∈ W s (X, Y )∩C ∞ ((0, ∞); Y ) : supp u is a bounded set in (0, ∞)} is dense in W s (X, Y ). In fact, since Y is con- tinuously embedded into X there exists a constant c > 0 such that y X ≤ c y Y for all y ∈ Y . Let u ∈ W s (X, Y ) ∩ C ∞ ((0, ∞); Y ). Let χ ∈ C ∞ ([0, ∞)) be such that 1 [0,1] ≤ χ ≤ 1 [0,2] . For all n ∈ N define χ n : [0, ∞) → R by χ n (t) = χ( t n ). Moreover, define u n = χ n u. Then u n ∈ W s (X, Y ). If n ∈ N, then t s (u − u n ) 2 L * 2 (Y ) = ∞ 0 t s (1 − χ n )(t) u(t) Y 2 dt t ≤ ∞ n t s u(t) 2 Y dt t . So lim n→∞ t s (u − u n ) L * 2 (Y ) = 0. Next, u ′ n = χ ′ n u + χ n u ′ for all n ∈ N. It follows similarly that lim n→∞ t s (u ′ − χ n u ′ ) L * 2 (X) = 0. We shall show that lim n→∞ t s χ ′ n u L * 2 (X) = 0. Let n ∈ N. Then t s χ ′ n u 2 L * 2 (X) = 1 n 2 ∞ 0 t s χ ′ ( t n ) u(t) X 2 dt t ≤ χ ′ 2 ∞ n 2 ∞ 0 t s u(t) 2 X dt t ≤ c 2 χ ′ 2 ∞ n 2 t s u 2 L * 2 (Y ) . So lim n→∞ t s χ ′ n u L * 2 (X) = 0 and hence lim n→∞ u n = u in W s (X, Y ). Then Step 2 follows by an application of Step 1. Step 3. We prove the two statements of Proposition 3.3. (b). This is a special case of Step 2. (a). Let u ∈ W s (X, Y ) ∩ C ∞ ((0, ∞); Y ) and suppose that supp u is a bounded set in (0, ∞). For all n ∈ N define u n : (0, ∞) → Y by u n (t) = u(t + 1 n ). Then u n ∈ C ∞ c ([0, ∞); Y ). Moreover, if n ∈ N, then t s u n 2 L * 2 (Y ) = ∞ 0 t 2s−1 u(t + 1 n ) 2 Y dt = ∞ 1 n (t − 1 n ) 2s−1 u(t) 2 Y dt ≤ ∞ 1 n t 2s−1 u(t) 2 Y dt ≤ t s u 2 L * 2 (Y ) ≤ u 2 Ws(X,Y ) , where we have used that 2s − 1 ≥ 0 in the first inequality. Similarly, t s u ′ n L * 2 (X) ≤ u Ws(X,Y ) for all n ∈ N. Hence the sequence (u n ) n∈N is bounded in W s (X, Y ). Therefore it has a subsequence which converges weakly in W s (X, Y ). So u is in the weak closure of C ∞ c ([0, ∞); Y ) in W s (X, Y ). Together with Step 2 it follows that C ∞ c ([0, ∞); Y ) is weakly dense in W s (X, Y ). Since C ∞ c ([0, ∞); Y ) is convex, it is then also norm dense in W s (X, Y ). Next, we want to specify our settings to Gelfand triples; i.e. we consider two Hilbert spaces H, V such that V d ֒→ H. Let i be the inclusion from V into H. Then the dual map i * is a continuous map from H ′ into V ′ , where H ′ and V ′ denote the antidual of H and V , respectively. Since i has dense image, the map i * is injective. Moreover, it also has a dense image. By the Riesz representation theorem one can identify H with H ′ . We call H the pivot space. Thus one has the chain V ֒→ H ≃ H ′ ֒→ V ′ which is known as a Gelfand triple. Therefore one has the following continuous and dense embeddings V d ֒→ H d ֒→ V ′ . Remark 3.8. By the spectral theorem up to unitary equivalence one can assume that H = L 2 (Γ, σ) for some measure space (Γ, Σ, σ), and V = L 2 (Γ, m dσ) for some measurable function m : Γ → [1, ∞). Then V ′ = L 2 (Γ, dσ m ) and the duality is given by f, g V ′ ,V = Γ f (x)g(x) dσ(x) for all f ∈ V ′ and g ∈ V . Thus f, g V ′ ,V is written in terms of the measure σ without weight. This is the reason for calling H the pivot space. In this unitary equivalent situation the complex interpolation space becomes [H, V ] s = L 2 (Γ, m s dσ) and [H, V ′ ] s = L 2 (Γ, m −s dσ). In particular, [H, V ] ′ s = [H, V ′ ] s and we have the new Gelfand triple [H, V ] s d ֒→ H d ֒→ [H, V ′ ] s , with again H as pivot space. The following integration by parts formula will be crucial for us. Proposition 3.9. Let 0 < s < 1. Let w ∈ W s (V ′ , H) and v ∈ W 1−s (H, V ). Then t → w ′ (t), v(t) V ′ ,V and t → w(t), v ′ (t) H are elements of L 1 ((0, ∞)). Moreover, − ∞ 0 w ′ (t), v(t) V ′ ,V dt = ∞ 0 w(t), v ′ (t) H dt + w(0), v(0) [H,V ′ ]s,[H,V ]s . Proof. Let w ∈ W s (V ′ , H) and v ∈ W 1−s (H, V ). By definition t s w ′ ∈ L * 2 (V ′ ) and t 1−s v ∈ L * 2 (V ). So t → w ′ (t), v(t) V ′ ,V is an element of L 1 ((0, ∞)). Similarly, t s w ∈ L * 2 (H) and t 1−s v ′ ∈ L * 2 (H). Consequently t → w(t), v ′ (t) H is an element of L 1 ((0, ∞)). Together with Proposition 3.2 it follows that the map ∞); H) and suppose that both supp w and supp v are bounded sets in (0, ∞). (w, v) → ∞ 0 w ′ (t), v(t) V ′ ,V + w(t), v ′ (t) H dt + w(0), v(0) [H,V ′ ]s,[H,V ]s is continuous from W s (V ′ , H) × W 1−s (H, V ) into C. Hence it suffices to show that ∞ 0 w ′ (t), v(t) V ′ ,V + w(t), v ′ (t) H dt + w(0), v(0) [H,V ′ ]s,[H,V ]s = 0 (3.1) for all (w, v) in a dense subset of W s (V ′ , H) × W 1−s (H, V ). Let w ∈ W s (V ′ , H) ∩ C ∞ ((0, ∞); H), v ∈ W 1−s (H, V ) ∩ C ∞ ((0,Then ∞ 0 w ′ (t), v(t) V ′ ,V + w(t), v ′ (t) H dt = lim ε↓0 ∞ ε w ′ (t), v(t) V ′ ,V + w(t), v ′ (t) H dt = lim ε↓0 ∞ ε w ′ (t), v(t) H + w(t), v ′ (t) H dt = lim ε↓0 − w(ε), v(ε) H . (3.2) We distinguish two cases. Case 1. Suppose that s ≥ 1 2 . Let w ∈ C ∞ c ([0, ∞); H) and v ∈ W 1−s (H, V ) ∩ C ∞ ((0, ∞); H) be such that supp v is a bounded set in (0, ∞). Then lim ε↓0 v(ε) = v(0) in H by Proposition 3.2. So lim ε↓0 w(ε), v(ε) H = w(0), v(0) H = w(0), v(0) [H,V ′ ]s,[H,V ]s . Hence (3.1) is valid by using (3.2). Since C ∞ c ([0, ∞); H) is dense in W s (V, H) by Proposition 3.3(b) and the space {v ∈ W 1−s (H, V ) ∩ C ∞ ((0, ∞); V ) : supp v is a bounded set in (0, ∞)} is dense in W 1−s (H, V ) by Proposition 3.3(a) , the proposition follows in this case. Case 2. Suppose that s < 1 2 . Obviously 1 − s ≥ 1 2 , so now the space C ∞ c ([0, ∞); V ) is dense in W 1−s (H, V ) by Proposition 3.3(b). Let v ∈ C ∞ c ([0, ∞); V ) and w ∈ W s (V, H)∩C ∞ ((0, ∞); V ) with supp w a bounded set in (0, ∞). Then Proposition 3.2 implies that lim ε↓0 w(ε) = w(0) in V ′ . Also lim ε↓0 v(ε) = v(0) in V since v ∈ C ∞ c ([0, ∞); V ). So lim ε↓0 w(ε), v(ε) H = lim ε↓0 w(ε), v(ε) V ′ ,V = w(0), v(0) V ′ ,V = w(0), v(0) [H,V ′ ]s,[H,V ]s . By (3.2) one deduces (3.1) and the density of Proposition 3.3 completes the proof in this case. The Dirichlet and Neumann problem The aim of this section is to prove well-posedness and regularity of solutions of a Dirichlet and a Neumann problem. Let V , H be Hilbert spaces such that V d ֒→ H and let E : V × V → C be a continuous and coercive sesquilinear form. So there are constants µ, M > 0 such that \|E(u, v)\| ≤ M u V v V and Re E(u, u) ≥ µ u 2 V for all u, v ∈ V . Denote by A ∈ L(V, V ′ ) the operator given by Au, v V ′ ,V = E(u, v) for all u, v ∈ V . Throughout the remainder of the paper, we shall use the notation E(u) := E(u, u). Let 0 < s < 1 be fixed throughout this section. We are interested in the equation u ′′ (t) + 1 − 2s t u ′ (t) − Au(t) = 0, t ∈ (0, ∞). (4.1) We shall see in Theorem 4.4 that the Sobolev space in the next definition is the correct space for the well-posedness of the Dirichlet problem. Definition 4.1. An (E, s)-harmonic function (or shortly s-harmonic function) is a function u ∈ W 1−s (H, V ) such that t 1−2s u ′ ∈ W s (V ′ , H) and −(t 1−2s u ′ ) ′ (t) + t 1−2s Au(t) = 0 in V ′ for a.e. t ∈ (0, ∞). (4.2) Note that both functions (t 1−2s u ′ ) ′ and t 1−2s Au are in L 2 (V ′ , t 2s dt t ) so that we actually obtain an identity in this space. Note also that (4.2) is equivalent to (4.1). If u is s-harmonic, then Proposition 3. exists in V ′ and is an element of [H, V ′ ] s . We consider this limit as an s-normal derivative. If s = 1 2 then it equals −u ′ (0). In this section we are interested in the following two problems. • Given x ∈ [H, V ] s , the Dirichlet problem consists in finding an s-harmonic function u such that u(0) = x. • Given y ∈ [H, V ′ ] s , the Neumann problem consists in finding an sharmonic function u such that y = − lim t↓0 t 1−2s u ′ (t). We will see that both problems are well-posed. We define the sesquilinear form b s : W 1−s (H, V ) × W 1−s (H, V ) → C by b s (u, v) := ∞ 0 u ′ (t), v ′ (t) H + E(u(t), v(t)) t 2(1−s) dt t . (4.4) Then b s is continuous and coercive. Lemma 4.2. Let u be s-harmonic. Write y := − lim t↓0 t 1−2s u ′ (t) in V ′ . Then b s (u, v) = y, v(0) [H,V ′ ]s,[H,V ]s (4.5) for all v ∈ W 1−s (H, V ). In particular, b s (u) = y, u(0) [H,V ′ ]s,[H,V ]s . (4.6) Proof. Note that u ∈ W 1−s (H, V ) since s-harmonic. Set w := t 1−2s u ′ . Then w ∈ W s (V ′ , H). Let v ∈ W 1−s (H, V ). Then Proposition 3.9 gives ∞ 0 w ′ (t), v(t) V ′ ,V dt = − ∞ 0 w(t), v ′ (t) H dt + y, v(0) [H,V ′ ]s,[H,V ]s = − ∞ 0 u ′ (t), v ′ (t) H t 1−2s dt + y, v(0) [H,V ′ ]s,[H,V ]s . Since w ′ (t) = t 1−2s Au(t) in V ′ for a.e. t ∈ (0, ∞), it follows that w ′ (t), v(t) V ′ ,V = t 1−2s E(u(t), v(t)) for a.e. t ∈ (0, ∞). This proves (4.5). Conversely, we may use the form b s to prove s-harmonicity using only a small space of test functions. H). We proved that u is s-harmonic. Lemma 4.3. Let u ∈ W 1−s (H, V ). Assume b s (u, v) = 0 for all v ∈ C ∞ c ((0, ∞); V ). Then u is s-harmonic. Proof. Let ϕ ∈ C ∞ c ((0, ∞)). For all v ∈ V defineṽ ∈ C ∞ c ((0, ∞); V ) byṽ(t) = ϕ(t)v. Then by assumption 0 = b s (u,ṽ) = ∞ 0 u ′ (t),ṽ ′ (t) H + E(u(t),ṽ(t)) t 1−2s dt = ∞ 0 u ′ (t), v H ϕ ′ (t)t 1−2s dt + ∞ 0 Au(t), v V ′ ,V ϕ(t)t 1−2s dt = ∞ 0 ϕ ′ (t) w(t), v V ′ ,V dt + ∞ 0 ϕ(t) t 1−2s Au(t), v V ′ ,V dt, where w = t 1−2s u ′ . Since v ∈ V is arbitrary, Definition 3.1 implies that −w ′ + t 1−2s Au = 0 in L 1 loc (V ′ ). Hence −(t 1−2s u ′ ) ′ (t) + t 1−2s Au(t) = 0 in V ′ for almost every t ∈ (0, ∞). Because u ∈ W 1−s (H, V ), one has t 1−s u ∈ L * 2 (V ), so t 1−s Au ∈ L ⋆ 2 (V ′ ). Hence t s (t 1−2s Au) ∈ L ⋆ 2 (V ′ ) and this implies that t s w ′ ∈ L ⋆ 2 (V ′ ). In addition t s w = t s t 1−2s u ′ = t 1−s u ′ ∈ L * 2 (H), since u ∈ W 1−s (H, V ). Therefore w ∈ W s (V ′ , We can now prove well-posedness of the Dirichlet problem and the Neumann problem. φ ∈ W 1−s (H, V ) such that φ(0) = x. Define L : W 0 1−s (H, V ) → C by Lv := b s (φ, v), where W 0 1−s (H, V ) := {v ∈ W 1−s (H, V ) : v(0) = 0}, which is a closed subspace of W 1−s (H, V ) . Then L is continuous and anti-linear. Since the form b s is coercive, there exists a unique The continuity of the first mapping follows from Proposition 3.2 and the continuity of the second follows from (4.7). The continuity of the inverses is a consequence of the closed graph theorem. w ∈ W 0 1−s (H, V ) such that b s (w, v) = Lv for all v ∈ W 0 1−s (H, V ). Let u := φ − w. Then u ∈ W 1−s (H, V ), u(0) = φ(0) = x and b s (u, v) = 0 for all v ∈ W We conclude this section by specifying to the case s = 1 2 , which is much simpler. Proof. By definition u is 1 2 -harmonic if and only if u ∈ W 1 2 (H, V ), u ′ ∈ W 1 2 (V ′ , H) and −u ′′ (t) + Au(t) = 0 in V ′ for a.e. t ∈ (0, ∞). The latter is equivalent to u ∈ L 2 ((0, ∞); V ), u ′ ∈ L 2 ((0, ∞); H), u ′′ ∈ L 2 ((0, ∞); V ′ ) and −u ′′ (t) + Au(t) = 0 in V ′ for a.e. t ∈ (0, ∞). By [10, Chapter 1, Proposition 2.2] one has the inclusion W 2,2 ((0, ∞); V ′ ) ∩ L 2 ((0, ∞); V ) ⊂ W 1,2 ((0, ∞); H). Then the proposition follows. Finally, we mention the following W 2,2 -regularity. ∞); H) ∩ L 2 ((0, ∞); D(A)) and −ũ ′′ + A 2ũ = 0. Therefore Proposition 4.8 implies that u is 1 2 -harmonic. Moreover,ũ(0) = φ(0) = u(0). Then Theorem 4.4(a) gives u =ũ ∈ W 2,2 ((0, ∞); H). This proves (i). → (u(0), u ′ (0)) maps W 2,2 ((0, ∞); X) ∩ L 2 ((0, ∞); Y ) into [X, Y ] 3 4 × [X, Y ] 1 4 and is surjective.= −φ ′′ +A 2 φ ∈ L 2 ((0, ∞); H). Since B D +A 2 is invertible there exists a w ∈ D(B D )∩ D(A 2 ) such that −w ′′ + A 2 w = f . Letũ = φ − w. Thenũ ∈ W 2,2 ((0, (iii) ⇒ (i). The proof is similar, but here we consider the negative Laplacian with Neumann boundary conditions B N on L 2 ((0, ∞); H); that is D(B N ) = {w ∈ W 2,2 ((0, ∞); H) : w ′ (0) = 0} and B N w = −w ′′ . This operator is associated with the closed form b N given by b N (w, v) = ∞ 0 w ′ (t), v ′ (t) H dt and D(b N ) = W 1,2 ((0, ∞); H). Then again by the Dore-Venni Theorem the operator B N + A 2 with usual domain D(B N + A 2 ) = D(B N ) ∩ D(A 2 ) is invertible, where A 2 is as above. By assumption we have u ′ (0) ∈ [H, D(A)] 1 4 . Then by Proposition 4.11 there exists a φ ∈ W 2,2 ((0, ∞); H) and we have shown (i). ∞); H) ∩ L 2 ((0, ∞); D(A)) such that φ ′ (0) = u ′ (0). Let f := −φ ′′ + A 2 φ ∈ L 2 ((0, ∞); H). Since B D + A 2 is surjective there exists a w ∈ D(B N ) ∩ D(A 2 ) such that −w ′′ + A 2 w = f . Letũ = φ − w. Thenũ ∈ W 2,2 ((0, ∞); H)∩L 2 ((0, ∞); D(A)) and −ũ ′′ + A 2ũ = 0. Moreover,ũ ′ (0) = φ ′ (0) = u ′ (0). Thusũ is the 1 2 -harmonic function satisfyingũ ′ (0) = u ′ (0). So u =ũ ∈ W 2,2 ((0, We notice that Property (i) in Proposition 4.10 is interesting. It implies that u ′′ (t) ∈ H and Au(t) ∈ H for almost all t > 0. This is a kind of maximal regularity. The fractional powers via the D-t-N operator We adopt the notation and assumptions of Section 4; that is V and H are Hilbert We call the operator B in Proposition 5.2 the operator associated with the pair (b, j). spaces, V d ֒→ H, the sesquilinear form E : V × V → C is continuous, coercive and A ∈ L(V, V ′ ) is given by Au, v V ′ ,V = E(u, v). We wish to apply Proposition 5.2 with W = W 1−s (H, V ) and b = b s the sesquilinear form given by (4.4). Recall that b s : W 1−s (H, V ) × W 1−s (H, V ) → C is given by b s (u, v) := ∞ 0 u ′ (t), v ′ (t) H + E(u(t), v(t)) t 2(1−s) dt t , the form b s is continuous and coercive. Define j : W 1−s (H, V ) → H by j(u) = u(0). Note that also j depends on s. Then j is linear, continuous with dense image (see Proposition 3.2). Recall that the operator −A generates a holomorphic C 0 -semigroup (e −tA ) t≥0 on H. In particular, the mapping t → e −tA is in C ∞ ((0, ∞); L(H)) and even in C ∞ ((0, ∞); D(A k )) for all k ∈ N if we provide D(A k ) with the norm x D(A k ) = A k x H . Moreover, e −tA L(H) ≤ e −µt for all t > 0, where µ > 0 is a coercivity constant of the form E. , v) = y, v(0) H for all v ∈ W 1−s (H, V ). Choosing in particular v ∈ C ∞ c ((0, ∞); V ), Lemma 4.3 shows that u is s-harmonic. Let z := D s x = − lim t↓0 t 1−2s u ′ (t) in V ′ . Then (4.5) gives y, v(0) H = b s (u, v) = z, v(0) [H,V ′ ]s,[H,V ]s for all v ∈ W 1−s (H, V ). Consequently, y, w H = z, w [H,V ′ ]s, Define the function U : [0, ∞) → L(H) by U(t) = 1 Γ(s) ∞ 0 e − t 2 4r r s e −rA dr r . (5.1) Then U ∈ C ∞ ((0, ∞); L(H)) ∩ C([0, ∞); L(H)). The function U has the following properties. U ′′ (t) + 1 − 2s t U ′ (t) = A U(t). (5.2) (b) U(0) = A −s . (c) Let x ∈ D(A s ). Define u ∈ C ∞ ((0, ∞); H) by u(t) := U(t)A s x. Then lim t↓0 −t 1−2s u ′ (t) = c s A s x in H. (d) There exist δ ∈ (0, µ) and M ≥ 0 such that U(t) L(H) ≤ M e −δt for all t > 0. u ′ (t) = − 1 2Γ(s) ∞ 0 e − t 2 4r tr s−1 e −rA A s x dr r . (5.3) Substituting τ = r t 2 gives −t 1−2s u ′ (t) = 1 2Γ(s) ∞ 0 e − 1 4τ τ s−1 e −t 2 τ A A s x dτ τ . Hence lim t↓0 −t 1−2s u ′ (t) = c s A s x, since 1 2Γ(s) ∞ 0 e − 1 4τ τ s−1 dτ τ = 1 2Γ(s) ∞ 0 e −r r 1−s 4 1−s dr r = 2 1−2s Γ(1 − s) Γ(s) = c s . (d). Using the fact that e −tA L(H) ≤ e −µt for all t ≥ 0, one obtains that U(t) L(H) ≤ 1 Γ(s) t 0 e −µr e − t 2 4r r s−1 dr + 1 Γ(s) ∞ t e −µr e − t 2 4r r s−1 dr ≤ e − t 2 4t Γ(s) t 0 e −µr r s−1 dr + e − µ 2 t Γ(s) ∞ t e − µ 2 r r s−1 dr ≤ µ −s e − t 4 + (2µ −1 ) s e − µ 2 t for all t > 0. (e). Write u = U(·)A s x. First we shall prove that t 1−s u ∈ L ⋆ 2 (V ). Observe that D(A) ֒→ V . Hence there exists a constant c > 0 such that z V ≤ c Az H for all z ∈ D(A). Since x ∈ D(A 2 ) and A U(t)A s x = U(t)A 1+s x, it suffices to show that ∞ 0 t 2(1−s) U(t)z 2 H dt t < ∞ for all z ∈ H. From part (d) we easily see that ∞ 1 t 2(1−s) U(t)z 2 H dt t < ∞ and 1 0 t 2(1−s) U(t)z 2 H dt t < ∞. Hence t 1−s u ∈ L ⋆ 2 (V ). Secondly, we show that t 1−s u ′ ∈ L ⋆ 2 (H). In order to prove this we first show that 1 0 t 2(1−s) u ′ (t) 2 H dt t ≤ c 2 1 0 t 2s dt t < ∞. It remains to show that As a consequence of these results, each s-harmonic function is a classical solution of the equation ∞ 1 t 2(1−s) u ′ (t) 2 H dt t < ∞. (5.4) We shall use (5.3). Let t ≥ 1. Observe that ∞ 0 e − t 2 4r tr s−1 e −rµ dr r ≤ e − t 8 t 0 e − t 2 8r tr s−1 e −rµ dr r + e − tµ 2 ∞ t e − t 2 4r tr s−1 e − rµ 2 dr r ≤ e − t 8 t ∞ 0 e − 1 8r r s−2 e −rµ dr + te − tµ 2 ∞ 1 r s−2 e − rµ 2 dr ≤ ce −εt for a suitable constant c > 0, where ε = 1 9 ∧ µ 4 . Hence it follows from (5.3) that u ′ (t) H ≤ c 2Γ(s) A s x H e −u ′′ (t) + 1 − 2s t u ′ (t) − Au(t) = 0, t ∈ (0, ∞). Now we want to rephrase the results for a concrete operator A. Then A is associated with the classical Dirichlet form E : W 1,2 0 (Ω) × W 1,2 0 (Ω) → C given by E(u, v) = Ω (∇u) · (∇v). It follows from Corollary 5.6 that each s-harmonic function can be identified with a function in C ∞ ((0, ∞) × Ω). More precisely, let u ∈ C ∞ ((0, ∞) × Ω). Then u is s-harmonic if and only if u(t, ·) ∈ W 1,2 0 (Ω) for all t > 0, ∞ 0 Ω \|∇ x u(t, x)\| 2 + \|∂ t u(t, x)\| 2 + \|u(t, x)\| 2 dx t 2(1−s) dt t < ∞, and ∂ 2 t u(t, x) + 1 − 2s t ∂ t u + ∆ x u(t, x) = 0 for all (t, x) ∈ (0, ∞) × Ω. Let u be an s-harmonic function. Then u(0, ·) := lim t↓0 u(t, ·) exists in L 2 (Ω) and is an element of [L 2 (Ω), W 1,2 0 (Ω)] s . Also w := − lim t↓0 t 1−2s ∂ t u(t, ·) exists in W −1,2 (Ω) := (W 1,2 0 (Ω)) ′ and is an element of [L 2 (Ω), W −1,2 (Ω)] s . Conversely, for each u 0 ∈ [L 2 (Ω), W 1,2 0 (Ω)] s there exists a unique s-harmonic function u such that u(0, ·) = u 0 . One has u 0 ∈ D(A s ) if and only if w := − lim t↓0 t 1−2s ∂ t u(t, ·) exists in W −1,2 (Ω) and w ∈ L 2 (Ω). In that case c s A s u 0 = w. This is Theorem 5.1 rephrased for the Dirichlet-Laplacian. We conclude this section commenting on the integral representation (5.5). Remark 5.9. Let ν ∈ R. The Modified Bessel's Equation t 2 w ′′ (t) + tw ′ (t) − (t 2 + ν 2 )w(t) = 0, (t > 0) has the modified Bessel function of second kind K ν as one of its solutions. An integral representation for K ν is given by K ν (t) = 1 2 t 2 ν ∞ 0 e −r e − t 2 4r r −ν dr r (5.7) for all t > 0, see for example [13, 10.32.10]. One has K ν (t) ∼ π 2t e −t as t → ∞. In our context 0 < s < 1 is given. Let λ > 0. Define ψ : (0, ∞) → (0, ∞) by ψ(t) = √ λt s K s ( √ λt). (5.8) Then ψ ′′ (t) + 1 − 2s t ψ ′ (t) − λψ(t) = 0 for all t > 0 as a direct computation shows. Using (5.7) and (5.8) one deduces that ψ(t) = 1 2 1+s λ s t 2s ∞ 0 e −r e − λt 2 4r r −s dr r (5.9) for all t > 0. Moreover, a substitution in (5.9) gives ψ(t) = λ s 2 s−1 ∞ 0 e −λr e − t 2 4r r s dr r (5.10) for all t > 0. Thus our approach (5.1) is a functional calculus which consists in replacing the parameter λ by the operator A in (5.10). Formula (5.10) is also used in [7]. The non-coercive case Up to now we used that the form E is coercive. In this section we wish to replace this by the much weaker condition that E is merely sectorial with vertex 0. In general, if a : D(a) × D(a) → C is a sesquilinear form, then we say that a is sectorial with vertex 0 if there exists a θ ∈ [0, π 2 ) such that a(u) ∈ Σ θ for all u ∈ D(a), where Σ θ = {re iα : r ∈ [0, ∞) and α ∈ [−θ, θ]}. In Theorem 6.1 we associate an m-sectorial operator to a densely defined sectorial form with vertex 0. There is even a j-version of it like in Proposition 5.2 that turns out to be very useful in this section. Proof. This is a special case of [3,Theorem 3.2]. Note that B is the same operator as in Proposition 5.2 if the domain D(a) is provided with a Hilbert space structure such that j is continuous and the form a is coercive and continuous. We call the operator B in Theorem 6.1 the operator associated with (a, j). In particular, if a is a densely defined sectorial form with vertex 0 in a Hilbert space H, then one can choose for j the identity map and we obtain an m-sectorial operator, which we call the operator associated with a. Now we extend the previous results for coercive forms to sectorial forms. Proof. It is easy to see that b is sectorial with vertex 0. For all n ∈ N define E n : V × V → C by E n (u, v) = E(u, v) + 1 n u, v V . Then E n is continuous and coercive. Let A n be the m-sectorial operator in H associated with E n . Then A n is sectorial with vertex 0. Moreover, lim n→∞ A n = A in the strong resolvent sense by [3,Corollary 3.9]. Hence lim n→∞ A s n = A s in the strong resolvent sense by the representation formula [18, (6) in Section IX.11]. For all n ∈ N define b n : W 1−s (H, V ) × W 1−s (H, V ) → C by b n (u, v) = ∞ 0 u ′ (t), v ′ (t) H + E n (u(t), v(t)) t 2(1−s) dt t = b(u, v) + 1 n ∞ 0 u(t), v(t) V t 2(1−s) dt t . Then b n is continuous and coercive. Let B n be the operator associated with (b n , j) as in Proposition 5.2. Then lim n→∞ B n = B in the strong resolvent sense again by [3,Corollary 3.9]. But B n = c s A s n for all n ∈ N by Theorem 5.1. Taking the limit as n → ∞ and using the uniqueness of the limit in the strong resolvent sense gives B = c s A s as required. Adopt the notation and assumptions as in Theorem 6.2. We suppose from now on in addition that E is H-elliptic, that is there exists a constant µ > 0 such that Re E(u) + u 2 H ≥ µ u 2 V (6.1) for all u ∈ V . In this case we can give an explicit description of the operator B and show that it is again a Dirichlet-to-Neumann map. For this we need quite some preparation. Recall that if X is a Banach space and −∞ < α < β < ∞, then W 1,1 ((α, β); X) ⊂ C([α, β]; X) and u(t) = u(α) + t α u ′ (r) dr for all u ∈ W 1,1 ((α, β); X) and t ∈ [α, β]. Conversely, if x ∈ X, v ∈ L 1 ((α, β); X) and u : (α, β) → X is given by u(t) = x + t α v(r) dr, then u ∈ W 1,1 ((α, β); X) and u ′ = v. For all −∞ ≤ a < b ≤ ∞ and p ∈ [1, ∞] we let W 1,p loc ((a, b); X) := {u : (a, b) → X : u\| (α,β) ∈ W 1,p ((a, β); X) for all α, β ∈ R with a < α < β < b} and if a = −∞, then we define L p loc ([a, b); X) = {u : [a, b) → X : u\| [a,β) ∈ L p ([a, β); X) for all β ∈ (a, b)}. We always identify u ∈ W 1,p loc ((a, b); X) with its continuous representative. Recall that 0 < s < 1. We define the space We provide W with the norm u 2 W = u(0) 2 H + ∞ 0 u ′ (t) 2 H + Re E(u(t)) t 2(1−s) dt t . We first prove that W is a Hilbert space. For the proof we need two lemmas. ′ , where V ′ denotes the antidual of the space V . Moreover, Au, v V ′ ,V = E(u, v) defines an operator A ∈ L(V, V ′ ), where L(V, V ′ ) is the space of all linear and bounded operators from V to V ′ . The part of A in H is the operator A. Given 0 < s < 1 we consider the Bessel kind of equation−u ′′ (t) − 1 − 2s t u ′ (t) + Au(t) = 0, t ∈ (0, ∞) (1.1)as in the papers mentioned above. We identify a precise function space and call the functions in the function space which solve(1.1) s-harmonic. Then we show that for each x ∈ [H, V ] s there is a unique s-harmonic function u satisfying u(0) = x. Moreover, this function has an s-normal derivative y in [H, V ′ ] s = [V ′ , H] 1−s (see (4.3) in Section 4 for more details). Here and throughout the paper, for all 0 < θ < 1 we denote by [H, V ] θ and [H, V ′ ] θ = [V ′ , H] 1−θ the complex interpolation spaces. The corresponding Dirichlet-to-Neumann operator D s which associates to x ∈ [H, V ] s the s-normal derivative y turns out to be an isomorphism from [H, V ] s to [H, V ′ ] s . The part of the operator D s in H is the multiple c s A s of the fractional power A s of A, where c s is an explicit constant depending only on s. For the proof we use a new version of the Kato-Lions method to associate a generator of a holomorphic semigroup to a sesquilinear form as it was established in 2 . 2Appetizer: the square root of a bounded operator Let H be a Hilbert space over C and let A ∈ L(H) := L(H, H) be coercive, i.e. there exists an α ∈ (0 W 1 , 2 12((0, ∞); H) ֒→ C 0 ([0, ∞); H) := {u ∈ C([0, ∞); H) : lim t→∞ u(t) H = 0}. Then W 2,2 ((0, ∞); H) ֒→ C 1 ([0, ∞); H), where W 2,2 ((0, ∞); H) := {u ∈ W 1,2 ((0, ∞); H) : u ′ ∈ W 1,2 ((0, ∞); H)}. Theorem 2 . 2 . 22We have that D = A Proposition 3 . 3 . 33Let s ∈ (0, 1). Then the following assertions hold.(a)If s ≥ 1 2 , then the space C 2 implies that u(0) := lim t↓0 u(t) exists in H and is an element of [H, V ] s . Similarly, Theorem 4. 4 . 4The following assertions hold.(a) (Dirichlet Problem). Let x ∈ [H, V ] s . Then there exists a unique sharmonic function u such that u(0) = x. (b) (Neumann Problem). Let y ∈ [H, V ′ ] s . Then there exists a unique sharmonic function u such that lim t↓0 −t 1−2s u ′ (t) = y. Proof. (a). By Proposition 3.2 there exists a (H, V ). It follows from Lemma 4.3 that u is s-harmonic. This proves existence. Uniqueness follows from Lemma 4.2. (b). Define L : W 1−s (H, V ) → C by Lv := y, v(0) [H,V ′ ]s,[H,V ]s . Then L is continuous and anti-linear by Proposition 3.2. By the Lax-Milgram Lemma there exists a unique u ∈ W 1−s (H, V ) such that b s (u, v) = Lv for all v ∈ W 1−s (H, V ). In particular, b s (u, v) = 0 for all v ∈ C ∞ c ((0, ∞); V ). It follows from Lemma 4.3 that u is s-harmonic. Let z := − lim t↓0 t 1−2s u ′ (t) in the sense of V ′ . Then z ∈ [H, V ′ ] s by Proposition 3.2. From Lemma 4.2 we deduce that y, v(0) [H,V ′ ]s,[H,V ]s = b s (u, v) = z, v(0) [H,V ′ ]s,[H,V ]s for all v ∈ W 1−s (H, V ). Hence y = z by the surjectivity in Proposition 3.2. This shows that u solves the Neumann problem. Uniqueness follows also from Lemma 4.2. Theorem 4.4 and Proposition 3.2 allow us to define the Dirichlet-to-Neumann operator D s in the following way. Definition 4 . 5 . 45Define D s : [H, V ] s → [H, V ′ ] s as follows. Let x ∈ [H, V ] s . Let u be the unique s-harmonic function satisfying u(0) = x. Then D s x = y, where y = − lim t↓0 t 1−2s u ′ (t) in V ′ . We call D s the Dirichlet-to-Neumann operator (with respect to s and E). Proposition 4 . 6 . 46The operator D s is an isomorphism from [H, V ] s onto [H, V ′ ] s . Proof. It follows from Theorem 4.4 that D s is linear and bijective. We show that D −1 s is continuous. Let y ∈ [H, V ′ ] s and set x := D −1 s y. Let u be the sharmonic function satisfying u(0) = x and − lim t↓0 t 1−2s u ′ (t) = y. Then b s (u) = y, u(0) [H,V ′ ]s,[H,V ]s by (4.6). By Proposition 3.2 there exists a constant c > 0 such that v(0) [H,V ]s ≤ c v W1−s(H,V ) for all v ∈ W s (H, V ). Let µ ∈ (0, 1] be a coercivity constant for E. Then µ u 2 W1−s(H,V ) ≤ Re b s (u) = Re y, u(0) [H,V ′ ]s,[H,V ]s ≤ y [H,V ′ ]s u(0) [H,V ]s ≤ c y [H,V ′ ]s u W1−s(H,V ) .Henceu W1−s(H,V ) ≤ cµ −1 y [H,V ′ ]s . (4.7)Thereforex [H,V ]s = u(0) [H,V ]s ≤ c u W1−s(H,V ) ≤ c 2 µ −1 y [H,V ′ ]s .This shows that D −1 s is continuous. Then also D s is continuous, by the bounded inverse theorem.The next proposition combines several results of this section. Proposition 4 . 7 . 47The set Har s := {u ∈ W 1−s (H, V ) : u is s-harmonic} is a closed subspace of W 1−s (H, V ). We provide Har s with the induced norm of W 1−s (H, V ). Then the mappings u → u(0) from Har s into [H, V ] s and u → − lim t↓0 t 1−2s u ′ (t) from Har s into [H, V ′ ] s are both isomorphisms. Proof. It follows from Lemmas 4.3 and 4.2 that Har s is a closed subspace. The surjectivity of both maps is proved in Theorem 4.4 and the injectivity in Lemma 4.2. Proposition 4. 8 . 8A function u is 1 2 -harmonic if and only if we have that u ∈ W 2,2 ((0, ∞); V ′ ) ∩ L 2 ((0, ∞); V ) and −u ′′ + Au = 0 in L 2 ((0, ∞); V ′ ). {u ∈ W 1,2 ((0, ∞); H) ∩ L 2 ((0, ∞); V ) : u is 1 2 -harmonic} is a closed subspace of W 1,2 ((0, ∞); H) ∩ L 2 ((0, ∞); V ). Moreover, the mappings u → u(0) from Har 1 2 into [H, V ] 1 2 and u → −u ′ (0) from Har 1 2 into [H, V ′ ] 1 2 are both isomorphisms. Denote by A the part of the operator A in H; i.e. D(A) = {x ∈ V, Ax ∈ H} and Ax = Ax. Then A is an m-sectorial operator with vertex γ > 0 in the sense of Kato [9, Subsection V.3.10]. Moreover, A has bounded imaginary powers, see for example [8, Corollary 7.1.8]. Hence [H, D(A)] θ = D(A θ ) for all θ ∈ (0, 1) by [17, Theorem 1.15.3]. Proposition 4 . 10 . 410Let u be 1 2 -harmonic. Then the following assertions are equivalent. The proof of Proposition 4.10 is based on the following properties of traces [10, Chapter 1, Theorems 3.1 and 3.2]. Proposition 4 . 11 . 411Let Y d ֒→ X, where X, Y are Hilbert spaces. Then the mapping u Proof of Proposition 4.10. Suppose (i) is valid. Since u ′′ = Au, it follows that Au(t) = u ′′ (t) ∈ H and hence u(t) ∈ D(A) for almost all t > 0. So u ∈ L 2 ((0, ∞); D(A)) ∩ W 2,2 ((0, ∞); H). Then the implications (i) ⇒ (ii) and (i) ⇒ (iii) follow directly from Proposition 4.11. (ii) ⇒ (i). Our proof is based on the Dore-Venni Theorem. We consider the negative Dirichlet Laplacian B D on L 2 ((0, ∞); H) given by D(B D ) = {w ∈ W 2,2 ((0, ∞); H) : w(0) = 0} and B D w = −w ′′ . This is a selfadjoint, positive operator. In fact it is associated with the closed form b D (w, v) = ∞ 0 w 0′ (t), v ′ (t) H dt and D(b D ) = W 1,2 0 ((0, ∞); H).The other operator in L 2 ((0, ∞); H) which we consider is the operator A 2 with domain D(A 2 ) = L 2 ((0, ∞); D(A)) given by (A 2 w)(t) = A(w(t)). Then the operators −B D and −A 2 generate bounded holomorphic C 0 -semigroups on L 2 ((0, ∞); H) which commute. Moreover, A 2 is invertible. It follows from a version of the Dore-Venni Theorem [6, Theorem 2.1] (see also[15, Theorem 8.4] and[12, Corollary 4.7]) that the operatorB D + A 2 with usual domain D(B D + A 2 ) = D(B D ) ∩ D(A 2 ) is invertible.By assumption we have u(0) ∈ [H, D(A)] 3 4 . Then by Proposition 4.11 there exists a φ ∈ W 2,2 ((0, ∞); H) ∩ L 2 ((0, ∞); D(A)) such that φ(0) = u(0). Let f : The number s ∈ (0, 1) is fixed and D s : [H, V ] s → [H, V ′ ] s is the Dirichlet-to-Neumann operator (see Definition 4.5). Note that [H, V ] s d ֒→ [H, V ′ ] s = [H, V ] ′ s . Let D s be the part of the operator D s in H. So D s is the operator in H given by D(D s ) := {x ∈ [H, V ] s : D s x ∈ H} and D s x = D s x. Therefore the graph of D s is given by graph(D s ) = {(x, y) ∈ H × H : there exists an s-harmonic map u such thatu(0) = x and y = − lim t↓0 t 1−2s u ′ (t) in V ′ }.Recall thatA is the part of the operator A in H. Denote by A s the fractional power of A. Our main result of this paper is the following. Define c s := 2 1−2s Γ(1−s) Γ(s) . Theorem 5.1. One has c s A s = D s . We first prove that D s is m-sectorial. For that we use the following result [3, Theorem 2.1]. Proposition 5.2. Let W be a Hilbert space and let b : W × W → C be a continuous sesquilinear form. Let H be a Hilbert space and j : W → H be a continuous, linear map with dense image. Suppose there exist µ > 0 and ω ∈ R such that µ u 2 W ≤ Re b(u) + ω j(u) 2 H for all u ∈ W . Then there exists a unique m-sectorial operator B on H such that graph(B) = {(x, y) ∈ H × H : there exists a u ∈ W such that j(u) = x and b(u, v) = y, j(v) H for all v ∈ W }. Proposition 5. 3 . 3The operator associated with (b s , j) is D s . In particular, D s is m-sectorial.Proof. Let B be the operator associated with (b s , j). Let (x, y) ∈ graph(B). Then there exists a u ∈ W 1−s (H, V ) such that u(0) = x and b s (u [H,V ]s for all w ∈ [H, V ] s by Proposition 3.2. This implies that z = y ∈ H. Hence x ∈ D(D s ) and D s x = Bx. We have shown that B ⊂ D s . Conversely, let x ∈ D(D s ). Write y = D s x ∈ H. Let u be s-harmonic such that u(0) = x. Then y = − lim t↓0 t 1−2s u ′ (t) in V ′ . Let v ∈ W 1−s (H, V ). Then (4.5) gives b s (u, v) = y, v(0) [H,V ′ ]s,[H,V ]s = y, v(0) H , where we have used that y ∈ H. Hence x = u(0) ∈ D(B) and Bx = y. This shows that D s ⊂ B. Proposition 5. 4 . 4The following assertions hold.(a)If t > 0, then U(t)x ∈ D(A) for all x ∈ H and Let x ∈ D(A 2 ). Then U(·)A s x is s-harmonic.Proof. (a). Since (e −tA ) t>0 is a holomorphic semigroup there exists a constantc 1 > 0 such that Ae −tA L(H) ≤ c 1 t −1 for all t ∈ (0, 1]. If t ∈ (1, ∞),then the semigroup property gives Ae −tA L(H) ≤ Ae −A L(H) e −(t−1)A L(H) ≤ c 1 e −µ(t−1) . Hence U(t)H ⊂ D(A) for all t > 0. It is straightforward to verify the identity (5.2). (b). This part follows from [2, (3.56)]. (c). Let t > 0. One has 1 0 1t 2(1−s) u ′ (t) 2 H dt t < ∞. In fact, by Statement (c) there exists a constant c > 0 such that t 1−2s u ′ (t) H ≤ c for all t ∈ (0, 1]. Hence εt and the estimate (5.4) is valid. So t 1−s u ′ ∈ L * 2 (H). Therefore u ∈ W 1−s (H, V ). Finally it follows from Statement (a) that b s (u, v) = 0 for all v ∈ C ∞ c ((0, ∞); V ). Hence u is s-harmonic by Lemma 4.3. Now we are able to prove Theorem 5.1. Proof of Theorem 5.1. Let x ∈ D(A 2 ). We shall show that x ∈ D(D s ) and D s x = c s A s x. In fact, by Proposition 5.4(e) the function u(·) := U(·)A s x is sharmonic and u(0) = x by Proposition 5.4(b). Moreover, c s A s x = − lim t↓0 t 1−2s u ′ (t) by Proposition 5.4(c). Thus, by the definition of D s , one has x ∈ D(D s ) and D s x = c s A s x. Since D(A 2 ) is a core of D(A s ) and D s is closed (as the operator D s is m-sectorial by Proposition 5.3), it follows that c s A s ⊂ D s . Because c s A s is m-sectorial and D s is sectorial one concludes that c s A s = D s . Theorem 5.1 has the following corollary. Corollary 5 . 5 . 55Let u be s-harmonic, x = u(0) and y = − lim t↓0 t 1−2s u ′ (t) in V ′ . Then x ∈ D(A s ) if and only if y ∈ H. Moreover Proposition 5.4(e) extends to the following representation formula. Corollary 5. 6 . 0 . 60Let u be s-harmonic. r s A s e −rA x dr r(5.5) for all t > 0, where x = u(0). In particular, u ∈ C ∞ ((0, ∞); H). Stronger, u ∈ C ∞ ((0, ∞); D(A k )) for all k ∈ N.Proof. Note that x ∈ [H, V ] s by Proposition 3.2. Since first D(A 2 ) is a core for A, secondly the domain D(A) with graph norm is densely and continuously embedded in V and thirdly V is densely and continuously embedded in [H, V ] s , it follows that D(A 2 ) is dense in [H, V ] s . Hence there exists a sequence (x n ) n∈N in D(A 2 ) such that x n → x in [H, V ] s . Now it follows from the second statement in Proposition 4.7 that u n → u in W 1−s (H, V ), where u n is the s-harmonic function satisfying u n (0) = x n for all n ∈ N. Since W 1−s (H, V ) ⊂ C((0, ∞); H), the closed graph theorem implies that u n (t) → u(t) in H as n → ∞ for all t > We r s e −rA A s x n dr r for all t > 0. Note that there exists a constant c > 0 such that A s e −rA y H ≤ c r s y H for all r ∈ (0, 1] and y ∈ H. Letting n → ∞, we get (5.5) by Lebesgue's Theorem. Example 5. 7 ( 7Fractional power of the Dirichlet Laplacian). Let Ω ⊂ R N be a bounded, open set and let A be the negative Dirichlet Laplacian on L 2 (Ω), that is D(A) = {u ∈ W 1,2 0 (Ω) : ∆u ∈ L 2 (Ω)} and Au = −∆u. Finally we specify the results for s = 1 2 . Let x ∈ [H, V ] 1 2 . By Theorem 4.4(a) there is a unique 1 2 -harmonic function u such that u(0) = x. Then y := −u ′ (0) ∈ [H, V ′ ] 1 2 . Then by definition D 1 2 x = y. Moreover, D 1 2 : [H, V ] 1 2 → [H, V ′ ] 1 2 is an isomorphism by Proposition 4.6. Also D 1 2 x ∈ H if and only if x ∈ D(A 1 2 ) by Theorem 5.1. In that case D 1 2 x = A 1 2 x, since c 1 2 = 1. Then the unique 1 2 -harmonic function u satisfying u(0) = x satisfies u(t) = e −tA There exist ε > 0 and M ≥ 1 such that e −tA 1 2 L(H) ≤ M e −εt for all t ≥ 0 (see for example [2, Theorem 5.1.12]). Let x ∈ D(A). Define u : (0, ∞) → H by u(t) = e −tA 1 2 x. Since D(A) is continuously embedded into V , there exists a constant c > 0 such that y V ≤ c Ay H for all y ∈ D(A). Then u(t) Ax H ≤ cM e −εt Ax H for all t > 0 and u ∈ L 2 ((0, ∞); V ). Moreover, u ∈ C 2 ([0, ∞); H) and u ′′ (t) = Au(t) = e −tA 1 2 1Ax for all t > 0. Henceu ∈ W 2,2 ((0, ∞); H) ⊂ W 2,2 ((0, ∞); V ′ ) and −u ′′ + Au = 0 in L 2 ((0, ∞); V ′ ).Then u is 1 2 -harmonic by Proposition 4.8. Now (5.6) follows from (5.5). Since D(A) is dense in D Theorem 6 . 1 . 61Let H be a Hilbert space, a : D(a) × D(a) → C a sectorial form with vertex 0 and j : D(a) → H a linear map with dense image. Then there exists a unique m-sectorial operator B in H such that graph(B) = {(x, y) ∈ H × H : there exists a sequence (u n ) n∈N in D(a) such that (i) lim n→∞ j(u n ) = x in H, (ii) sup{Re a(u n ) : n ∈ N} < ∞, and (iii) lim n→∞ a(u n , v) = y, j(v) H for all v ∈ D(a)}. Theorem 6. 2 . 2Let H, V be Hilbert spaces such that V d ֒→ H and let E : V × V → C be a continuous sectorial form with vertex 0. Let A be the operator associated with E. Further, let s ∈ (0, 1) and define b :W 1−s (H, V ) × W 1−s (H, V ) → C by b(u, v) = ∞ 0 u 0′ (t), v ′ (t) H + E(u(t), v(t)) t 2(1−s) dt t . Define j : W 1−s (H, V ) → H by j(u) = u(0). Then b is sectorial with vertex 0. Let B bethe operator associated with (b, j). Then B = c s A s , where c s = 2 1−2s Γ(1−s) Γ(s) . W := {u ∈ C([0, ∞); H) ∩ W 1,2 loc ((0, ∞); H) ∩ L 2 loc ((0, ∞); V ) : Lemma 6. 3 . 3Let u ∈ W . Then u ′ ∈ L 1 loc ([0, ∞); H). Moreover, t 0 u ′ (r) H dr ≤ u W Acknowledgements. The first author is most grateful for a stimulating stay at the University of Puerto Rico, Rio Piedras Campus. The second and third authors are most grateful for the hospitality extended to them during their fruitful stay at the University of Ulm. The research of the third author is partially supported by the AFOSR Grant FA9550-15-1-0027. Part of this work is supported by an NZ-EU IRSES counterpart fund and the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. Part of this work is supported by the EU Marie Curie IRSES program, project 'AOS', No. 318910.for all t > 0.Proof. Let u ∈ W and t > 0. Then ≤ u W · 1 2s t 2s1 2.This shows(6.2)and that u ′ ∈ L 1 loc ([0, ∞); H). If a ∈ (0, t), thenLetting a ↓ 0 givesMoreover, (6.4) and(6.2)giveand (6.3) follows.Recall that µ is defined in (6.1).Lemma 6.4. If u ∈ W and T ≥ 1, thenProof. By H-ellipticity and (6.3) one estimatesand the lemma follows.Proposition 6.5. The space W is a Hilbert space.Proof. Let (u n ) n∈N be a Cauchy sequence in W . Then x := lim n→∞ u n (0) exists in H. Moreover, there exists a v ∈ L 2 ((0, ∞); H, t 2(1−s) dt t ) such that lim n→∞ u ′ n = v in L 2 ((0, ∞); H, t 2(1−s) dt t ). It follows from (6.2) that (u ′ n \| (0,T ) ) n∈N is a Cauchy sequence in L 1 ((0, T ); H) for all T > 0. Hence v\| (0,T ) ∈ L 1 ((0, T ); H) and, more-Let ε > 0. There exists an N 0 ∈ N such that u n − u m 2 W ≤ ε for all n, m ∈ N with n, m ≥ N 0 . Let n ∈ N with n ≥ N 0 . Then [9, Lemma VIII.3.14a] and Fatou's lemma giveHence u n −u ∈ W and u ∈ W . So u n −u 2 W ≤ ε for all n ≥ N 0 and lim n→∞ u n = u in W . We have shown that the space W is complete.We need one more lemma before we can give a Dirichlet-to-Neumann type description for the operator c s A s .2n]. Define u n := η n u ∈ W 1−s (H, V ). We shall show that sup n∈N u n W < ∞. ObviouslyIt remains to show that sup n∈NHence sup n∈N u n W < ∞.Passing to a subsequence if necessary, we have that there exists a w ∈ W such that lim n→∞ u n = w weakly in W . Then (6.3) implies that lim n→∞ u n (t) = w(t) in H for almost all t > 0. So u = w ∈ W . We have shown that u is in the weakNow we are able to show that the operator B in Theorem 6.2 is a Dirichlet-to-Neumann map if E is H-elliptic. Theorem 6.7. Adopt the assumptions and notation as in Theorem 6.2. Moreover, assume that E is H-elliptic. Define the formb :Let x, y ∈ H. Then the following assertions are equivalent.There exists a u ∈ W such that u(0) = x andb(u, v) = y, v(0) H for all v ∈ W .Proof. Definej : W → H byj(u) = u(0). Thenb is continuous andfor all u ∈ W . Moreover,j is continuous and has dense image. Obviouslyb andj are extensions of b and j, respectively. In addition, W is complete and W 1−s (H, V ) is dense in W by Proposition 6.5 and Lemma 6.6. Hence by[3,Proposition 3.3]it follows that B is the operator associated with (b,j) in the sense of Proposition 5.2. Then the equivalence follows immediately from Theorem 6.2 and the definition of the graph of B in Proposition 5.2.Letb be as in Theorem 6.7 and let u ∈ W be as in Condition (ii) in Theorem 6.7. Then u ∈ W 2,2 loc ((0, ∞); V ′ ) andu ′′ (t) + 1 − 2s t u ′ (t) − Au(t) = 0 in V ′ for a.e. t ∈ (0, ∞),where A : V → V ′ is given by Aw, v V ′ ,V = E(w, v) for all w, v ∈ V . Linear Functional Analysis. An application-oriented introduction. Translated from the 6th German edition by Robert Nürnberg. H W Alt, SpringerLondonH.W. Alt. Linear Functional Analysis. An application-oriented introduction. Translated from the 6th German edition by Robert Nürnberg. Springer, London, 2016. Vector-valued Laplace Transforms and Cauchy Problems. W Arendt, C J K Batty, M Hieber, F Neubrander, Birkhäuser/Springer96Basel AG, BaselSecond edition. Monographs in MathematicsW. Arendt, C.J.K. Batty, M. 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Berlin-New YorkSpringer-VerlagSixth editionK. Yosida. Functional Analysis. Sixth edition. Grundlehren der Mathematischen Wis- senschaften 123. Springer-Verlag, Berlin-New York, 1980. D-89069 Ulm (Germany) E-mail address: wolfgang.arendt@uni-ulm. de18Wolfgang Arendt, Institute of Applied Analysis, University of Ulm. HelmholtzstrWolfgang Arendt, Institute of Applied Analysis, University of Ulm. Helmholtzstr. 18, D-89069 Ulm (Germany) E-mail address: [email protected] A F M Ter, Elst , Private bag 92019. Auckland 1142 (New Zealand) E-mail address: [email protected]. Department of Mathematics, University of AucklandA.F.M, ter Elst, Department of Mathematics, University of Auckland. Private bag 92019. Auckland 1142 (New Zealand) E-mail address: [email protected] Mahamadi Warma, University Of Puerto, Rico, College of Natural Sciences, Department of Mathematics. Rio Piedras CampusPO Box 70377 San Juan PR 00936-8377 (USA) E-mail address: [email protected], [email protected] Warma, University of Puerto Rico (Rio Piedras Campus), College of Nat- ural Sciences, Department of Mathematics, PO Box 70377 San Juan PR 00936-8377 (USA) E-mail address: [email protected], [email protected]	[]
[ "A FULLY CONVOLUTIONAL NEURAL NETWORK BASED STRUCTURED PREDICTION APPROACH TOWARDS THE RETINAL VESSEL SEGMENTATION", "A FULLY CONVOLUTIONAL NEURAL NETWORK BASED STRUCTURED PREDICTION APPROACH TOWARDS THE RETINAL VESSEL SEGMENTATION" ]	[ "Avijit Dasgupta \nIndian Institute of Technology Kharagpur West Bengal\n721302India\n", "Sonam Singh \nIndian Institute of Technology Kharagpur West Bengal\n721302India\n" ]	[ "Indian Institute of Technology Kharagpur West Bengal\n721302India", "Indian Institute of Technology Kharagpur West Bengal\n721302India" ]	[]	Automatic segmentation of retinal blood vessels from fundus images plays an important role in the computer aided diagnosis of retinal diseases. The task of blood vessel segmentation is challenging due to the extreme variations in morphology of the vessels against noisy background. In this paper, we formulate the segmentation task as a multi-label inference task and utilize the implicit advantages of the combination of convolutional neural networks and structured prediction. Our proposed convolutional neural network based model achieves strong performance and significantly outperforms the stateof-the-art for automatic retinal blood vessel segmentation on DRIVE dataset with 95.33% accuracy and 0.974 AUC score.	10.1109/isbi.2017.7950512	[ "https://arxiv.org/pdf/1611.02064v2.pdf" ]	335,714	1611.02064	5e52c24e54f9ac3e2609800526f4a012930668d2	A FULLY CONVOLUTIONAL NEURAL NETWORK BASED STRUCTURED PREDICTION APPROACH TOWARDS THE RETINAL VESSEL SEGMENTATION Avijit Dasgupta Indian Institute of Technology Kharagpur West Bengal 721302India Sonam Singh Indian Institute of Technology Kharagpur West Bengal 721302India A FULLY CONVOLUTIONAL NEURAL NETWORK BASED STRUCTURED PREDICTION APPROACH TOWARDS THE RETINAL VESSEL SEGMENTATION Index Terms-Computer-aided diagnosisretinal ves- selsconvolution neural networksimage segmentation Automatic segmentation of retinal blood vessels from fundus images plays an important role in the computer aided diagnosis of retinal diseases. The task of blood vessel segmentation is challenging due to the extreme variations in morphology of the vessels against noisy background. In this paper, we formulate the segmentation task as a multi-label inference task and utilize the implicit advantages of the combination of convolutional neural networks and structured prediction. Our proposed convolutional neural network based model achieves strong performance and significantly outperforms the stateof-the-art for automatic retinal blood vessel segmentation on DRIVE dataset with 95.33% accuracy and 0.974 AUC score. INTRODUCTION Segmentation and localization of retinal blood vessels serve as an important cue for the diagnosis of opthalmological diseases such as diabetes, hypertension, microaneurysms and arteriochlerosis [1]. However, manual segmentation of blood vessels is both tedious and time consuming. Thus, the focus of this paper is on automatic segmentation of retinal blood vessels from fundus images. The task of automatic segmentation of blood vessels is challenging due to their abrupt variations in branching patterns. This task becomes even more challenging due the presence of noisy background and tortuosity. Related Work: Previous attempts of blood vessels segmentation can be broadly divided into two categories. The first group used unsupervised methods which includes vessel tracking [2], adaptive thresholding [3], and morphology based techniques [4] etc. The second group utilized the supervised machine learning algorithms which make use of hand-labeled images (i.e. ground truth) for learning models. Most of the * equal contribution. Avijit Dasgupta is affiliated with Electronics and Communication Dept. Sonam Singh is affiliated with Advanced Technology Development Centre Email: [email protected], [email protected] Project website:https://avijit9.github.io/my posts/FCN Retina.html supervised methods extract hand-crafted features e.g. ridge features, Gabor at different scales and degrees etc. from the fundus images and classify them using Nearest Neighbour, Bayesian, Gaussian Mixture Models, Support Vector Machine, Artificial Neural Networks or their variants [5,6,7]. Recently, Deep Learning (DL) has gained a lot of interest due to their highly discriminative representations that has outperformed many state-of-the-art techniques in the field of computer vision and natural language processing. Recently, it has also attracted medical imaging research community. In 2016, Liskowski et al. [8] proposed a deep convolutional neural network architecture for vessel segmentation in fundus images. Maji et al. [9] proposed an ensemble of 12 deep convolutional neural networks and take the mean of the outputs of all networks as the final decision. Lahiri et al. [10] proposed an architecture which is based on an ensemble of stacked denoising autoencoders (SDAE). The final decision is the combination of all SDAEs outputs passed through a softmax layer. Contribution: In this paper, we propose a fully convolutional neural network architecture for blood vessel segmentation. As suggested by [8], we formulate the vessel segmentation problem as a multi-label inference problem which is learnt by joint loss function. In this way, we can learn about the class label dependencies of neighboring pixels which play an important role in segmentation of anatomical structures. To the best of our knowledge, our work is the first of its kind to leverage the combined advantage of fully convolutional neural network and structured prediction approach for retinal blood vessel segmentation in fundus images. The rest of the paper is organized as follows: Section 2 defines the problem statement more formally and describes the proposed methodology in detail. In Section 3 we show the experimental results on publicly available DRIVE [5] dataset which validate our claims. Finally, in Section 4 we conclude our paper with a summary of our proposed methodology and future scope. Problem Statement Given a color fundus image I M ×N ×3 and the intensity value at (x, y) is denoted by I(x, y). Let us denote the neighborhood of the pixel at position (x, y) by N (x, y). Our task is to classify each and every pixel contained in the neighborhood N (x, y) into either of the classes denoted by ω = {vessels, background}. Hence, by training the CNN we learn a function H(ω\|I, N (x, y)). We will start with a brief introduction of convolutional neural networks (CNN) followed by the proposed technique. Convolutional Neural Networks Convolutional neural networks (CNN) are a special type of neural network where neurons are arranged in 3-dimensional grid (width, height and depth). Every layer of a CNN takes a 3D input volumes and tranforms them into 3D output volumes. There are four main types of layer in CNN architectures: Convolutional layer, Pooling layer, Upsampling and Fully-connected layer. A CNN architecture can be made by stacking these layers. Each convolutional layer transforms input representation using convolution operation. Mathematically, if W l i denotes the weights of i-th filter of l-th convolutional layer, g l−1 denotes the inputs coming from previous layer, and g l i be the non-linearity applied on that layer, then the output can be written as follows: y l i = g l i (W l i ⊗ g l−1 ),(1) where '⊗' denotes convolution operation. A pooling layer simply performs spatial downsampling of input feature maps while the upsampling layer does the exact opposite. Preprocessing and Data Preparation Given a RGB fundus image, I, we extract the green channel image, I g , as the blood vessels manifest high contrast in green channel [11]. Then, we normalize the images by using the following formula- I g = I g − µ σ ,(2) where µ and σ denote the mean and standard deviation of the data. Contrast limited adaptive histogram equalization [12] and gamma adjustment is applied on normalized images. Finally, the intensity values are scaled to have a minimum value of 0 and a maximum value of 1 to get the preprocessed image denoted byÎ. Fig. 1 shows some pre-processed images alongwith the original image from DRIVE [5] dataset. The Proposed Architecture Each layer of CNN learn task dependent hierarchical features. The input to the first convolutional layer in the proposed architecture is a 1 × 28 × 28 patch extracted from the preprocessed imageÎ. The proposed CNN architecture has the same layer organization as shown in Fig. 2. Each of the first and second convolutional layers ( C1 and C2) contain 32 filters with padding for same size. The third layer (M1) is a max-pooling layer with a pooling window of 2 × 2. The fourth and fifth layers (C3 and C4) are convolutional layers with 64 filters in each layer. The sixth layer (U1) is an upsampling layer to increase spatial dimension for structured output. The seventh and eighth layers (C5 and C6) are convolutional layers with same size padding and 32 filters each. The output is of dimension 1 × 28 × 28. Kernel size of 3 × 3 is used in all convolutional layers. Rectified Linear Unit (ReLU) activation is used in the whole model except the last layer where softmax is used. Dropout with probability 0.7 is used after each convolutional layer. In multi-label learning problem we learn to predict a vector instead of predicting a scalar value. In our proposed architecture, we use cross-entropy loss which is defined as: J CE (y,ŷ) = − y i logŷ i + (1 − y i )log(1 −ŷ i ),(3) where both y i andŷ i are ground truth and predicted vectors respectively. Both have the same dimension as the neighborhood of pixel at location (x, y) i.e. N (x, y) inÎ. Fig. 2: The proposed fully convolutional architucture for structured prediction desgined to segment retinal blood vessels from fundus images. RESULTS AND DISCUSSIONS We have evaluated the performance of our proposed method on a very popular and publicly available DRIVE [5] dataset. Training Parameters and Evaluation Metrics Throughout the experiments, we have fixed the learning rate to be 0.0001 and RMSprop [13] optimization algorithm is used with momentum fixed at 0.7. Our model is trained for 60 epochs with a batch size of 32. We perform the evaluation in terms of Precision, Sensitivity, Specificity, Accuracy and Area under the ROC curve (AUC). Experimental results In Table 1, we demonstrate significant improvement in performance with our proposed method against other state-ofthe-art results from recent works. Method Precision Sensitivity Specificity Accuracy AUC Orlando et al. [14] 0 CONCLUSION Deep neural networks can learn hierarchical feature representations from the raw pixel data without any domainknowledge. This has tremendous potential in medical imaging where handcrafting features can be tedious. In this paper, we propose a fully convolutional architecture capable of structured prediction for retinal vessel segmentation task. We demonstrated state-of-the-art performance of our proposed architecture on DRIVE database. Fig. 1 : 1Visualization of the preprocessing step on the images taken from DRIVE dataset: (a) Original RGB images, (b) Preprocessed images. It can be clearly seen that the vessels are more prominent in preprocessed image than original images. Fig. 3 Fig. 3 : 33shows the qualitative outputs of our proposed method. More visualizations of results and intermediate results can be found at project website 1 . 1 https://avijit9.github.io/my posts/FCN Retina.html Visualization of the prediction made by our proposed technique on three samples randomly taken from the DRIVE dataset: (a) Original preprocessed image (b) Corresponding ground truth and (c) Segmented output. Table 1 : 1Quantitative comparison of our proposed method on the DRIVE dataset with other existing state-of-the-art methods..7854 0.7897 0.9684 - - Lahiri et al. [10] - 0.7500 0.9800 0.9480 0.9500 Maji et al. [9] - - - 0.9470 0.9283 Fu et al. [15] - 0.7294 - - 0.9470 Dai et al. [16] - 0.7359 0.9720 0.9418 - Soares et al. [6] - 0.7283 0.9788 0.9466 0.9614 Zhang et al. [17] - 0.7120 0.9724 0.7120 - Niemeijer et al. [18] - 0.6793 0.9725 0.9416 0.9294 Vega et al. [19] - 0.7444 0.9600 0.9414 - Fathi et al. [20] 0.8205 0.7152 0.9768 0.9430 - Fraz et al. [21] 0.8112 0.7302 0.9742 0.9422 - Proposed method 0.8498 0.7691 0.9801 0.9533 0.9744 Clinical ophthalmology: a systematic approach. J Jack, Brad Kanski, Bowling, Elsevier Health SciencesJack J Kanski and Brad Bowling, Clinical ophthalmol- ogy: a systematic approach, Elsevier Health Sciences, 2011. A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering. Y A Tolias, S M Panas, IEEE TMI. 172Y. A. Tolias and S. M. Panas, "A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering," IEEE TMI, vol. 17, no. 2, pp. 263-273, 1998. Adaptive local thresholding by verification-based multithreshold probing with application to vessel detection in retinal images. X Jiang, D Mojon, IEEE TPAMI. 251X. Jiang and D. Mojon, "Adaptive local thresholding by verification-based multithreshold probing with applica- tion to vessel detection in retinal images," IEEE TPAMI, vol. 25, no. 1, pp. 131-137, 2003. 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Ensemble of deep convolutional neural networks for learning to detect retinal vessels in fundus images. D Maji, A Santara, P Mitra, D Sheet, abs/1603.04833CoRR. D. Maji, A. Santara, P. Mitra, and D. Sheet, "Ensem- ble of deep convolutional neural networks for learning to detect retinal vessels in fundus images," CoRR, vol. abs/1603.04833, 2016. Deep neural ensemble for retinal vessel segmentation in fundus images towards achieving label-free angiography. A Lahiri, A G Roy, D Sheet, P K Biswas, EMBC 2016. A. Lahiri, A. G. Roy, D. Sheet, and P. K. Biswas, "Deep neural ensemble for retinal vessel segmentation in fun- dus images towards achieving label-free angiography," in EMBC 2016, IEEE, Aug 2016, pp. 1340-1343. Vessel extraction from non-fluorescein fundus images using orientation-aware detector. B Yin, H Li, B Sheng, X Hou, Y Chen, W Wu, P Li, R Shen, Y Bao, W Jia, Medical image analysis. 261B. Yin, H. Li, B. Sheng, X. Hou, Y. Chen, W. Wu, P. Li, R. Shen, Y. Bao, and W. Jia, "Vessel extraction from non-fluorescein fundus images using orientation-aware detector," Medical image analysis, vol. 26, no. 1, pp. 232-242, 2015. Contrastlimited adaptive histogram equalization. Eugene Stephen M Pizer, James P Johnston, Bonnie C Ericksen, Keith E Yankaskas, Muller, Publ by IEEE. Stephen M Pizer, R Eugene Johnston, James P Ericksen, Bonnie C Yankaskas, and Keith E Muller, "Contrast- limited adaptive histogram equalization," in Publ by IEEE, 1990. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. Tijmen Tieleman, Geoffrey Hinton, COURSERA: Neural Networks for Machine Learning. 4Tijmen Tieleman and Geoffrey Hinton, "Lecture 6.5- rmsprop: Divide the gradient by a running average of its recent magnitude," COURSERA: Neural Networks for Machine Learning, vol. 4, no. 2, 2012. A discriminatively trained fully connected conditional random field model for blood vessel segmentation in fundus images. 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Peishan Dai, Hanyuan Luo, Hanwei Sheng, Yali Zhao, Ling Li, Jing Wu, Yuqian Zhao, Kenji Suzuki, PloS one. 106127748Peishan Dai, Hanyuan Luo, Hanwei Sheng, Yali Zhao, Ling Li, Jing Wu, Yuqian Zhao, and Kenji Suzuki, "A new approach to segment both main and peripheral reti- nal vessels based on gray-voting and gaussian mixture model," PloS one, vol. 10, no. 6, pp. e0127748, 2015. Retinal vessel extraction by matched filter with firstorder derivative of gaussian. Bob Zhang, Lin Zhang, Lei Zhang, Fakhri Karray, Computers in biology and medicine. 404Bob Zhang, Lin Zhang, Lei Zhang, and Fakhri Karray, "Retinal vessel extraction by matched filter with first- order derivative of gaussian," Computers in biology and medicine, vol. 40, no. 4, pp. 438-445, 2010. Comparative study of retinal vessel segmentation methods on a new publicly available database. Meindert Niemeijer, Joes Staal, Marco Bram Van Ginneken, Michael D Loog, Abramoff, Medical Imaging. Meindert Niemeijer, Joes Staal, Bram van Ginneken, Marco Loog, and Michael D Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in Medical Imaging 2004. ISOP, 2004, pp. 648-656. Retinal vessel extraction using lattice neural networks with dendritic processing. Roberto Vega, Gildardo Sanchez-Ante, Luis E Falcon-Morales, Humberto Sossa, Elizabeth Guevara, Computers in biology and medicine. 58Roberto Vega, Gildardo Sanchez-Ante, Luis E Falcon- Morales, Humberto Sossa, and Elizabeth Guevara, "Retinal vessel extraction using lattice neural networks with dendritic processing," Computers in biology and medicine, vol. 58, pp. 20-30, 2015. Automatic wavelet-based retinal blood vessels segmentation and vessel diameter estimation. 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[ "Superconductivity above 500 K in conductors made by bringing n-alkane into contact with graphite", "Superconductivity above 500 K in conductors made by bringing n-alkane into contact with graphite" ]	[ "Yasushi Kawashima [email protected] \nDepartment of Precision Engineering\nSchool of Engineering\nTokai University\n259-1292HiratsukaKanagawaJapan\n" ]	[ "Department of Precision Engineering\nSchool of Engineering\nTokai University\n259-1292HiratsukaKanagawaJapan" ]	[]	In 1986, a cuprate superconductor (Ba-La-Cu-O system) having a critical temperature which goes over the BCS limit (~30 K) was discovered and then a cuprate superconductor (Y-Ba-Cu-O system) with a critical temperature higher than 77 K was discovered. Furthermore, a Hg-based cuprate with a critical temperature of 133 K was found. The 133 K is still the highest critical temperature of conventional superconductors under atmospheric pressure.We have shown that materials obtained by bringing n-alkanes into contact with graphite are capable of conducting electricity with almost no energy loss at room temperature. We here report that the sudden jump in resistance showing a phase transition is observed in the materials during heating by two-probe resistance measurement. The measured critical temperatures of the materials consisting of pitch-based graphite fibers and n-alkanes having 7-16 carbon atoms range from 363.08 to 504.24 K and the transition widths range between 0.15 and 3.01 K. We also demonstrate that superconductors with critical temperatures beyond 504 K are obtained by alkanes with 16 or more carbon atoms.	null	[ "https://export.arxiv.org/pdf/1612.05294v1.pdf" ]	118,886,606	1612.05294	cae32cfe4116f8863f42f385a8f1f80c8f2d369a	Superconductivity above 500 K in conductors made by bringing n-alkane into contact with graphite Yasushi Kawashima [email protected] Department of Precision Engineering School of Engineering Tokai University 259-1292HiratsukaKanagawaJapan Superconductivity above 500 K in conductors made by bringing n-alkane into contact with graphite 1new superconductorsgraphitealkaneroom temperature 2 In 1986, a cuprate superconductor (Ba-La-Cu-O system) having a critical temperature which goes over the BCS limit (~30 K) was discovered and then a cuprate superconductor (Y-Ba-Cu-O system) with a critical temperature higher than 77 K was discovered. Furthermore, a Hg-based cuprate with a critical temperature of 133 K was found. The 133 K is still the highest critical temperature of conventional superconductors under atmospheric pressure.We have shown that materials obtained by bringing n-alkanes into contact with graphite are capable of conducting electricity with almost no energy loss at room temperature. We here report that the sudden jump in resistance showing a phase transition is observed in the materials during heating by two-probe resistance measurement. The measured critical temperatures of the materials consisting of pitch-based graphite fibers and n-alkanes having 7-16 carbon atoms range from 363.08 to 504.24 K and the transition widths range between 0.15 and 3.01 K. We also demonstrate that superconductors with critical temperatures beyond 504 K are obtained by alkanes with 16 or more carbon atoms. Introduction In 1911 Heike Kamerlingh Onnes discovered that mercury becomes suddenly zero resistance at temperature close to 4K [1]. This is the discovery of the first superconductor. Gold, silver and copper, which are the best conductors at room temperature, do not become superconductive at all. As temperature is lowered, their resistivity decrease gradually, but they never show a sudden drop. Thereafter, superconductivity was observed for pure metals such as Pb, Nb, and V and alloys such as Nb3Al, Nb3Sn, and V3Ga [2]. The discovery of superconductors with higher critical temperature became the targets of many researchers. It was mysterious for a long time why superconductivity occurred, but the BCS theory which explains superconductivity was established by J. Bardeen, L. Cooper and R. Schrieffer in 1957 [3]. A superconductor with the critical temperature beyond ~30 K was predicted not to exist according to this BCS theory. However, metallic superconductor MgB2 with a critical temperature of 39 K was discovered in 2001 [4]. This temperature is the highest critical temperature of the metallic superconductor up to now. A cuprate superconductor (La-Ba-Cu-O ceramics) with the critical temperature of 35 K which goes over the BCS limit (~30 K) has been discovered by Bednorz and Müller in 1986 [5], and subsequently, in 1987, the 93 K cuprate superconductor (Y-Ba-Cu-O system) having a critical temperature higher than the boiling point of liquid nitrogen (77 K) was found by M. K. Wu and P. W. Chu [6]. A Hg-based cuprate whose critical temperature is 133 K has been found in 1993 [7], but a cuprate superconductor with a critical temperature higher than that has not been found until now. Although iron-based superconductors were discovered in 2006 [8], the highest critical temperature of iron-based superconductors remains at 56 K up to this time [9]. In 2015, it has been reported that hydrogen sulfide exhibits superconductivity at 203 K under a high pressure of 150 Gpa [10]. This critical temperature is the highest critical temperature currently obtained in conventional superconductors. However, the high pressures exceeding 100 GPa cannot be generated without using a diamond anvil high pressure cell of which the sample chamber is extremely small. On the other hand, Kopelevich et al. reported ferromagnetic and superconducting-like magnetization hysteresis loops in some HOPG samples below and above room temperature suggesting the local superconductivity in graphite in 2000 [11,12]. Furthermore, in 2012, it was reported that a room temperature superconductor can be made by mixing the water into the graphite particles [13]. However, the authors did not confirm that macroscopic superconducting current passes through this material. Moreover, its critical temperature has not been measured. Recently, Precker1 and Esquinazi et al. identified a transition to zero-resistance state at ∼350 K in highly ordered natural graphite sample by resistance measurement, whereas the transition width was ∼40 K [14]. We found a possible room-temperature superconductor material obtained by bringing alkanes into contact with the graphite materials [15][16][17][18]. We showed that ring current in a ring-shaped container into which n-octane-soaked thin graphite flakes were compressed did not decay for 50 days at room temperature, suggesting that the material is capable of conducting electricity without energy loss at room temperature [15][16][17][18]. But the critical temperatures of these materials have not been measured. Therefore, in this study, we attempt to measure the critical temperature of the room temperature superconductor obtained by bringing alkane into contact with graphite material. However, since the above-mentioned material to be measured is an inhomogeneous material, the four-probe method cannot be applied to its resistance measurement. The reason is that there is a possibility that the measurement current path does not necessarily pass through a voltage-measurement terminal in inhomogeneous materials and therefore even if the potential difference between the two voltage-measurement terminals becomes zero, it does not necessarily mean that the resistance becomes zero. Thus, the resistance measurement using four-probe method for inhomogeneous materials causes misunderstanding [19]. The transition from the normal to the superconducting state or from the superconducting to the normal conducting state is accompanied by abrupt change in resistance. Although the result obtained by the resistance measurement using the two-probe method includes contact resistance, sudden jump in resistance at critical temperature can be discriminated by the two-probe method. It has been confirmed that the mixture obtained by bringing the alkane into contact with the graphite material has almost zero resistance at room temperature [15]. If the mixture obtained by bringing alkane into contact with graphite material is gradually heated from room temperature, when it reaches the critical temperature, the resistance of the mixture will jump suddenly. In this study, the critical temperatures of mixtures consisting of graphite materials and alkanes are measured by the two-probe method. In this research, a pitch-based graphite fiber was used as the graphite material. The sample for critical temperature measurement was prepared by packing the graphite fiber in a polytetrafluoroethylene (PTFE) tube and then injecting alkane into the tube with a syringe. Since the pitch-based graphite fiber is brittle, the fiber is sometimes broken into pieces when packing it in the PTFE tube. Therefore, the resistance of packed pitch-based graphite fibers before injecting alkane into the PTFE tube has a wide range of values. By using the pitch-based graphite fiber, it becomes possible to measure the critical temperature of the sample in which the packed graphite fibers are broken into pieces. In addition, the measurements of the critical temperature are also performed for the samples where alkanes of various carbon numbers are used. On the basis of the experimental results, we will discuss the effect of graphite basal plane surface on superconductivity in the materials. Methods The mixed materials which consist of pitch-based graphite fibers and various alkanes were packed in the PTFE tube shown in Fig. 1 (a). As shown in Fig. 1 (a), the tube has outer diameter of 1.7 mm, an inner diameter of 0.9 mm, a length of 32 mm, and at both ends of the tube, two internal screws of M1 and pitch 0.25 are cut. First, pitch-based graphite fibers (Nippon Graphite Fiber Co., Ltd., XN-100-25Z, no sizing agent, average fiber diameter: 10μm, average fiber length: 25 mm, true density: 2.2 g, resistivity: 1.5 × 10 ─4 Ωcm) were packed in the PTFE tube using metal wire. Secondly, one alkane was chosen from n-heptane, n-octane, n-nonane, n-decane, n-dodecane, n-tridecane and n-hexadecane. Only the alkane was injected into the graphite fibers packed in the PTFE tube by syringe. The experiments were done by using all the alkanes mentioned above. Finally, as shown in Fig. 1 Ωcm). Weights of graphite fibers packed in the tube range from 0.013 g to 0.018 g. The bulk density of the graphite material packed in the PTFE tube is calculated from the volume of the container (φ0.09 × 2 cm) (see Fig. 1(b)), which ranges from 1.02 g / cm 3 to 1.41 g / cm 3 . A copper wire of diameter 0.4 mm is brazed by solder of a high-melting point (the melting point: 586 K) to the heads of the two external screws. In this report, a sample is defined as consisting of the mixture of the graphite fibers and n-alkane packed in the PTFE tube, the two external screws (see Fig. 1 have zero resistance at room temperature on the basis of the previous experiments [15], it is considered that the resistance of the sample at room temperature is almost due to contact resistance between the graphite fibers soaked with n-alkane and the two tellurium copper screws. AC resistance of the sample was measured using LCR meter (E4980A, Keysight Technologies) with Kelvin clip leads (16089B, Keysight Technologies). Measuring frequency was 1 kHz, and a measurement electric current was 10 mA. In order to heat the PTFE tube in which the mixture of the graphite fibers and n-alkane is inserted (see Fig. 1 (b)), the PTFE tube was immersed in n-heptadecane (boiling point: 574 K) which was put in a glass vessel placed on a hot plate (see Fig. 2). As shown in Fig. 2, during heating the vessel by the hot plate, the n-heptadecane was agitated by brass blades installed in the motor to make the temperature of n-heptadecane in the vessel uniform. The tip of K type thermocouple used to measure the temperature of the mixture of the graphite fibers and alkane was brought into contact with the PTFE tube. The K type thermocouple was connected to a digital multimeter (34470A, Keysigh Technologies) through cold junctions (see Fig. 2). The LCR meter and the digital multimeter were linked to a computer by GPIB cables. The alternating-current resistance and thermal data measured by the LCR meter and the digital multimeter were written in the computer memory at intervals of one second. Results and Discussion The relationships between the temperature and the alternating-current resistance of the samples, which were obtained by injecting n-heptane, n-octane, n-nonane, n-decane, n-dodecane, n-tridecane and n-hexadecane separately into the graphite fibers packed in the PTFE tube, are shown in Figs. 3-9. All these figures show sudden and complete rise in resistance at temperatures above room temperature, suggesting that the mixtures of graphite fibers and n-alkane in all the samples undergo phase transitions from one state of matter to another. It has been confirmed by the previous experiment [15] that the mixture of graphite fibers and n-alkane should have zero resistance at room temperature. The observed transition shows that the mixtures of graphite fibers and n-alkane were maintained in the superconducting state until the sudden jump in resistance occurs. Therefore, the sudden jump in resistance shows that a phase transition from the superconducting to the normal conducting state occurs in the mixture of graphite fibers and n-alkane. It can be seen from Figs 3-9 that as the temperature increases, the resistance of the sample, i.e., the contact resistance gradually increases from room temperature. This increase in resistance can be explained as follows. The thermal expansion coefficient of PTFE is of the order of 10 -4 K -1 while that of graphite is of the order of 10 -6 K -1 . Since thermal expansion coefficient of PTFE is approximately two orders of magnitude greater than that of graphite, the PTFE tube expands larger than the graphite materials due to the temperature rise. Therefore, as the temperature increases, the contact pressure decreases and consequently the contact resistance, i.e., the resistance of the sample increases. Thus, the gradual increase in resistance of the samples can be considered to be caused only by the decrease in contact pressure arising from the difference in thermal expansion coefficient between PTFE and graphite. The sudden increase in resistance of the sample shows that the mixture of graphite fibers and n-alkane inserted in the PTFE tube was maintained constant at zero until the phase transition occurs. Accordingly, the temperatures at which the resistances of the samples suddenly rise can be considered to be the critical temperatures. Figure 3 shows the relationship between the temperature and the alternating-current resistance of the sample which were obtained by injecting n-heptane into the graphite fibers packed in the PTFE tube. Figure 3 shows that the resistance suddenly increases from 363.08 K and the sharp rise in resistance is completed at 364.97 K. From this, it can be seen that the resistance of the sample steeply increased by 6.50 Ω with a temperature rise of 1.89 K and the critical temperature is 363.08 K. Figure 7 (b) shows that while the temperature rises 2.33 K from 442.94 K, the resistance suddenly increases by 0.856 Ω, suggesting that the critical temperature is 442.94 K (see inset in Fig.7(b)). It can be seen from Figure 9 (a) shows that the resistance increases suddenly by 11.92 Ω while the temperature increases from 470.81 K to 472.00 K, showing that the critical temperature is 470.81 K. Figure 9 (b) shows that while the temperature rises by 1.2 K from 498.98 K, the resistance steeply increases 2.97 Ω (see inset in Fig. 9(b)). This shows that the critical temperature is 498.98 K. Fig. 9 (c) shows that the resistance makes a sudden jump at 504.24 K (see inset in Fig. 9 (c)) and it increases by 0.78 Ω due to a temperature rise of 0.32 K from 504.24 K. This suggests that the critical temperature is 504.24 K. Figures 9 (a) Fig. 9 that as the amount of increase in resistance during the phase transition decreases, the critical temperature becomes higher. The mixture of graphite fibers and n-alkane inserted in the PTFE tube is considered to have the zero-resistance until the resistance suddenly jumps up. Therefore, the amount of change in resistance during the phase transition is considered to be nearly equal to the resistance of the sample where only the graphite fibers are packed before injecting n-alkane into the PTFE tube. Accordingly, it is found that the smaller the resistance of the graphite fibers packed in the PTFE tube before injecting alkane into the tube, the higher the critical temperature. Figure 10 shows the relationships between the critical temperature and the amount of increase in resistance during the transition in the cases of using n-octane and n-hexadecane. Figure 10 indicates that there exists a nearly linear relationship between the critical temperature and the amount of increase in resistance during the phase transition. Figure 11 shows the relationship between the carbon number (n) of alkanes and the critical temperature (Tc). In Fig. 11, these critical temperatures are values when the amount of increase in resistance during the phase transition is 2.553 Ω. As can be seen from Fig. 10, since it can be assumed that there exists a near-linear relationship between the critical temperature and the amount of change in resistance during the phase transition, the critical temperatures at 2.553 Ω in the case of alkanes with 8-10, 12, 16 carbon atoms were obtained by linear interpolation. Figure 11 shows that as the carbon number of n-alkane increases, the critical temperature becomes higher. Figure 11 Relationship between the critical temperature and the carbon number of alkanes. A line is drawn to make the relationship easier to see. The resistances of the graphite fibers packed in the PTFE tube before injecting alkanes into the tube, which are obtained from the amount of change in resistance during the phase transition, range from 0.27 to 48.6 Ω. Since the dimensions of the sample container of PTFE tube is the diameter of 0.09 cm, length 2 cm, the range of apparent resistivity of the sample calculated from the above resistances was between 1.54 × 10 ─1 Ωcm and 8.59 × 10 ─4 Ωcm. The resistivity of the graphite fiber used is 1.5 × 10 ─4 Ωcm. Even taking into account the bulk density of the graphite material packed in the PTFE tube (1.02~1.41 g / cm 3 ), the apparent resistivity of 1.54 × 10 ─1 Ωcm is considered to be too high. The pitch-based carbon fiber used is very brittle because of its highly developed graphitizability [20]. Therefore, the high resistance of the samples is attributable to the breakage of the graphite fibers packed in the TPFE tube. If the graphite fibers break down into shorter fibers, the resistance of the sample will become higher accordingly. The surface of graphite fiber almost consists of basal plane surface. However, it can be considered that in the case of high resistance, the graphite fibers crack finely and therefore the ratio of basal surface to edge surface is reduced. Thus, it can be deduced that as the ratio of basal surface to edge surface becomes smaller, the critical temperature becomes lower. This suggests that basal surface plays an important role in the superconductivity in the mixture of graphite and alkane. We believe that in the superconductors obtained by bringing alkanes into contact with graphite material, superconductivity can be ascribable to protons abstracted by the graphite basal surface from n-alkanes [21]. We appreciate that all protons causing superconductivity should move coherently without activation energy on the basal surface [16][17][18]21]. As the ratio of basal plane surface to edge surface becomes smaller, the proton is more easily separated from the basal surface by the influence of heat, so that the superconducting state collapses. Therefore, the smaller the ratio of basal surface to edge surface, the lower the critical temperature. Figure 11 shows that the greater the number of carbon atoms of alkane, the higher the critical temperature. The boiling points of n-alkanes increase regularly with the increase in the number of carbon atoms. Furthermore, as the temperature approaches the boiling point of n-alkane, the thermal motion of the n-alkane molecules grows more intense. It can be considered that the superconducting state in the mixture of graphite materials and alkane may be broken by the thermal motion of alkane molecules. Figure 11 indicates that when n-alkanes with 16 or more carbon atoms are used, the critical temperature exceed 500 K. Conclusions We have observed a sharp rise in resistance of mixture of graphite fibers and alkanes inserted in the PTFE tube during heating them, using the two-probe-resistance measurements. This observation shows that a phase transition from the superconducting to the normal conducting state occurred in the mixture of graphite fibers and n-alkane. In other words, this indicates that the mixture of graphite fibers and n-alkane packed in the PTFE tube remains superconductive until its resistance rapidly rises, and the temperature at which the resistance suddenly rises is the critical temperature. We have noticed that when the pitch-based graphite fibers are packed in the PTFE tube, they may be broken into pieces depending on how they are packed into the tube. The finer the graphite fibers break, the greater the resistance of the sample before injecting alkane into the PTFE tube packed with the graphite fiber. From the relationship between the amount of change in resistance at the phase transition and the critical temperature, we found that the critical temperature decreases as the fiber is finely divided. That is, the critical temperature decreases, as the ratio of the basal plane surface to the edge plane surface decreases. This fact suggests that the basal plane plays an important role in superconductivity. Furthermore, we have found that the greater the carbon number of alkane, that is, the higher the boiling point of alkane, the higher the critical temperature. We have demonstrated that superconductors having critical temperatures exceeding 500 K can be obtained by using n-alkanes having 16 or more carbon atoms. (b)), and the two copper wires brazed to the heads of the screws. Resistance measurement of the sample was performed through the two cooper lead wires. Excluding the case of hexadecane, the resistances of the samples at room temperature ranged from 0.912 Ω to 1.98 Ω, and in the case of hexadecane the resistances of samples were 5.326 Ω, 8.575 Ω and 21.373 Ω. Since the combined resistance of the two copper lead wires and the two screw was 0.0136 Ω, and besides the mixture of graphite material and n-alkane packed in the PTFE tube should Figure 1 1PTFE tube container for packing the sample and PTFE tube into which graphite fiber and alkane are packed by using two external screws. (a) Two internal screws of M1 and pitch 0.25 are cut at both ends of the PTFE tube. (b) A mixture of graphite fiber and alkane was crammed into the PTFE tube (a) by using two external screws of M1.2 and pitch 0.25. Figure 2 2Schematic of a device for measuring a relationship between the AC resistance and temperature of the graphite fibers and alkane mixture packed in PTFE tube. Figure 3 3Resistance versus temperature of a sample consisting of graphite fibers and n-heptane. Critical temperature: 363.08 K; Amount of change in resistance during the phase transition: 6.501 Ω; the transition width: 1.89 K. Figure 4 4Resistance versus temperature of a sample consisting of graphite fibers and n-octane. (a) Critical temperature: 367.77 K; amount of change in resistance during the phase transition: 14.997 Ω; the transition width: 2.39 K. (b) Critical temperature: 379.41 K; amount of change in resistance during the transition: 4.827 Ω; the transition width: 0.15 K. (c) Critical temperature: 386.38 K; amount of change in resistance during the transition: 0.27 Ω; the transition width: 0.54 K. In (c), the inset shows the magnified view of jump in resistance near the critical temperature. Figure 5 5Resistance versus temperature of a sample consisting of graphite fibers and n-nonane. (a) Critical temperature: 379.34 K; amount of change in resistance during the phase transition: 23.895 Ω; the transition width: 1.09 K. (b) Critical temperature: 406.73 K; amount of change in resistance during the transition: 1.147 Ω; the transition width: 0.27 K. In (b), the inset shows the magnified view of jump in resistance near the critical temperature. Figure 6 6Resistance versus temperature of a sample consisting of graphite fibers and n-decane. (a) Critical temperature: 387.22 K; amount of change in resistance during the phase transition: 48.596 Ω; the transition width: 2.83 K. (b) Critical temperature: 424.95 K; amount of change in resistance during the transition: 3.393 Ω; the transition width: 3.01 K. Figure 7 7Resistance versus temperature of a sample consisting of graphite fibers and n-dodecane. (a) Critical temperature: 402.90 K; amount of change in resistance during the phase transition: 39.984 Ω; the transition width: 2.55 K. (b) Critical temperature: 442.94 K; amount of change in resistance during the transition: 0.856 Ω; the transition width: 2.33 K. In (b), the inset shows the magnified view of jump in resistance near the critical temperature. Figure 8 8Resistance versus temperature of a sample consisting of graphite fibers and n-tridecane. Critical temperature: 453.97 K; amount of change in resistance during the phase transition: 2.553 Ω; the transition width: 1.61 K. The inset shows the magnified view of jump in resistance near the critical temperature. Figure 9 9Resistance versus temperature of a sample consisting of graphite fibers and n-hexadecane. (a) Critical temperature: 470.81 K; amount of change in resistance during the phase transition: 11.925 Ω; the transition width: 1.19 K. (b) Critical temperature: 498.98 K; amount of change in resistance during the transition: 2.967 Ω; the transition width: 1.2 K. (c) Critical temperature: 504.56 K; amount of change in resistance during the transition: 0.782 Ω; the transition width: 0.32 K. In (b) and (c), insets show the magnified view of jump in resistance near the critical temperatures. Figures 4 ( 4a), (b), and (c) show the relationships between the temperature and the alternating-current resistance of samples which were obtained by injecting n-octane to the graphite fibers packed in the PTFE tube.Figure 4(a) shows that the resistance suddenly increases by 15.0 Ω while the temperature rises from 367.77 K to 370.16 K, showing that the critical temperature is 367.77 K.Figure 4(b) shows that the resistance suddenly increases by 4.828 Ω while the temperature rises by 0.15 K from 379.41 K. This suggests that the critical temperature is 379.41 K.Figure 4(c) shows that while the temperature rises by 0.54 K from 386.38 K, the resistance suddenly increases by 0.27 Ω (see inset inFig. 4(c)). This suggests that the critical temperature is 386.38 K. Theamounts of increase in resistance during the phase transitions obtained from Figs. 4 (a), (b), and (c) are 15.0, 4.828, and 0.27 Ω, respectively.Figures 5 (a) and (b) show the relationships between the temperature and the alternating-current resistance of samples which were obtained by injecting n-nonane to the graphite fibers packed in the PTFE tube. Figure 5 (a) shows that the resistance makes a sudden jump at 379.34 K and it increases by 23.90 Ω while the temperature rises from 379.34 K to 380.43 K, showing that the critical temperature is 379.34 K. Figure 5 ( 5b) shows that the resistance suddenly increases by 1.15 Ω while the temperature rises by 0.27 K from 406.73 K, showing that the critical temperature is 406.73 K (see inset in Fig. 5(b)). Therefore, it is seen from Figs 5 (a) and (b) that when the critical temperatures are 379.34 K and 406.73 K, the amounts of change in resistance during the phase transition are 23.90 and 1.15 Ω, respectively.Figures 6 (a) and (b) show the relationships between the temperature and the alternating-current resistance of samples which were obtained by injecting n-decane to the graphite fibers packed in the PTFE tube.Figure 6 (a) shows that the resistance makes a sudden jump at 387.22 K and it increases by 48.60 Ω while the temperature rises from 387.22 K to 390.05 K, showing that the critical temperature is 387.22 K. Figure 6 ( 6b) shows that the resistance steeply increases by 3.39 Ω while the temperature rises by 3.01 K from 424.95 K, showing that the critical temperature is 424.95 K. It is found from Figs. 6 (a) and (b) that when the critical temperatures are 387.22 K and 424.95 K, the amounts of increase in resistance during the transition are 48.the relationships between the temperature and the alternating-current resistance of the samples which were obtained by injecting n-dodecane into the graphite fibers packed in the PTFE tube.Figure 7 (a)shows that the resistance makes a sudden jump at 402.90 K and it increases by 39.98 Ω while the temperature rises from 402.90 K to 405.445 K, showing that the critical temperature is 402.90 K. Figs 7 (a) and (b) that when the critical temperatures are 402.90 K and 442.94, the amounts of increase in resistance during the transition are 39.98 Ω and 0.856 Ω, respectively. Figure 8 8shows the relationship between the temperature and the alternating-current resistance of a sample which was obtained by injecting n-tridecane to the graphite fibers crammed in the PTFE tube.Figure 8shows that while the temperature rises from 453.97 K to 455.58 K, the resistance steeply increases by 2.553 Ω, showing that the critical temperature is 453.97 K and furthermore the amount of increase in resistance during the transition is 2.553 Ω (see inset inFig. 8).Figures 9 (a), (b), and (c) show the relationships between the temperature and the alternating-current resistance of samples which were obtained by injecting n-hexadecane into the graphite fibers packed in the PTFE tube. Figure 10 10Relationships between the critical temperature and the amount of change in resistance during the transition in the cases of using n-octane and n-hexadecane. Straight lines in the figure were obtained by the least squares method. 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[ "Anderson Localisation for periodically driven systems", "Anderson Localisation for periodically driven systems" ]	[ "Raphael Ducatez \nCEREMADE\nUniversité Paris-Dauphine\nFrance\n", "François Huveneers \nCEREMADE\nUniversité Paris-Dauphine\nFrance\n" ]	[ "CEREMADE\nUniversité Paris-Dauphine\nFrance", "CEREMADE\nUniversité Paris-Dauphine\nFrance" ]	[]	We study the persistence of localization for a strongly disordered tight-binding Anderson model on the lattice Z d , periodically driven on each site. Under two different sets of conditions, we show that Anderson localization survives if the driving frequency is higher than some threshold value that we determine. We discuss the implication of our results for recent development in condensed matter physics, we compare them with the predictions issuing from adiabatic theory, and we comment on the connexion with Mott's law, derived within the linear response formalism.	10.1007/s00023-017-0574-1	[ "https://arxiv.org/pdf/1607.07353v1.pdf" ]	119,161,568	1607.07353	dc2ea76058fc1e94cb68dd6f1e5b70d95dac0f72	Anderson Localisation for periodically driven systems February 19, 2018 Raphael Ducatez CEREMADE Université Paris-Dauphine France François Huveneers CEREMADE Université Paris-Dauphine France Anderson Localisation for periodically driven systems February 19, 2018 We study the persistence of localization for a strongly disordered tight-binding Anderson model on the lattice Z d , periodically driven on each site. Under two different sets of conditions, we show that Anderson localization survives if the driving frequency is higher than some threshold value that we determine. We discuss the implication of our results for recent development in condensed matter physics, we compare them with the predictions issuing from adiabatic theory, and we comment on the connexion with Mott's law, derived within the linear response formalism. Introduction In this paper, we study the fate of Anderson localization in periodically driven systems. Let H 0 be the tight-binding Anderson Hamiltonian on the lattice Z d . At strong enough disorder, it is well known that all eigenstates of H 0 are exponentially localized (see [6] [15] [5] as well as [13] for more references). Let us then consider a periodic time-dependent Hamiltonian of the form H(t) = H 0 + gH 1 (t)(1) with H 1 (t) = H 1 (t + T ) for some period T , and with g some coupling constant. We assume that H 1 (t) acts everywhere locally: there exists R such that \|(x, H 1 (t)y)\| = 0 for all x, y ∈ Z d , and all time t, as soon as \|x − y\| > R (with the notation (x, Ay) = (δ x , Aδ y ) = A(x, y) for an operator A). The time-evolution of an initial wave function ψ(0) is governed by the time-dependent Schrödinger equation: i dφ(t) dt = H(t)φ(t). The long time properties of the solutions of this equation are best understood through the Floquet eigenstates of H(t) [19]. The question addressed in this paper can then be rephrased as follows: Under suitable regularity conditions on the time-dependence of H 1 (t), is there a range of values for g and T such that the structure of the eigenfunctions of H 0 is only weakly affected by the periodic potential H 1 (t), so that the the Floquet eigenstates of H(t) are themselves localized? We answer this question positively in Theorem 1 below, for two different regularity conditions on H 1 , leading to different allowed values for g and T . Localization and Floquet physics. The above question has already received some attention in the mathematical physics community. The connection with with the discrete non-linear Schrödinger equation (DNLS) constituted a first motivation, see [7] [27]. In this context, the more general case of a quasi-periodic driving shows up naturally: In a first approximation, the non-linearity in the DNLS equation can be replaced by a quasi-periodic perturbation. On the other hand, in this perspective, it is natural to restrict oneself to spatially localized perturbations ((x, H 1 (t)y) decays fast as x or y goes to infinity and not only as \|x − y\| goes to infinity as we consider); indeed, stability results for the DNLS equation all deal with originally localized wave packets. More recently, periodically driven Hamiltonian systems have been studied intensively in condensed matter theory. For two reasons at least: First, from a theoretical perspective, driven systems constitute the first examples of dynamics out-of-equilibrium systems, lacking even energy conservation. The natural question that arrises is whether the system will absorb energy until it reaches an infinite temperature state (i.e. a state with maximal entropy), as it would be the case for a chaotic system, or whether some extensive effectively conserved quantity emerges, forbidding energy absorption after some transient regime [9][10] [2][3] [4] [1]. For non-interacting particles on a lattice, as we consider in this paper, this issue becomes trivial and fully independent of the issue of Anderson localization, once the driving frequency becomes higher than the bandwidth of individual particles, see [2] [4]. Nevertheless, thanks to the Anderson localization phenomenon, our results guarantee the existence of an effective extensive conserved quantity for frequencies much below this trivial threshold, see Proposition 2 below. Second, from a more practical point of view, driven systems furnish a way to engineer topological states of matter [26] [24]. Though this possibility is not apriori related to the phenomenon of Anderson localization, it turns out that, for interacting many-body systems, localization makes it possible to "lift" phase transitions from the ground state to the full spectrum [20]. This observation is at the heart of very recent investigations of new phases of matter inside the many-body localized phase [21][28] [29]. Hence, in view of the increasing role played by localized Floquet systems in modern condensed matter physics, it appeared useful to bring some firm mathematical foundations to the theory of Anderson localization in periodically driven systems, even though the need for mathematical rigor forces us to restrict the setup to non-interacting particles. Results in this direction already appeared in [18], where the localization for some random unitary operators is established; this question is directly related to ours since the long time evolution of a periodically system is governed by the spectral properties of the unitary U (T ), where U (t) solves idU (t)/dt = H(t)U (t). However, for a Hamiltonian as in (1), we do not recover the particular form for U studied in [18]. Before stating our results, we now introduce two more specific aspects that deserved clarification and motivated the present article. Adiabatic Theory. Time-dependent Hamiltonian systems varying smoothly and slowly enough with time can be described through the use of adiabatic theory. Here, adapting the analysis from [3], we argue that localization emerges when level crossings in the system become typically nonadiabatic, and we determine the threshold frequency above which this happens. Let us first remind the theory of the Landau-Zener effect for a time-dependent two-levels Hamiltonian G(t) [23] [31]. To make the connection with our problem, let us assume that G(t) is of the form G(t) = P H(t)P where P projects on two eigenstates of H 0 . Moreover, we assume that G(t) varies smoothly on the scale of one period, i.e. we can write G(t) =G(νt) for some smooth 2π-periodic functionG and ν = 2π T . It is then convenient to move to the basis of the eigenstates of H 0 , i.e. the basis where P is diagonal, and to decompose G(t) = G dia (t) + G off (t), as a sum of the diagonal and off-diagonal part. We notice that the time-dependent part of G dia (t) is of order g. We set (1, G off (t)2) =: g , where g depends mainly on the distance between the two localization centers of the two states projected on by P , and is typically much smaller than g. Finally we assume that the two levels of P H 0 P are close enough (g-close in fact) to each others so that the system undergoes an avoided crossing as time evolves: At some time, the levels of G dia (t) cross, while G off (t) induces level repulsion, leading to an avoided crossing for G(t). If the system is initially (i.e. before the crossing) prepared in an eigenstate of G dia (t), Landau-Zener theory tells us that, after the crossing, the state in which the systems ends up depends on the value of \|(1, G off 2)\| 2 v 12 ∼ (g ) 2 gν ,(2) where v 12 is the rate of change in the energy of H dia (t) at the crossing. At high frequency, when this value is much smaller than 1, the crossing is non-adiabatic and the system remains in the original state; at intermediate frequency, when this value is of order 1, the system ends up in a superposition of the eigenstates of G dia (t); and finally at low frequency, when this value is much smaller than 1, the crossing is adiabatic and the system ends up in the other eigenstate of G dia (t). The above scenario, valid for a two level systems, may be seen as a caricature of the localization-delocalization transition: non-adiabatic crossings do not entail hybridization of unperturbed eigenstates, while intermediate and adiabatic crossings, present at low enough frequency, allow the system to move from one state to the other, and constitute a possible mechanism for delocalization. Based on this picture, let us try to determine a critical value of ν above which localization survives. Let us fix g in (1) as well as W characterizing the strength of the disorder. Let us then pick a point a ∈ Z d . We first determine a minimal length L * so that there is typically at least one crossing between the state centered around a and an other state with localization center in a ball of radius L * around a. Since the probability of finding a crossing in a ball of radius L is of the order of L d g W , we find L * ∼ W g 1/d . The effective coupling between a state centered around a and a state at a distance L of a, corresponding to g in (2) is of the order of g ∼ ge −L/ξ , where ξ is the localization length of H 0 . Hence, from (2), we find that localization will survive if g 2 e −2L * /ξ gν 1 ⇔ ν g e − 2 ξ ( W g ) 1/d .(3) In Theorem 1 below, for a smooth driving (condition (C1)), we prove localization for ν larger than some threshold value comparable to what we obtain in (3). We notice that the Landau Zener theory proceeds through non-perturbative arguments. Instead, our proof is based on the multiscale analysis developed in [15], which is mainly a perturbative approach. It is thus somehow remarkable that the same upshot can be recovered in two a priori very different ways. Finally we notice that the approach through adiabatic theory outlined above is only expected to work for H 1 (t) depending smoothly on time. Unfortunately, both in theoretical and experimental physics works, it is a common protocol to just shift between two Hamiltonians periodically. This leads obviously to a non-smooth time-dependence. As we wanted to cover this case as well, we also derived a result for H 1 being only in square-integrable in time; see Theorem 1 below with the condition (C2). The lack of smoothness forced us to increase significantly the threshold on ν with respect to (3). Mott's law. Mott's law asserts that the ac-conductvity of an Anderson insulator behaves as σ(ν) ∼ ν 2 log(1/ν) d+1 as ν → 0 ( [25], see also [17] for the case of interacting electrons). An upper bound on σ(ν) was rigorously established in [22] (with d + 1 replaced by d + 2). The conductivity σ(ν) is derived within the linear response (LR) formalism; in our set-up, this corresponds to fixing ν and taking the limit g → 0 while observing the dynamics over a time of order ν/g 2 . In such a regime, the hypotheses of Theorem 1 below are satisfied (we consider a monochromatic perturbation with frequency ν so that condition (C1) holds): The dynamics is localized for g small enough once ν has been fixed. 1 It may thus come as a surprise that still σ(ν) > 0. This puzzling behavior was recently analyzed in details for many-body systems in [16]. As it was pointed out to us by [12], the conductivity σ(ν) is computed for a system in equilibrium at zero or finite temperature. Moreover, as can be expected from its definition, for g > 0, LR should in general furnish only an accurate description of the dynamics for a transient regime in time of order ν/g 2 . It is true though that, for "generic" or "ergodic" systems, it is reasonable to think that the predictions from LR remain valid for much longer time scales: While heating, the system remains approximately in equilibrium and LR can be applied iteratively until the infinite temperature state is reached. This is manifestly not true for localized systems as long as g is small enough compared to ν: The conductivity σ(ν) > 0 represents mainly the Rabi oscillation of rare resonant spots ("cat states") in the Hamiltonian H 0 , but these oscillations do not need to entail delocalization on the longes time scales described by the Floquet physics. Organisation of the paper. The precise definition of the model studied in this paper together with our results are presented in Section 2. The main steps of the proof of our main theorem are contained in Section 3, while some more technical intermediate results are shown in Sections 4 to 6. The two corollaries are shown in Section 7. In several places, the proof of our results proceeds through a straightforward adaptation of delicate but well-known methods; as much as possible, we choose to describe in details only the steps where some significant amount of new material was required. Models and results The models We consider a lattice model on Z d and we note \|x\| = sup i=1..d \|x i \|. Our results could be of course extended to more general lattices. We are interested in the long time behavior of the Schrödinger equation: i d dt φ(t) = H(t)φ(t),(4) where the function φ(t) is defined on L 2 (Z d ) for any t, and the Hamiltonian H(t) is a periodic function with frequency ν = 2π/T . The operator H(t) is an idealized version of (1): We move to the basis where H 0 is diagonal and we replace it by an uncorrelated random potential V ω , while we assume that H 1 (t) is still a nearest-neighbor hopping (Anderson model): H(t) = −g∆(t) + V ω .(5) Here −∆(t) is hermitian operator for any t such that −∆(t)(x, y) = 0 if \|x − y\| > 1 and − ∆(t)(x, y) L 2 ([0;T ]) ≤ 1(6) for any x, y. We use the notation −∆ because in the usual time-independent Anderson model, −∆(t) is the usual discrete Laplacian on (Z d ) −∆φ(x) = 1 2d \|y−x\|=1 φ(y), There exists a unitary operator U (t), with U (0) = Id such that φ(t) = U (t)φ(0) and satisfying i d dt U (t) = H(t)U (t),(7) Existence and uniqueness of solution of (4) and (7) can be proved using a usual fixed point technique. (RP) Potential regularity. We assume the following form for the random potential which are widely used in the literature: V ω = x∈Z d v x δ x(8) where v x are i.i.d. random variables, with a bounded density ρ, such that ρ ∞ < ∞ defined on a bounded support [−M ; M ]. We choose units such that ρ ∞ = 1. Furthermore we will assume that the density ρ is piecewise C 1 . The time-dependent term −g∆(t) is considered to be a perturbation of order g 1, usually referred to as the strong disorder regime. We treat this model in two particular cases. (C1) Smooth driving. We suppose that −∆(t)(x, y) is a monochromatic signal: For any x and y, − ∆(t)(x, y) = a x,y + b x,y cos(νt) + b x,y sin(νt)(9) with a x,y = a y,x , b x,y = b y,x and b x,y = b y,x . In this regime, we are able to prove localization for frequencies ν up to a threshold comparable to the one given in (3). Moreover, we claim that the result can then be extended to a hopping −∆(t) with Fourier coefficients that decay fast enough, but we focus on the case of single Fourier mode for simplicity. (C2) L 2 driving. We only assume (6). In this case, a much larger threshold value for ν is needed, actually ν ≥ 1. We refer to [3] for the optimality of this condition. Remark 1. Between these two extreme cases, one could obviously consider intermediate regularity cases, depending on the decay of the Fourier coefficients of −∆(t). This should lead to other conditions on ν that are not investigated in this paper. The Floquet operator We will work in the Fourier space instead of the time-domain, and we denote byx = (x, k) a point of Z d × Z. Let's introduce the central object of our paper: Definition 1. LetĤ = −g∆ +V ω (10) be a Hamiltonian on Z d × Z, with −∆ψ(x, k) = − \|y−x\|≤1 k ∆ x,y (k )ψ(y, k − k ) (11) where∆ x,y (k) = 1 T T 0 ∆ x,y (t)e −iνkt dt and V ω = V ω + kν.(12) In the mono-chromatic case (C1), the Laplacian −∆ is explicitly given by −∆ψ(x, k) = \|y−x\|≤1 a x,yψ (y, k) + b x,y + ib x,y 2ψ (y, k + 1) + b x,y − ib x,y 2ψ (y, k − 1) We remark that it is a local operator, meaning it connect only sitesx,ŷ such that \|x −ŷ\| = 1 in the space-Fourier graph Z d × Z. In the general L 2 case (C2), this is no longer true. Indeed, points (x, k), (y, k ) could be connected with \|k − k \| arbitrary large. The new HamiltonienĤ gives the evolution of the "finite time Fourier series" of φ(t) defined as followsφ (x, k, t) = 1 T t+T t φ(x, u)e −iνku du.(13) We get formally a time-independent Schrödinger equation governed by the HamiltonianĤ: Proposition 1. i∂ tφ (x, k, t) =Ĥφ(x, k, t)(14) Proof. i∂ tφ (x, k, t) = 1 T t+T t i∂ u φ(x, u)e −iνku du = 1 T t+T t kν + H(u) φ(x, u)e −iνku du = 1 T t+T t (kν + V ω )φ(x, u)e −iνk + g \|y−x\|≤1 k (−∆ x,y (k ))φ(y, u)e −iν(k−k )u du = (V ω + kν)φ(x, k, t) + g \|y−x\|≤1 k (−∆ x,y (k ))φ(y, k − k , t) =Ĥφ(x, k, t). The time evolution ofφ is deduced from the eigenvectors ofĤ: λψ = − g∆ +V ω ψ(15) Looking for the eigenvectors ofĤ is equivalent to the search of solution of the form φ(t) = e iλt ψ(t) with ψ a T -periodic function (Floquet theory). Indeed, in the Fourier variables, (4) is equivalent to (15). In particular, as we will see, localization forĤ implies the absence of diffusion for φ. Remark 2. Because ψ(t)e iλt = ψ(t)e −inνt e i(nν+λ)t , if (ψ,λ) is a solution then (ψ(t)e −inνt , nν + λ) is a solution as well for any n ∈ Z. Hence it is enough to consider the caseλ ∈ [0; ν]. Results Our main theorem states Anderson localisation forĤ. Theorem 1. There exists > 0 such that, if g < , and if ν ≥ e −g − 1 4p+8d for some p > 2d under the condition (C1), or if ν ≥ 1 under the condition (C2), thenĤ exhibits localization :Its spectrum is pure point and its eigenvectors decay exponentially in space, P a.s. Remark 3. Under (C1), we will see that the eigenvectors are also deterministically exponentially localized along the frequency axis. The two following corollaries do not logically follow from Theorem 1, but rather from a refinement of its proof. The first one shows the absence of diffusion for solutions of (4) (dynamical localization): Corollary 1. There exist > 0 and q > 0 (and one may take q → ∞ as → 0) such that, if g < and ν ≥ e −g − 1 4p+8d for some p > 2d under (C1), or ν ≤ 1 under (C2), then E sup t>0 x∈Z d \|x\| q \|φ(x, t)\| 2 < ∞(16) for any solution φ(x, t) of (4) with initial condition φ(x, 0) defined on a bounded support. The second one deals with the existence of a local effective Hamiltonian, i.e. an Hamiltonian H ef f such that U (T ) = e −iT H ef f and such that H ef f (x, y) decays fast as \|x − y\| → ∞. Under the conditions of Theorem 1, given λ ∈ [0, ν[ and a corresponding eigenfunctionψλ(k, x) ofĤ, and given t ∈ R, let us denote by P ψλ(·,t) the projector L 2 (Z d ) → L 2 (Z d ), f → ψλ(·, 0), f ψλ(·, t). The representation U (t) = λ ∈[0,ν[ e −iλt P ψλ(·,t) holds. Hence, since the functions ψλ(·, t) are T -periodic in time, we may set H ef f = λ ∈[0,ν[λ P ψλ(·,0) ,(17) which defines an operator on L 2 (Z d ). Under condition (C1), we have a more 2 Corollary 2. There exist > 0 and q > 0 (and one may take q → ∞ as → 0) such that, if g < , ν ≥ e −g − 1 4p+8d for some p > 2d, and under condition (C1), then E \|x − y\| q \|H ef f (x, y)\| < ∞. with H ef f as defined by (17). Proof of Theorem 1 We will prove that the HamiltonianĤ reveals localisation by applying the classical tools of the multi-scale analysis (MSA). Thanks to the huge literature on MSA, it we will be enough for us to prove a probability estimate, usually referred to as Wegner estimate, and the initialization of the MSA to show the localisation (as well as some extra technical results when dealing with the L 2 case, i.e. under assumption (C2)). We start with the Wegner estimate. Below we call columns sets of the form Λ 0 × I ⊂ Z d × Z, for some finite spatial box Λ 0 and some frequency interval I. Given Λ ⊂ Z d × Z and given H ∈ L 2 (Z d × Z), we denote by H \|Λ the operator acting on L 2 (Λ) such that H \|Λ (x,ŷ) = H(x,ŷ) for allx,ŷ ∈ Λ. Proposition 2 (Wegner Estimate). Let Λ 0 ⊂ Z d be finite. Then 1. (The finite column case) For any K ∈ N, k 0 ∈ Z so that Λ 0 × [k 0 − K; k 0 + K] ⊂ Z d × Z, we have ∀E, P(∃λ eigenvalue ofĤ \|Λ0×[k0−K;k0+K] :λ ∈ [E − , E + ]) ≤ 2π (2K + 1)\|Λ 0 \|\|\|ρ\|\| ∞ .(18) 2. (The infinite column case) There exists a constant C which depends only on ρ L ∞ and ρ L ∞ , such that for Λ 0 × Z ⊂ Z d × Z, we also have P(∃λ eigenvalue ofĤ \|Λ0×Z :λ ∈ [E − , E + ]) ≤ 2π √ \|Λ 0 \|\|\|ρ\|\| ∞ max(1, M ν ).(19) The proof of this proposition will be carried over in section 4. Part 1. will be needed to establish Theorem 1 under the assumption (C1) and part 2. under the assumption (C2). The crucial property that allows to show the second part of this proposition is contained in Remark 2: Ifψ(x, k) is an eigenvector with eigenvalueλ ofĤ \|Λ0×Z , thenψ(x, k − k 0 ) is also an eigenvector with eigenvalueλ + νk 0 for any k 0 ∈ Z. Therefore the eigenvalue are of the form {λ i : i = 1, . . . , \|Λ 0 \|} + νZ, allowing to use \|Λ 0 \| in the rhs of (19) instead of the cardinal of the column which in this case is infinite. The second ingredient in the MSA consists in proving the exponential decay of the resolvent (Ĥ − λ) −1 with high probability for a given λ ∈ R. We will follow [13]. To initialize the MSA, we need to show that, given a pointx ∈ Z d × Z, there exists with high probability a finite domain aroundx, called "good box", where the resolvent decay exponentially. From now on we fix some λ ∈ [0, ν]. Indeed, it is enough to consider values of λ in this interval, because of the symmetry described in Remark 2. For Λ ⊂ Z d × Z, we will write ∂ in Λ = {x ∈ Λ : ∃ŷ / ∈ Λ,∆(x,ŷ) = 0}(20)∂ ext Λ = {x / ∈ Λ : ∃ŷ ∈ Λ,∆(x,ŷ) = 0}(21) Smooth driving (C1) Definition 2 (Good box). Under the assumption (C1), we say that ( x+[−L, L] d )×[k 0 −K, k 0 + K] is a µ-good box, for some µ > 0, if, for any (y, k) ∈ ∂ in x + [−L, L] d ) × [k 0 − K, k 0 + K] , \| (x, k 0 ), Ĥ \|(x+[−L,L] d )×[k0−K,k0+K] − λ −1 (y, k))\| ≤ e −µ\|(x,k1)−(y,k2)\| (22) where \|(x, k 1 ) − (y, k 2 )\| = \|k 2 − k 1 \| + d i=1 \|x i − y i \| . The difference between our model and the classical Anderson model is the absence of independence along the frequency axis. However we have the following proposition. Proposition 3. If \|k 0 \| > M + √ g ν + K then for any Λ 0 ⊂ Z d , Λ 0 × [k 0 − K; k 0 + K] is a − ln(2(d + 1)g) good box. The proof of this proposition will appear as a simple case of the proof of Proposition 4 below (see Section 5 after the proof of Proposition 6). Thanks to this proposition, it is now enough then to study boxes close to the k = 0 axis. Once we restrict ourselves to such boxes, non-intersecting boxes are stochastically independent, and we can proceed with the usual MSA approach. So the idea of the proof is to show initialization of the MSA for boxes like Λ 0 × [− 2(M + √ g) ν ; 2(M + √ g) ν ] . Remark 4. For any x ∈ Z d , there exists k such that \|V (x, k) − λ\| ≤ ν Hence, there is no way avoiding a resonance of order ν for all x, and we cannot look for good boxes as free of any resonances. Nevertheless, we prove that good boxes appears with high probability when g 1. Let p > d. Proposition 4 (Initialisation of the MSA under the assumption (C1)). Assume that (C1) holds. For any µ > 0, L * ∈ N, there exist > 0 and L ≥ L * such that for any g < , such that if ν > exp(− 1 g 1 8d+4p ) then P(B L (x) is a µ-goog box) > 1 − 1 L 2p (23) where B L (x) = x + [−L; L] d × [− M ν ; M ν ]. L 2 driving (C2) A new problem appears here: For which distance on Z d × Z should we prove the exponential decay? In the smooth case,∆ was a local operator, so the usual distance on works fine. But because g(k − k) is non-zero for k − k large if the driving is only in L 2 ([0, T ]), the operator∆ connects now points (x,ŷ) that are not close to each other in Z d × Z and there is no exponential decay along the frequency k. In order to prove exponential decay on Z d , we introduce a new decay function on Z d × Z, which can actually easily be used in the "random walk expansion" that appears in the MSA. Definition 3. Let G : Z d × Z 2 → R such that for all anyx 0 ∈ Z d × Z, G(x 0 , .) ∈ L 1 (Z d × Z) with G(x 0 , .) L 1 < 1/2. We define the decay function d G by d G (x,ŷ) = − ln C(x→ŷ) i \|G(ẑ i ,ẑ i+1 )\|(24) for anyx,ŷ ∈ Z d × Z ifx =ŷ and 0 otherwise, where C(x →ŷ) is the set of all finite sequences of the type (x =ẑ 0 ,ẑ 1 ,ẑ 2 , . . . ,ẑ k =ŷ) (or "paths" fromx toŷ). Let P : Z d × Z → R be defined by P ((x, k)) = 1/ √ g if kν ∈ [−M − √ g, M + √ g], 1 ν(\|k\|−1)−M if kν / ∈ [−M − √ g, M + √ g]. We say thatx is a resonant site if \|V ω (x) − λ\| < √ g. We have defined the function P (x) such that if there is no resonant site on x × Z, then P (x) > 1 \|Vω(x)−λ\| . Definition 4. Under assumption (C2), we say that C L (x) = (x + [−L, L] d ) × Z ⊂ Z d × Z is a µ-good column if there exists a decay functiond G such that \|(x, (Ĥ \|C L (x) − λ) −1ŷ )\| ≤ P (x)e −d G (x,ŷ) for allŷ ∈ ∂ in C L (x), and such that ŷ∈∂ in C L (x) e −d G (x,ŷ) < e −µL . Proposition 5 (Initialisation of the MSA under the assumption (C2)). Assume that (C2) holds. For any µ > 0, L * ∈ N, there exist > 0 and L ≥ L * such that for any g < , such that if ν > 1 then P C L (x) is a µ-good column > 1 − 1 L 2p .(25) As in the smooth case (C1), Theorem 1 will follow from the Wegner estimate (Eq. (19) in Proposition 2) the initialization of the MSA (Proposition 5), and the stochastic independence of distinct columns (obvious here). But there is still one difference : the MSA has to be performed with infinite columns. This issue will be addressed in Section 6.4, where we explain the technicals adaptations to perform in the proof in [13]. Wegner Estimate In this Section, we prove Proposition 2 (Wegner estimate). For (18) (finite column), we closely follow [30], while for (19) (infinite column), we follow [14] (see also [8]). Thanks to the resolvent formula, we have the Shur formula : for any P projector and B = P BP , then P (A + B) −1 P = ((P A −1 P ) −1 + B) −1(26) Where the two last "· −1 " in the right hand side correspond to the inverse for operators restricted to Im(P ). Proof of (18). We follow the proof from [13]. Let Λ ⊂ Z d × Z, E ∈ R. Let P x , x ∈ Z d the projectors on the subspace {x} × [k 0 − K, k 0 + K] and Λ 0 ⊂ Z d the projection of Λ on its first parameters. P(∃λ eigenvalue ofĤ \|Λ :λ ∈ [E − , E + ]) ≤ E(Tr(1 [E− ,E+ ] (Ĥ \|Λ ))) ≤ E(2 (Tr(Ĥ \|Λ − E − i ) −1 )) = E 2 x∈Λ0 Tr P x (Ĥ \|Λ − E − i ) −1 P x ) = 2 x∈Λ0 E Tr (P x (Ĥ \|Λ − v x P x − E − i ) −1 P x ) −1 + v x P x −1 = 2 x∈Λ0 E Vy:y =x Tr (P x (Ĥ \|Λ − v x P x − E − i ) −1 P x ) −1 + v x P x −1 ρ(x)dv x = 2 x∈Λ0 E Vy:y =x µi∈σ((Px(Ĥ \|Λ −vxPx−E−i ) −1 Px) −1 ) ( µ i + v x −1 )ρ(x)dv x ≤ 2 x∈Λ0 E Vy:y =x π ρ ∞ (2K + 1) ≤ 2π (2K + 1)\|Λ 0 \| ρ ∞ , where, to get the last equality, we used that P x acts as the identity on the subspace generated by P x . Proof of (19). Let Λ 0 be a finite subset of Z d . We make a change of variable for the potential α = 1 \|Λ0\| x∈Λ0 V ω (x). As in [14] (see also [8]), the conditional probability of α knowingṼ (x) = V ω (x) − α for all x ∈ Λ 0 , admits a density ξṼ (α) and there exists a constant C such that, on a set U belonging to the sigma-algebra generated byṼ (x) for all x ∈ Λ 0 , and with probability larger that 1 − C √ , ξṼ ∞ ≤ C √ (2M ) 1/2 ρ ∞ + (2M ) 3/2 ρ ∞(27) Because of the symmetry described in Remark 2, for any realization (Ṽ , α 0 ), there exist λ 1 , ..,λ \|Λ0\| ∈ [0, ν] such that σ(ĤṼ ,α0 ) = {λ 1 , . . . ,λ \|Λ0\| }+νZ. Now, keepingṼ fixed and changing α, one gets σ(ĤṼ ,α ) = {λ 1 + (α − α 0 ), . . . ,λ \|Λ0\| + (α − α 0 )} + νZ. Then, for any E ∈ R, P(d(σ(Ĥ), E) < ) ≤ C √ + P({d(σ(Ĥ), E) < } ∩ U ) ≤ C √ + EṼ 1 U Λ0 i=1 k∈Z 1(\|λ i + kν + (α − α 0 ) − E\| < )ξṼ (α)dα ≤ C √ + 2 1 √ C (2M ) 1/2 ρ ∞ + (2M ) 3/2 ρ ∞ K 0 where K 0 is the maximum number of eigenvalueλ in σ(ĤṼ ,α0 ) such that there exists α ∈ [−M, M ] such that \|λ + α − α 0 − E\| < with non-zero probability. In particular we have K 0 ≤ 2\|Λ 0 \| M ν + 1. The key tool for the MSA is the following formula : (v 0 , (Ĥ − λ) −1ẑ ) = û∈∂ in Λ,v∈∂ ext Λ (v 0 , (Ĥ \|Λ − λ) −1û )(û, g∆v)(v, (Ĥ − λ) −1ẑ )(28) for anyv 0 ∈ Λ andẑ / ∈ Λ, and Λ ⊂ Z d × Z with z / ∈ Λ, which is a direct application of the well known resolvent formula. We will repeat it as many times as we can, replacing v for v 0 and choosing correctly the new Λ. The next subsection deals with this question. Resonant sites, security box and propagation decay Remind thatv = (x, k) ∈ Z d × Z is a resonant site if \|V ω (v) − λ\| = \|v x + νk − λ\| < √ g. Obviously, for any x there exits a segment K x ⊂ Z so that (x, k) is a resonant site for k ∈ K x , where K x is of the form K x = Z∩[k 0 − √ g/ν, k 0 + √ g/ν] for some k 0 that depends on V ω (x) (Figure 5.1). Around each segment of resonant sites K x , we define a security box Λ Kx = {z ∈ Z d × Z : d(z, K x ) < N }, where N is an integer that will be defined later, and d is the usual graph distance on Z d × Z. We will say that a set of the form Λ 0 ×I ⊂ Z d ×Z is not strongly resonant if d(σ(Ĥ \|Λ0×I ), λ) > ν 2 α(g), where α(g) is a function which will be defined at the end of the proof of Proposition 7 below. ∂ in (x + [−L, L] d ), k 1 ,k 2 ∈ Z, (x, k 1 ), Ĥ \|(x+[−L,L] d )×[k0−K,k0+K] − λ −1 (y, k 2 )) ≤ 2( √ g N 2 ) n0 (ν 2 α(g)) 2(29) where n 0 = d((x,k1),(y,k2)) 2N . In particular this proposition implies that ( x + [−L, L] d ) × [k 0 − K, k 0 + K] is a µ-good box with µ = − ln(g) 4 − 2 ln(ν 2 α(g)) L . Proof. For this proof, we work inside the the space L 2 ((x + [−L, L] d ) × [k 0 − K, k 0 + K]) and we write simplyĤ instead ofĤ \|(x+[−L,L] d )×[k0−K,k0+K] . Iterating (28), we obtain the usual random walk expansion for the resolvent (see e.g. [13]): Givenx,ŷ ∈ Z d × Z, we get (x, (Ĥ − λ) −1ŷ ) = ûi∈∂ in i Λ(vi−1),vi∈∂ ext Λ(vi−1) (x, (Ĥ \|Λ(v0) − λ) −1 ,û 1 )(û 1 , g∆v 1 )(v 1 , (Ĥ \|Λ(v1) − λ) −1û 2 )(û 2 , g∆v 2 ) . . . (v n , (Ĥ − λ) −1ŷ ). (30) In this writing, we need to specify when we stop iterating (28) and how Λ(v i−1 ) is defined. The following choice will guarantee the desired exponential decay: 1. If \|v −ŷ\| ≤ N , we stop iterating (28). ifv is \|(x, (Ĥ − λ) −1ŷ )\| ≤ (x, (Ĥ \|Λ(v0) − λ) −1 ,û 1 )(û 1 , g∆v 1 ) (v 1 , (Ĥ \|Λ(v1) − λ) −1û 2 )(û 2 , g∆v 2 ) . . . (Ĥ − λ) −1 .(31) The factors in each term in this sum are bounded in two different ways, depending on whether they are resonant or not: 1. Ifv i = (x, k) is not a resonant site, then (Ĥ \|Λ − λ) = (v x + kν − λ)δ (x,k) so that (x, k), (Ĥ \|Λ(vi) − λ) −1 (x, k) (x, k), g∆(x , k ) ≤ (x, k), g∆(x , k ) √ g ≤ √ g. (32) 2. Ifv i = (x, k) belongs to K x , then (x, k), (Ĥ \|Λ(vi) − λ) −1 (x , k ) (x , k ), g∆(x , k ) ≤ g d(σ(Ĥ \|Λ Kx ), λ) .(33) The sum in (31) will be small, if for every path joiningx toŷ, the number n of non resonant sites is large enough to dominate the resonant terms (indexed by J), i.e. (2(d + 1) √ g) n j∈J d(σ(Ĥ \|Λj ), λ)(34) We can now understand the reason why we have introduced the security boxes: Assuming that no security boxes intersect one to another, then u i is a resonant site implies that u i+1 is not resonant and its distance to any resonant sites is at least larger than N . From this we can deduce that for any path joiningx toŷ, every resonant term is followed by at least N non resonant ones. Let N ∈ N such that N d−1 ((2N + √ g ν ))(2d + 2) N +1 ( √ g) N −1 2 ν 2 α(g) < 1(35) Then, ifû i is resonant, and assuming that, there is no strongly resonant security box, and no intersecting security boxes, we find that the following product of N + 1 consecutive factors can be bounded as (v i , (Ĥ \|Λ(vi) −λ) −1û i+1 )(û i+1 , g∆v i+1 ) . . . (v i+N , (Ĥ \|Λ(v i+N ) −λ) −1û i+N +1 )(û i+N +1 , g∆v i+N +1 ) ≤ ( √ g) N d(σ(Ĥ \|Λ Kx ), λ) ≤ ( √ g) N +1 2 N d−1 ((2N + √ g ν ))(2(d + 1)) N . Hence, for a path connectingx toŷ in l = k(N + 1) + s steps (s < N + 1), we obtain (x, (Ĥ \|Λ(vi) − λ) −1û 1 )(û 1 , g∆v 1 ) . . . (v i+l−1 , (Ĥ \|Λ(v l−1 ) − λ) −1û l )(û l , g∆v l ) ≤ ( √ g) N +1 2 N d−1 ((2N + √ g ν ))(2(d + 1)) N k ( √ g) s−1 ν 2 α(g) . We can now conclude the proof. Indeed, any path connectingx toŷ contains at least (d((x, k 1 ), (y, k 2 )) − N )/2 steps. Denoting by A l the set of paths connectingx toŷ in l steps, we find \|(x, (Ĥ − λ) −1ŷ )\| ≤ ∞ l=(d((x,k1),(y,k2))−N )/2 \|A l \| ( √ g) N +1 2 N d−1 ((2N + √ g ν ))(2(d + 1)) N k ( √ g) s−1 ν 2 α(g) 1 ν 2 α(g) ≤ ∞ l=(d((x,k1),(y,k2))−N )/2 √ g l/2 1 (ν 2 α(g)) 2 ≤ ( √ g N 2 ) n0 (1 − √ g)(ν 2 α(g)) 2 Proof of Proposition 3. For anyx ∈ Λ 0 × [k 0 − K; k 0 + K], \|V (x) − λ\| ≥ √ g. One can now do the random walk development as previously with no resonant term. Proposition 7. The probability of the event "there is no strongly resonant security box, and no intersecting security boxes" is smaller than 1/L 2d when g goes to 0 assuming N = O( ln(ν) ln(g) ), L = m 1 N , with m 1 a fixed large integer and \| ln(ν)\| ≤ g − 1 8d+4p . Proof. To deal with the strongly resonant boxes, we use the Wegner type estimate (18) with = ν 2 α(g): P(Λ Kx is strongly resonant ) ≤ M/ν k0=−M/ν P(Λ Kx is strongly resonant and K x = Z ∩ [k 0 − 1/(ν √ g), k 0 + 1/(ν √ g)]) ≤ M/ν k0=−M/ν P(Λ Z∩[k0−1/( √ gν),k0+1/( √ gν)] is strongly resonant) ≤ 2M ν 2πν 2 α(g)(N d ( 2 √ g ν + 2N )) ρ ∞ ≤ 4M (N d ( 2 √ g ν + 2N ))να(g)(36) We deal now with the probability of non intersecting security boxes: For any x, y ∈ [−L, L] d , Λ Kx ∩ Λ Ky = ∅. This will be satisfied if there is no \|k\| ≤ 2N such that \|v x − v y + kν\| ≤ √ g. If ν ≤ √ g, the probability P of intersecting security boxes is bounded by: P ≤ (2L) d (2L) d − 1 2 P \|v x − v y \| < 2(N ν + √ g) ≤ 2(2L) d (2L) d − 1 (N ν + √ g) ρ ∞(37) and in any case (when ν > √ g) by P ≤ 2(2L) d (2L) d − 1 (N + 1) √ g ρ ∞(38) From (37) (or (38)) and Proposition 6 we conclude the proof of our theorem. We need:      4M (N d ( 2 √ g ν + 2N ))να(g) ≤ 1 2L 2p 2(2L) d (2L) d − 1 (N ν + √ g) ρ ∞ ≤ 1 2L 2p −( ln(g) 4 − 2 ln(ν 2 α(g)) L ) > µ(39) or (when ν > √ g)      4M (N d ( 2 √ g ν + 2N ))να(g) ≤ 1 2L 2p 2(2L) d (2L) d − 1 (N + 1) √ g\|\|ρ\|\| ∞ ≤ 1 2L 2p −( ln(g) 4 − 2 ln(ν 2 α(g)) L ) > µ(40) and (35). We set α(g) = 1 in case of ν < √ g and α(g) = g in case of ν > √ g. 1. N = n 1 ln(ν) √ g with n 1 > 7. 2. L = m 1 N with m 1 a large enough integer. We have then −( ln(g) 4 − ln(ν 2 α(g)) L ) > \| ln(g)\|( 1 4 − 1 m1 ). Then assume \| ln(ν)\| ≤ g − 1 8d+4p . So we get L 4d+2p √ g = O(g 1/4 ). Finally the three conditions of (39) are satisfied in the limit g → 0 and this is the end of the proof of 4. L 2 driving (C2) We now consider the case of an L 2 driving. In this set up, we will work on infinite columns C L (x) = (x + [−L, L] d ) × Z, so that distinct column are independent with respect to the disorder. Instead, one should be careful in the random walk expansion since infinite sums appear. That this is not a problem comes from the decay of the Green function at the large frequencies: 6.1 Decay of the Green function along the frequency axes Proposition 8. Letφ be an eigenfunction ofĤ with eigenvalueλ. Then x,k \|\|kν − λ\|φ(x, k)\| 2 ≤ (g + M ) 2 .(41) In particular \|φ(x, k)\| ≤ 1 + M + g 1 + \|kν −λ\|(42) for any x. Proof. We use the time representation ofφ. Recall that φ(t) = e iλt ψ(t) with ψ solution of (4). Since the evolution is unitary, for all t ∈ [0, T ], φ(t) = ψ(t) = ψ(0) = φ(0) . So x,k \|\|kν − λ\|φ(x, k)\| 2 = 1 T T 0 (i∂ t −λ)φ(t) 2 dt = 1 T T 0 (−g∆(t) + V )φ(t) 2 dt ≤ 1 T T 0 (g∆(t) + V ) 2 dt ≤ g 2 + 2M 1 T T 0 (g∆(t) dt + M 2 ≤ (g + M ) 2 , and we deduce that (1 + (\|kν − λ\|))φ(x, k) is square integrable. From this we can deduce an estimate for the resolvent : Proposition 9. There exist a constant C depending only on ν so that we have \|(ẑ, (Ĥ \|C L (x) − λ) −1ŷ )\| ≤ (2L + 1) d/2 (2 + M )P (x) 1 + \|k z − k y \| sup i 1 \|λ −λ i \| + C for anyẑ = (z, k z ),ŷ = (y, k y ) ∈ C L (x), whereλ i are the eigenvalue ofĤ \|C L (x) . Proof. We decomposeĤ \|C L (x) into its eigenvectors and we apply Cauchy Schwartz. The eigenvalues ofĤ \|C L (x) are all of the formλ i + kν, where we can assume thatλ i are such that \|λ i + kν − λ\| ≥ ν/2 if k = 0. Then (ẑ, (Ĥ \|C L (x) − λ) −1ŷ ) = \|Λ\| i=1 k∈Z 1 λ i + νk − λ φλ i+νk (ẑ)φλ i+νk (ŷ) ≤ \|Λ\| i=1 k∈Z (1 + \|λ i + ν(k − k z )\|) 2 \|φλ i+νk (ẑ)\| 2 1/2 . \|Λ\| i=1 k∈Z 1 \|λ i + νk − λ\| 2 1 (1 + \|λ i + ν(k − k z )\|) 2 \|φλ i+νk (ŷ)\| 2 1/2 = \|Λ\| i=1 k∈Z (1 + \|λ i + ν(k − k z )\|) 2 \|φλ i (z, k z − k)\| 2 1/2 . \|Λ\| i=1 k∈Z 1 \|λ i + νk − λ\| 2 1 (1 + \|λ i + ν(k − k z )\|) 2 \|φλ i+νk (ŷ)\| 2 1/2 We use now (41) to control the first factor, and (42) to get an estimate on \|φλ i+νk (ŷ)\| in the second one: (ẑ, (Ĥ \|C L (x) − λ) −1ŷ ) ≤ (1 + M + g) 2 \|Λ\| i=1 k∈Z 1 \|λ i + νk − λ\| 2 1 (1 + \|λ i + ν(k − k z )\|) 2 1 (1 + \|λ i + ν(k − k y )\|) 2 1/2 = (1 + M + g) 2 \|Λ\| i=1 1 \|λ i − λ\| 2 1 (1 + \|λ i + ν(k − k z )\|) 2 1 (1 + \|λ i + νk y \|) 2 + \|Λ\| i=1 k∈Z * 1 \|λ i + νk − λ\| 2 1 (1 + \|λ i + ν(k − k z )\|) 2 1 (1 + \|λ i + ν(k − k y )\|) 2 1/2 ≤ \|Λ\| 1/2 (1 + M + g) 2 (sup i 1 \|λ −λ i \| + C)P (ẑ) 1 (1 + \|k z − k y \|) , where the last inequality comes from the estimate of the integral dk 1 1 + k 2 1 1 + (k − k z ) 2 1 1 + (k − k y ) 2 ∼ 1 (1 + \|k z \|) 2 1 (1 + \|k z − k y \|) 2 . Definition 5. We say that C L (x) is not strongly resonent if inf λi∈σ(Ĥ \|C L (x) ) {\|λ i − λ\|} > e − √ L .(43) In particular, if C L (x) is not strongly resonant, we have \|(ẑ, (Ĥ \|C L (x) − λ) −1ŷ )\| ≤ CL d/2 P (ẑ) 1 + \|k z − k y \| e √ L where C is a constant. The decay function If Anderson localization is most of the time studied over Z d , the problem could be raised on any set of point X. It is indeed easy to define a random potential V (x), x ∈ X and a "Laplacian" ∆(x 1 , x 2 ) without assuming a particular geometry of the system. But to recover the decay, one should then first define a decay function, and ∆ is the only object that we can use to construct such a decay function. We first give a general definition. Definition 6. Let G : X × X → R + , for anyx,ŷ ∈ X, d G (x,ŷ) = − ln C(x→ŷ) i \|G(ẑ i ,ẑ i+1 )\|(44) ifx =ŷ and 0 otherwise, where C(x →ŷ) is the set of all pathsx =ẑ 0 ,ẑ 1 ,ẑ 2 , ...,ẑ k =ŷ fromx toŷ. Proposition 10. If for any z ∈ X, z1 \|G(z, z 1 )\| < 1/2, then d G is positive and satisfies the triangle inequality. Proof. We first check that d G is positive. Letx,ŷ C(x→ŷ) i \|G(ẑ i ,ẑ i+1 )\| ≤ ŷ C(x→ŷ ) i \|G(ẑ i ,ẑ i+1 )\| ≤ n>0 n i=0 max zi ẑi+1∈X \|G(ẑ i ,ẑ i+1 )\| = n>0 max x ŷ∈X \|G(x,ŷ)\| n = maxx ŷ∈X \|G(x,ŷ)\| 1 − maxx ŷ∈X \|G(x,ŷ)\| < 1. We now check the triangle inequality. Letẑ be another point in X. d G (x,ŷ) + d G (ŷ,ẑ) = − ln C(x→ŷ) i \|G(ẑ i ,ẑ i+1 )\| − ln C(ŷ→ẑ) j \|G(ẑ j ,ẑ j+1 )\| = − ln C(x→ŷ) C(ŷ→ẑ) i \|G(ẑ i ,ẑ i+1 )\| j \|G(ẑ j ,ẑ j+1 )\| ≥ − ln C(x→ẑ) i \|G(ẑ i ,ẑ i+1 )\| = d G (x,ẑ). initialisation of the multiscale Proof of Proposition 5. Proposition 5 follows from Propositions 12 and 13 below. Definition 7. We will use d G with X = Z d × Z and G(x,ŷ) = g\|∆(x,ŷ)P (ŷ)\|(45) Remark that we also have ∆(ẑ, .)P (.) ∈ L 1 because ∆(ẑ, .) ∈ L 2 and P (.) ∈ L 2 . We will write G 1 max = sup x y G(x, y). This quantity goes to zero as g → 0. The decay function is related to usual distance on Z d through the following proposition: Proposition 11. For anyx = (x, k x ), z:\|x−z\|=L k e −d G ((x,kx),(z,k)) ≤ e L ln(( G 1 max )−ln(1− G 1 max ) (46) in particularẑ = (z, k z ), \|x − z\| > L. d G (x,ẑ) ≥ L(− ln( G 1 max )) + ln(1 − G 1 max )(47) Proof. Because no path of length smaller than L connectx with the boundary of {(z, k) : \|x−z\| > L}, C(x→ẑ) i \|G(ẑ iẑi+1 )\| ≤ n>L G n 1 max ≤ G L 1 max 1 − G 1 max . (48) So d G (x,ẑ) ≥ −L ln(( G 1 max ) + ln(1 − G 1 max ). Proposition 12. If there is no resonant site at all in C L (x), and ifĤ \|C L (x) has no eigenvaluē λ i with \|λ i − λ\| ≤ √ g, then C L (x) is a (µ ,d G ) good column Proof. We use here again the resolvent formula: (x, (Ĥ \|C L (x) − λ) −1ŷ ) = ẑ g∆(x,ẑ) V (x) − λ (ẑ, (Ĥ \|C L (x) − λ) −1ŷ ). Applying it several times yields the usual random walk expansion: (x, (Ĥ \|C L (x) − λ) −1ŷ ) = ẑ,ẑ1,ẑ2,...,ẑn g∆(x,ẑ 1 ) V (x) − λ g∆(ẑ 1 ,ẑ 2 ) V (ẑ 1 ) − λ . . . g∆(ẑ n−1 ,ẑ n ) V (ẑ n−1 ) − λ (ẑ n , (Ĥ \|C L (x) − λ) −1ŷ ) Because there is no resonant site, 1 V (ẑ)−λ ≤ P (ẑ) for anyẑ ∈ C L (x). So \|(x, (Ĥ \|C L (x) − λ) −1ŷ )\| = P (x) ẑ,ẑ1,ẑ2,...,ẑn \|g∆(x,ẑ 1 )P (ẑ 1 )g∆(ẑ 1 ,ẑ 2 ) . . . P (ẑ n−1 )g∆(ẑ n ,ẑ n−1 )(ẑ n , (Ĥ \|C L (x) − λ) −1ŷ )\| ≤ CP (x) ẑ,ẑ1,ẑ2,...,ẑn \|g∆(x,ẑ 1 )P (ẑ 1 )g∆(ẑ 1 ,ẑ 2 ) . . . P (ẑ n−1 )g∆(ẑ n ,ẑ n−1 )P (ẑ n )\| L d/2 √ g ≤ CL d/2 P (x) √ g C(x→y) i g\|∆(ẑ i ,ẑ i+1 )\|P (ẑ i+1 ) where the first inequality is obtained through Proposition 9 and the hypothesis on the eigenvalues λ i . So one has \|(x, (Ĥ \|C L (x) − λ) −1ŷ )\| ≤ CL d/2 P (x) √ g e −d G (x,ŷ) Proposition 13. The probability of the event "there is no resonant site at all in C L (x), and H \|C L (x) has no eigenvalue λ i with \|λ i − λ\| ≤ √ g" goes to 0 with g → 0 . Proof. First, P(there is no resonant site in C L (x)) ≤ \|\|ρ\|\| ∞ 2M ν (2L + 1) d 2g.(49) Next, thanks to Wegner estimate, P(C L (x) is not strongly resonant ) ≤ \|\|ρ\|\| ∞ 2M ν (2L + 1) d 2g.(50) This gives the proposition for g → 0. Technical results for the iteration of the MSA We have proved that for a fixed L, C L (x) is a good column with high probability. MSA induces that the property is valid for all L k with L k+1 = L α k , L 0 = L, but some adaptations with wrt. [13] are needed, due to the long range hopping along the frequency axis. It turns out that only Theorems 10.14 and 10.20 need to be re-investigated. Here we prove Proposition 15 below that will play the role of Theorem 10.14 in [13] (the equivalent of Theorem 10.20 in [13] can then be obtained without any new idea). Thanks to the estimates on Green function obtained in Section 6.1, we obtain Proposition 14. sup x,y ky ẑ 1 1 + \|k x − k y \| \|∆(ŷ,ẑ)P (ẑ)\| < ∞(51) In particular G(x, .) = ky 1 1+\|kx−ky\| \|∆(ŷ, .)P (.)\| is in L 1 uniformly in x. Proof. We have \|∆(ŷ, .)\| ∈ L 4 , with a norm that can be bounded uniformly inŷ, 1 1+\|.\| ∈ L 1 1 + \|k x − k y \| \|∆(ŷ,ẑ)P (ẑ)\| ≤ sup x,y,ẑ ky 1 1 + \|k x − k y \| \|∆(ŷ,ẑ)\| sup y ẑ \|∆(ŷ,ẑ)\|P (z) ≤ 1 1 + \|.\| L 4 3 \|∆(ŷ, .)\| L 4 \|∆(ŷ, .)\| L 4 P (.) L 4 3 < ∞ Proposition 15. If there is no two distinct small scale columns C L k (y) ⊂ C L k+1 (x) which are not µ-good, and there is no columns C 2L k (y ) ⊂ C L k+1 (x) that are strongly resonant and C L k+1 (x) is not strongly resonant, then C L k+1 (x) is µ good with µ > µ − 3L k L k+1 . Proof. Let d G the decay function used for the small scale good boxes. In the case of C L k is a bad column, we use the resolvent development twice \|(x, (Ĥ \|C L k+1 (x) − λ) −1ŷ )\| ≤ ẑ1∈∂ in C L 2k (x), z2∈∂ ext C L 2k (x) \|(x, (Ĥ \|C 2L k (x) − λ) −1ẑ 1 )g∆(ẑ 1 ,ẑ 2 )(ẑ 2 , (Ĥ \|C L k+1 (x) − λ) −1ŷ )\| ≤ ẑ1∈∂ in C L 2k (x) z2∈∂ ext C L 2k (x) ẑ3∈∂ in C L k (z2) z4∈∂ ext C L k (z2) \|(x, (Ĥ \|C 2L k (x) − λ) −1ẑ 1 )g∆(ẑ 1 ,ẑ 2 ) (ẑ 2 , (Ĥ \|C L k (x) − λ) −1ẑ 3 )g∆(ẑ 3 ,ẑ 4 )(ẑ 4 , (Ĥ \|C L k+1 (x) − λ) −1ŷ )\| ≤ P (x) ẑ1∈∂ in C L 2k (x) z2∈∂ ext C L 2k (x) ẑ3∈∂ in C L k (z2) z4∈∂ ext C L k (z2) e √ L k C(2L k ) d/2 1 + \|kx − kẑ 1 \| \|g∆(ẑ 1 ,ẑ 2 )\| P (ẑ 2 )e −d G (ẑ2,ẑ3) g\|∆(ẑ 3 ,ẑ 4 )(ẑ 4 , (Ĥ \|C L k (x) − λ) −1ŷ )\| So let us define G as follows: G (x,ŷ) = e −d G (x,ŷ) if C L k (x) is a µ good box andŷ ∈ ∂ ext C L k (x), and G (x,ŷ) = ẑ1∈∂ in C L 2k (x) z2∈∂ ext C L 2k (x) z3∈∂ in C L k (z2) e √ L k C(2L k ) d/2 1 + \|kx − kẑ 1 \| \|g∆(ẑ 1 ,ẑ 2 )\|P (ẑ 2 )e −d G (ẑ2,ẑ3) \|g∆(ẑ 3 ,ŷ)\|P (ŷ) if C L (x) is a bad box. Thanks to Proposition 14, there is a constant C independent of L k such that for the second case : G L 1 ≤ C e 2 √ L k e −µL k . We can then recover the usual tools, using that e −µL k dominate the other terms for L k large. In particular because for any path from x to ∂C L (x) there is at least ( L k+1 L k − 3) µ good boxes. So, with the same argument as in the proof of Proposition 11, ŷ∈∂ in C L (x) e −d G (x,ŷ) ≤ e −µ(L k+1 −3L k )−ln(1− G 1 max ) Proof of the corollaries As said, Corollaries 1 and 2 do not follow logically from Theorem 1; instead one should go trough the MSA once again and refine several estimates. This work has been carried over in [11], and one indicates here only the main steps as well as the few needed extra adaptations. Let us start with Corollary 1. Proposition 16. there exist p > 0 (and one can take p → ∞ as → 0) such that: E(sup t>0 x∈Z d k \|x\| p \|φ(x, k, t)\| 2 ) < ∞(52) for anyφ(x, k, 0) defined on a bounded support. Proof. Thanks to the MSA carried over in this paper, one can check that the results of [11] holds; in particular the assumptions of Theorem 3.1 in [11] are satisfied. In order to recover φ fromφ we use the following proposition. Remind that, thanks to (6) Proof. Let's separate ψ(0) = 1 x=x0 ψ(0) + 1 x =x0 ψ(0). Because the A(t) is hermitian, there exists U (t) unitary such that ψ(t) = U (t)ψ(0) = U (t)(1 x=x0 ψ(0)) + U (t)(1 x =x0 ψ(0)) (55) Calling ψ 1 = U (t)(1 x=x0 ψ(0)), ψ 2 = U (t)(1 x =x0 ψ(0)) we have (ψ 1 , ψ 2 ) = 0 and ψ 1 2 + ψ 2 2 = 1. Because 1 \|z−x0\|<R is a projector, ψ 1 + ψ 2 , 1 \|z−x0\|<R (ψ 1 + ψ 2 ) = ψ 1 , 1 \|z−x0\|<R ψ 1 + ψ 2 , 1 \|z−x0\|<R ψ 2 + 2 ψ 1 , 1 \|z−x0\|<R ψ 2 ≥ ψ 1 , 1 \|z−x0\|<R ψ 1 − 2\| ψ 2 , 1 \|z−x0\|≥R ψ 1 \| ≥ ψ 1 2 − 1 \|z−x0\|≥R ψ 1 2 − 2\|(ψ 2 , 1 \|z−x0\|≥R ψ 1 )\| ≥ ψ 1 2 − 1 \|z−x0\|≥R ψ 1 2 − 2 1 \|z−x0\|≥R ψ 1 We now proof that the locality of A(t) implies that 1 \|z−x0\|≥R ψ 1 2 is small. i d dt ψ 1 (y, t) = A(t)ψ 1 (y, t) = \|y −y\|≤1 A y,y (t)ψ 1 (y , t). Hence d dt \|ψ 1 (y, t)\| ≤ \|y −y\|≤1 \|A y,y (t)\|\|ψ 1 (y , t)\| ≤ A(t) \|y −y\|≤1 \|ψ 1 (y , t)\| Let now a(y, t) solution of the system d dt a(y, t) = A(t) \|y −y\|≤1 a(y , t) a(y, 0) = \|ψ 1 (x 0 , 0)\|1 y=x0 (56) We have then for any (y, t) \|ψ 1 (y, t)\| ≤ a(y, t) We can evaluate a with the following remark : Let X(t) be the classical markovian random walk on Z of variable rate A(t) and starting at point x 0 . Its generator is d dt P x0 (X(t) = y) = A(t) \|y −y\| (P x0 (X(t) = y ) − P x0 (X(t) = y)) (58) and then we have e −(2d+1) t 0 A(u) du a(y, t) = a(x 0 , 0)P x0 (X(t) = y) We can then deduce y≥R a(y, t) ≤ a(x 0 , 0)e (2d+1) t 0 A(u) du P(N 2d t 0 A(u) du ≥ R) where N 2d t 0 A(u) du is the Poisson process of parameter 2d t 0 A(u) du. So for any t ≤ T y≥R a(y, t) ≤ a(x 0 , 0)e C k≥R (2dC) k k! We can now conclude \|z−x0\|<R \|ψ(z, t)\| 2 = ψ 1 + ψ 2 , 1 \|z−x0\|<R (ψ 1 + ψ 2 ) ≥ ψ 1 2 − 1 \|z−x0\|≥R ψ 1 2 − 1 \|z−x0\|≥R ψ 1 ≥ \|ψ(x 0 , 0)\| 2 − \|ψ(x 0 , 0)\| 2 (e C k≥R (2dC) k k! ) 2 − \|ψ(x 0 , 0)\|(e C k≥R (2dC) k k! ) The above proposition and the dynamical localisation ofφ enable us to conclude: Proposition 18. For any > 0, there exist some constants C , D such that C x∈Z d k \|x\| p \|φ(x, k, t)\| 2 + D ≥ x0∈Z d \|x 0 \| p− \|φ(x 0 , t)\| 2(62) Proof. Let > 0. Let now x 0 → R(x 0 ) be such that x0∈Z d \|x 0 \| p k≥R(x0) (2dC) k k! < ∞(63) and such that, for all x 0 ∈ Z d , e C k≥R(x0) (2dC) k k! < 1 2(64) moreover that \|x − x 0 \| < R(x 0 ) then \|x − x 0 \| < (1 + )R(x), and such there is constant C such that \|x−x0\|≤(1+ )R(x) \|x 0 \| p− ≤ C \|x\| p(65) for \|x 0 \| > 1. For example we could have chosen R(x) = ln(x) 2 for large x. ≥ 1 C E(sup t>0 x0∈Z d \|x 0 \| p− 1 T t+T t \|x−x0\|≤R(x0) \|φ(x, u)\| 2 du) ≥ 1 C E(sup t>0 x0∈Z d \|x 0 \| p− 1 T t+T t \|ψ(x 0 , t)\| 2 − \|ψ(x 0 , t)\| 2 (e C k≥R (2dC) k k! ) 2 − \|ψ(x 0 , t)\|(e C k≥R (2dC) k k! )du ≥ 1 2C E sup t>0 x0∈Z d \|x 0 \| p− \|ψ(x 0 , t)\| 2 − e C 1 C x0∈Z d \|x 0 \| p k≥R(x0) (2dC) k k! So E(sup t>0 x0∈Z d \|x 0 \| p− \|ψ(x 0 , t)\| 2 ) < ∞(66) Let us now come to Corollary 2: Proof of Corollary 2. Since ψ λ (·, 0) = k∈Zψ (·, k), Again, thanks to the MSA shown in this paper, and the deterministic exponential decay along the frequency axis of the eigenfunctions under Assumption (C1), we can reuse the methods leading to Theorem 3.1 in [11], to get our result. The proof of this proposition will be carried over in Section 5. For the usual Anderson model, Theorem 1 would follow from (see Theorem 8.3 in [13]): 1. MSA initialisation (Theorem 11.1 in [13]), 2. Wegner estimate (Theorem 5.23 in [13]),3. Independence of these two properties for two distinct boxes (obvious in the usual model).As already said, the only peculiarity of our model under assumption (C1) is the special form of the potential. In our case, it will thus be enough to then obtained as Theorem 8.3 in[13]. Figure 1 Proposition 4 . 14Proposition 4 is deduced from Proposition 6 and Proposition 7 below. Proposition 6 . 6Let L ∈ N. If no security boxes intersect, if no security box is strongly resonant, and if (x + [−L, L] d ) × Z is not strongly resonant, then for any y ∈ not a resonant site, we choose Λ(v) = {v}. There are then at most 6d + 2 points in ∂ ext Λ(v). 3. ifv is a resonance site, we choose Λ(v) = Λ Kx . There are at most CdN d−1 (N + √ g/ν) points in ∂ ext Λ(v) for some numerical constant C > 0. See Figure 5.1 for a typical path fromx toŷ. From (30), we obtain Figure 2 : 2A typical path fromx toŷ. In red the resonant sites and in yellow the security boxes with N = 2. , we have H(t) L 1 [0;T ] ≤ √ T H(t) L 2 [0;T ] . Proposition 17. Let ψ(t) ∈ L 2 (Z d ) satisfying ψ(t) L 2 = 1 for all t ∈ R be a solution of i∂ t ψ(t) = A(t)ψ(t)(53)where for any t A(t) is hermitian,C = A(.) L 1 ([0,T ]) < ∞ and (x, A(t)y) = 0 if \|x − y\| > 1.For any t ∈ [0, T ] and any x 0 ∈ Z d , we have 0 \| p− \|φ(x, u)\| 2 du) η : R → R, s → η(s) = 1 [0,ν[ (s)s. 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[ "A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS", "A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS" ]	[ "Rupert M ", "Callum " ]	[]	[]	Suppose that ∆, ∆ ′ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of ∆ generated by a nonempty open set of chambers, and whose codomain is ∆ ′ , is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups.	10.1515/advgeom-2012-0029	[ "https://arxiv.org/pdf/1009.4249v12.pdf" ]	119,569,071	1009.4249	fcc58948d39dc57a894678d4d526cf742136eac3	A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS 23 Jul 2012 Rupert M Callum A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS 23 Jul 2012topological geometrylocal homomorphismtopological buildingBorel-Tits the- orem Mathematicsl Subject Classification 2000: 51H10 Suppose that ∆, ∆ ′ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of ∆ generated by a nonempty open set of chambers, and whose codomain is ∆ ′ , is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups. Introduction Throughout the history of Lie theory there has been a notion of "local isomorphism". Indeed, the original notion of Lie groups considered by Sophus Lie in [9] was an essentially local one. Suppose that G is an algebraic group defined over a Hausdorff topological field k. One may consider the notion of a local k-isogeny from the group G to a group G ′ defined over k of the same dimension. This is a mapping defined on a nonempty open neighbourhood of the identity of G(k) in the strong k-topology, (see Definition 1.10), which "locally" acts as a k-isogeny, in the sense that it is a local k-homomorphism and its range is Zariski dense. As far as I know this notion has not been investigated systematically. In this paper we shall produce a local version of the Borel-Tits theorem which hints at the possibility that it may be fruitful to investigate this notion. The Borel-Tits result was used by Mostow [12] to prove his "strong rigidity theorem", and also underpins the work of Margulis on "superrigidity" [10]. More recently, using Tanaka's theory of prolongations of maps of filtered structures on manifolds [16], Yamaguchi [21] showed that smooth local maps preserving the fibrations in a Tits building in the real case must arise form the the action of the associated semisimple group. This suggests that there might be a "local rigidity theorem" for buildings which does not require any assumptions of smoothness, or even continuity, and in this paper we give a local-to-global result for the buildings of semisimple algebraic groups which establishes this. We now proceed to the description of the results. Definition 1.4. Suppose that ∆ is a spherical building of type (W, S), considered as a simplicial complex. Suppose that for each s ∈ S we are given a Hausdorff topology on the set of vertices of type s. We thereby obtain, for each J ⊆ S, a Hausdorff topology on the simplices of type J, by viewing the simplices of type J as J-tuples of vertices and taking the subspace topology arising from the product topology. Given any two subsets J, K ⊆ S, consider the set D J,K := {(X, Y ) \| X has type J, Y has type K, and there exist opposite chambers containing X and Y } and the mapping D J,K → Cham ∆, (X, Y ) → proj X Y . Suppose that all of these mappings are continuous. Then the building is said to be a topological spherical building. We will prove in Section 2 that if G is a semisimple algebraic group over a Hausdorff topological field k, then the building ∆ := ∆(G, k) is always a topological spherical building. Now, suppose that ∆ is a topological spherical building arising from a semisimple algebraic group in this way, with Coxeter system (W, S), and suppose that the Coxeter diagram of ∆ has no isolated nodes, so that the building is strictly Moufang in the sense of [1]. We will now define a certain base for the topology on the set of chambers. Definition 1.5. Suppose that C, C ′ are two opposite chambers in ∆. For each s ∈ S let U s be the root group corresponding to the root of the apartment containing C and C ′ which does not contain C but which is attached to C along a boundary panel of cotype {s}. There is a topology on U s arising from the topology on the set of all chambers which are attached to the root along this panel. For each s ∈ S, let N s be an open neighbourhood of the identity in U s with respect to this topology. Let w 0 be the longest word in the Coxeter group (W, S), with a fixed reduced decomposition s 1 s 2 . . . s n . Let U be the set of all chambers which arise from C under the action of the set N s 1 N s 2 . . . N sn . Then U is said to be a basic open set, or a basic open set with respect to the pair of opposite chambers (C, C ′ ). The empty set is also said to be a basic open set. We will prove in Section 2 that every basic open set is indeed an open set and that in this way a base for the topology on the set of chambers is defined. Definition 1.6. Suppose that U is an open set of chambers in ∆. Suppose that, given any two chambers C, D ∈ U, there exists a sequence (U 1 , U 2 , . . . U r ) of basic open subsets of U, such that for each integer i such that 1 ≤ i < r we have U i ∩ U i+1 = ∅, and C ∈ U 1 , D ∈ U r . Then U is said to be quasi-connected. In the case where the field k from which the building ∆ arises is equal to R or C, it can be shown that every open connected set is quasi-connected. We can now state our result. Theorem 1.7 (The Main Theorem). Suppose that G (resp. G ′ ) is a semisimple algebraic group defined over a field k (resp. k ′ ). Suppose that the k-rank of G is equal to the k ′ -rank of G ′ . Suppose that k is a non-discrete Hausdorff topological field. Let ∆ := ∆(G, k), ∆ ′ := ∆(G ′ , k ′ ). Suppose that the Coxeter diagrams of ∆, ∆ ′ have no isolated nodes. Suppose that U is a nonempty open quasi-connected subset of Cham ∆ and ∆(U) is the subcomplex of ∆ consisting of all faces of members of U. Then an injective chamber map ∆(U) → ∆ ′ has a unique extension to an injective chamber map ∆ → ∆ ′ . This theorem generalises results given in [11]. 1 Some of the geometries considered in [11] are unitals rather than buildings. It would appear that the result generalises to this context as well and we hope to explore that further in future work. In the case of the general linear group, a version of the theorem requiring smoothness of the mapping has appeared in [2], and a version requiring continuity in the real and complex case has appeared in [15]. A version which does not require continuity appears in [19]. Related to Theorem 1.7, a result about topological buildings, is a group-theoretic result; a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups, which appears in [4]. In that paper Borel and Tits prove the following result. First we give the definition of a special isogeny. 1 It should be noted that Chapter 6 of [11] used a different definition of "quasi-connected", and the theorems of Chapter 6 of [11] in which that term appears are actually not true as they stand. For example, if we consider the disjoint union of the interiors of two squares separated by a distance of one unit, and if the slopes of the line segments are allowed to range from -1 to 1, then it is easy to see that a mapping defined on that domain which preserved line segments with slope in that range could be a disjoint union of two distinct projective transformations. But if some requirement were made on the set of slopes that it should match the set of points in appropriate way, which would ensure that the corresponding set of chambers in the building is gallery connected, as is clearly the case with our "quasi-connected" sets defined here, then the theorems would come out true. Definition 1.8. If G, G ′ are two absolutely almost simple algebraic groups defined over a field k and ϕ : G → G ′ is an isogeny, and T is a maximal torus of G and T ′ = ϕ(T ), then the restriction of ϕ to T induces a homomorphism ϕ * : X * (T ′ ) → X * (T ) of the character group of T ′ into that of T , and the isogeny ϕ is said to be special if ϕ * maps the short roots of G ′ onto roots of G. Now we give the statement of the Borel-Tits theorem. Theorem 1.9 (Borel-Tits theorem). Suppose that G (resp. G ′ ) is a connected affine algebraic group over a field k (resp. k ′ ), and that G is absolutely almost simple and G ′ is absolutely simple and adjoint. Denote by G + the subgroup of G(k) generated by all krational points of unipotent radicals of k-parabolic subgroups. Then, given an abstract group homomorphism α : G + → G ′ (k ′ ) whose range is Zariski dense in G ′ , there exists a field homomorphism ϕ : k → k ′ and a special isogeny β : ϕ G → G ′ such that α = β • ϕ • \| G + . Now we shall describe the local version of this theorem. First of all we need to give a definition of the strong k-topology on the set of k-rational points of a variety over a Hausdorff topological field k. This is discussed in the algebraically closed case in Chapter I §10 of [13], where the main properties are left as an exercise for the reader. It is also discussed in the case k = C in Appendix III of [20]. Definition 1.10. Suppose that V is an affine variety over a Hausdorff topological field k, and suppose that we are given a k-embedding of V in an affine space A n (k), where k is an algebraic closure of k, where the affine space is given the obvious k-structure. Then the topology on k induces a topology on the set of k-rational points A n (k) of the affine space A n (k), and thereby induces a topology on V (k), the set of k-rational points of the variety V . This topology on V (k) does not depend on the choice of embedding in an ambient affine space, and is called the strong k-topology on V (k). The definition can be generalised to non-affine varieties in the obvious way. Suppose that G, G ′ and G + are as in the statement of the Borel-Tits theorem. We can give G + a topology arising from the strong k-topology on G(k). We need to define a base for this topology. Note that this definition is similar to Definition 1.5 except in the context of groups rather than buildings. Definition 1.11. Let S be a maximal k-split torus in G. Let Φ k (G, S) be the set of k-roots of G with respect to S and let Ψ = {α ∈ Φ k (G, S) \| 1 2 α / ∈ Φ k (G, S)}. For each α ∈ Ψ let U α be the subgroup of G + consisting of all k-rational points of the unipotent radical of the k-parabolic subgroup of G containing S corresponding to α, and let N α be an open neighbourhood of the identity in U α in the strong k-topology on U α . As observed in [17], given a system of simple roots Π ⊆ Ψ there is a natural one-to-one correspondence between conjugacy classes of k-parabolic subgroups and subsets of Π. Select a system of simple roots Π and identify it with the set of generators S for the Coxeter system (W, S) of the building of G over k. Pick a longest word w 0 ∈ W and let s 1 s 2 . . . s n be a reduced decomposition. For each i such that 1 ≤ i ≤ n, let α i be the element of Π corresponding to s i . We say that N −α 1 N −α 2 . . . N −αn N α 1 N α 2 . . . N αn and any right coset thereof is a basic open subset of G + . We also say that the empty set is a basic open subset with respect to the torus S. We will prove in Section 2 that every basic open subset is indeed open and that a base for the strong k-topology on G + is defined in this way. Theorem 1.12 (The local Borel-Tits theorem). Suppose that G, G ′ , and G + are as in the statement of the Borel-Tits theorem and that k is a non-discrete Hausdorff topological field, and give G(k) the strong k-topology. Then a local abstract group homomorphism α : U ⊆ G + → G ′ (k ′ ), where U is an nonempty basic open neighbourhood of the identity in G + , whose range is Zariski dense in G ′ , extends to a global group homomorphism. A result related to a special case of this is already known for simply connected almost simple complex Lie groups. There is a classical theorem which says that, given a simply connected Hausdorff topological group G and an open symmetric neighbourhood U of the identity, any local homomorphism from this neighbourhood into an abstract group H extends to a global group homomorphism (see Corollary A2.26 of [6]). A special case of the local Borel-Tits result follows from this. In Section 2 we review the basic definitions of concepts from the theory of buildings and topological spherical buildings, and prove that the building of a semisimple algebraic group G over a Hausdorff topological field k is always a topological spherical building. We also present once again the definition of the notion of a "quasi-connected" set, and the notion of a basic open subset in the context of both buildings and groups, and prove some of the basic properties of these notions. In Section 3 we present the proof of the local-to-global result for buildings and the local version of the Borel-Tits theorem. In Section 4 we examine a corollary dealing with nilpotent Lie groups which is a version of a result due to Yamaguchi not requiring the hypothesis of continuity. This paper was written partly while I was employed as Research Intensive Academic at the Australian Catholic University, partly while I was an Adjunct Lecturer at the University of New South Wales, and partly while I held a post-doctoral position at the University of Münster. I am grateful to these institutions for their support. I would like to thank Bill Franzsen and Norman Wildberger for proofreading preliminary drafts, and Michael Cowling for giving helpful suggestions with the Introduction. I am particularly grateful to Linus Kramer for taking an interest in my work and providing me with much helpful guidance about the theory of topological buildings. Basic Definitions In this section we review the definitions of the basic concepts of the theory of buildings. The main references are [18] and [1]. For convenience we also review the definition of topological spherical building given in [8], as we did in Section 1. We will give a proof that the building of a semisimple algebraic group G over a Hausdorff topological field k is always a topological spherical building. We also repeat the definition of the term "quasi-connected" and prove some of its basic properties. Definition 2.1. A complex is a partially ordered set (P, ≤) such that (1) for all v ∈ P , the set {w ∈ P \| w ≤ v} is order isomorphic to (P(S), ⊆) for some set S, P(S) being the powerset of S. In this paper the set S will always be finite. Such a partially ordered set is called a simplex. (2) if A, B ∈ P then A and B have a greatest lower bound denoted by A ∩ B. If A ≤ B we say that A is contained in B or is a face of B. A complex has just one minimal element called 0. The elements which are minimal nonzero elements are called vertices and the number of vertices contained in a given element of a complex is called its rank. The rank of the complex is the supremum of the ranks of all the elements. As remarked earlier, in this paper we will only be considering complexes of finite rank. Given an element A of a complex (∆, ≤), the set {B ∈ ∆ \| A ≤ B}, with the order relation induced from ∆, is called St A or the star of A. It is also a complex. Given B ∈ St A, the rank of B in St A is called the codimension of A in B. The greatest lower bound of two elements C, D in a complex (which always exists) is denoted C ∩ D, and the least upper bound (when it exists) is denoted C ∪ D. See Appendix A of [1] for further discussion of the notion of a simplicial complex. Definition 2.2. We say that a complex ∆ is a chamber complex if every element is contained in a maximal element and if, given two maximal elements C, C ′ , there exists a finite sequence C = C 0 , C 1 , . . . C m = C ′ , called a gallery of length m, such that for all integers i such that 1 ≤ i ≤ m, the codimension of C i−1 ∩ C i in either C i−1 or C i is at most one. The maximal elements are called chambers. We write Cham ∆ for the set of chambers of ∆. An element of a chamber complex has the same codimension in any chamber which contains it; this quantity is called the codimension of the element of the complex. A chamber complex is called thick (respectively, thin) if every element of codimension one is contained in at least three (respectively, exactly two) chambers. The diameter of a chamber complex is the supremum of the lengths of all the minimal galleries connecting two chambers of the chamber complex. A flag complex is a complex in which any family of elements any two of which has an upper bound has an upper bound. Definition 2.3. A morphism of chamber complexes ∆, ∆ ′ , also called a chamber map, is a mapping φ : ∆ → ∆ ′ such that the restriction of φ to the simplex of all faces of any given element A ∈ ∆ is an isomorphism from the ordered set of all faces of A to the ordered set of all faces of φ(A), and φ maps chambers onto chambers. A subcomplex of a chamber complex is a chamber subcomplex if the inclusion mapping is a morphism of chamber complexes. An endomorphism of a thin chamber complex φ : ∆ → ∆ is said to be a folding if it is idempotent and every chamber in φ(∆) is the image of exactly two chambers by φ. The range of a folding is called a root. A complex Σ is said to be a Coxeter complex if it is a thin chamber complex, and if for every pair (C, C ′ ) of adjacent chambers, there exists a folding of Σ which maps C ′ onto C. There is a natural one-to-one correspondence between Coxeter complexes and Coxeter groups, whose definition we discuss below. Suppose that I is a finite set and that (m ij ) is a matrix indexed by I × I with entries in N ∪{∞}, such that m ij ≥ 2 if i = j, and m ii = 1 for all i. The corresponding Coxeter system is (W, S), where W is a group with generating set S = {s i \| i ∈ I} and relations (s i s j ) m ij = 1 for all i, j. For J ⊆ I let W J be the subgroup of W generated by the elements s j for j ∈ J. The mapping J → W J is an isomorphism of posetes. The left cosets of the groups W J for all J ⊆ I with the reverse of inclusion as the ordering form the Coxeter complex Σ(W, S) corresponding to the Coxeter system (W, S). All Coxeter complexes up to isomorphism arise in this way ( [1], Chapter 3). Definition 2.4. If ∆ is a chamber complex and A is a family of chamber subcomplexes of ∆ called apartments then we say that (∆, A) is a building if (1) Any simplex of codimension one meets exactly three chambers (that is, the building is thick); (2) The elements of A are Coxeter complexes; (3) Any two elements of ∆ belong to an apartment; (4) If two apartments Σ and Σ ′ contain two elements A, A ′ ∈ ∆, there exists an isomorphism of Σ onto Σ ′ which leaves invariant A, A ′ and all their faces. It is clear that all the apartments of a building are isomorphic. The isomorphism class of the apartments is called the Coxeter complex of the building. The rank of the Coxeter complex is also called the rank of the building. If the Coxeter complex is isomorphic to the Coxeter system (W, S), then we say that the building has Coxeter system (W, S) or that it is a building of type (W, S). Given that we are working with simplicial complexes of finite rank, it can be shown that a building is always a flag complex ([1], Exercise 4.50). If the Coxeter complex is finite, then the geometric realisation of the Coxeter complex is homeomorphic to a sphere, ([1], Propostion 1.108), and the building is called spherical. It should be noted that the star of any element of a building is again a building. ( [1], Proposition 4.9). If a complex ∆ admits a set of apartments A which makes it into a building then the union of all such sets also makes the complex ∆ into a building; hence the complex ∆ has at most one "maximal building structure" ([1], Section 4.5). If the diameter of ∆ is finite, then the set of apartments A is unique when it exists. For these reasons we sometimes by abuse of notation speak of "the building ∆". Definition 2.5. Suppose that G is a semisimple algebraic group defined over a field k. Then ∆(G, k) is defined to be the set of all k-parabolic subgroups of G with the reverse of inclusion as the order relation. It can be shown that this is a simplicial complex. For each maximal k-split torus of G we define the apartment corresponding to this torus to be the set of all k-parabolic subgroups containing this torus. The complex ∆(G, k) with this collection of apartments is a spherical building [18]. This is proved in detail in § §20-21 of [3]. Given a building ∆ and a chamber C ∈ ∆ there exists a unique retraction λ C of ∆ onto the simplex of all faces of C. Two elements of the building are said to be of the same type if their image by λ C is the same; this does not depend on the choice of C. If we take the quotient of the building by the equivalence relation "A and A ′ have the same type" then we obtain the typical simplex of the building typ ∆, and there is a canonical mapping typ : ∆ → typ ∆. If the building has Coxeter system (W, S) then there is a canonical bijection typ ∆ → P(S). Given a Coxeter complex Σ = Σ(W, S), where S is indexed by I, one may define a Winvariant double-coset-valued distance function on Σ × Σ as follows: (1) δ : Σ × Σ → {W J \ W/W K , J, K ⊆ I}, δ(uW J , vW K ) = W J u −1 vW K . It can be shown that if ∆ is a building with Coxeter system (W, S), and Σ is an apartment of ∆, then the above function δ on Σ × Σ does not depend on the choice of isomorphism of Σ with Σ(W, S). It can also be shown that if ∆ is a building then there exists a well-defined double-coset-valued distance function (2) δ : ∆ × ∆ → {W J \ W/W K , J, K ⊆ I} whose restriction to any apartment of the building is the function δ given above ([1], Section 4.8). Next, for convenience, we repeat the definition of topological spherical building taken from [8]. Suppose that C, D are chambers in a building ∆ with Coxeter system (W, S), and that δ(C, D) = w, and that w = s i 1 s i 2 . . . s ir is a reduced expression for w in terms of the generating set S. Then there exists a unique minimal gallery C = C 0 , C 1 , . . . C r = D such that δ(C k−1 , C k ) = s i k holds for k = 1, 2, . . . r. Suppose that X is an element of the bulding ∆ and C is a chamber. Then there exists a unique chamber E ∈ St X denoted by proj X C, such that for every chamber D ∈ St X, and for every minimal gallery γ from C to D, the first element of γ contained in St X is E. If Y ∈ ∆ is arbitrary, there exists a unique Z which is contained in some chamber in St X, such that Cham St Z=proj X Cham St Y , and we denote this element Z by proj X Y . Definition 2.6. Suppose that ∆ is a spherical building of type (W, S), considered as a simplicial complex. Suppose that for each s ∈ S we are given a Hausdorff topology on the set of vertices of type s. We thereby obtain, for each J ⊆ S, a Hausdorff topology on the simplices of type J, by viewing the simplices of type J as J-tuples of vertices and taking the subspace topology arising from the product topology. Given any two subsets J, K ⊆ S, consider the set D J,K := {(X, Y ) \| X has type J, Y has type K, there exist opposite chambers containing X and Y } and the mapping D J,K → Cham ∆, (X, Y ) → proj X Y . Suppose that all of these mappings are continuous. Then the building is sad to be a topological spherical building. If k is a Hausdorff topological field and G is an semisimple algebraic group defined over k, then the building ∆(G, k) is a spherical building. Let us denote its Coxeter system by (W, S). For each subset J ⊆ S, the simplicies of type J form a projective variety over k. We can give this variety the strong k-topology as defined in Definition 1.10. It is easy to see that this is compatible with the way of deriving the topology on the set of simplicies of a given type of rank greater than one from the topologies on the sets of vertices of each type. We must now show that this makes ∆(G, k) into a topological spherical building. What we must prove is that the projection maps are continuous. It will be sufficient to show that they are k-morphisms of quasiprojective k-varieties. Definition 2.7. Suppose that a group G acts on a building ∆ of type (W, S) in a way that preserves the Weyl distances of chambers and suppose further that G acts transitively on {(C, D) ∈ Cham ∆× Cham ∆ \| δ(C, D) = w} for each fixed w ∈ W . Then the action of G is said to be Weyl transitive. Lemma 2.8. Suppose that G is a semisimple algebraic group defined over a Hausdorff topological field k. Then the building ∆ := ∆(G, k) is a topological spherical building. Proof. Suppose that we are given a fixed w ∈ W with reduced decomposition s 1 s 2 . . . s n . Given a pair of chambers (C, D) such that δ(C, D) = w, we may consider the projection of D onto the panel of C with type s 1 , which we denote by E. It is sufficient to prove that the mapping defined on {(C, D) \| δ(C, D) = w} sending each pair (C, D) to the chamber E defined in this way is a k-morphism. Given any chamber C, let us denote by G C the subgroup of G which fixes C. Now, given a pair (C, D) such that δ(C, D) = w, the set {(C, D) \| δ(C, D) = w} is the set of k-rational points of a quasiprojective variety which may be identified with (G/(G C ∩ G D )) k (see [7], Section 34.2), because the group G acts Weyl transitively on ∆ (see for example [1], Chapter 6). We have G C ∩ G D ⊆ G E and the mapping we are considering may be identified with the projection morphism G/(G C ∩ G D ) → G/G E . Since the groups G C , G D and G E are k-subgroups this is clearly a k-morphism. In Chapter 5 of [18], Jacques Tits classifies the isomorphisms ∆(G, k) → ∆(G ′ , k ′ ), where k and k ′ are infinite fields and G (resp. G ′ ) is a semisimple algebraic group defined over k (resp. k ′ ), such that the buildings ∆(G, k), ∆(G ′ , k ′ ) have Coxeter diagrams with no isolated nodes. Later we shall show how to modify his argument slightly so as to obtain a classification of the injective chamber maps between two of these buildings of the same rank; the only change necessary is that one allows field homomorphisms which are not necessarily surjective rather than field isomorphisms. Definition 2.9. Suppose that α is a root of an apartment of the building ∆. By the root group U α we mean the group of all type-preserving automorphisms of ∆ which fix every chamber in the root α and also every chamber adjacent to a chamber in α via a panel which is not in the boundary of α. Definition 2.10. Suppose that a thick spherical building ∆ has the property that for each root α in some apartment of ∆, the root group U α acts simply transitively on the set of apartments containing α. Then the building is said to be strictly Moufang (see [1], Chapter 7). We now review the notion of "quasi-connected" from Section 1. First we must define a certain base for the topology on the set of chambers of the building of a semisimple algebraic group over a Hausdorff topological field. Recall that the definition of basic open set can be found in Definition 1.5. We must now show that every basic open set is indeed an open set and that in this way a base for the topology on the set of chambers is defined. Then there exist u 1 ∈ N s 1 , u 2 ∈ N s 2 , . . . u n ∈ N sn such that D = u 1 u 2 . . . u n C. Some of the u's may be equal to the identity. Thus the gallery (C, u 1 C, u 1 u 2 C, . . . u 1 u 2 . . . u n C) is a gallery joining C to D, possibly a stammering gallery. We may find an apartment A containing a non-stammering gallery ( E, v 1 C, v 1 v 2 E, . . . v 1 v 2 . . . v n E) where v i ∈ U s i for all i such that 1 ≤ i ≤ n, and further, if i is an integer such that 0 ≤ i ≤ n, then u 1 u 2 . . . u i C is opposite v 1 v 2 . . . v i E. Then the gallery from C to D may be co-ordinatised by the panels of the apartment A via the procedure described in Section 7.2 of [8], and the co-ordinatisation map is a homeomorhpism. This shows that there is an open neighbourhood of D contained in U, and the second part of the lemma is also now clear. Recall the definition of "quasi-connected" given in Definition 1.6. Next, we review the definition of a basic open subset of G + , where G + is as in the statement of the Borel-Tits theorem. Definition 2.12. Let S be a maximal k-split torus in G. Let Φ k (G, S) be the set of k-roots of G with respect to S and let Ψ = {α ∈ Φ k (G, S) \| 1 2 α / ∈ Φ k (G, S)}. For each α ∈ Ψ let U α be the subgroup of G + consisting of all k-rational points of the unipotent radical of the k-parabolic subgroup of G containing S corresponding to α, and let N α be an open neighbourhood of the identity in U α in the strong k-topology on U α . As observed in [17], given a system of simple roots Π ⊆ Ψ there is a natural one-to-one correspondence between conjugacy classes of k-parabolic subgroups and subsets of Π. Select a system of simple roots Π and identify it with the set of generators S for the Coxeter system (W, S) of the building of G over k. Pick a longest word w 0 ∈ W and let s 1 s 2 . . . s n be a reduced decomposition. For each i such that 1 ≤ i ≤ n, let α i be the element of Π corresponding to s i . We say that N −α 1 N −α 2 . . . N −αn N α 1 N α 2 . . . N αn and any right coset thereof is a basic open subset of G + . We also say that the empty set is a basic open subset with respect to the torus S. By considering the simply transitive action of G + on pairs of opposite chambers, it follows from Lemma 2.11 that this does indeed define a base for the strong k-topology on G + . Our goal in this paper is the proof of Theorems 1.7 and 1.12. We shall give these in the next section. Proof of the Main Theorems Suppose that G (resp. G ′ ) is a semisimple algebraic group defined over a field k (resp. k ′ ). Suppose that k is a non-discrete Hausdorff topological field. Let ∆ := ∆(G, k), ∆ ′ := ∆(G ′ , k ′ ). Suppose that the Coxeter diagrams of ∆, ∆ ′ have no isolated nodes. The building ∆ is a topological spherical building, and we wish to prove that, given a nonempty open quasi-connected subset U of Cham ∆ and an injective chamber map ϕ : ∆(U) → ∆ ′ , there is a unique extension of ϕ to an injective chamber map ∆ → ∆ ′ . It is sufficient to prove that if U ⊆ V are basic open sets and ∆(U) is the domain of the injective chamber map ϕ, then there is a unique extension of ϕ to an injective chamber map with domain ∆(V ). Proof of Theorem 1.7. Suppose that U is a basic open set with respect to the pair of opposite chambers (C, C ′ ). We denote by G + the subgroup of G(k) generated by all the k-rational points of unipotent radicals of k-parabolic subgroups, and similarly for G ′+ . First we must show how to associate to the mapping ϕ a local group homomorphism ψ : W ⊆ G + → G ′+ , where W is a basic open neighbourhood of the identity in G + . Let A be an apartment containing C which is wholly contained in U. This exists because for each decomposition w 0 = s 1 s 2 . . . s k of the longest word w 0 ∈ W we may let U ′ be the set of ends of of a open set U ′′ of non-stammering galleries starting at C of type (s 1 , s 2 , . . . s k ) containing the constant gallery, with the property that every member of every gallery in U ′′ is contained in U. Taking the intersection of all such U ′ for every such decomposition, we obtain an open set containing C contained in U. A chamber opposite C contained in this set determines together with C an apartment wholly contained in U. Let D be the chamber of A opposite C. Suppose that the building ∆ is of type (W, S). For each s ∈ S let V s be the root group corresponding to the root R s of the apartment A which does not contain C but which is attached to C along the panel of cotype {s}. There exists an open neighbourhood of the identity W s ⊆ V s , such that if g ∈ W s , then g maps an open neighbourhood U 1 ⊆ U of C onto some other open set U 2 ⊆ U, and furthermore U 1 may be chosen to be a basic open set with respect to the pair of opposite chambers (C, D). From this we obtain a mapping g * : ϕ(U 1 ) → ϕ(U 2 ) given by g * = ϕ•g•ϕ −1 . Let V ′ s be the root group in G ′ (k ′ ) corresponding to the root ϕ(R s ) of the apartment ϕ(A). Lemma 3.1. If g ∈ W s and U 1 , U 2 are as above, then g * is the restriction to ϕ(U 1 ) of an element of V ′ s . Proof. The rigidity theorem, Theorem 1.2, says that an injective chamber map is determined by its action on E 1 (D) ∪{C} where (C, D) are a pair of opposite chambers. The proof of this theorem given in [1], Section 5.9, can be modified to show that if U 1 is a basic open set with respect to (C, D), then g \| U 1 is determined by its action on (E 1 (D) ∩ U 1 ) ∪ {C}. Thus any chamber map U 1 → U 2 that agrees with g on (E 1 (D)∩U 1 )∪{C} must be equal to g. It follows that any chamber map ϕ(U 1 ) → ϕ(U 2 ) that agrees with g * on (E 1 (ϕ(D)) ∩ ϕ(U 1 ) ∪ {ϕ(C)} must be equal to g * . Since ∆ ′ is strictly Moufang there is exactly one element of V ′ s whose restriction to ϕ(U 1 ) has this property, so it follows that g * is the restriction to ϕ(U 1 ) of an element of V ′ s . We respectively are the root groups corresponding to −R s in G(k), G ′ (k ′ ) respectively. These mappings extend to a local group homomorphism ψ : W ⊆ G + → G ′+ , where W is a basic open neighbourhood of the identity in G + with respect to the maximal k-split torus corresponding to the apartment A. This group homomorphism also has the property that root groups are mapped into root groups, for any root, not just the simple ones. In fact more can be said: if we let k Φ be the set of k-roots of G relative to the maximal k-split torus corresponding to the apartment A, and we let U α be the set of k-rational points of the unipotent k-group corresponding to α for each α ∈ k Φ, and similarly for U ′ β for a k ′ -root β of G ′ with respect to the maximal k-split torus corresponding to the apartment ϕ(A), then there is a bijection f between the relative root systems of each group such that U α ∩ W is mapped into U ′ f (α) . This follows from the fact that the "refined RGD-system" structure is determined by the geometry of the building; see Section 7.9.3 of [1], and also Theorem 7.116. Supose that α ∈ k Φ, 1 2 α / ∈ k Φ. By U (α) we mean the group generated by U kα for all positive integers k such that kα ∈ k Φ. Consider the mapping ψ : W ∩ U (α) , U (−α) → U ′ (f (α)) , U ′ (−f (α)) . We must try to show that this extends to a global group homomorphism on U (α) , U (−α) . There exists a k-split torus S of dimension one, contained in the maximal k-split torus corresponding to the apartment A, such that S ∩ α −1 (k * 2 ) ⊆ U (α) , U (−α) . Consider first the case where 2α / ∈ k Φ. The action of S on U α and U −α by conjugation makes them into vector spaces over k. (The group operations on U α and U −α provide the vector space addition in both cases, and the action of S by conjugation gives the scalar multiplication.) Let W ∩ S ∩ α −1 (k * 2 ) be equal to S ∩ α −1 (W * ) where W * is an open neighbourhood of 1 in k 2 . By considering the action of S on U α and U α we can show that we can extend ψ from S ∩ α −1 (W * ) to S ∩ α −1 (R) where R is the ring generated in k by W * . This ring includes an open neighbourhood of 0. We now see that from ψ we can obtain a local field homomorphism k → k ′ . (From the action of ψ on S we obtain a mapping defined on k * in a neighbourhood of 1, but this can be transferred to a neighbourhood of 0.) But local field homomorphisms extend uniquely to global field homomorphisms. It is now easy to see that there is a unique extension of ψ to a global group homomorphism. The argument is similar if 2α ∈ k Φ, except that we consider the action of S on U (α) /U 2α , U 2α , U (−α) /U −2α , U −2α . Since the restriction of ψ to W ∩ U (α) , U (−α) extends to a unique global group homomorphism on U (α) , U (−α) for each α ∈ k Φ, it follows that ψ extends to a unqiue global group homomorphism on G + . From this extension we can now construct an extension of ϕ to an injective chamber map ϕ : ∆ → ∆ ′ , or to a unique injective chamber map ϕ : ∆(V ) → ∆ ′ where V is a basic open set containing U. Proof of Theorem 1.12. We saw above that it is sufficient to prove that the root groups of the refined RGD system are mapped into root groups. The proof of this given in [4] works in the local case with no difficulties. A corollary dealing with the rigidity of nilpotent groups In this section we look at a corollary to the main result of the foregoing section which deals with fibrations of nilpotent Lie groups. Specifically, we prove the following theorem. Theorem 4.1. Suppose that N is a nilpotent real or complex Lie group with Lie algebra n. Suppose that the Lie algebra n occurs in the Iwasawa decomposition of a semisimple Lie algebra s = k ⊕ a ⊕ n whose corresponding Lie group Iwasawa decomposition is S = KAN. Suppose that n 1 and n 2 are two linearly disjoint Lie subalgebras of n which are each the direct sum of a set of root spaces in n and which generate n as a Lie algebra. Suppose that for all X ∈ n 1 there exists a Y ∈ n 2 such that [X, Y ] = 0, and for all X ∈ n 2 there exists a Y ∈ n 1 such that [X, Y ] = 0. Suppose that N 1 and N 2 are the analytic groups corresponding to n 1 and n 2 . Suppose that U is a nonempty open connected subset of N and that φ : U → N is a mapping whose range is Zariski dense in N such that, for all x ∈ U, φ(xN 1 ∩U) ⊆ φ(x)N 1 ∩U and φ(xN 2 ∩ U) ⊆ φ(x)N 2 ∩ U. Then, if we let k = R or C depending on whether we are in the real or complex case, φ is a mapping induced by the left action of an element of S · C * 1 on N = S/KA, possibly composed with a mapping induced by a field homomorphism ψ : k → k. Proof. Under the hypotheses of the theorem, there must exist two parabolic subgroups of S, P 1 = KAN 1 and P 2 = KAN 2 , such that φ induces a mapping from an open connected set in the family of left cosets of P 1 into the family of left cosets of P 1 , and similarly with P 2 . Now, by the condition on commutators of elements of n 1 and n 2 , the image of any left coset of a parabolic subgroup contained in P 1 under projection to S/P 2 can be obtained as the intersection of the projections of a finite family of sets of the form gP 1 to S/P 2 , and similarly with the roles of P 1 and P 2 reversed. We may conclude that φ induces a mapping from an open connected subset of S/KA to itself whose range is Zariski dense and which preserves the images in S/KA of left cosets of parabolic subgroups. The result now follows from theorem 1.7. A version of this theorem appears in [14] which requires smoothness. This result can be used to obtain a classification-free and unified version of many of the rigidity results in Yamaguchi [21]. Lemma 2 . 211. A basic open set is an open set, and the basic open sets form a base for the topology on the set of chambers. Proof. Let all notations be as in the definition of a basic open set. 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Differential systems associated with simple graded Lie algebras. K Yamaguchi, Adv. Stud. Pure Math. 22University of MünsterEinsteinstrasse. 48149 Münster, Germany E-mail address: [email protected]. Yamaguchi. Differential systems associated with simple graded Lie algebras. Progress in Differential Geometry, Adv. Stud. Pure Math., Vol. 22, pages 413-494, 1993. University of Münster, Einsteinstrasse 62, 48149 Münster, Germany E-mail address: [email protected]	[]
[ "High-energy astroparticle physics with CALET", "High-energy astroparticle physics with CALET" ]	[ "Paolo Maestro [email protected] \nDept. of Physics\nUniversity of Siena\nVia Roma 5653100SienaItaly\n" ]	[ "Dept. of Physics\nUniversity of Siena\nVia Roma 5653100SienaItaly" ]	[]	The CALorimetric Electron Telescope (CALET) will be installed on the Exposure Facility of the Japanese Experiment Module (JEM-EF) on the International Space Station (ISS) in 2014 where it will measure the cosmic-ray fluxes for five years. Its main scientific goals are to search for dark matter, investigate the mechanism of cosmic-ray acceleration and propagation in the Galaxy and discover possible astrophysical sources of high-energy electrons nearby the Earth. The instrument, under construction, consists of two layers of segmented plastic scintillators for the cosmic-ray charge identification (CHD), a 3 X0-thick tungsten-scintillating fiber imaging calorimeter (IMC) and a 27 X0-thick lead-tungstate calorimeter (TASC). The CHD can provide single-element separation in the interval of atomic number Z from 1 to 40, while IMC and TASC can measure the energy of cosmic-ray particles with excellent resolution in the range from few GeV up to several hundreds of TeV. Moreover, IMC and TASC provide the longitudinal and lateral development of the shower, a key issue for good electron/hadron discrimination. In this paper, we will review the status of the mission, the instrument configuration and its expected performance, and the CALET capability to measure the different components of the cosmic radiation.	10.1088/1742-6596/409/1/012026	[ "https://arxiv.org/pdf/1302.1257v1.pdf" ]	118,691,737	1302.1257	e81ae1ca1222a62a8aaf5e002032f4864b09db15	High-energy astroparticle physics with CALET 6 Feb 2013 Paolo Maestro [email protected] Dept. of Physics University of Siena Via Roma 5653100SienaItaly High-energy astroparticle physics with CALET 6 Feb 2013* on behalf of the CALET collaboration The CALorimetric Electron Telescope (CALET) will be installed on the Exposure Facility of the Japanese Experiment Module (JEM-EF) on the International Space Station (ISS) in 2014 where it will measure the cosmic-ray fluxes for five years. Its main scientific goals are to search for dark matter, investigate the mechanism of cosmic-ray acceleration and propagation in the Galaxy and discover possible astrophysical sources of high-energy electrons nearby the Earth. The instrument, under construction, consists of two layers of segmented plastic scintillators for the cosmic-ray charge identification (CHD), a 3 X0-thick tungsten-scintillating fiber imaging calorimeter (IMC) and a 27 X0-thick lead-tungstate calorimeter (TASC). The CHD can provide single-element separation in the interval of atomic number Z from 1 to 40, while IMC and TASC can measure the energy of cosmic-ray particles with excellent resolution in the range from few GeV up to several hundreds of TeV. Moreover, IMC and TASC provide the longitudinal and lateral development of the shower, a key issue for good electron/hadron discrimination. In this paper, we will review the status of the mission, the instrument configuration and its expected performance, and the CALET capability to measure the different components of the cosmic radiation. Introduction CALET (CALorimetric Electron Telescope) is a space-based detector developed by a Japanese led international collaboration to directly measure the high-energy cosmic radiation on the International Space Station (ISS). CALET is scheduled to be launched in 2014 by the Japanese rocket HTV (H-IIA Transfer Vehicle) and robotically installed on the Japanese Experiment Module Exposure Facility (JEM-EF) on ISS. The CALET mission will address many of the outstanding questions of High-Energy Astrophysics, such as the origin of cosmic rays, the mechanism of CR acceleration and galactic propagation, the existence of dark matter and nearby CR sources, by the observations of CR electrons, γ rays and nuclei in a wide energy window from few GeV up to the TeV region [1,2]. The CALET instrument and its performance The CALET instrument consists of a Total AbSorption Calorimeter (TASC), a finely segmented pre-shower IMaging Calorimeter (IMC), and a CHarge Detector (CHD) (Fig. 1). The TASC is a homogeneus calorimeter made of 192 Lead Tungstate (PWO) "logs" (20×19×320 mm 3 ) arranged in 12 layers. The logs in the top layer are readout by photomultiplier tubes (PMTs), while a dual photodiode/avalanche-photodiode system is used for the readout of the remaining layers. The TASC can determine the energy of the incident particle with excellent energy resolution: ∼2% for e ± and γ rays above 100 GeV, ∼40% for 1 TeV protons and ∼30% for nuclei at 50 GeV/amu, as estimated from simulations. Moreover, exploting its shower imaging capabilities, a proton rejection >10 5 can be achieved, sufficient to keep the proton contamination below a few percent in the observation of CR electrons in the TeV region [3]. The IMC consists of 7 tungsten plates interleaved with double layers of 1 mm 2 scintillating fibers (SciFi), arranged in belts along orthogonal direction and readout by multianode PMTs, and is capped by an additional SciFi layer pair. Its surface area is 45×45 cm 2 and its total thickness ∼3 radiation lengths (X 0 ). The IMC fine granularity allows to measure precisely the incident particle trajectory (with angular resolution better than 1 • ), determine the starting point of the shower and separate the incident from backscattered particles. The charge of the CR nuclei is measured via the Z 2 dependence of the specific ionization loss in a double layered, segmented, plastic scintillator array (CHD) positioned above the IMC. Each layer is composed of 14 scintillator paddles (3.2×1.0×44.8 cm 3 ) each readout by a PMT. Taking advantage of its excellent charge resolution (∼0.1 electron charge units (e) for B, ∼0.2e for Fe) [4], CHD can resolve individual chemical elements from Z=1 to Z=40. The total thickness of the instrument is equivalent to 30 X 0 and 1.3 proton interaction length. The effective geometrical factor of CALET for high-energy electrons and nuclei is ∼1200 cm 2 sr. CALET science goals It is generally accepted that CRs are accelerated in shock waves of supernova remnants (SNRs), which are the only galactic candidates known with sufficient energy output to sustain the CR flux. Recent observations of electron synchrotron and gamma-ray emission from SNRs proved that high-energy charged particles are accelerated in SNR shocks up to energies beyond 100 TeV [5]. Unlike the hadronic component of CRs, the electrons, during their diffusion in the Galaxy, suffer radiative energy losses proportional to their squared energy. Thus TeV electrons observed at Earth likely originated in sources younger than 10 5 years and <1 kpc far from the Solar System. Since the number of such nearby SNRs is limited (e.g.: Vela, Monogem, Cygnus Loop remnants, and few others), the electron energy spectrum around 1 TeV could exhibit spectral features and, at very high energies, a significant anisotropy in the electron arrival directions would be expected. Thanks to its excellent energy resolution and capability to discriminate electrons from hadrons, CALET will be able to investigate possible spectral structures by detecting very high-energy electrons and possibly provide the first experimental evidence of the presence of a nearby CR source. Additional information on the CR acceleration mechanism might be obtained by directly measuring, besides electrons, the energy spectra of individual CR nuclei up to the PeV scale. Possible charge-dependent high-energy spectral cutoffs, hypothesized to explain the CR "knee" [6], or spectral hardening due to non-linear acceleration mechanisms [7], could only be investigated by a space experiment with long enough exposure to extend the direct measurement of CR nuclei spectra to unprecedented energies. CALET will be able to identify CR nuclei with individual element resolution and measure their energies in the range from a few tens of GeV to several hundreds of TeV. In five years of data taking on the ISS, it is expected to extend the proton energy spectrum up to ∼900 TeV, the He spectrum up to 400 TeV/amu (Fig. 2) and measure the energy spectra of the more abundant heavy nuclei C, O, Ne, Mg, Si and Fe, with sufficient statistical precision up to ∼20 TeV/amu (Fig. 3). It will also investigate precisely possible spectral features, like a hardening above 200 GeV/amu recently reported by CREAM [8], or deviations from a pure power-law spectrum. Moreover, exploiting the CHD particle identification capability, CALET should measure the ultra-heavy ions at few GeV/amu in the 26<Z≤40 charge range with an expected statistics ∼5 times larger the one collected by the balloon experiment TIGER [9]. Figure 2. Expected CALET measurement of the energy spectra of proton and He after 5 years of observation, compared with previous data [10,11,12,13,14,15,16]. The relative abundances of CR secondary-to-primary elements (like B/C or sub-Fe/Fe) are known to decrease, following a power-law in energy E −δ , where δ is a key parameter in the description of the CR diffusion in the Galaxy at high energies [22]. At several TeV/amu, the available data suffer from statistical limitations and large systematic errors, due to the residual atmospheric overburden at balloon altitude, and has not allowed so far to place a stringent experimental constraint on the value of δ. Taking advantage of its long exposure in space and the absence of atmosphere, CALET will provide new data to improve the accuracy of the present measurements above 100 GeV/amu and extend them above 1 TeV/amu. Besides studying the CR sources and diffusion, CALET will also conduct a sensitive search for signatures of dark matter candidates (e.g.: Weakly Interacting Massive Particles (WIMPs), Kaluza-Klein particles, etc.) in both the electron and gamma-ray spectra. With its excellent energy resolution and long exposure in space, it will be able to detect possible lines due to WIMP decays in the gamma-ray spectrum above few hundreds of GeV, and shed light on the controversial anomalous excess in the electron spectrum recently reported by the balloon experiments ATIC [23] but not confirmed by FERMI [24]. . Expected CALET measurement of the energy spectra of the more abundant heavy nuclei after 5 years of observation, compared with previous data [17,18,19,20,21]. Finally, additional CALET science objectives are the detailed study of the solar modulation by the measurement of the electron spectrum time evolution below 10 GeV, and the detection of gamma-ray bursts and X-ray transients by means of a dedicated scintillator-based Gamma-ray Burst Monitor associated to the main CALET telescope. Figure 1 . 1Schematic view of CALET. The picture of a simulated shower is superimposed. Figure 3 3Figure 3. Expected CALET measurement of the energy spectra of the more abundant heavy nuclei after 5 years of observation, compared with previous data [17, 18, 19, 20, 21]. K Yoshida, Proc. of 31 st ICRC (Lodz). of 31 st ICRC (Lodz)6360Yoshida K et al. 2011 Proc. of 31 st ICRC (Lodz) vol 6 p 360 . S Torii, Nucl. Instr. And Meth. A. 63055Torii S et al. 2011 Nucl. Instr. And Meth. A 630 55 . Y Akaike, Adv. Space Res. 45690Akaike Y et al. 2010 Adv. Space Res. 45 690 . P Marrocchesi, Nucl. Instr. And Meth. A. 659477Marrocchesi P S et al. 2011 Nucl. Instr. And Meth. 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[ "Time-reversal symmetry breaking surface states of d-wave superconductors induced by an additional order parameter with negative T c", "Time-reversal symmetry breaking surface states of d-wave superconductors induced by an additional order parameter with negative T c" ]	[ "Takeshi Tomizawa \nDepartment of Physics\nKobe University\n657-8501KobeJapan\n\nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan\n", "Kazuhiro Kuboki \nDepartment of Physics\nKobe University\n657-8501KobeJapan\n" ]	[ "Department of Physics\nKobe University\n657-8501KobeJapan", "Department of Physics\nNagoya University\n464-8602NagoyaJapan", "Department of Physics\nKobe University\n657-8501KobeJapan" ]	[]	Surface states of d x 2 −y 2 -wave superconductors are studied using the Ginzburg-Landau (GL) theory. For a [110] surface it has been known that the time-reversal symmetry (T ) breaking surface state, (d±is)-wave state, can occur if the bare transition temperature of the s-wave order parameter (OP) is positive. We show that even if this bare Tc is negative, it is possible to break T because the coupling to the spontaneously generated magnetic field may induce the s-wave OP. The T -breaking state is favored when the GL parameter κ is small.	10.1103/physrevb.78.094519	[ "https://arxiv.org/pdf/0806.0209v3.pdf" ]	118,601,144	0806.0209	9742c622e8191bbcdae49ef8692df3e8a09de481	Time-reversal symmetry breaking surface states of d-wave superconductors induced by an additional order parameter with negative T c 30 Sep 2008 (Dated: September 30, 2008) Takeshi Tomizawa Department of Physics Kobe University 657-8501KobeJapan Department of Physics Nagoya University 464-8602NagoyaJapan Kazuhiro Kuboki Department of Physics Kobe University 657-8501KobeJapan Time-reversal symmetry breaking surface states of d-wave superconductors induced by an additional order parameter with negative T c 30 Sep 2008 (Dated: September 30, 2008)arXiv:0806.0209v3 [cond-mat.supr-con] PACS numbers: 74.20.De, 74.20.Rp Surface states of d x 2 −y 2 -wave superconductors are studied using the Ginzburg-Landau (GL) theory. For a [110] surface it has been known that the time-reversal symmetry (T ) breaking surface state, (d±is)-wave state, can occur if the bare transition temperature of the s-wave order parameter (OP) is positive. We show that even if this bare Tc is negative, it is possible to break T because the coupling to the spontaneously generated magnetic field may induce the s-wave OP. The T -breaking state is favored when the GL parameter κ is small. I. INTRODUCTION Superconducting (SC) states of high-T c cuprates are known to have d x 2 −y 2 -wave symmetry. 1,2 Since the pair wave function of such an unconventional SC state has strong angular dependence, the effects of the presence of surfaces, impurities are different from those in conventional s-wave superconductors. For example, it is possible to break the time-reversal symmetry (T ) near a surface or a Josephson junction by inducing the second component of the SC order parameter (OP) 2,3,4,5,6,7,8,9,10,11,12,13,14 with a nontrivial phase difference between the two OPs. In the case of a Josephson junction it may occur when the surface has [110] orientation, because the second SCOP induced by the tunneling process can have phase difference ±π/2 leading to a T -breaking state. 8,9 For a [110] surface faced to a vacuum the necessary condition to break T seems to be that the bare transition temperature (T c ) of the second OP is positive. 7,10,11,12,14 In this paper we examine the possibility to have a Tbreaking surface state near the [110] surface of a d x 2 −y 2wave superconductor when the bare T c of the additional OP is negative, namely, the second OP will not occur in the bulk even at zero temperature. We take an swave SCOP as the second component, since d x 2 −y 2 -wave and extended s-wave symmetries are natural candidates for superconducting states in the models with nearestneighbor interactions (e.g., the t − J model). We will show that this kind of T -violation is possible, and that both the SCOPs and the magnetic field (vector potential) should be treated self-consistently in order to describe this situation correctly. It also turns out that the T violation may occur for a relatively small GL parameter κ (i.e., of the order of 10), when T c of the second OP is negative. Then the present mechanism may not be relevant to the T -violation in hole-doped cuprates in which κ ∼ 100. However, we expect the surface states of electron-doped cuprates may be described by the present theory, because some of the latter systems have much smaller κ values. 15,16 II. GINZBURG-LANDAU EQUATION We consider a superconductor with tetragonal symmetry and assume only a d x 2 −y 2 -wave SCOP, ∆ d , is present in the bulk. An s-wave SCOP, ∆ s , is taken into account as a possible second component when ∆ d is suppressed near the surface. For such a system the Ginzburg-Landau (GL) free energy is given as 3 F = dr µ=d,s α µ \|∆ µ \| 2 + β µ 2 \|∆ µ \| 4 + K µ \|D∆ µ \| 2 + γ 1 \|∆ d \| 2 \|∆ s \| 2 + γ 2 {∆ 2 d (∆ * s ) 2 + (∆ * d ) 2 ∆ 2 s } + K ds (D x ∆ d )(D x ∆ s ) * − (D y ∆ d )(D y ∆ s ) * + c.c + 1 8π (∇ × A) 2 (1) where A is the vector potential and D = ∇ − (2πi/Φ 0 )A is the gauge invariant gradient with Φ 0 = hc/2e being the magnetic flux quantum. Coefficients α µ (∝ T − T cµ ), β µ , K µ , γ 1 , γ 2 and K ds are real, and we assume T cd > 0, while T cs can be both positive and negative. The γ 2 is one of the terms which determine the relative phase of OPs, φ ds (≡ φ d − φ s ; ∆ µ = \|∆ µ \| exp(iφ µ )). We take γ 2 > 0, because this choice would lead to the (d ± is)state (φ ds = ±π/2) instead of the (d ± s)-state (φ ds = 0, π). In the former case the nodes of the d-wave state are removed and the more condensation energy can be gained. It is also to be noted that γ 1 ± 2γ 2 is positive in usual weak-coupling model, since two OPs compete each other. Now we rewrite F in the dimensionless unit 17 to see the parameter dependence of the model more clearly, F = H 2 c ξ 3 d 4π dr − \|η d \| 2 + 1 2 \|η d \| 4 + \|Dη d \| 2 +α s \|η s \| 2 +β s 2 \|η s \| 4 +K s \|Dη s \| 2 +γ 1 \|η d \| 2 \|η s \| 2 +γ 2 η 2 d (η * s ) 2 + (η * d ) 2 η 2 s +K ds (D x η d )(D x η s ) * − (D y η d )(D y η s ) * + c.c. +(∇ × a) 2 ,(2) where η µ = ∆ µ /∆ 0 (µ = d, s) with ∆ 0 = \|α d \|/β d being the bulk d-wave OP. r was rescaled using the coherence length for the d-wave OP, ξ d (= K d /\|α d \|), as r → r/ξ d , andD ≡ ∇ − ia/κ. Here a = A/( √ 2H c ξ d ), and the magnetic field is measured in units of √ 2H c , where H c = 4πα 2 d /β d is the thermodynamic criti- cal field. κ = λ d /ξ d is the GL parameter with λ d = φ 0 /(2 √ 2πH c ξ d ) being the penetration depth for the bulk d-wave superconductor. The parameters in Eq.(2) are defined asα s = α s /\|α d \|,β s = β s /β d ,K s = K s /K d , γ 1 = γ 1 /β d ,γ 2 = γ 2 /β d andK ds = K ds /K d . Usually the surface effect is described by the second-order surface GL free energy, F sf = sf dS µ,ν=d,s g µν η * µ η ν , where integration is carried out on the surface. Using the symmetry argument we find g ds = g sd = g 0 cos 2θ where θ is the angle between the surface and the crystal a-axis with g 0 being a constant. This term could also determine φ ds , and it leads to the (d ± s)-state in the case of a [100] surface (θ = 0), since the γ 2 term is higher order than the g ds term. However, g ds vanishes for a [110] surface (θ = 45 • ) which we consider in the following. The g µµ term will represent the suppression of η µ near the surface. Instead of using g dd we impose the condition η d = 0 at the [110] surface, because the d x 2 −y 2 -wave SCOP should vanish there. Since the s-wave SCOP is only little affected by the presence of the surface, we take g ss = 0. (In numerical calculations we have checked that taking small positive g ss will not change the results qualitatively.) In order to consider the [110] surface we transform the coordinate system, (x, y, z) → (x,ỹ, z). Here x (y) is parallel to the crystal a (b) axis (z is parallel to the surface), andx andỹ axes are perpendicular and parallel to the surface, respectively. (See Fig.1.) In the free energy density only theK ds term is changed under this transformation to 2K ds κ aỹIm(η * s ∂xη d − η d ∂xη * s )(3) where we have assumed that the system is uniform along the surface, and the gauge freedom was taken as a = aỹ(x)eỹ. The expression for the supercurrent is obtained by varying the electronic part of F (i.e., except the last term) with respect to a. Since the surface is faced to the vacuum, thex component, Jx, should obviously vanish. (We have numerically checked that Jx actually vanishes.) Theỹ component, Jỹ, and that in the the dimensionless unit, jỹ, are given as jỹ = Jỹ √ 2H c c ξ d = − 1 4π 1 κ 2 aỹ(\|η d \| 2 +K s \|η s \| 2 ) +K ds κ Im(η * s ∂xη d − η d ∂xη * s ) .(4) III. SURFACE STATE AND SPONTANEOUS CURRENT We numerically solve the problem by employing the quasi-Newton method 18 to minimize the free energy F under the condition η d (x = 0) = 0. We minimize F with respect to all variables, i.e., η d , η s and aỹ. Note that the Maxwell's equation is taken into account in this procedure, and we call this as "fully self-consistent calculation". For the sake of comparison we will also show the results by treating only η d and η s self-consistently. First let us consider the case ofα s < 0 (i.e., T < T cs ). In this case, we would get finite η s if η d were absent. However, for T cd > T cs the stability condition for η s in the bulk is given as,α s + (γ 1 − 2γ 2 )\|η d \| 2 < 0, so the transition temperature of η s is lower than the bare one, T cs , and η s would be totally suppressed if T cd ≫ T cs . Near the surface or impurities the situation can be different. There η s may be finite because the dominant SCOP, η d , is suppressed. In Fig.2 the spatial variations of the SCOPs near the surface are shown. η s gets finite near the surface while η d is suppressed. The relative phase φ ds will be determined byγ 2 andK ds terms, and the former favors φ ds = ±π/2 as mentioned. From Eq.(3) we see that theK ds term also favors φ ds = ±π/2, and aỹ will be spontaneously generated. (We take η d to be real and aỹ = 0 in the bulk, i.e.,x → ∞.) Numerical calculations show that η d is real for allx, and that φ ds = ±π/2 where η s is finite. This indicates that a T -violating (d + is)wave surface state with a spontaneous magnetic field b z (= ∂xaỹ) and a supercurrent jỹ occurs near the surface. The spatial distributions of b z and jỹ are presented in Fig.3. In order to see the role played by the vector potential, we investigate the same problem by setting aỹ = 0 everywhere. Namely we treat only SCOPs self-consistently. When aỹ is set to zero, the spontaneous current jỹ has contributions from only the spatial variations of SCOPs (ı.e., the last line of Eq.(4)), and we calculate the magnetic field from jỹ using Maxwell's equation, jỹ( x) = − 1 4π ∂bz (x) ∂x . Forα s < 0, the results for the SCOPs look similar as in the fully self-consistent calculations. The Tbreaking (d + is)-state occurs as shown in Fig.2. On the contrary, the behaviors of b z and jỹ are different in that jỹ always has the same sign, and that b z is a monotonous function ofx. These results are not correct even qualitatively as well as in a quantitative sense. Integration of the Maxwell's equation with the boundary condition b z (±∞) = 0 leads to ∞ −∞ dxjỹ(x) = 0, implying that the averaged current should vanish. 9 This is the case for the fully self-consistent calculation but not in the case where the magnetic field is not treated self-consistently, because of the absence of the screening effect in the latter. Next we consider the case ofα s > 0, i.e., T > T cs . Note that T cs may be negative, in which case η s will not occur in the bulk at T = 0 even when η d is absent. The results for the SCOPs are depicted in Fig.4. (Here the GL parameter is taken to be κ = 16.) It is seen that finite Im(η s ) is obtained, though we naively expect η s = 0. This is because theK ds term couples ∂xRe(η d ) bilinearly to aỹIm(η s ). It may induce the state with Im(η s ) = 0 and b z = 0, but the state with η s = 0 and b z = 0 may also be a self-consistent solution. Numerical calculations show that the former one has the lower energy, and thus the time-reversal symmetry is violated spontaneously. Here \|α s \|,β s andK s were taken to be much smaller than those in Fig.2. Otherwise the T -violation will not occur, because these terms cost the energy forα s > 0 and the In the case ofα s > 0, the results with or without treating the vector potential self-consistently are completely different. If we do not take into account the aỹ term, η s will never appear, since there is no mechanism to derive finite η s . Thus neither the spontaneous current nor the spontaneous field can occur. It implies that the Tviolation near the surface cannot be described in this kind of simplified treatment for the superconductors in which the second SCOP has negative T c . In order to see the dependence on κ we show the results for a larger κ (κ = 19) in Fig.6 and 7. It is seen that \|η s \|, \|b z \| and \|jỹ\| are much smaller than those for κ = 16. This κ dependence can be understood as follows. η d is suppressed in the region near the surface (x ξ d ), and η s and aỹ would be finite there if T is broken. On the other hand the magnetic field b z would be finite in the regionx λ d . When κ is large, the loss of energy due to finite b z in the large region (ξ d x λ d ) overwhelms the energy gain coming from theK ds term which acts only in the small regionx ξ d . Thus for large κ the T -violation is not favored. If the larger value ofK ds is taken, the T -breaking state can occur for larger κ. But the natural assumption seems to be K ds ≤ K d (K ds ≤ 1), so that the T -violation may occur for κ of the order of 10. (On the contrary the T -violation may occur for much larger κ in the case ofα s < 0, because the energy can be gained by not onlyK ds but alsoα s term.) It implies that the present mechanism may not be relevant to holedoped cuprates in which κ ∼ 100, but it may describe the surface states of electron-doped cuprates which have smaller κ. If we assume H c =1T, the maximum values of \|B z \| and \|Jỹ\| are 2.5 × 10 −1 T and 3.7 × 10A/cm 2 , respectively, for κ = 16. For κ = 19 they are 8.6 × 10 −2 T and 1.2 × 10A/cm 2 , respectively. These values rapidly decrease as κ increases, and the T -breaking state disappears as κ exceeds 19 for the parameters used here. If we compare these values with experiments, it should be noted that surface roughness will reduce \|B z \| and \|Jỹ\|, because Tviolation is most favored in the case of θ = 45 • . 12 (When θ = 45 • , g ds will be finite and the T -violation is not favored.) IV. SUMMARY We have examined the role played by the vector potential concerning the occurrence of surface states with spontaneously broken T in d x 2 −y 2 -wave superconductors. It has been known that the T -breaking state may naturally appear if the bare T c of the additional OP is positive. For the Josephson junction composed of d x 2 −y 2wave and other superconductors, tunneling may induce second component of SCOP and thus T may be broken. In these cases the T -breaking states may be described without treating the vector potential self-consistently. In this paper it was shown that the surface state of a d x 2 −y 2wave superconductor may break T even when the bare T c of the second SCOP is negative. However, to describe this situation correctly not only the SCOPs but also the vector potential must be treated on an equal footing. In the present mechanism the T -violation may occur for rather small values of the GL parameter κ ( 20), so that it may not be relevant to hole-doped cuprates. We expect the present theory may be used to describe the surface states of electron-doped high-T c cuprates, because their κ are much smaller than those of hole-doped systems. FIG. 1 : 1Schematic of a [110] surface of a d x 2 −y 2 -wave superconductor with tetragonal symmetry. x and y are parallel to the crystal a and b axes, respectively. FIG. 2 : 2Spatial variations of SCOPs forαs = −0.2,βs = 0.2,Ks = 0.5,γ1 = 0.5,γ2 = 0.1,K ds = 0.3 and κ = 16. Note that all SCOPs are normalized by the bulk d-wave OP, and x = 0 corresponds to the surface faced to the vacuum. (a) Reη d and (b) Imηs in the fully self-consistent calculation. (c) Reη d and (d) Imηs in the simplified one without treating aỹ self-consistently. FIG. 3 :FIG. 4 : 34Spatial variations of bz and jỹ. Parameters used are the same as in Fig.2. (a) jỹ and (b) bz in the fully selfconsistent calculation. (c) jỹ and (d) bz in the simplified one without treating aỹ self-consistently. Note bz and jy are in the dimensionless unit. energy gain is solely coming from theK ds term. The spatial variations of b z and jỹ are shown in Fig.Spatial variations of SCOPs forαs = 0.01,βs = 0.01,Ks = 0.04,γ1 = 0.5,γ2 = 0.1,K ds = 1.0 and κ = 16. (a) Reη d and (b) Imηs in the fully self-consistent calculation. (c) Reη d in the simplified one without treating aỹ selfconsistently. FIG. 5 : 5Spatial variations of bz and jỹ. Parameters used are the same as inFig.4. (a) jỹ and (b)bz in the fully selfconsistent calculation. FIG. 6 : 6Spatial variations of SCOPs. Parameters used are the same as in Fig.4 except κ = 19. 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[ "Correlations between nuclear landscape boundaries and neutron-rich r-process abundances", "Correlations between nuclear landscape boundaries and neutron-rich r-process abundances" ]	[ "Q Z Chai \nSchool of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n", "Y Qiang \nSchool of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n", "J C Pei \nSchool of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n" ]	[ "School of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina", "School of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina", "School of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina" ]	[]	Motivated by the newly observed 39 Na in experiments, systematic calculations of global nuclear binding energies with seven Skyrme forces are performed. We demonstrate the strong correlation between the two-neutron separation energies (S2n) of 39 Na and the total number of bound nuclei of the whole nuclear landscape. Furthermore, with calculated nuclear masses, we perform astrophysical rapid-neutron capture process (r-process) simulations by using nuclear reaction code TALYS and nuclear reaction network code SkyNet. r-process abundances from ejecta of neutron star mergers and core-collapse supernova are compared. Prominent covariance correlations between nuclear landscape boundaries and neutron-rich r-process abundances before the third peak are shown. This study highlights the needs for further experimental studies of drip-line nuclei around 39 Na for better constraints on nuclear landscape boundaries and r-process.	10.1103/physrevc.105.034315	[ "https://arxiv.org/pdf/2105.13043v1.pdf" ]	235,212,289	2105.13043	119030556aa24bd2abd26d82b4cdda320fec8131	Correlations between nuclear landscape boundaries and neutron-rich r-process abundances Q Z Chai School of Physics State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina Y Qiang School of Physics State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina J C Pei School of Physics State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina Correlations between nuclear landscape boundaries and neutron-rich r-process abundances numbers: 2110Re2160Cs2160Ev Motivated by the newly observed 39 Na in experiments, systematic calculations of global nuclear binding energies with seven Skyrme forces are performed. We demonstrate the strong correlation between the two-neutron separation energies (S2n) of 39 Na and the total number of bound nuclei of the whole nuclear landscape. Furthermore, with calculated nuclear masses, we perform astrophysical rapid-neutron capture process (r-process) simulations by using nuclear reaction code TALYS and nuclear reaction network code SkyNet. r-process abundances from ejecta of neutron star mergers and core-collapse supernova are compared. Prominent covariance correlations between nuclear landscape boundaries and neutron-rich r-process abundances before the third peak are shown. This study highlights the needs for further experimental studies of drip-line nuclei around 39 Na for better constraints on nuclear landscape boundaries and r-process. I. INTRODUCTION It is well known that studies of exotic nuclei close to drip lines are precious for understandings of the origin of elements in nature [1,2]. The rapid neutron capture process (r-process) involving high neutron densities is responsible for producing half of the elements heavier than iron and all elements beyond bismuth. The actual astrophysical sites for the occurrence of r-process is not definitely determined yet [1,3,4]. The developments of newgeneration rare isotope beam facilities around the world provide unprecedent opportunities to access the nuclear drip lines. For example, the Facility for Rare Isotope Beams (FRIB) is expected to be fully operational in 2022 and will be able to reach the neutron drip line up to nuclei with charge number Z = 40 [5]. However, it is almost impossible to reach the neutron drip line in heavy nuclear mass region by terrestrial experiments. Therefore, the examination on correlations between existing experimental evidences and theoretical predictions is crucial for better exportations. In a very recent experiment performed in RIKEN, 31 F and 34 Ne were reconfirmed to be the drip-line nuclei [6]. Surprisingly, this experiment also observed one event of 39 Na [6], indicting it is weakly bound and mostly likely it is the drip line of sodium. This is an exciting progress to reach the neutron drip line since the last observation of 40 Mg in 2007 [7]. It is known that different theoretical models can have remarkably divergent predictions about the neutron drip lines [8][9][10][11]. In contrast, the proton drip line has much reduced uncertainties. Therefore, the newly observed 39 Na provides a great opportunity to constrain theoretical models. It is interesting to know how small discrepancies in drip lines of light nuclei propagate to large uncertainties in drip lines of heavy nuclei. * [email protected] Consequently, the total number of bound nuclei in the nuclear landscape can be more accurately estimated. So far it was known that binary neutron star mergers (NSM) [12,13] and ejecta from core collapse supernova (CCSN) [14][15][16] are possible scenarios for r-process. Following the gravitational wave event GW170817 of NSM, the r-process kilonova electromagnetic transient was observed, resulting from the ejection of ∼0.05 solar masses of neutron-rich material [17]. These observations are becoming increasingly precise. NSM provides a much higher neutron density scenario to support a strong version of r-process to reach heavy elements such as uranium and thorium, while CCSN is associated with a larger electron faction Y e . Therefore it is expected that r-process via NSM is more sensitive to properties of neutron drip lines compared to that via ejecta of CCSN. There have been extensive studies of the impacts of uncertainties of nuclear inputs for r-process abundances in the literature [2,[18][19][20][21]. The r-process involves the neutron capture reactions, β-decays and fission properties. The fission is essential for appearance of the second peak (A∼160) in elemental abundance [22,23]. The uncertainties in (n, γ) reactions play a sensitive role. It was found that mass variations of ±0.5 MeV can result in up to an order of magnitude change in the final abundance [18]. The (n, γ) reaction is mainly determined by the nuclear masses and thus reliable predictions of nuclear mass by self-consistent microscopic framework are crucial. In principle, the ab initio calculations of nuclear drip lines are more reliable but it is problematic for heavy nuclei due to tremendous computing costs [24]. Semimicroscopic and phenomenological models can be precise for known nuclei but could be less reliable for explorations. As a suitable tool, the density functional theory with high-precision effective interactions are versatile for reasonable descriptions of global finite nuclei and neutron stars, including exotic structures and dynamics of halo nuclei, and nuclear fission [8,[25][26][27][28][29][30][31][32]. Previously the properties of 39 Na and neighboring drip line nuclei have been studied [32]. The subsequent combined constrains on the whole nuclear landscape and r-process are expected. Compared to earlier studies of r-process by focusing on the impact of uncertainties of nuclear inputs [18], the aim of this work is to examine the correlations between theoretical discrepancies in 39 Na, nuclear landscape boundaries, and r-process abundances based on several effective nuclear forces. Firstly, the global nuclear masses are calculated with the Skyrme Hartree-Fock-Bogoliubov (HFB) framework [33]. In particular, the results are evaluated with the existing evidence of the drip line nucleus 39 Na. This results in very different total number of bound nuclei and r-process paths. With the calculated nuclear masses, the (n, γ) reaction rates are obtained with TALYS [34]. The updated reaction rates are merged into REACLIB database [35], and then the r-process simulations are performed with SkyNet [36] which interfaces with REACLIB. Finally, the covariance correlations between r-process abundances and nuclear landscape boundaries are analyzed. II. THE THEORETICAL FRAMEWORK Firstly, the global nuclear masses at ground states are calculated by the Skyrme HFB approach with the parallel scheme. The HFB calculations are performed with the HFBTHOv3.00 solver [33], in which wavefunctions are presented by the basis expansion of 22 harmonic oscillator shells. The default oscillator length b 0 = /mω 0 , where ω 0 = 1.2 × 41/A 1/3 . For each nucleus, the ground state is determined by computing several quadrupole deformations from β 2 =−0.5 to 0.5, in case shape coexistence present. In HFB calculations, seven Skyrme type effective forces have been adopted. SkM * force has good surface properties and has been widely applied in descriptions of fission [37]. SLy4 force has been widely used in descriptions of neutron-rich nuclei and neutron stars [38]. SLy4 force is a refitted force that improves global descriptions of binding energies compared to the original SLy4 [30]. UN-EDF0 has been well optimized for descriptions of global binding energies with a high precision [39]. In addition, we speculate that a single density dependent term in standard Skyrme forces is not sufficient for the Skyrme force to simulate many-body correlations. The extended SLy4E global [30], SkM * ext1 and UNEDF0 ext1 forces [28] with an additional high-order density-dependent term are also adopted. In the particle-particle channel, a densitydependent pairing interaction has been adopted [29,40], V pair (r) = V 0 1 − η ρ(r) ρ 0 γ ,(1) where ρ 0 is the nuclear saturation density and we adopt the constants as η = 0.8 and γ = 0.7. The pairing strengths V 0 are fitted to the neutron gap of 120 Sn of 1.245 MeV for different Skyrme forces. The pairing gaps could be very different towards drip lines by using different pairing interaction forms. The resulted pairing gaps at the neutron drip lines are between the surface pairing and the mixed pairing [29]. This is a reasonable choice because the pairing gaps obtained with the surface pairing interaction are too large toward the neutron drip lines if the pairing strength is invariant for stable and weakly bound nuclei. The global binding energies of odd-A and odd-odd nuclei are obtained by the average pairing gap method after even-even nuclei are calculated with the HFB approach [8,10,41]. Secondly, we compute neutron capture rates with the TALYS code [34] and the calculated nuclear masses. The neutron capture rate is sensitive to the neutron separation energy [18]. Calculated masses are used in TALYS when no experimental masses are available. The reaction rates are calculated at 24 temperatures ranging from T 9 = 0.1 to 10 GK. The reaction rates λ are converted to coefficients a 0 ∼ a 6 in REACLIB format [35], λ = exp(a 0 +a 1 T −1 9 +a 2 T −1/3 9 +a 3 T 1/3 9 +a 4 T 9 +a 5 T 5/3 9 +a 6 InT 9 ),(2) where a 0 ∼ a 6 are obtained by the least square fitting method and next we updates the REACLIB data. The inverse (γ, n) reaction rates are calculated with the detailed balance [36]. In this work, we replaced 3825 (n, γ) reaction rates for targets with 10 ≤ Z ≤ 83 and 2453 (n, γ) reaction rates for targets with 84 ≤ Z ≤ 112 in REA-CLIB. It was reported that r-process abundances are less sensitive to uncertainties of β-decay rates compared to neutron capture rates [20]. The present r-process nucleosynthesis calculation includes 7836 nuclear species and 95051 reactions rates. In SkyNet, the nuclear statistical equilibrium (NSE) is adopted for all strong reactions when T 9 ≥ 7.0 GK [36]. The NSE is calculated with a given temperature, density and Y e in SkyNet. This is different from WinNet and XNet in which inverse rates taken from REACLIB are not completely consistent with NSE [36]. Finally, the abundance evolution is calculated with SkyNet, which actually solves the reaction network equations, i.e., the coupled first-order non-linear ordinary differential equations, with a given set of rates [36]. The initial NSE abundances are obtained with given temperature T , entropy S and Y e . The initial density ρ is related to entropy that is proportional to T 3 /ρ. After the numerical convergence is obtained at the evolution time of 10 9 s (T ≈3×10 5 K, Y e ≈0.4), the final abundance are obtained by the sum over all reaction species. In this work, for the ejecta of NSM, the initial temperature is taken as 7.1 GK; Y e is taken as 0.03 (within ranges suggested in [13]); and initial density is taken as 2.2×10 11 g cm −3 (S=2.8 k B /baryon). For the ejecta of CCSN, the initial temperature is taken as 10 GK; Y e is taken as 0.2 according to [16,42]; and initial density is taken as 2.0×10 8 g cm −3 (S=10 k B /baryon). The density expansion timescales of the ejecta are taken as 1 ms and 20 ms for NSM and CCSN respectively. The combination of very low Y e and rapid expansion timescale guarantees the occurrence of a strong r-process [1]. It is difficult to reproduce the solar r-process abundances by only one r-process scenario. SkyNet is a flexible modular library and has been successfully used for nucleosynthesis calculations in all astrophysical scenarios [36]. For example, very recently, Jin et al. have investigated that the enhanced triple-α reaction reduces proton-rich nucleosynthesis in supernovae using SkyNet [43]. The recent experiment on 39 Na has attracted great interests for theorists [32,45]. 39 Na has a magic neutron number of N =28 but has a well prolate deformation and a deformed halo structure [32]. The observation of 39 Na provides a good opportunity for examination of various nuclear mass models. In Fig.1, the two-neutron separation energies S 2n of 39 Na are calculated with seven Skyrme-type forces. One can see SkM * , SkM * ext1 , UN-EDF0 and UNEDF0 ext1 forces could reproduce the existence of 39 Na, while three SLy4-class forces obtain negative S 2n . SkM * gives the largest S 2n of 39 Na and predicts that 41 Na is the drip line nucleus. Correspondingly, we performed global calculations of nuclear binding energies from Z=8 to Z=120 with seven Skyrme forces. The total number of bound nuclei of the nuclear landscape from Z=8 to Z=120 ranges from 7105 to 8761 with different Skyrme forces. Generally, we see that the Skyrme force obtains a large S 2n of 39 Na also predicts a large number of bound nuclei. The S 2n of 39 Na is strongly correlated with the total number of bound nuclei N b , with a correlation r=0.947. The linear regression gives N b ∼ N (b + aS 2n , σ 2 ), in which a=638.5, b=7789.7, and σ=199.9. Once we know the experimental S 2n of 39 Na, we can immediately get a stringent prediction of the total number of bound nuclei of the nuclear landscape according to this linear regression. Fig.2 displays the calculated nuclear landscape boundaries with seven Skyrme forces. Large uncertainties in nuclear landscape boundaries are shown in neutron drip lines. Furthermore, it can be clearly seen that uncertainties of boundaries in light and medium mass region are small but propagate to remarkable uncertainties in boundaries of heavy and superheavy mass region. The consistency between the S 2n of 39 Na and landscape boundaries is shown. SkM * results in the furthest extension of neutron drip line, while SLy4 results in the nearest boundaries. UNEDF0 results are close to that of SkM * . SkM * ext1 boundaries are between SLy4 and SkM * , UNEDF0 results. The recent Bayesian mixing of eleven mass models infers that the total number of bound nuclei is 7708±534 [41]. This Bayesian-mixing inference is very close to the SkM * ext1 prediction of 7671 as constrained by newly observed 39 Na. Present calculations employ the HO basis while coordinate space calculations should be more accurate but are too costly. For example, with SkM * ext1 , S 2n of 39 Na by calculations in HO basis [33] and in coordinate space [46] are 0.23 MeV and 0.27 MeV, respectively. It should be noted that SkM * systematically overestimates binding energies of neutron-rich nuclei [8,28]. Thus SkM * is expected to overestimate the extension of neutron drip line and its prediction can be seen as an upper limit of nuclear landscape boundaries. In the literature, similar conclusions can be obtained that SkM * gives the largest number of bound nuclei while SLy4 gives the smallest number of bound nuclei [41]. The symmetry energy a sym at the saturation density ρ 0 may paly a role. However, the extended SkM * ext1 has a close a sym to that of SkM * . It has been pointed out that the symmetry energy at 2 3 ρ 0 (0.11 fm −3 ) is strongly correlated with the neutron drip line location [10]. Indeed, the symmetry energies at subsaturation (0.11 fm −3 ) are 26.90, 26.54, 26.49, 25.69, 24.70, 24.37, 24.31 MeV for SLy4 , SLy4 global , SLy4, SkM * ext1 , UNEDF0, UNEDF0 ext1 , SkM * , respectively. These are strongly correlated with the total number of bound nuclei N b , with a correlation r=−0.989. This exactly verified that the total number of bound nuclei is correlated with symmetry energy at subsaturation. We pointed out that SkM * ext1 is a very reasonable force to describe the drip line nuclei around 39 Na and the nuclear landscape boundaries. The associated r-process paths vary with different models, which is defined as S 2n ≈2.0 MeV [8,10]. The kink patterns of r-process paths and boundary lines appear around neutron magic shells. Generally, the boundary lines have strong odd-even effects. For each isotope, the number of bound nuclei N drip can be determined as a function of charge number Z. In Fig.2, for different Skyrme forces, the uncertainties in N drip (Z) are particularly large just after the neutron magic number while . The black squares denote stable nuclei and green squares denote the experimental observed nuclei according to [44]. become much reduced towards the next neutron magic number. This feature is expected to impact the r-process uncertainties. The calculated final r-process abundances from ejecta of NSM and ejecta of CCSN are displayed in Fig.3, based on seven Skyrme forces. It was known that the prominent abundance peaks around A∼130 and A∼195 are related to the neutron shells at N =82 and 126 respectively [47]. Generally, the resulted uncertainties in NSM scenario are considerably larger than that from CCSN. In Fig.3(a), the most significant uncertainties appear around A∼184. The series of SLy4 forces result in a deep valley in the abundance. On the other hand, SkM * produce highest abundance around A∼184. It was explained that this valley is related to nuclear shape transitions in SLy4 calculations [2]. In our results, it seems that the valley varies systematically corresponding to nuclear boundary extensions of nuclear forces. In other regions, SLy4 gives slightly larger abundances than others around A∼140 but smaller abundances around A∼130. In CCSN scenario, the abundances are larger than that of NSM for A<120, but much smaller in the region A>195. This is reasonable that high neutron density scenario is required to produce the heavy and superheavy elements. In both NSM and CCSN cases, the position of the peak around A=195 is not well reproduced but shifted to slightly heavier masses. In addition, the peaks around A=195 are all overestimated in NSM. Similar features of the third peak have also been shown in other r-process simulations [2,22]. For detailed analysis of r-process evolutions, the abundances during freeze-out are also displayed in Fig.4. The abundances at 1.2 s of NSM and abundances at 0.72 s of CCSN are shown. In NSM abundance, the significant uncertainties around A∼182 present in the early phase, indicating that the dominate cause is from neutron capture rates close to neutron drip lines. It can be seen that the position of third peak in NSM is reproduced at 1.2 s but shifted slightly to heavier masses in Fig.3. Indeed, it has been pointed out that the late neutron captures have a direct effect on the final position of the third peak, with neutrons released from fission of heavier nuclei [22]. In NSM, the first peak is not yet produced at freeze-out and late fission fragments are essential to reproduce the first peak around A∼130. Note that both N =82 neutron shell and Z=50 proton shell play a role in the first peak. The evolution analysis indicates that Z=50 shell is responsible for the overestimated abundances around A∼135 in NSM. The role of Z=50 proton shell is less significant in CCSN since less heavier neutron-rich nuclei beyond 132 Sn contributed. The CCSN can reach freeze-out more quickly with less neutron seeds and the role of fission is not significant. Generally the freeze-out abundances are much irregular and have strong odd-even effects in contrast to final abundances. The β decays and β-delayed decays in late phases would smooth out the abundances. Finally, the statistical analysis is performed to look into the correlations between neutron drip lines and rprocess abundances. The covariance correlation matrix is shown in Fig.5. In the correlation analysis, the logarithm of abundances log(Y (A)) in terms of nuclear mass A is adopted. For the other side, the relative value N R (Z)=N b (Z)/Z is used, where N b (Z) is the number of bound isotopes of each charge number Z. The relative uncertainties emphasize the correlations between drip-line light nuclei and r-process since drip-line heavy nuclei are not likely accessible. The correlation matrix is calculated as, Corr[logY (i), N R (j)] = 1 6 7 k=1 [logY (i, k) − logY (i)] · [N R (j, k) − N R (j)] σ[logY (i)] 2 σ[N R (j)] 2 where k denotes the results of seven Skyrme forces and σ 2 denotes the variance. In the NSM case, we found strong positive correlations between the A∼180 abundance and neutron drip lines. This demonstrated that SLy4 with least extended nuclear boundaries would result in the underestimated r-process abundance around A∼180. This correlation is not an accident. The correlation matrix points out that boundaries of some isotopes are especially important. For example, the drip lines at Z=11-13 are important, and the next is Z=18. The analysis of evolution movies (see Supplemental Material) shows distinct features between SkM * and SLy4 in the early phase. The r-process with SkM * runs very quickly to heavy masses and considerable abundances are already accumulated just at the left side of the neutron magic number. It is understandable that SkM * with most extended boundaries has large early neutron capture rates. In contrast, SLy4 obtains much less abundances before the third peak due to much less early abundances just before N =126. The in-between SkM * ext1 obtains reasonable abundance in the NSM scenario. In all cases, it is surprising to see that there is no correlation between rprocess abundance and neutron drip lines around proton shells at Z=50 and 82. The early abundances of NSM has similar but smaller correlations in Fig.5(b) due to larger variances, compared to that of final NSM abundance in Fig.5(a). Fig.5(c, d) shows that CCSN cases have no significant correlations between neutron drip lines and abundances around A∼180 in the less neutron-rich environment. There are some negative correlations in the transitional region around A=150∼160. For some regions, such as the peaks around 130 and 195, there is no strong statistical correlations, since shell effects are dominated. It is encouraging that the statistical analysis can provide reasonable clues. In fact, r-process evolution is so complex that a big data net analysis is inspiring from a different perspective [48]. IV. SUMMARY In summary, we studied S 2n of 39 Na with seven Skyrme forces to constrain the neutron drip lines. We found strong linear correlation between S 2n of 39 Na and the total number of bound nuclei. The in-between SkM * ext1 predicts 7671 bound nuclei of the nuclear landscape, which is very close to the recent Bayesian mixing result. Our key motivation is to study the uncertainty propagation from neutron drip lines of light nuclei to heavy nuclei, which is crucial for r-process simulations but not accessible by terrestrial experiments. Based on obtained nuclear masses with different Skyrme forces, r-process abundances from ejecta of NSM and CCSN are calculated using the reaction rate code TALYS and reaction network code SkyNet. We see large uncertainties in NSM abundances before the third peak. Further covariance analysis indicate that the abundance uncertainties are strongly correlated with the extension of neutron drip lines. SLy4 predicts the least extended nuclear boundaries and results in the valley in abundances before the third peak. The statistical analysis shows that neutron drip lines of some isotopes are especially important to constrain r-process in neutron-rich environments. Our study highlights the further experimental study of S 2n of 39 Na would be very needed. In contrast, the r-process in CCSN is not sensitive to the neutron drip lines. Currently, the understandings of r-process still need comprehensive and accurate nuclear inputs, in particular, reliable fission predictions. The statistical analysis can provide reasonable clues and big data analysis is a promising perspective. It is reciprocal to develop highly accurate effective nuclear forces for consistent modelings of Equation of State, drip line nuclei, and nuclear reactions, for better exportations of nuclear astrophysics at extreme conditions. FIG. 1 . 1Calculated S2n of 39 Na with seven Skyrme forces and the corresponding total number of bound nuclei from Z=8 to 120. The shadows show the confidential interval at 95%. FIG. 2 . 2The calculated nuclear landscape boundaries with seven Skyrme forces (see text) FIG. 3 . 3Calculated r-process final abundances as a function of nuclear mass A with seven Skyrme forces (see text for calculation details). The r-process abundances of NSM scenario (a) and CCSN scenario (b) are displayed. The solar r-process abundance in black dots is also shown for comparison. FIG. 4 . 4Similar to Fig.3 but for r-process abundance before freeze-out. For NSM (a) and CCSN (b), the abundances are obtained at 1.2 s and 0.72 s, respectively. FIG. 5 . 5The calculated correlation matrix between the rprocess abundance and the number of bound isotopes. The results for NSM final abundance (a), NSM freeze-out abundance (b), CCSN final abundance (c), CCSN freeze-out abundance (d) are shown. In the correlation matrix, the logarithm of abundances log(Y (A)) and the relative value of N b (Z)/Z are used, where N b (Z) is the number of bound isotopes for each Z. ACKNOWLEDGMENTSWe are grateful for useful discussions with J. Lippuner, L.W. Chen, Y.S. Chen and F.R. Xu. . J J Cowan, C Sneden, J E Lawler, A Aprahamian, M Wiescher, K Langanke, G Martínez-Pinedo, F K Thielemann, https:/link.aps.org/doi/10.1103/RevModPhys.93.015002Rev. Mod. Phys. 9315002J. J. Cowan, C. Sneden, J. E. Lawler, A. Aprahamian, M. Wiescher, K. Langanke, G. Martínez-Pinedo, F. K. Thielemann, Rev. Mod. Phys. 93 (2021) 015002. . D Martin, A Arcones, W Nazarewicz, E Olsen, https:/link.aps.org/doi/10.1103/PhysRevLett.116.121101Phys. Rev. Lett. 116121101D. Martin, A. 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[ "Supersymmetric Higgses beyond the MSSM: An update with flavour and Dark Matter constraints", "Supersymmetric Higgses beyond the MSSM: An update with flavour and Dark Matter constraints" ]	[ "F Boudjema ", "G Drieu ", "La Rochelle ", "\nLAPTh †\nUniv. de Savoie\nCNRS\nB.P. 110, Annecy-le-VieuxF-74941France\n", "\nUMR 5108 du CNRS, associéeà l'Université de Savoie\n\n" ]	[ "LAPTh †\nUniv. de Savoie\nCNRS\nB.P. 110, Annecy-le-VieuxF-74941France", "UMR 5108 du CNRS, associéeà l'Université de Savoie\n" ]	[]	Spurred by the discovery of a boson resonance at the LHC as the result of the search for the Standard Model Higgs, we pursue our investigation of the properties and signatures of Higgses in an effective supersymmetric scenario that goes beyond the usual MSSM. Such scenarios were first introduced to alleviate the naturalness problem of the MSSM Higgs and are found to have a very rich phenomenology that allows departures from the Standard Model in the production rate of the Higgs in many of the search channels. We now include the constraints from flavour observables in particular the rare decays B → X * s γ and Bs → µ + µ − including the recent measurement from LHCb. We also address the issue of Dark Matter and its impact on Higgs physics. In particular, we incorporate the latest data from XENON100 on the spin independent direct detection rates. These turn out to be powerful constraints, especially if one also imposes that the observed thermal relic density is obtained. We also study models with a low abundance that can more easily evade the direct detection rates. We study the impact of the flavour and Dark Matter observables on the production rates of the Higgs at the LHC, and their correlations in the diphoton, diphoton+jets and 4 leptons. We also comment on the other channels.	10.1103/physrevd.86.115007	[ "https://arxiv.org/pdf/1208.1952v1.pdf" ]	119,198,304	1208.1952	e2692b3384dba3a24830bc6d4fb35032e2bfdd60	Supersymmetric Higgses beyond the MSSM: An update with flavour and Dark Matter constraints May 2, 2014 9 Aug 2012 F Boudjema G Drieu La Rochelle LAPTh † Univ. de Savoie CNRS B.P. 110, Annecy-le-VieuxF-74941France UMR 5108 du CNRS, associéeà l'Université de Savoie Supersymmetric Higgses beyond the MSSM: An update with flavour and Dark Matter constraints May 2, 2014 9 Aug 2012 Spurred by the discovery of a boson resonance at the LHC as the result of the search for the Standard Model Higgs, we pursue our investigation of the properties and signatures of Higgses in an effective supersymmetric scenario that goes beyond the usual MSSM. Such scenarios were first introduced to alleviate the naturalness problem of the MSSM Higgs and are found to have a very rich phenomenology that allows departures from the Standard Model in the production rate of the Higgs in many of the search channels. We now include the constraints from flavour observables in particular the rare decays B → X * s γ and Bs → µ + µ − including the recent measurement from LHCb. We also address the issue of Dark Matter and its impact on Higgs physics. In particular, we incorporate the latest data from XENON100 on the spin independent direct detection rates. These turn out to be powerful constraints, especially if one also imposes that the observed thermal relic density is obtained. We also study models with a low abundance that can more easily evade the direct detection rates. We study the impact of the flavour and Dark Matter observables on the production rates of the Higgs at the LHC, and their correlations in the diphoton, diphoton+jets and 4 leptons. We also comment on the other channels. Introduction The July 4th 2012 announcement by both the ATLAS [1,2] and CMS [3] Collaborations of a 5σ resonance as a result for the search of the standard model Higgs boson may well correspond to the discovery of the last missing piece of the standard model, SM. The SM should then be elevated to the status of a theory especially in view of the fact that the mass of this resonance is in accord with the indirect limits from precision measurements. If this particle turns out to be indeed the Standard Model Higgs with perhaps no new particles being discovered, the naturalness argument that has motivated the construction of so many beyond the SM (BSM) models will remain a mystery. These BSM constructions also aimed at providing a dark matter (DM) candidate for which the LHC might have provided some circumstantial evidence. Probing the nature of the newly discovered resonance will certainly take time. Moreover the present data, though compatible with a SM Higgs interpretation when all analyses are combined, seems to deviate from the prediction of the SM in some channels. In particular the 2γ final state channel points to a signal rate that is higher than what is predicted in the SM. Other couplings, like the crucial coupling to W W/ZZ require more data taking. As soon as the July results were made public, there has been a flurry of analyses aiming at fitting the Higgs couplings in a model independent way [4]. Other analyses concentrated on specific models. Most prominent among the latter analyses was the status of the MSSM [5,6] (MSSM for the Minimal Supersymmetric Standard Model). Unfortunately the mass of the resonance, ∼ 125 GeV, is very difficult to reconcile with naturalness in the MSSM, see for example [7]. Moreover, unless one appeals to quixotic [8] choices of the parameters it is impossible for the 2γ rate to be higher in the MSSM than in the SM. Yet, it is hard to give up the idea of supersymmetry, not only because of the DM candidate. One must then seek models beyond the MSSM. The NMSSM (Next to MSSM) for example has been shown to fit better the data [9,10,11,6]. Specific extended versions with ultraviolet (UV) completion have also been proposed. It is therefore important to follow an effective theory approach that encapsulates the effects of a large class of specific UV completed scenarios beyond the MSSM and to parametrise the implications of some unknown model based on the symmetry of the low lying theory, namely the MSSM. The main motivation for such an approach that keeps the same field content as the MSSM, has been that the addition of a few operators in the Higgs sector [12,13,14,15,16,17,18] alleviates very easily the finetuning and naturalness problem. One no longer requires heavy stops to have the lightest Higgs, h, weigh 125 GeV. In fact, before the LHC Higgs data of 2011 these generic BMSSM models accommodated a lightest Higgs as heavy as 250 GeV. The phenomenology of these models is very rich, since the properties of the Higgses can change drastically. Such a set-up could serve for an analysis of the two Higgs doublet model [19,20,21]. Nonetheless the set-up is more restrictive, not only because of the contribution of the higher dimensional operators to Higgs observables but also because the superfield implementation means that the Higgsino sector is affected. There are then implications on dark matter observables and, considering the link between Higgs and heavy fermions, on flavour physics, in particular the rare B-decays B s → µ + µ − and B → X * s γ. LHCb [22] has for example set new stringent constraints on B s → µ + µ − . As far as DM detection is concerned, July 2012 has also seen XENON100 [23] set unprecedented bounds on the rate of direct detection, while the last few years have witnessed a measurement of the relic density of DM that has reached a precision of 3%. One has therefore, no doubt, entered an exciting era in probing the details of symmetry breaking and confront them with models that provide at the same time a dark matter candidate. In view of the LHC results and the improvements that are expected in the coming months and years on the reconstruction of the Higgs properties, an effective theory approach that generalises the usual MSSM and may encompass specific manifestations (extra singlets [24,25], extra triplets [26], U(1)', · · · ) is warranted. This paper is an update on our recent detailed analyses [17,18] that took into consideration all of the constraints on Higgs physics including the LHC 2011 data [27,28] the electroweak indirect precision measurements and other constraints such as t → bH + . Like in our previous analyses we do not aim at finding the best fits to the 2012 Higgs search/signal data for the effective parameters of the higher order operators, not because they are numerous, but because we consider such an exercise, considering the experimental uncertainty, to be rather premature. Instead, in the signal region with an alleged Higgs of 125 GeV, we will compute the possible signal strengths of the different channels together with the correlations between the different search channels. Although the mass for the alleged signal has narrowed around 125 GeV, we still present our results for the range 122 − 128 GeV. In this update we include the impact of the rare decays B s → µ + µ − and B → X * s γ on the signal strengths. We will then address the issue of dark matter, in particular the impact of the spin-independent direct detection constraints and then the relic density. Though some model dependence is introduced with DM, we will see that the constraints can shed important light on the nature of DM. In a first stage we will assume that the BMSSM lightest supersymmetric particle (LSP) accounts for all of DM and look how direct detection, in particular XENON100, restricts the parameter space and what consequences on the Higgs signals it brings. In a second stage we investigate which of these scenarios do indeed provide the observed relic abundance. In a third stage we review models where the abundance is low. Although these models can not account for all of DM, the direct detection rates can be more easily evaded. For such configurations we review the Higgs signal strengths. Very little has so far been done as regards flavour and Dark Matter in the BMSSM and certainly not from the point of view of the Higgs signal. Prior to the Higgs signal results, flavour observables in the BMSSM have been studied in [29]. Dark Matter, in particular the relic density, has been considered in [30,31,32,33]. In all these studies only the case of dim-5 operators has been considered. In our study we include the full set of operators up to dim-6. The paper is organised as follows : in section 2 a brief description of the model and the prominent experimental features are presented. Some technical issues having to do with the calculation of the different observables are reviewed in this section. Section 3 implements the constraints from B s → µ + µ − and B → X * s γ and study the consequences on the Higgs observables. With the flavour constraints taken into account, we make the link with DM in section 4. First, we look at the effect of direct detection as set by XENON100 (2012) assuming the model accounts for all of DM. We then impose the bound set by the observed relic density. We finally consider models with an abundance which is lower than what is observed. Section 5 summarises our conclusions. 2 Description of the model Since the interested reader will certainly learn the details of the model in [17] and the references therein, we will only sketch a quick overview. The model is within the effective theory approach where the effects of extra degrees of freedom beyond those that describe the MSSM are taken into account through higher order operators. The scale, M , that enters these higher order operators is the heavy scale of the New Physics (beyond the MSSM). We set this scale at M = 1.5 TeV,(1) Since our low energy theory is supersymmetry (precisely the MSSM), L low energy = L MSSM , these effective operators will be products of superfields. Given that we are not assuming anything on the UV completion of the theory, those operators can be any gauge and super Poincaré invariant product of superfields. Since we concentrate on Higgs phenomenology, we only consider operators involving the Higgs superfields. In any case, in the same way that radiative corrections to the Higgs in the MSSM have a very big impact, we suspect that these operators can change the Higgs phenomenology in an important way. As we [17,18], and others [12,13,14,15,16], have done so far, we include the dim-5 (superpotential) and dim-6 (Kähler) operators. They are the following : SU (2) gaugino mass) to 300 GeV, M 1 (the U (1) bino mass) is fixed by the universal gaugino mass relation M 1 = 5 3 tan 2 θ W M 2 M 2 /2, and M 3 = 800 GeV (the SU (3) gaugino mass), with cos 2 θ W = M 2 W /M 2 Z . When including DM observables we will keep M 2 = µ = 300 GeV fixed but will scan on M 1 in the range M 1 ∈ [50, 300] GeV in order to generate all possible mixtures of higgsino-bino for the neutralino LSP. Indeed the nature of the LSP (bino, wino, higgsino) is a key ingredient for dark matter observables. The difference between scenario A and B is in the third generation of squarks. • In model A we take M u3 R = M d3 R = M Q3 = 400 GeV, these are respectively the soft masses for the up, down singlet and the doublet. The tri-linear stop mixing is A t = 0. This is a benchmark where the stops are light (as dictated by naturalness considerations) but do not affect much the Higgs loop couplings to gg and γγ since they are mass degenerate. • In model B we consider the case mt 1 = 200 GeV together with mt 2 = 600 GeV and a maximal mixing in the stop sector with sin 2θ t = s 2θt = 1. This scenario exemplifies the role of stops in modifying the Higgs couplings to gg and γγ as compared to the SM expectations. In [18] we covered a larger spectrum oft 2 and considered also s 2θt = −1. Fits to recent LHC data including flavour constraints in a MSSM set up with light stops have just appeared, see [35]. Note however, as we already pointed out in the introduction, such a natural set up has some tension with the rather heavy Higgs mass, which is not the case in the BMSSM. Snapshot of the BMSSM Once the effective operators are plugged in the Kähler potential, the superpotential and the susy-breaking potential we can derive the actual alterations to the interactions of the physical fields themselves. The most salient feature that has been discussed thoroughly in the literature (see [12,36,34]) has to do with the substantial increase in the lightest Higgs mass, m h , compared to the MSSM case. In these scenarios m h can be raised up to 250 GeV. Although such high masses are no longer an issue in view of the latest LHC results, this does show that contrary to the MSSM a mass of 125 GeV for the lightest Higgs can be very easily attained, in particular without demanding too much from the stops. What is important and is still of crucial importance in view of the latest trends from the LHC is that the mixing and couplings of the BMSSM Higgses can differ substantially from those of the Standard Model. As we show in [18], these deviations from the SM are however not haphazard despite the relatively large number of parameters that the BMSSM introduces. There exists strong correlations between different searches and signal channels. Some technicalities on the computation The impact of these operators on flavour physics and Dark Matter has previously only been addressed for the dim-5 operators arising from the superpotential, see [29] for the flavour observables and [30,31,32,33] for DM observable. However their impact on a Higgs with mass 125 GeV was not studied. The implementation of both dim-5 and dim-6 operators on flavour physics as well as the relic density and direct detection is performed here for the first time. As we outline in a previous publication, the implementation of the effect of all the higher order operators is quite tricky. We perform this with an automated tool from the superfield level to the physical states. We then pass the newly created model file to external codes such as micrOMEGAs [37,38,39,40] for the dark matter observables for example. For our study on the Higgs observables all these changes were a rescaling of the standard model couplings. Flavour Observables For the flavour observables all what affects the Yukawa sector is of relevance and hence the importance of these new contributions. For the calculation of observables which involve loop calculations one needs to be careful that these higher order contributions do not generate ultraviolet divergences. One could imagine that all contributions from the higher order operators could be naively counted as of a non renormalisable type. For example, in the calculation of B s → µ + µ − enters theχ − j φχ + i vertex which contributes to the penguin diagram, χ ± are the chargino fields and φ = h, H, A 0 . At first sight the contribution from the higher order operators exhibits a new Lorentz structure containing derivatives on the chargino field. However, it can be shown that these new structures can be removed by using the equations of motion. At the end, the net effect is fully taken into account by a rescaling of the MSSM coupling. This then permits to easily adapt the calculation performed in the MSSM, in this case [41,42]. In general all our calculations of the flavour and Dark Matter observables take the codes implemented in micrOMEGAs as a skeleton. Dark Matter Since our formulation stems from extra contributions involving Higgs superfields, the neutralino and chargino sector will be directly affected. This has an impact on the calculation of Dark Matter observables in particular within a higgsino configuration. The usual computation of the relic density and direct detection in the BMSSM is particularly affected when higgsino and Higgs are affected. For instance, for the relic density, processes involving Higgs final states such as hA 0 that occur when the neutralino is a mixed bino-higgsino get corrected by as much as 30% compared to the same MSSM point. In the Higgsino case, the new operators allow for larger mass splitting between the neutralino and the chargino. This helps evade more easily the LEP constraint on the chargino. Another possibility that opens up is that co-annihilation with a stop are more plausible than in the MSSM due to the fact that very heavy stops are no longer necessary to obtain a Higgs with mass 125 GeV. For other novelties in the computation of the relic density which however do not have a bearing on our study of the Higgs see [33,31,32]. Higgs observables at the LHC In order to use the results from the ATLAS and CMS collaborations, we have used the following ratios R XX = σ pp→h→XX σ SM pp→h→XX and R exclusion XX = σ pp→H→XX σ excluded 95% pp→H→XX ,(6) where XX denotes a particular final state (say the inclusive 2γ). σ excluded 95% stands for the 95% C.L. excluded cross-section reported by the collaborations with the 2011 data ([43, 44, 45]) : the reason why 2012 data for exclusion has not been used so far is that the most sensitive channels (notably φ →τ τ, φ = A 0 , h, H) have not been updated yet. In practice the R XX will be used in the signal case, to compare with the best fitμ -of the so called signal strength µ-given by the experiments. In eq. 6, h in the BMSSM will refer either to the lightest or heaviest CP-even Higgs. R exclusion XX will be used in the no-signal case as a measure of the sensitivity of the search, here H stands for all Higgses not contributing to a signal in the mass range 122 − 128 GeV. For R XX the most important channels so far are the inclusive 2γ, ZZ → 4l and the exclusive 2γ + 2jets. We simulate the ratio R γγ+2j as R γγ+2j = 0.15 σ VBF + 0.005 σ gg→h 0.15 σ SM VBF + 0.005 σ SM gg→h × BR γγ BR SM γγ (7) We checked that this parametrisation of σ γγ+2 jets when folded in with the SM cross sections for the LHC at 7 TeV [46] and taking into account the luminosity quoted by CMS reproduced quite exactly the number of selected events given by CMS [47]. We assume that this parametrisation that was verified to be excellent for m h = 120 GeV still holds to a very good degree in the range 122 < m h < 128 GeV. Note that with the 2012 data there exist three different 2γ + 2 jets channels (one for ATLAS and two for CMS) which correspond to three different efficiencies. Given the high statistical uncertainty we will however only consider one set of efficiencies as a representative of the channel. We must note that we impose constraints from the electroweak precision data. Therefore the slightest SU(2) custodial symmetry breaking effect is wiped out. The models have therefore R W W /R ZZ = 1. Impact of flavour on Higgs signals We first consider the impact of B s → µ + µ − , B → X * s γ and (g − 2) µ and look at how these observables can constraint the rates for Higgs production in the different channels •B s → µ + µ − We apply the latest bounds on B s → µ + µ − B s → µ + µ − < 4.7 10 −9 LHCb [22] •B → X * s γ For B → X * s γ we take B → X * s γ = 3.55 ± 16 ± 9 × 10 −4 Heavy Flavour Averaging Group [48] where we have required the prediction to stay within two sigma deviations from the mean value. We have taken the SM prediction to be BR(B → X * s γ) = 3.27 10 −4 (see [49]). Note that any extra contribution beyond the SM is rather small. [50,51]. •(g − 2) µ . ∆a µ = a exp. µ − a th. µ = (2.8 ± 0.8) 10 −9 With the values of the MSSM parameters that we have taken (heavy sleptons) there is no effect from (g−2) µ , either in terms of constraining the parameter space or alleviating the apparent 2σ discrepancy with the SM. For B s → µ + µ − , the effect is sensitive to quite high values of t β , t β ≥ 20 and small values of M A 0 , M A 0 ≤ 150 GeV. This set of parameter space is constrained by B s → µ + µ − is in fact no longer allowed by the Higgs exclusion limits set by the LHC itself in the analysis φ →τ τ , φ = h, H, A 0 [45], see later. Since the latter are folded in our Higgs analysis, B s → µ + µ − does not add much. Note also that the effect of the dim-5 operators in particular are more important for small values of t β , therefore the BMSSM does not impact much more than the MSSM. B → X * s γ is much more sensitive to the stop sector. We have observed that in our case most of the supersymmetric corrections are brought by the Wilson operators O 7 = (s L σ µν b R )F µν : it is driven by the charged Higgs loop on one side and the stops-charginos loop on the other. The former depends on the value of M A 0 and to a lesser extent on t β . The latter shows a t β -enhanced term whose size is driven by s 2θt and ∆mt = mt 2 −mt 1 . It is sensitive to the sign of µ. In this study we take µ > 0. Since the experimental value of B → X * s γ is close to the SM prediction, it means that the supersymmetric contribution must be quite small. This will drive us either to a small t β region, a small mass splitting ∆m between stops or a small mixing s 2θt 0. The last two instances characterise model A. In model B, where we have a light stop mt 1 = 200 GeV and a heavier one with mt 2 = 600 (GeV), we expect B → X * s γ to be more constraining. Impact on Higgs observables The signal and correlations that we will show have 0 < R γγ < 4. This is a very generous band which allows to read the predictions in a most transparent manner. The reader can easily select a particular range. We prefer not to select a narrow range since the measurements on the different rates will evolve and get more precise. In the case of model A with degenerate stops with mass of 400 GeV, the correlations between the signal strengths in the γγ, ZZ and γγ + 2j are unaffected by the flavour constrained. Nor is the range of the signal strengths further reduced by the flavour constraints. We note that in this particular case of degenerate stops the signals are all strongly correlated with R γγ ∼ R ZZ and R γγ+2j < R γγ . The correlations are shown in Fig. 1. For R γγ ∼ 2 we can obtain the following ranges for the orther channels: R ZZ = 2 − 2.05 and R γγ+2j = 1.6 − 1.8, R bb = 0.5 − 0.7, R τ τ = 0.6 − 1. It is important to point out that for this particular value, R γγ ∼ 2, the τ τ and bb need not be dramatically reduced. Much higher signal rates in the γγ channels are only possible with very much reduced of the latter two channels. Model A facing flavour What is not visible in the projections of Fig. 1 is the fact that the flavour observables do eliminate quite a few configurations. Indeed we obtain the constraint M A 0 > 200 GeV and t β < 20, see Fig. 2. These bounds come from the B → X * s γ observable. Indeed at low M A 0 the contribution of the charged Higgs loop is enhanced. This can in principle be cancelled by t β -enhanced terms from the stop sector whose contribution has an opposite sign to the one of the charged Higgs loop, however since there is practically no effect of the stop sector in the model A the t β dependence of B → X * s γ is very mild. Thus, in order to reproduce the correct value for B → X * s γ with M A 0 < 200, one would require t β > 20, a value which is already excluded by the φ →τ τ search at the LHC. Since A 0 is pushed to M A 0 > 200 GeV. This means that the hypothesis of the heavy CP-even Higgs boson generating the signal is disfavoured. Such possibility was entertained prior to applying the flavour constraints [18]. Fig. 2 reveals that for t β ∼ 2 we get R γγ ∼ 1, while for t β ∼ 5 we obtain 0.5 < R γγ < 2. For higher values of t β a much large range for R γγ opens up. We therefore see that, despite the many higher order operators, precision measurements on the Higgs combined with flavour measurements can give a good measure of t β . We also see that in model A, values of R γγ ∼ 2 are possible for any value of M A 0 > 250 GeV, as long as t β > 5. R γγ < 0.5 would mean that the pseudo scalar mass is lighter than 400 GeV. Figure 3: Model A. Allowed points in the M A 0 − t β plane after imposing the Higgs searches constraints (red), the B s → µ + µ − bound (orange) and the B → X * s γ limit (black). Fig. 3 is very instructive. It reveals that B s → µ + µ − does not restrict the parameter space once the LHC limit on the search A 0 → τ τ has been imposed. On the other hand, B → X * s γ carves out a significant region of parameter space. Model B has been introduced in order to obtain the hierarchy R γγ+2j > R γγ > R ZZ by decreasing the contribution of the gluon fusion with respect to the W W fusion. This is obtained through strong mixing in the stop sector and with one of the stops relatively light. The underlying strong Yukawa coupling of the stops can therefore have an important impact on B observables in particular B → X * s γ. To illustrate this scenario we take the case of maximal mixing with s 2θt = 1, more moderate effects are obtained with smaller values of s 2θt . The effect of the B → X * s γ constraint is quite different from what we observed in model A. Indeed, while the contribution of the charged Higgs is still the same, with an important contribution for small M A 0 , there is now a significant contribution from the stop-chargino loop (since s 2θt = 1 and ∆mt is non-zero) which is moreover t β enhanced. Since the latter has an opposite sign to the Standard Model contribution, it will tend to decrease BR(B → X * s γ) as t β grows. The conclusion is twofold : first, the contribution of the charged Higgs can be cancelled by the effect of the stop-chargino loop. Therefore, in model B, on the one hand flavour constraints do not provide a lower bound for M A 0 , which is backed up by the results shown in Fig. 4. On the other hand, observe that M A 0 does not extend beyond M A 0 > 400 GeV, otherwise the compensation between the charged Higgs contribution and the stop contribution in B → X * s γ will not be effective. Second, for the cancellation in B → X * s γ to be effective and in order to control the stop-chargino loop, t β is restricted to be small (t β < 5) for any value of M A 0 . This is what is conveyed by Fig. 4. The range of t β is very much reduced compared to what we found in Model A. Moreover for the largest allowed values of M A 0 R γγ < 1. The fact that in this decoupling regime one does not recover the SM value is due to the reduction in σ(gg → h) brought up by the stops. This restriction to small t β makes it difficult to obtain a maximal suppression of the g hbb coupling that would lead to an increase in R γγ , this is why in Fig. 5, where we plot the allowed points in scenario B, we have much fewer points with R γγ > 2 than before applying the flavour constraints. We note however that solutions with R γγ > 2 can not be obtained with M A 0 > 250 GeV. With large values of M A 0 decoupling will set in. Still, there are configurations with R γγ ∼ 2 which exhibit the hierarchy R γγ+2j > R γγ > R ZZ/W W . With M A 0 < 250 GeV we can attain R γγ = 2 together with R ZZ = 1.6 − 1.8, R γγ+2j = 1.9 − 2.4, R bb = 0.3 − 0.6 R τ τ = 0.2 − 0.5, see Fig. 6. Observe that R γγ ∼ 2 corresponds to much lower rates for the bb and τ τ channels than in Model A, see Fig. 3. With M A 0 > 250 GeV, the increase in the bosonic final states (γγ, ZZ, γγ + 2j) is reduced. While the correlations are maintained with R γγ+2j /R γγ ∼ 1.3, R ZZ /R γγ = 0.8, we now have R γγ < 1.5 (for M A 0 > 250 GeV). Figure 5: Allowed regions in scenario B with s 2θt = 1 before (left panels) and after (right panels) applying the flavour constraints. We distinguish the case of heavy (M 0 A > 250 GeV) and light (M 0 A < 250 GeV) pseudoscalar masses. We plot here the features of a signal with m h = 125 GeV, that is to say the signal strength in the following channel : γγ (x-axis), ZZ (red points) and γγ + 2 jets (blue points). The results we have presented so far depend quite crucially on the value of µ, in particular the expansion in the effective operators is based on the ratio µ/M . The values of M 1,2 that determine the nature of the neutralino LSP have almost no impact on the Higgs observables we have studied. With M 2 ∼ 2M 1 and with M 1 > 70 GeV, invisible decays to dark matter neutralinos is not possible and the contribution of charginos to h → γγ is negligible. Direct detection has a more direct connection with Higgs physics, due to the contribution of Higgs exchange. The relic density requires the knowledge of more parameters. For instance, lighter sleptons would have an important impact. This is the reason why we first consider the constraint of direct detection on the Higgs observables. In doing so we will, in a first stage, assume that the density of Dark Matter is totally accounted for by neutralinos. Even if the relic density turns out to be outside what is measured by WMAP, Ωh 2 ∼ 0.1, one can always appeal to non thermal scenarios which can bring the relic density to the desired experimental value [52,53]. We evoke this possibility only as a way out not to include the impact of the relic density at this stage. In a second stage we compute the relic density within a standard thermal cosmological model and ask which scenarios can indeed be compatible with the correct relic density and pass the direct detection constraint. Models that do not pass the cold dark matter relic density constraint but give a value that is smaller than what is measured are acceptable at the expense that the neutralino does not account for the totality of DM in the Universe. Such possibilities are reviewed in a third stage. In this case, given a spin-independent cross section, the direct detection rate is smaller due to the smaller neutralino halo fraction. We assume that this fraction is the same as the one on cosmological scales which is set by the relic density. The observed relic density Ωh 2 is the result of the fit of ΛCDM whereas the direct detection rate is proportional to the number density of the neutralinos passing through the detector Ω χ 0 1 . In this case we reweight the result on the spin-independent cross section and look at the effect on the Higgs signal strength. Let us at this point recall some important differences between model A and model B as regards the DM candidate. In model B, the lightest stop weighs 200 GeV, therefore the LSP must be lighter. As a consequence, in these scenarios M 1 , M 2 can not be taken very high, with µ = 300 GeV we can not have µ M 1 , M 2 . Model B facing flavour Direct Detection We have used the latest (July 2012) XENON 100 [23] results on the spin independent cross-section of the dark matter candidate on the nucleus. We have used the routines of micrOmegas-2.4 [38]. In the framework we have chosen with squarks of the first and second generation being heavy and with the possibility that the Higgses of the model, including the pseudoscalar, can have mass below 300 GeV, the rates for direct detection can provide a further constraint on the parameters in the Higgs sector. The interplay between direct detection and flavour in the context of the MSSM was emphasized in [54]. The impact of direct detection depends also on the composition of the neutralino, of course. This composition is determined by the values of the µ parameter and the U (1) (M 1 ) and SU (2) (M 2 ) gaugino masses. The latter played practically no role in the properties of the Higgs and the rates at the LHC. What determines the cross section is the coupling of the higgses to the LSP neutralino and the coupling of the Higgses to the quarks. These effects will naturally be more important if the exchanged Higgs is not too heavy. For the coupling to quarks, high t β give the largest effect. We therefore expect that adding the direct detection limit will constrain the Higgses couplings to the LSP. The latter requires mixing between the Higgsino and the gaugino (M 1,2 , in our case essentially M 1 ) components. We first stick to the values of M 1 and M 2 that define models A and B. In this case with M 1 ∼ M 2 /2 = 150 GeV giving mχ0 1 ∼ 146 GeV for t β = 20, our benchmark points barely make it. We find that very few points pass the new direct detection test for both models. In fact as will be seen shortly, our benchmark choice for M 1,2 is borderline. This is not difficult to explain. Indeed, although the bino component is large (90%), there is nonetheless about 10% higgsino component. With the latest limits from XENON100, such combined mixing and therefore such configurations are almost ruled out both for model A and model B. The message is that XENON100 is now providing a very strong constraint (also on many MSSM implementations). We expect that in these scenarios reducing the amount of higgsino-gaugino mixing will help. M 1 needs to be much further removed from µ. We will therefore scan on M 1 so as to allow smaller values than our benchmark M 1 = 150 GeV. In this study we do not entertain the possibility of M 1,2 µ. In model B with mt 1 = 200 GeV a DM candidate is not possible, while for model A, M 1 would not extend above 400 GeV and therefore we will be in a quite mixed bino-higgsino configuration anyhow. With M 1 < µ, our scan covers M 1 : 70 − 300 GeV (we keep µ = M 2 = 300 GeV). The lower value of M 1 is taken so as to avoid possible invisible decays of the Higgs that we have not taken into account for our analysis (see however our paper [17]). We first assume that neutralinos account for all of dark matter. The flavour constraints are taken into account. Looking at Fig. 7 we see that we can find, in both model A and model B, points that pass the XENON100 (2012) but only for neutralino lighter than about 150 GeV. Our benchmark point with M 1 = 150 GeV was indeed borderline. Many configurations of the parameters including M A 0 , t β , are therefore excluded. This shows that assuming that the models account for the bulk of DM, the new limit from XENON100 (2012) are now extremely powerful. No doubt that future XENON1T [55] which will improve the sensitivity by at least an order of magnitude will either soon discover such models or will exclude all of them. This is a conclusion that should apply to all natural susy models with small enough µ (see for example [56]). The correlations between the different Higgs signal channels are, of course, unchanged. What is important to check is whether the signal strengths are affected. We find that the only change concerns Model A where direct detection now imposes R γγ > 0.5, see Fig. 7. After inspection we have found that direct detection now cuts on the small values of M A 0 in particular those with largest t β . In this case the couplings to bb of the Higgs are not so large and hence the reduction in R γγ is more modest. In model A , R γγ > 1 is possible (we even obtain R γγ ∼ 2) for all values of the neutralino mass in the considered range 60 − 150 GeV. This is not the case of Model B, where in the range 120 < mχ0 1 < 150 GeV, an enhancement of R γγ is not easy to find. Relic Density Combining the results of the 7-year WMAP data [57] on the 6-parameter ΛCDM model, the baryon acoustic oscillations from SDSS [58] and the most recent determination of the Hubble constant [59] one [60] arrives at Ωh 2 = 0.1126 ± 0.0036, where Ω is the density of cold dark matter normalised to the critical density, and h is the Hubble constant in units of 100 km s −1 Mpc −1 . This experimental results is very precise with only 3% uncertainty. However, it has been shown, in particular in supersymmetry, that such a precision was difficult to match on the theoretical side. Indeed the loop corrections can easily be higher than 10% on different processes. Despite the efforts to account for those contributions (see references [61,62,63,64]), it remains a challenge and is so far not implemented in the code we have used, micrOmegas-2.4. We will thus impose the value of the relic density within 15% uncertainty : Ωh 2 = 0.1126 ± 0.016.(8) Let us briefly sketch the main channels that enter the computation of the relic density in our scenarios: •χ 0 1χ 0 1 → ff : this is the most frequent case when the lightest neutralino is mainly bino-like. Though the cross section of this process is usually too small to respect the relic density constraint, it can be enhanced by an A 0 resonance, requiring M A 0 ∼ 2mχ0 1 . •χ 0 1t1 co-annihilation is also possible, in particular for Model B. We now investigate whether in all scenarios we studied and that pass the direct detection constraint one could still obtain the correct relic density within a standard thermal cosmological model. We will start with model B where it is easier to illustrate why we perform scans in steps of 10 GeV over M 1 . Model B with the correct abundance The good news is that it is possible to reproduce the correct thermal relic density and be in accord with the latest measurement from XENON100 (2012) over the whole range 60 < mχ0 1 < 150 GeV that passed the XENON100 (2012) limit, Fig. 8. It is important to observe that a scan over M 1 returns values for the relic density that span a range over orders of magnitude in Ωh 2 : 10 −4 − 10, with a small subset that leads to the correct relic density and includes configurations in accord with direct detection, see Fig. 8. Among the points that have passed the previous direct detection limit some are associated either with an overabundance or an underabundance. The figure does not include configurations where the spin-independent cross section is too large but which indeed corresponds to an underabundance. We will deal with these scenarios next. Note at this point that for mχ0 1 > 160 GeV, all scenarios represent underabundance. The figure illustrates the fact that as the mass of the neutralino increases, and therefore the higgsino component increases, annihilation of higgsino dominated LSP becomes more and more efficient and the relic density drops. The scan we have performed was done in steps of varying M 1 in order to reveal an important feature. As the value of M 1 increases and we enter the higgsino domain, the strips in M 1 /m χ 0 1 become wider for the highest values (though still small) of the relic density. The pole-like structure (without a "head") for low values of M 1 corresponds in fact to the contribution of a Higgs resonance, M A 0 ∼ 2mχ0 1 . Around these tuned resonant regions the value of the relic density fluctuates vastly. An example of this is shown in Fig. 9. Obtaining the correct relic density could then be considered fine-tuned with M A 0 ∼ 2mχ0 1 . As we reach the higgsino domain, these resonant processes become irrelevant since other channels are more efficient. This explains the "pole with the head" structure, see Fig. 8. These regions correspond to underabundance. The relic density constraint does not change the conclusions concerning the signal rates. We find that in the range 60 < mχ0 1 < 120 GeV we can have R γγ > 1 (R γγ ∼ 2 is possible here) while for the rest of the allowed mass range 120 < mχ0 1 < 150 GeV we have R γγ < 1. Because the models which are retained are those with M A 0 ∼ 2mχ0 1 due to imposing the relic density constraint, these ranges can be converted to ranges over M A 0 . Fig. 4 confirms the behavior of the 2-photon Higgs signal strength. Model B with an underabundance, a reappraisal of the direct detection If neutralinos do not make up all of the dark matter, one can reconsider those scenario for which the spin-indepent cross section seemed to high. Naturally configurations with mχ0 1 > 160 GeV are now possible since the annihilation cross section for higgsinos are large and do not require the contribution of a Higgs resonance contrary to scenarios with mχ0 1 < 160 GeV. Fig. 10 shows how R γγ is affected. Up to mχ0 1 ∼ 160 GeV we again observe a gradual decline of the di-photon rate. In particular in the range of neutralino masses between 120-160 GeV, this rate drops below that of the SM narrowing around a value of 0.5. These values can be interpreted in terms of the dependence of the R γγ as a function of M A 0 , see Fig. 4 taking into account the fact that for these configurations M A 0 ∼ 2mχ0 1 . On the other hand as soon as we enter the higgsino regime (and also co-annihilation with stops), any value of M A 0 will do to give a small enough relic density of neutralinos. In this case the di-photon rate is spread over a wide range, in particular large enhancements are now possible. Figure 10: Model B : points with underabundance of the relic density for which the modified XENON100 limit is respected. Model A with the correct thermal abundance Many of the arguments that were detailed in the previous two sections for Model B can be invoked when looking at the impact of the relic density on Model A. One common feature shared by the two models is that the dominance of the higgsino component in the calculation of the relic density kicks in at about the same value of mχ0 1 , i.e. mχ0 1 ∼ 160 GeV. We have extended the range of mχ0 1 to about 250 GeV, because the lightest stop is much heavier in Model A. At around mχ0 1 ∼ 200 GeV, we do not have the added contribution of the stop co-annihilation. As Fig. 11 shows, the maximum value of the relic density drops steadily as the neutralino mass increases. The pole like structures, indicative of an annihilation through a resonance, are still present. Recall however that contrary to Model B, the flavour constraints have imposed M A 0 > 200 GeV, while allowing the larger range 2 − 17 for t β . Yet we see a resonance like contribution that is much thinner around mχ0 1 ∼ 60 GeV. This in fact is due to precisely the lightest Higgs. This Figure 11: Model A. Values of the relic density (red/light grey) are superimposed with those that have passed the XENON100 (2012) limit(blue/dark grey). The WMAP bound is shown. said, although some of these configurations pass the XENON100 (2012) constraint they do not simultaneously provide the standard relic density, in fact these correspond to overabundances. Insisting on producing the present abundance while abiding by the XENON limit, the masses of allowed neutralinos are in a narrower range than in Model B : 100 − 160 GeV. Model A with an underabundance, a reappraisal of the direct detection Figure 12: . Model A : points with underabundance for which the modified XENON100 limit is respected. We now allow that the neutralinos of Model A do not account for all of DM and assess which neutralino masses are possible after imposing the flavour constraint and the direct detection limit, see Fig. 12. The range ∼ 100 GeV to ∼ 250 GeV is now open, together with the small island around mχ0 1 ∼ 60 GeV that corresponds to annihilation through h. For the latter, the di-photon rate is SM like. For the range mχ0 1 = 100 − 250 GeV we span 0 < R γγ < 3.5. Contrary to model B, where the relic density proceeds through the pseudoscalar resonance for mχ0 1 : 100 − 160 GeV corresponding to M A 0 > 200 GeV, we reproduce 0 < R γγ < 3.5 as could have been guessed from Fig. 2. For mχ0 1 > 160 GeV, the situation is similar to what we have seen with Model B. Conclusions The results that the LHC Collaborations have announced in July 2012 are most probably pointing to the discovery of a Higgs boson. If this is the SM Higgs boson, the naturalness argument upon which one justified, for decades, the construction of new models for better explanation of symmetry breaking would be most baffling. It is therefore important to seek whether the Higgs signals could be incorporated within a natural set-up, see for example the recent arguments in [65]. Supersymmetry would be an ideal framework that could provide also a solution to the DM problem. However the relatively heavy mass of the resonance discovered at the LHC suggests that the much studied MSSM will not be as natural as wished, moreover if the excess in the di-photon signal is established, the MSSM will have to be abandoned. Even before the LHC started taking data, there were signs of tension between the MSSM and naturalness. Keeping the supersymmetric framework but allowing for a more general set-up had been advocated through the BMSSM to alleviate the problem. The series of analyses we have been performing is to investigate whether the BMSSM is a viable alternative in the light of the new data. Despite the large number of new parameters these effective models are rather constrained and predictive. The study we have performed here aimed at reviewing what predictions and correlations for the different signals of the Higgs are possible. While we eagerly await more precision on many of the Higgs rates, it is important that one confronts these predictions with measurements concerning flavour and those that address the DM observables. In this paper we considered two sets of scenarios. A BMSSM model with degenerate stops at 400 GeV (Model A) and a strongly mixed scenario in the stop sector with a lightest stop at 200 GeV and a heavier one at 600 GeV, Model B. We find that the heavy flavour observables, in particular B → X * s γ (and to a lesser extent the new constraint from B s → µ + µ − ) delimit in an important way the parameter space of the general BMSSM. For Model A, the pseudoscalar mass is restricted to M A 0 > 200 GeV while allowing for a relatively large range for t β , t β < 20. Such restrictions exclude the possibility that the signal at the LHC could originate from the heavier CP-even Higgs. Despite these constraints, we still have scenarios for a 125 GeV Higgs with rates higher in the di-photon signal than in the SM. Model A can still give for example R γγ ∼ R ZZ ∼ 2 while R γγ+2j ∼ 1.6 while R bb ∼ 0.7 and R τ τ ∼ 0.7. In Model B, the flavour constraints could be considered even stronger. Indeed, for the maximal scenarios we have taken M A 0 < 400 GeV and t β < 5. While the hierarchy R γγ+2j > R γγ > R ZZ ∼ 2 is maintained it is possible to have R γγ ∼ 2. The BMSSM models are also natural in the sense that the usual Higgs mixing parameter µ is not large. In our study this was set at µ = 300 GeV, justifying the approach of new operators associated with a new physics at a scale 1.5 TeV. This parameter is also important in defining the nature of the DM candidate through the composition of the neutralino LSP. This composition depends on the weak gaugino parameters M 1 , M 2 which have, contrary to µ within the BMSSM, little impact on Higgs observables. By looking at various M 1 versus µ hierarchies, we imposed the newly published XENON100 limits. These are very strong limits. We first assumed that the BMSSM LSP neutralino accounts for all of DM. In both Model A and Model B, we find that only neutralinos with as little as possible higgsino component pass the new limit. Therefore the LSP can not have mass above 160 GeV. The projected XENON1T will exclude all configurations with this assumption on the abundance. We then ask whether the configurations that do indeed pass the XENON100 test have the correct relic density as set by WMAP. We find that this is possible to achieve only if annihilation occurs through the pseudoscalar resonance with M A 0 ∼ 2mχ0 1 . This could be considered as fine tuned. In turn, in model B, for 120 < mχ0 1 < 150 GeV even a small enhancement of the di-photon rate is difficult to achieve. For model A, R γγ > 1 is possible over the whole allowed range 60 < mχ0 1 < 150 GeV, in fact the additional direct detection constraint imposes R γγ > 0.5. We have then asked how some configurations can be rescued if we instead also accept scenarios with underabundance and underdensity in the halo that lead to smaller direct detection rates even for large spin-independent cross section. Masses of neutralinos up to the lightest stop mass are now possible. Apart from neutralinos with mass in the range 120 − 160 GeV in Model B where the Higgs signal strength is small, in all other configurations we now span a large range of R γγ . The drawback in many of these scenarios is that the BMSSM does not provide all of the observed DM, at least within a standard cosmological scenario. Although we have not explored all the possibilities within the BMSSM implementations (e.g. we could have considered larger values of µ together with a larger scale M , study more implementations of the stop and stop mixings or the effect of a wino component), this study shows the importance of a global study including Higgs, flavour and DM especially that new powerful data is pouring in. We eagerly await the results from the projected XENON1T. Above all we keep a very close eye on the upcoming analyses of the Higgs at the LHC. These include better measurements of as many channels in the signal region and also further investigations of other mass regions that can probe the other Higgses of these two doublet models. We stress once more that, despite the addition of many new operators beyond the usual MSSM, there are strong correlations between the Higgs observables in the BMSSM. Direct searches for the stops will also bring important information in these scenarios. Figure 1 : 1Allowed regions in scenario A after applying the flavour constraints for a signal with m h = 125 GeV. We plot here the signal strengths and the correlations for a) Left panel: γγ (x-axis), ZZ (red points) and γγ + 2 jets (blue points), b) right panel: τ τ and bb. Figure 2 : 2Allowed region in M A 0 versus R γγ and t β versus R γγ in scenario A after applying flavour constraints. Figure 4 : 4Allowed region in M A 0 versus R γγ and t β versus R γγ in scenario B after applying the flavour constraints. Figure 6 : 6As in Fig. 5 but for the signal strengths for τ τ andbb channels after flavour constraints are imposed. Figure 7 : 7We show the allowed parameter space in m χ 0 1 -σ SI after imposing XENON100 (2012) (left panels) and the impact on R γγ (right panels). The upper (lower) plots are for model A (B). W W/ZZ : This occurs when the higgsino component is highest and the channels are open. With µ = 300 GeV this takes place when M 1 ≥ µ. W H/ZH/hA 0 : this channel only opens up for high masses, that is mχ0 1 > 200 GeV. Figure 8 :Figure 9 : 89Model B. Values of the relic density (red/light grey) are superimposed with those that have passed the XENON100 (2012) limit(blue/dark grey). The WMAP bound is shown. The scan has been done by increasing the value of M 1 in steps, rather than randomly (see text of why this was done) Model B. We show the result of a scan on M A 0 on the relic density as a function for M 1 = 93 GeV (left panel) and M 1 = 180 GeV right panel. GeV. All trilinear couplings are set to 0, except for the stop sector. The MSSM parameters t β , M A 0 will be varied in the ranget β ∈ [2, 40], M A 0 ∈ [50, 450] (GeV).t β is the ratio between the expectation values in the Higgs doublets. M A 0 is the mass of the pseudoscalar Higgs, A 0 . The CP-even Higgses will be denoted as h (the lightest) and H (the heaviest). The gaugino masses will only play a role when studying the DM. 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[ "Unsupervised Domain Adaptation for Semantic Segmentation with GANs", "Unsupervised Domain Adaptation for Semantic Segmentation with GANs" ]	[ "Swami Sankaranarayanan \nUMIACS\nUniversity of Maryland\nCollege ParkMD\n", "Yogesh Balaji \nUMIACS\nUniversity of Maryland\nCollege ParkMD\n", "Arpit Jain \nGE Global Research\nNiskayunaNY\n", "Ser Nam Lim \nGE Global Research\nNiskayunaNY\n", "Rama Chellappa \nUMIACS\nUniversity of Maryland\nCollege ParkMD\n" ]	[ "UMIACS\nUniversity of Maryland\nCollege ParkMD", "UMIACS\nUniversity of Maryland\nCollege ParkMD", "GE Global Research\nNiskayunaNY", "GE Global Research\nNiskayunaNY", "UMIACS\nUniversity of Maryland\nCollege ParkMD" ]	[]	Visual Domain Adaptation is a problem of immense importance in computer vision. Previous approaches showcase the inability of even deep neural networks to learn informative representations across domain shift. This problem is more severe for tasks where acquiring hand labeled data is extremely hard and tedious. In this work, we focus on adapting the representations learned by segmentation networks across synthetic and real domains. Contrary to previous approaches that use a simple adversarial objective or superpixel information to aid the process, we propose an approach based on Generative Adversarial Networks (GANs) that brings the embeddings closer in the learned feature space. To showcase the generality and scalability of our approach, we show that we can achieve state of the art results on two challenging scenarios of synthetic to real domain adaptation. Additional exploratory experiments show that our approach: (1) generalizes to unseen domains and (2) results in improved alignment of source and target distributions.	null	[ "https://arxiv.org/pdf/1711.06969v1.pdf" ]	11,247,316	1711.06969	d822222081bd3a711eb9a3b28bbe872cc42e594f	Unsupervised Domain Adaptation for Semantic Segmentation with GANs Swami Sankaranarayanan UMIACS University of Maryland College ParkMD Yogesh Balaji UMIACS University of Maryland College ParkMD Arpit Jain GE Global Research NiskayunaNY Ser Nam Lim GE Global Research NiskayunaNY Rama Chellappa UMIACS University of Maryland College ParkMD Unsupervised Domain Adaptation for Semantic Segmentation with GANs Visual Domain Adaptation is a problem of immense importance in computer vision. Previous approaches showcase the inability of even deep neural networks to learn informative representations across domain shift. This problem is more severe for tasks where acquiring hand labeled data is extremely hard and tedious. In this work, we focus on adapting the representations learned by segmentation networks across synthetic and real domains. Contrary to previous approaches that use a simple adversarial objective or superpixel information to aid the process, we propose an approach based on Generative Adversarial Networks (GANs) that brings the embeddings closer in the learned feature space. To showcase the generality and scalability of our approach, we show that we can achieve state of the art results on two challenging scenarios of synthetic to real domain adaptation. Additional exploratory experiments show that our approach: (1) generalizes to unseen domains and (2) results in improved alignment of source and target distributions. Introduction Deep Convolutional Neural Networks (DCNNs) have revolutionalized the field of computer vision, achieving the best performance in a multitude of tasks such as Image Classification [12], Semantic Segmentation [19], Visual Question Answering [22], etc. This strong performance can be attributed to the availability of abundant labeled training data. While annotating data is relatively easier for certain tasks like Image Classification, they can be extremely laborious and time-consuming for others. Semantic segmentation is one such task that requires great human effort as it involves obtaining dense pixel-level labels. The annotation time for obtaining pixel-wise labels for a single image from the CITYSCAPES dataset is about 1 hr., highlighting the level of difficulty ( [4], [26] Figure 1: Characterization of Domain Shift and effect of the proposed approach in reducing the same collecting the data: While natural images are easier to obtain, there are certain domains like medical imaging where collecting data and finding experts to precisely label them can also be very expensive. One promising approach that addresses the above issues is the utility of synthetically generated data for training. However, models trained on the synthetic data fail to perform well on real datasets owing to the presence of domain gap between the datasets. Domain adaptation encompasses the class of techniques that address this domain shift problem. Hence, the focus of this paper is in developing domain adaptation algorithms for semantic segmentation. Specifically, we focus on the hard case of the problem where no labels from the target domain are available. This class of techniques is commonly referred to as Unsupervised Do-main Adaptation. Traditional approaches for domain adaptation involve minimizing some measure of distance between the source and the target distributions. Two commonly used measures are Maximum Mean Discrepancy (MMD) ( [9], [20] [21]), and learning the distance metric using DCNNs as done in Adversarial approaches ( [7], [29]). Both approaches have had good success in the classification problems; however, as pointed out in [31], their performance improvement does not translate well to the semantic segmentation problem. This motivates the need for developing new domain adaptation techniques tailored to semantic segmentation. The method we present in this work falls in the category of aligning domains using an adversarial framework. Among the recent techniques that address this problem, FCN in the wild [13] is the only approach that uses an adversarial framework. However, unlike [13] where a discriminator operates directly on the feature space, we project the features to the image space using a generator and the discriminator operates on this projected image space. Adversarial losses are then derived from the discriminator. We observed that applying adversarial losses in this projected image space achieved a significant performance improvement as compared to applying such losses directly in the feature space (ref. Table 5). The main contribution of this work is that we propose a technique that employs generative models to align the source and target distributions in the feature space. We first project the intermediate feature representations obtained using a DCNN to the image space by training a reconstruction module using a combination of L 1 and adversarial losses. We then impose the domain alignment constraint by forcing the network to learn features such that source features produce target-like images when passed to the reconstruction module and vice versa. This is accomplished by employing a series of adversarial losses. As training progresses, the generation quality gradually improves, while at the same time, the features become more domain invariant. Related Work Semantic segmentation is a well studied problem in computer vision. The Fully Convolutional Networks (FCN) by Shelhamer et al [19] signified a paradigm shift in how to fully exploit the representational power of CNNs for the pixel labeling task. While performance has been steadily improving for popular benchmarks such as PAS-CAL VOC [6] and MS-COCO [17], they do not address the challenges of domain shift within the context of semantic segmentation. Domain adaptation has been widely explored in computer vision primarily for the classification task. Some of the earlier approaches involved using feature reweighting techniques [5], or constructing intermediate representations using manifolds [11] [10] or dictionaries [24]. Since the advent of deep neural networks, emphasis has been shifted to learning domain invariant features in an end-to-end fashion. A standard framework for deep domain adaptation involves minimizing a term that measures domain discrepancy along with the task being solved. Some approaches use Maximum Mean Discrepancy and its kernel variants for this task [20] [21], while others use adversarial approaches. We focus on adversarial approaches since they are more related to our work. Revgrad [7] performs domain adaptation by applying adversarial losses in the feature space, while PixelDA [2] and CoGAN [18] operate in the pixel space. While these techniques perform adaptation for the classification task, there are very few approaches aimed at semantic segmentation. To the best of our knowledge, [13] and [31] are the only two approaches that address this problem. FCN in the wild [13] proposes two alignment strategies -(1) global alignment which is an extension to the domain adversarial training proposed by [7] to the segmentation problem and (2) local alignment which aligns class specific statistics by formulating it as a multiple instance learning problem. Curriculum domain adaptation [31] on the other hand proposes curriculum-style learning approach where the easy task of estimating global label distributions over images and local distributions over landmark superpixels is learnt first. The segmentation network is then trained so that the target label distribution follow these inferred label properties. One possible direction to address the domain adaptation problem is to employ style transfer or cross domain mapping networks to stylize the source domain images as target and train the segmentation models in this stylized space. Hence, we discuss some recent work related to the style transfer and unpaired image translation tasks. The popular work of Gatys et al. [8] introduced an optimization scheme involving backpropagation for performing content preserving style transfer, while Johnson et al. [14] proposed a feedforward method for the same. CycleGAN [32] performs unpaired image-to-image translation by employing adversarial losses and cycle consistency losses. In our experiments, we compare our approach to some of these style-transfer based data augmentation schemes. Method In this section, we provide a formal treatment of the proposed approach and explain in detail our iterative optimization procedure. Let X ∈ R M ×N ×C be an arbitrary input image (with C channels) and Y ∈ R M ×N be the corresponding label map. Given an input X, we denote the output of a CNN asŶ ∈ R M ×N ×Nc , where N c is the number of classes.Ŷ (i, j) ∈ R Nc is a vector representing the class probability distribution at pixel location (i, j) output by the CNN. The source(s) or target (t) domains are denoted by a superscript such as X s or X t . First, we provide an input-output description of the different network blocks in our pipeline. Next, we describe separately the treatment of source and target data, followed by a description of the different loss functions and the corresponding update steps. Finally, we motivate the design choices involved in the discriminator (D) architecture. Description of network blocks Our training procedure involves alternatively optimizing the following network blocks: • The base network, whose architecture is similar to a pre-trained model such as VGG-16, is split into two parts: the embedding denoted by F and the pixel-wise classifier denoted by C. The output of C is a label map up-sampled to the same size as the input of F . • The generator network (G) takes as input the learned embedding and reconstructs the RGB image. • The discriminator network (D) performs two different tasks given an input: (a) It classifies the input as real or fake in a domain consistent manner (b) It performs a pixel-wise labeling task similar to the C network. Note that (b) is active only for source data since target data does not have any labels during training. Treatment of source and target data Given a source image and label pair {X s , Y s } as input, we begin by extracting a feature representation using the F network. The classifier C takes the embedding F (X s ) as input and produces an image-sized label mapŶ s . The generator G reconstructs the source input X s conditioned on the embedding. Following recent successful works on image generation, we do not explicitly concatenate the generator input with a random noise vector but instead use dropout layers throughout the G network. As shown in Figure 3, D performs two tasks: (1) Distinguishing the real source input and generated source image as source-real/source-fake (2) producing a pixel-wise label map of the generated source image. Given a target input X t , the generator network G takes the target embedding from F as input and reconstructs the target image. Similar to the previous case, D is trained to distinguish between real target data (target-real) and the generated target images from G (target-fake). However, different from the previous case, D performs only a single task i.e. it classifies the target input as target-real/target-fake. Since the target data does not have any labels during training, the classifier network C is not active when the system is presented with target inputs. Fig. 3 shows various losses used in our method. We begin by describing these losses, and then describe our iterative optimization approach. Iterative optimization The different adversarial losses used to train our models are shown in Table. 1. In addition to these adversarial losses, we use the following losses: (1) L seg and L auxpixel-wise cross entropy loss used in standard segmentation networks such as in FCN and (2) L rec -L 1 loss between input and reconstructed images. The directions of flow of information across different network blocks are listed in Figure 2. In each iteration, a randomly sampled triplet (X s , Y s , X t ) is provided to the system. Then, the network blocks are updated iteratively in the following order: (1) D-update: For source inputs, D is updated using a combination of within-domain adversarial loss L s adv,D and auxiliary classification loss L s aux . For target inputs, it is updated using only the adversarial loss L t adv,D . The overall loss L D is given by L D = L s adv,D + L t adv,D + L s aux . (2) G-update: In this step, the generator is updated using a combination of an adversarial loss L s adv,G + L t adv,G intended to fool D and a reconstruction loss L rec . The adversarial loss encourages realistic output from the generator. The pixelwise L 1 loss is crucial to ensure image fidelity between the generator outputs and the corresponding input images. The overall generator loss is given as: Classify fake source input as real target (tgt-real) L t adv,F Classify fake target input as real source (src-real) Table 1: Within-domain and Cross-domain adversarial losses that are used to update our networks during training. G and D networks are updated using only the within-domain losses while F is updated only using the cross domain loss. All these adversarial losses originate from the D network. L adv,X implies that the gradients from the loss function L are used to update X only, while the other networks are held fixed. L G = L s adv,G + L t adv,G + L s rec + L t rec .(3) shift is captured. The parameters of F are updated using a combination of several loss terms: Table 1, the adversarial loss terms used to update F account for the domain adaptation. More specifically, the iterative updates described here can be considered as a min-max game between the F and the G-D networks. During the D update step discussed earlier, the adversarial loss branch of D learns to classify the input images as real or fake in a domain consistent manner. To update F , we use the gradients from D that lead to a reversal in domain classification, i.e. for source embeddings, we use gradients from D corresponding to classifying those embeddings as from target domain (L s adv,F ) and for target embeddings, we use gradients from D corresponding to classifying those embeddings as from source domain (L t adv,F ). Note that, this is similar to the min-max game between the G-D pair, except in this case, the com-petition is between classifying the generated image as from source/target domains instead of them being real/fake. L F = L seg + α L s aux + β (L s adv,F +L t adv,F ). As illustrated in Motivating design choice of D • In traditional GANs that are derived from the DC-GAN [25] implementations, the output of the discriminator is a single scalar indicating the probability of the input being fake or drawn from an underlying data distribution. Recent works on image generation have utilized the idea of Patch discriminator in which the output is a two dimensional feature map where each pixel carries a real/fake probability. This results in significant improvement in the visual quality of their generator reconstructions. We extend this idea to our setting by using a variant of the Patch discriminator, where each pixel in the output map indicates real/fake probabilities across source and target domains hence resulting in four classes per pixel: src-real, src-fake, tgt-real, tgt-fake. • In general, GANs are hard to train on tasks which involve realistic images of a larger scale. One promising approach to training stable generative models with the GAN framework is the Auxiliary Classifier GAN (AC-GAN) approach by Odena et al. where they show that by conditioning G during training and adding an auxiliary classification loss to D, they can realize a more stable GAN training and even generate large scale images. Inspired by their results on image classification, we extend their idea to the segmentation problem by employing an auxiliary pixel-wise labeling loss to the D network. Both these components prove crucial to our performance. The ablation study performed in Section 5.3 shows the effect of the above design choices on the final performance. Specific details about the architectures of these network blocks can be found in the supplementary material. Experiments and Results In this section, we provide a quantitative evaluation of our method by performing experiments on benchmark datasets. We consider two challenging synthetic datasets available for semantic segmentation: SYNTHIA and GTA-5. SYNTHIA [27] is a large dataset of photo-realistic frames rendered from a virtual city with precise pixellevel semantic annotations. Following previous works ( [13], [31]), we use the SYNTHIA-RAND-CITYSCAPES subset that contains 9400 images with annotations that are compatible with cityscapes. GTA-5 is another large-scale dataset containing 24966 labeled images. The dataset was curated by Richter et al. [26] and is generated by extracting frames from the computer game Grand Theft Auto V. We used CITYSCAPES [4] as our real dataset. This dataset contains urban street images collected from a moving vehicle captured in 50 cities around Germany and neighboring countries. The dataset comes with 5000 annotated images split into three sets -2975 images in the train set, 500 images in the val set and 1595 images in the test set. In all our experiments, for training our models we used labeled SYNTHIA or GTA-5 dataset as our source domain and unlabeled CITYSCAPES train set as our target domain. We compared the proposed approach with the only two contemporary methods that address this problem: FCN in the wild [13] and Curriculum Domain adaptation [31]. Following these approaches, we designate the 500 images from CITYSCAPES val as our test set. Architecture In all our experiments, we used FCN-8s as our base network. The weights of this network were initialized with the weights of the VGG-16 [28] model trained on Imagenet [16]. The architectures we used for D and G networks along with the hyper-parameter settings are described in the supplementary material. Implementation details In all our experiments, images were resized and cropped to 1024 × 512. We trained our model for 100, 000 iterations using Adam solver [15] with a batch size of 1. Learning rate of 10 −5 was used for F and C networks, and 2 × 10 −4 for G and D networks. While evaluating on CITYSCAPES dataset whose images and ground truth annotations are of size 2048 × 1024, we first produce our predictions on the 1024 × 512 sized image and then upsample our predictions by a factor of 2 to get the final label map, which is used for evaluation. We will make our models and code publicly available. SYNTHIA -> CITYSCAPES In this experiment, we use the SYNTHIA dataset as our source domain, and CITYSCAPES as our target domain. We randomly pick 100 images from the 9400 labeled images of SYNTHIA dataset and use it for validation purposes, the rest of the images are used for training. We use the unlabeled images corresponding to the CITYSCAPES train set for training our model. In order to ensure fairness of experimental results, we followed the exact evaluation protocol as specified by the previous works ( [13], [31]): The 16 common classes between SYNTHIA and CITYSCAPES are chosen used as our labels. The predictions corresponding to the other classes are treated as belonging to void class, and not backpropagated during training. The 16 classes are: sky, building, road, sidewalk, fence, vegetation, pole, car, traffic sign, person, bicycle, motorcycle, traffic light, bus, wall, and rider. Table 2a reports the performance of our method in comparison with [13] and [31]. The source-only model which corresponds to the no adaptation case i.e. training only using the source domain data achieves a mean IOU of 25.7. The target-only values denote the performance obtained by a model trained using CITYSCAPES train set (supervised training), and they serve as a crude upper bound to the domain adaptation performance. These values were included to put in perspective the performance gains obtained by the proposed approach. We observe that our method achieves a mean IOU of 34.8, thereby improving the baseline by 9.1 points, thus resulting in a higher performance improvement compared to other reported methods. GTA5 -> CITYSCAPES In this experiment, we adapt from the GTA-5 dataset to the CITYSAPES dataset. We randomly pick 1000 images from the 24966 labeled images of GTA-5 dataset and use it for validation purpose and use the rest of the images for training. We use the unlabeled images corresponding to the CITYSCAPES train set for training our model. In order to ensure fairness of experimental results, we followed the exact evaluation protocol as specified by the previous works ( [13], [31]): we use 19 common classes between GTA-5 and CITYSCAPES as our labels. The results of this experiment are reported in Table. 2b. Similar to the previous experiment, our baseline performance (29.6) is higher than the performance reported in [13], due to difference in network architecture and experimental settings. On top of this, the proposed approach yields an improvement of 7.5 points to obtain a mIOU of 37.1. This performance gain is higher than that achieved by the other compared approaches. Note regarding different baselines: The baseline numbers reported by us do not match with the ones reported in [31] and [13] due to different experimental settings (this mismatch was also reported in [31]). However, we would like to point out that we improve over a stronger baseline compared to the other two methods in both our adaptation experiments. In addition, [31] uses additional data from PASCAL-CONTEXT [23] dataset to obtain the superpixel segmentation. In contrast, our approach is a single stage end-to-end learning framework that does not use any additional data and yet obtains better performance improvement. Discussion In this section, we perform several exploratory studies to give more insight into the functionality and effectiveness of the proposed approach. similar to the previous section, all the evaluation results are reported on the CITYSCAPES val set, unless specified otherwise. We denote this set as the test set. We would like to note that owing to space constraints, we have added example results such as label predictions and images sampled from generator network etc in the supplementary material. Effect of Image Size The datasets considered in this paper consists of images of large resolution which is atleast twice larger than the most commonly used Segmentation benchmarks for CNNs i.e. PASCAL VOC (500×300) and MSCOCO (640×480). In this setting, it is instructive to understand the effect of image size on the performance of our algorithm both from a quantitative and computational perspective. Table 3 presents the results of our approach applied over three different image sizes along with the training and evaluation times. It should be noted that the Curriculum DA approach [31] used a resolution of 640×320. By comparing with our main results in Table 2a, we see that our approach provides a higher relative performance improvement over a similar baseline. For computational efficiency, the remaining experiments in this section are run with the image size of 640×320. Comparison with direct style transfer Generative methods for style transfer have achieved a great amount of success in the recent past. A simple approach to performing domain adaptation is to use such approaches as a data augmentation method: transfer the images from the source domain to target domain and use the provided source ground truth to train a classifier on the combined source and target data. In order to compare the proposed approach with this direct data augmentation procedure, we used a state of the art generative approach (Cycle-GAN [32]) to transfer images from source domain to target domain. As can be observed from the results, using generative approaches solely as a data augmentation method provides a relatively small improvement over the source-only baseline and clearly suboptimal compared to the proposed approach. By augmenting the feature learning process with gradients from the G-D pair, our method achieves superior performance and is more reliable in cases where approaches based on pure generation may fail to improve. This experiment highlights the difficulty in achieving domain adaptation by performing a direct style transfer. Component-wise ablation In this experiment, we show how each component in our loss function affects the final performance. We consider the following cases: (a) Ours(full): the full implementation of our approach (b) Ours w/o auxiliary pixel-wise loss: Here, the output of the D network is a single branch classifying the input as real/fake. This corresponds to α = 0 in the F -update step. Note that, setting both α and β as zero corresponds to the source-only setting in our experiments. Setting only β = 0 does not improve over the source-only baseline as there is no cross domain adversarial loss. (c) Ours w/o Patch discriminator: Instead of using the D network as a Patch discriminator, we used a regular GAN-like discriminator where the output is a 4-D probability vector that the input image belongs to one of the four classes -srcreal, src-fake, tgt-real and tgt-fake. (d) Feature space based D: In this setting, we remove the G-D networks and apply an adversarial loss directly on the embedding. This is similar to the global alignment setting in the FCN-in-the-wild approach [13]. The mean IoU results on the test set are shown in Table. 5. It can be observed that each component is very important to obtain the full improvement in performance. Cross Domain Retrieval A crucial aspect of domain adaptation is in finding good measures of domain discrepancy that provide a good illustration of the domain shift. While there exist several classical measures such as A-distance [1] and M M D [9] for the case of image classification, the extension of such measures for a pixel-wise problem such as semantic segmentation is non-trivial. In this section, we devise a simple experiment in order to illustrate how the proposed approach brings source and target distributions closer in the learnt embedding space. We start with the last layer of the F network, which we label as the embedding layer, whose output is a spatial feature map. We perform an average pooling to reduce this spatial map to a 4096 dimensional feature descriptor for each input image. We begin the cross domain retrieval task by choosing a pool of N = N src +N tgt images from the combined source and target training set. Let X denote these set of images and F X denote the set of the feature descriptors computed for X. Then, we choose two query sets, one consisting of source images (S) and the other consisting of target images (T ), each disjoint with X. Let the corresponding feature sets be denoted as Q S and Q T . We retrieve k-NN lists for each item in the query set from the combined feature set F X . For each query point in Q S , we count the number of target samples retrieved in the corresponding k-NN list. \|A k \| indicates the average number of target samples retrieved over the entire source query set Q S . For each query point in Q T , we count the number of source samples retrieved in the corresponding k-NN list. \|B k \| indicates the average number of source samples retrieved over the entire target query set Q T . We used cosine similarity as a metric to compute the k-NN lists. If more target samples are retrieved for a source query point (and vice-versa), it suggests that source and target distributions are aligned well in the feature space. For this experiment, the sizes of query sets and the feature set F X are as follows: N src = N tgt = 1000, \|Q S \| = 1000, \|Q T \| = 1000. The mean average precision (mAP) was computed across the entire query sets for the respective cross domain tasks. Figure 4 shows the plot of the quantities \|A k \| (Fig.4b) and \|B k \| (Fig.4a) for a range of values of k. It can be observed from the plots in both the tasks that for any given rank k, the number of cross domain samples retrieved by the adapted model is higher than the sourceonly model. This effect becomes more clear as k increases. This observation is supported by better mAP values for the adapted model as shown in Figure 4. While this by itself is not a sufficient condition for better segmentation performance, however this along with the results from Table 2 imply that the proposed approach performs domain adaptation in a meaningful manner. Owing to the difficulty in visualizing the mapping learned for segmentation tasks, a cross domain retrieval experiment can be seen as a reasonable measure of how domain gap is reduced in the feature space. Generalization to unseen domains A desirable characteristic of any domain adaptation algorithm is domain generalization i.e. improving performance over domains that are not seen during training. To test the generalization capability of the proposed approach, we test the model trained for the SYNTHIA → CITYSCAPES setting on the CamVid dataset [3]. We choose to evaluate our models on the 10 common classes among the three datasets. More details regarding the choice of classes is given in the supplementary material. Table 6 shows the mean IoU values computed for the source-only baseline and the adapted model. The proposed approach yields a raw improvement of 8.3 points in performance which is a significant improvement considering the fact that CamVid images are not seen by the adapted model during training. This experiment showcases the ability of the proposed approach to learn do-(a) Target → Source, \|B k \| (vs) k (b) Source → Target, \|A k \| (vs) k Figure 4: Illustration of Domain Adaptation achieved by the proposed approach. The plot compares the average number of retrieved sampled for the cross domain retrieval task described in Section 5.4 between the source-only model and the model adapted using the proposed approach. Target → Source implies that the query set used belongs to target domain (Q T ) and items queried for from the set X belong to the source domain and vice-versa for Source → Target. In general, the values plotted on the y-axis corresponds to the number of samples retrieved from the set X that belong to the opposite domain as to that of the query set. main invariant representations in a generalized manner. Conclusion and Future Work In this paper, we have addressed the problem of performing semantic segmentation across different domains. In particular, we have considered a very hard case where abundant supervisory information is available for synthetic data (source) but no such information is available for real data (target). We proposed a joint adversarial approach that transfers the information of the target distribution to the learned embedding using a generator-discriminator pair. We have shown the superiority of our approach over existing methods that address this problem using experiments on two large scale datasets thus demonstrating the generality and scalability of our training procedure. Furthermore, our approach has no extra computational overhead during evaluation, which is a critical aspect when deploying such methods in practice. As future work, we would like to extend this approach to explicitly incorporate geometric constraints accounting for perspective variations and to adapt over temporal inputs such as videos across different domains. Figure 2 : 2The directions of data flow solid arrows during the forward pass and gradient flow dotted arrows during the backward pass of our iterative update procedure. Solid blocks indicate that the block is frozen during that update step while dotted block indicate that it is being updated. Red denoted source information and Blue denotes target information. Figure 3 : 3During training, the F and C networks are trained jointly with the adversarial framework(G-D pair). F is updated using a combination of supervised loss and an adversarial component. In the bottom right, we show the test time usage. Only the F and C network blocks are used. There is no additional overhead during evaluation compared to the base model. source input as src-real; fake source input as src-fakeWithin-domain L s adv,G Classify fake source input as src-real L t adv,D Classify real target input as tgt-real; fake target input as tgt- ). The other challenge lies in * First two authors contributed equallyModel Trained on Synthetic Data (F s ) Extreme Domain Shift Supervised Synthetic data and Unsupervised Real Data Proposed Approach Model Trained on Real Data (F r ) Test on Synthetic Data High Accuracy on Synthetic Data Model Trained on Synthetic Data (F s ) Test on Real Data , , Our trained model (F ours ) Proposed GAN based Training GAN Due to Domain Shift Performance on Real Data F r F s Reduces Domain Gap Performance on Real Data F r F s F ours Test on Real Data F-update: The update to the F network is the critical aspect of our framework where the notion of domainReal Source image Real Target image 1 F network G network C network Fake Source image Fake Target image D network Source real Source fake Target real Target fake 2 3 4 1 2 3 4 : Pixelwise classification loss : Pixelwise reconstruction loss : Pixelwise adversarial loss : Auxiliary segmentation loss Supervised classification component Adversarial component F network C network Test phase Table 2 : 2Results of Semantic Segmentation by adapting from (a) SYTNHIA to CITYSCAPES and (b) GTA-5 to CITYSCAPES. We compare with two approaches that use two different base networks. To obtain a fair idea about our performance gain, we compare with the Curriculum DA approach that uses the same base network as ours. The Target-only training procedure is the same for both the settings since in both cases the target domain is CITYSCAPES. However, the results in (a) are reported over the 16 common classes while the results in (b) are reported over all the 19 classes. Table 3 : 3Mean IoU values and computation times across different image size on the SYNTHIA → CITYSCAPES setting. The numbers in bold indicate the absolute improvement in performance over the Source-only baseline. The reported training and evaluation times are for the proposed approach and are averaged over training and evaluation runs.Image size 512 × 256 640 × 320 1024 × 512 mIOU-Source-only 20.5 22.2 25.7 mIOU-Ours 29.3 (+8.8) 32.1 (+9.9) 34.8 (+9.1) Train time (per image) 1.5s 2.1s 2.9s Eval time (per image) 0.16s 0.19s 0.3s Table 4 : 4Comparison of semantic segmentation performance on SYNTHIA → CITYSCAPES setting when using a GAN based approach as data augmentation. We use Cy-cleGAN[32] as the cross domain generation procedure.Method mean IoU Source-only 22.2 Source + CycleGAN-augmented 25.6 Ours 32.1 Table 5 : 5Ablation study showing the effect of each component on the final performance of our approach on the SYN-THIA → CITYSCAPES settingMethod mean IoU Source-only 22.2 Feature space based D 25.3 Ours w/o Patch Discriminator 28.3 Ours w/o auxiliary loss (α = 0) 29.2 Ours 32.1 Table 6 : 6Mean IoU segmentation performance measured on a third unseen domain (CamVid dataset) for the models corresponding to the SYNTHIA → CITYSCAPES settingMethod mean IoU Source-only 36.1 Ours 44.4 Analysis of representations for domain adaptation. S Ben-David, J Blitzer, K Crammer, F Pereira, Advances in neural information processing systems. S. Ben-David, J. Blitzer, K. Crammer, and F. Pereira. Analysis of representations for domain adaptation. 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[ "COMMENSURABILITY CLASSES CONTAINING THREE KNOT COMPLEMENTS", "COMMENSURABILITY CLASSES CONTAINING THREE KNOT COMPLEMENTS" ]	[ "Neil Hoffman " ]	[]	[]	This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.	10.2140/agt.2010.10.663	[ "https://arxiv.org/pdf/0905.1672v1.pdf" ]	17,586,372	0905.1672	174d765f2e968370284b02e6a11b6e38fa2deaf3	COMMENSURABILITY CLASSES CONTAINING THREE KNOT COMPLEMENTS 11 May 2009 Neil Hoffman COMMENSURABILITY CLASSES CONTAINING THREE KNOT COMPLEMENTS 11 May 2009 This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes. Introduction The study of the commensurability classes of hyperbolic knot complements that contain other knot complements has attracted some recent interest (see [BBW], [CD], [GHH] [HS], [MM], [NR1], [Re], [RW]). A particularly interesting set of examples results from cyclic surgeries on hyperbolic knot complements, since the cyclic surgeries give rise to cyclic covers by other knot complements (see [GW]). Moreover, The Cyclic Surgery Theorem [CGLS] shows that there are at most two non-trivial cyclic surgeries on a hyperbolic knot complement and so a hyperbolic knot complement has at most two non-trivial, finite sheeted covers which are other knot complements. Similarly, if a hyperbolic knot complement, S 3 − k 1 is covered by another knot complement, S 3 − k 2 , then S 3 − k 1 admits a cyclic surgery. There are known examples of hyperbolic knot complements with exactly three knot complements in their commensurability classes. For example, the (−2, 3, 7) pretzel knot of [FS] famously admits two non-trivial cyclic surgeries and is therefore covered by two other hyperbolic knot complements. An infinite family of pairs of commensurable hyperbolic knot complements was constructed by W. Neuman. For a discussion of this construction, see [GHH]. Finally, two hyperbolic knot complements can be commensurable if they both have hidden symmetries. This property is equivalent to both knot complements Date: May 11, 2009. non-normally covering the same orbifold (see § 2.2). The dodecahedral knots of [AR] admit the only known examples of non-arithmetic knot complements with hidden symmetries (see [NR1]) and the figure 8 knot complement is the only arithmetic knot complement (see [Re]). This discussion motivates the following conjecture of Reid and Walsh (see [RW,Conj 5.2]). Conjecture. Let S 3 − K be a hyperbolic knot complement. Then, there are at most two other knot complements in its commensurability class. It has been announced by Boileau,Boyer,and Walsh ([BBW,Thm 1.3]) that the conjecture holds for knot complements without hidden symmetries. In their paper, they show that if a hyperbolic knot complement does not admit hidden symmetries, then any commensurable hyperbolic knot complement will cover a common orbifold. Furthermore, this orbifold admits a finite cyclic surgery for each knot complement that covers it. This paper presents a family of such orbifolds that are covered by exactly three hyperbolic knot complements. Specifically, the main theorem of this paper is the following (see § 2 for definitions): Theorem 1.1. Let n ≥ 1 and (n, 7) = 1. For all but at most finitely many pairs of integers (n, m), the result of (n, m) Dehn surgery on the unknotted cusp of the Berge manifold is a hyperbolic orbifold with exactly three knot complements its commensurability classes. The infinite family of orbifolds described by Theorem 1.1 which we refer to as β n,m (see §2) also has the property that for n = 1, each knot complement covering β n,m admits an n-fold symmetry which does not fix any point on the cusp. In particular, even when n = 2, this symmetry is not a strong involution. By [WZ], such a knot complement cannot admit a lens space surgery and so, by the above discussion, is not covered by any other knot complement. The paper is organized as follows. In addition to some background material and definitions, § 2 we prove a lemma about possible orbifold quotients of the Berge manifold. In § 3, we show that the orbifolds β n,m are shown to admit three cyclic surgeries, and the proof of the main theorem is contained in §4. In § 5, we provide a partial classification of commensurability classes containing three knot complements. Preliminaries 2.1. Two hyperbolic 3-orbifolds, H 3 /Γ 1 and H 3 /Γ 2 , are said to be commensurable if they share a common finite sheeted cover. In terms of groups, ∃g ∈ P SL(2, C) so that Γ 1 and gΓ 2 g −1 have a common subgroup which is finite index in both groups. Let Comm + (Γ) = {g ∈ P SL(2, C)\|[Γ : Γ ∩ gΓg −1 ] < ∞ and [gΓg −1 : Γ ∩ gΓg −1 ] < ∞} and N + (Γ) be the normalizer of Γ in P SL(2, C). We say that a group Γ has hidden symmetries if [Comm + (Γ) : N + (Γ)] > 1. A hyperbolic orbifold, M, has hidden symmetries if π orb 1 (M ) has hidden symmetries. For this discussion, we consider only orientable manifolds and orbifolds. 2.2. When a hyperbolic knot group has hidden symmetries the associated knot complement non-normally covers some orbifold with a rigid cusp i.e. the cusp is C × [0, ∞) where C is S 2 (2, 3, 6), S 2 (3, 3, 3) or S 2 (2, 4, 4) (see [Re,Lemma 4]). By [NR1,Prop 2.7], the cusp field of a hyperbolic orbifold is a subfield of the invariant trace field. Thus, if a hyperbolic orbifold has a S 2 (3, 3, 3) or S 2 (2, 3, 6) cusp, Q( √ −3) must be a subfield of the orbifold's invariant trace field and if the cusp is S 2 (2, 4, 4), Q(i) must be a subfield of the orbifold's invariant trace field (see [NR1, Proof of Thm 5.1(iv)]). Proposition 2.1. Let p : O 1 → O 2 be a covering of orbifolds such that O 1 has a rigid cusp C 1 . Then, O 2 has a rigid cusp C 2 such that p(C 1 ) = C 2 and if x ∈ C 2 then \|p −1 (x) ∩ C 1 \| = n 2 for some integer n unless C 1 is S 2 (3, 3, 3) and C 2 is S 2 (2, 3, 6) then \|p −1 (x) ∩ C 1 \| = 2n 2 for some integer n. Proof. First consider the case where C 1 is an S 2 (2, 4, 4). In this case, C 2 must also be a S 2 (2, 4, 4) cusp. The peripheral subgroup corresponding to C 2 is P 2 ∼ = (Z × Z) ⋊ φ Z/4Z, and so P 2 has an element of order 4 acting on the cusp. Thus, φ : Z/4Z → Aut(Z × Z) is a faithful representation. Let P 1 ⊂ P 2 be the peripheral subgroup corresponding to C 1 . So P 1 ∼ = (nZ × mZ) ⋊ φ Z/4Z. However, the order 4 automorphism switches the two generators for the Z × Z subgroup of P 2 . Thus, n = m and the degree of the covering is n 2 . A similiar proof carries through if C 1 and C 2 are both either S 2 (3, 3, 3) or S 2 (2, 3, 6) cusps. In the case, where C 1 is a S 2 (3, 3, 3) and C 2 is a S 2 (2, 3, 6) cusp, the Z/3Z subgroup of P 1 is index 2 in the Z/6Z subgroup of P 2 . Hence, the covering degree is 2n 2 . 2.3. For n ≥ 1 and (n, 7) = 1, let β n,m be the orbifold obtained by (n, m) Dehn surgery on the unknotted cusp of the Berge manifold (see Figure 1) using a standard framing on the cusps of this link complement as in [Ro]. we will obtain the (−2, 3, 7) pretzel knot (see [FS]). Also, if we drill out a solid solid torus along the unknotted cusp of the manifold we would obtain the one of the knots in the solid torus that admits three D 2 × S 1 fillings (see [Be,Cor 2.9]). Furthermore, if we perform Dehn surgery along the (1, r) slope and then drill along the core of the surgered torus, we would also obtain a knot complement in D 2 × S 1 that admits three D 2 × S 1 surgeries. In fact, by the above mentioned corollary, these are the only knots in solid tori with this property. The above constuction shows that Dehn surgery along a (1, r) slope of the unknotted cusp of the Berge manifold produces knot complements that admit three lens space surgeries. In fact, it is well known that the (1, 0), (18, 1) and (19, 1) surgery slopes on the (−2, 3, 7) pretzel knot admit lens space surgeries (see [FS]). By drilling out the unknotted cusp of the Berge manifold, these are also the surgery slopes that produce a solid torus filling. Since the linking number of the knotted cusp and the unknotted cusp is 7, the longitude gets sent to the curve (49r, 1) after (1, r) Dehn surgery on the unknotted cusp while the meridian (1, 0) remains fixed (see [Ro,Sect 9.H]). So the (1, 0), (18, 1), and (19, 1) surgery parameters get sent to (1, 0), (49r + 18, 1), and (49r + 19, 1) respectively after (1, r) Dehn surgery on the unknotted cusp. Furthermore, we can use the surgery paramters to compute the homology of the manifolds resulting from lens space surgeries on the knot complements. In fact, we see that for these knots we obtain S 3 and two lens spacesone with fundamental group of order \|49r + 18\| and another of order \|49r + 19\|. More generally, if we allow Dehn surgery along any (p, q) slope of the unknotted cusp of the Berge manifold where (p,q)=1, and either (1, 0), (18, 1), or (19, 1) Dehn surgery on the knotted cusp, we will also get lens spaces. Again, by [Ro,Sect 9.H], we see that the (1, 0) surgery slope corresponds to a lens space of order \|p\|, (18, 1) surgery slope corresponds to a lens space of order \|49q + 18p\|, and (19, 1) surgery slope corresponds a lens space of order \|49q + 19p\|. 2.4. Denote v 0 ≈ 1.01494146 as the volume of the regular ideal tetrahedron. The Berge manifold is comprised of four such tetrahedra and therefore its volume is 4v 0 . Denote by Γ L as the fundamental group of the Berge manifold. Since the complement of the Berge manifold is comprised of four regular ideal tetrahedra, Γ L ⊂ Isom + (T) ∼ = P GL(2, O 3 ), where T is a tesselation of H 3 by regular ideal tetrahedra. Hence, the Berge manifold is arithmetic. The proof of the following lemma takes advantage of the fact that the Berge manifold has relatively low volume in order to show that it cannot cover an orbifold with a torus cusp and a rigid cusp. Where necessary, we consider all groups as subgroups in P SL(2, C). (see Prop 2.1), we see that the covering degree of such a map would be 3l + n 2 or 3l + 2n 2 for some integers l, n (In the later case, l must be even). Thus, the covering of Q M by the Berge manifold is of order d = 3k(3l + n 2 ) or d = 3k(3l + 2n 2 ). Now, d ≤ 48 (see [Me]) and since k, l, n ≥ 1, we have that d ≥ 12. Hence, vol(Q M ) ≤ v 0 /3 if Q M has a S 2 (3, 3, 3) cusp and vol(Q M ) ≤ v 0 /6 if Q M has a S 2 (2, 3, 6) cusp. It follows that this orbifold must appear on the lists in [A,Thm 3.3,4.2] and [NR2]. However, none of the orbifolds with S 2 (3, 3, 3) cusps appearing on these lists correspond to maximal groups commensurable with the Berge manifold, so we may assume that Q M has a S 2 (2, 3, 6) cusp. After combining the above restrictions on the degree of a cover and the restrictions from Adams' list, there are two possiblities for Q M : either Q M has volume v 0 /6 and a S 2 (2, 3, 6) cusp (here k = 1, l = 2, n = 1) or Q M has volume v 0 /12 and a S 2 (2, 3, 6) cusp (here k = 2, l = 2, n = 1). First, consider the case where Q M has volume v 0 /6. By noting that π orb 1 (Q M ) has an index 2 subgroup Γ :=< x, y, z\|x 2 , y 2 , z 3 , (yz −1 ) 2 , (zx −1 ) 6 , (xy −1 ) 3 > and π orb 1 (Q M ) =< Γ, w > where w is the order 2 rotation on the fundamental domain of Γ, we obtain a presentation for π orb 1 (Q M ) (see [NR1], [MR] and Figure 2). Thus, we obtain the following presentation π orb 1 (Q M ) =< w, x, y, z\|x 2 , y 2 , z 3 , w 2 , (yz −1 ) 2 , (zx −1 ) 6 , (xy −1 ) 3 , (wx) 2 , wywyz −1 > . However, using GAP, the above group does not have any index 8 subgroups. Thus, there can be no orbifold Q T . In Thus, we may assume that π orb 1 (Q T ) ⊂ P SL(2, O 3 ) and deduce that there is a unique subgroup Λ of index 2 in π orb 1 (Q T ) such that Λ ⊂ P SL(2, O 3 ). By covolume considerations Λ has index 8 in P SL(2, O 3 ). Also, H 3 /Λ has a torus cusp and an S 2 (3, 3, 3) cusp. Since H 3 /P SL(2, O 3 ) has an S 2 (3, 3, 3) cusp, the degree of the covering p : H 3 /Λ → H 3 /P SL(2, O 3 ) has to be 3l + n 2 (see Prop 2.1), which is never 8. This completes the proof. Cyclic Surgeries on β n,m In this section, we show that for fixed n and m, β n,m admits three finite cyclic surgeries. We also show directly it is covered by three knot complements if n = 7. Proof. For a fixed β n,m , let r = (n, m) and consider β n,m as the union of the complement of a knot in a solid torus, T 1 and a solid torus with core a singular locus of order r, T 2 (see Figure 3). Thus, β n,m admits three Dehn surgeries that are homeomorphic to T 2 and a solid torus glued together along their boundaries. Each orbifold O j (j ∈ {1, 2, 3}) resulting from one of these Dehn surgeries has underlying space a lens space with π orb 1 (O j ) finite cyclic. In fact, \|π orb 1 (O j )\| is distinct for each choice of j. To see this we observe, as noted above, that O j is an orbifold with underlying space a lens space. Moreover, this underlying space is a lens space with fundamental group of order either n r , \|49 m r + 18 n r \|, or \|49 m r + 19 n r \| depending on the choice of surgery on T 1 (see § 2). Splitting O j into a solid torus coming from the Dehn surgery on T 1 and T 2 the solid torus core a singular curve, we can compute π orb 1 (O j ) using van Kampen's theorem. Thus, the orders of the each fundamental group increase by a factor of r and \|π orb 1 (O j )\| is either n, r · \|49 m r + 18 n r \| or r · \|49 m r + 19 n r \| which take on three distinct values for fixed n, m and r. In addition, by the Orbifold Theorem (see [BP,Thm 2]) and the above argument that π orb 1 (O j ) is finite cyclic, each O j has S 3 as its universal cover. Denote this covering map φ j : S 3 → O j . We may view O j as the union of the solid torus torus coming from the cusp Dehn filling of β n,m and the complement of this solid torus, which we denote by B. Hence φ −1 j (B) is a knot or link exterior in S 3 . Since (n, 7) = 1 and the singular set of T 2 has linking number 7 with the knotted cusp of β n,m , the boundary of φ −1 j (B) is connected. Hence, if (n, 7) = 1, β n,m will be covered by three knot complements in S 3 . Also, since the orders of \|π orb 1 (O j )\| are distinct, the covering degree of φ j will take on a distinct value for each j. Remark 3.2. When n = 1, the classification of exceptional Dehn surgeries in [MP,Table A.1,Rem A.3] shows that β n,m is hyperbolic. Hence, β 1,m is a hyperbolic knot complement that admits three cyclic surgeries. Proof of The Main Theorem In this section, we prove Theorem 1.1. Also for this section, we consider Ω n,m , ∆ n,m , and Ω L as subgroups of P SL(2, C). Proof of Theorem 1.1. Using Lemma 3.1, each β n,m is covered by three knot complements such that the covers are of distinct degrees. Also, the Hyperbolic Dehn Surgery Theorem [Th,Thm 5.8.2] shows that all but at most finitely many of the β n,m are hyperbolic. For the rest of the proof we only consider those β n,m that are hyperbolic. Given this condition, each β n,m we consider is covered by three distinct knot complements. By [BBW,Thm 1.3], to prove Theorem 1.1 it suffices to show that the knot complements covering β n,m do not have hidden symmetries. Suppose an infinite number of the hyperbolic knot complements that cover β n,m admit hidden symmetries. By the discussion in §2.2, every such a knot complement will non-normally cover an orbifold Q n,m with a rigid cusp. Furthermore, on passage to a subset of the β n,m , we can assume that the orbifolds Q n,m have the same type of rigid cusp, C. Let Ω n,m = π orb 1 (β n,m ), ∆ n,m = π orb 1 (Q n,m ) and let P ⊂ P SL(2, C) be the peripheral subgroup of ∆ n,m . We may assume that each Ω n,m is conjugated so that P has a fixed representation in P SL(2, C). Since β n,m has one cusp, notice that ∆ n,m = P · Ω n,m By Thurston's Hyperbolic Dehn Surgery Theorem [Th,Thm 5.8.2], the volumes of the β n,m are bounded from above by the volume of the Berge manifold. In addition, the minimum volume of a non-compact oriented hyperbolic 3-orbifold is v0 12 (see [Me]). Hence, vol(Q n ) ≥ v0 12 . Thus, we can further subsequence to arrange that β n,m covers Q n,m , that the Q n,m 's have the same type of rigid cusp, and that the covering degree is fixed, say d. Since β n,m is obtained by Dehn surgery on the Berge manifold, the Ω n,m will converge algebraically and geometrically to Ω L , the fundamental group of the Berge manifold (see [Th,Thm 5.8.2]). As P was a fixed group in our construction, ∆ n,r also converges algebraically and geometrically to P · Ω L . We have the following diagram: Remarks The following theorem provides a partial classification of hyperbolic orbifolds covered by three knot complements. It can be seen as a direct corollary to a result of [BBW]. However, a proof is provided below for completeness. (1) hyperbolic, Now, assume that γ has two components γ 1 and γ 2 . M = T 2 × I − K ′ , where K ′ is a knot. Each of the three finite cyclic on O − K corresponds M admitting a T 2 × I filling. Hence, Dehn filling along the cusp corresponding to γ 1 will produce a knot complement in D 2 × S 1 with three D 2 × S 1 fillings. Denote l 1 to be the linking number of γ 1 and K ′ and l 2 to be the linking number of γ 2 and K ′ . If l 1 is zero, K ′ would be a knot in a solid torus that is not a 1-braid Figure 4. The K(7,5,2,-1) after (1, 0) on γ 2 but has two non-trivial S 1 × D 2 fillings. This contradicts [Be,Cor 9.1]. Hence, we may assume l 1 = 0 and l 2 = 0. Also, (1, n) surgery on γ 2 will produce a knot K ′′ in a solid torus that has linking number l 2 + n · l 1 with γ 2 . In particular for large enough n l 2 + n · l 1 = 7. Hence, in cannot be in the family of knots that admit two non-trivial S 1 × D 2 fillings. One might hope to relax condition (4) above. However, Brandy Guntel pointed out that the K(7, 5, 2, −1) knot complement (see Figure 4) is hyperbolic and admits two non-trivial cyclic surgeries. The fundamental group of one of these lens spaces is of order 32. By our original discussion in §2.3, knot complements obtained by Dehn surgery on the unknotted cusp of the Berge manifold have lens spaces of order \|49r − 18\| and \|49r − 19\| neither of which can be 32. Hence, the K(7, 5, 2, −1) complement is not one of the β n,m . However, since the invariant trace field of the K(7, 5, 2, −1) is an odd degree extension of Q, we see that this knot complement does not admit hidden symmetries and the K(7, 5, 2, −1) has exactly three knot complements in its comensurability class (see [RW,Cor 5.4]). As mentioned above (1, m) surgery on the unknotted cusp of the Berge manifold produces Berge knots. It seems natural to ask if any hyperbolic Berge knots can have hidden symmetries. More generally, we might ask if any hyperbolic knot complements can have hidden symmetries and admit non-trivial lens space surgeries. As discussed in § 1, there are three hyperbolic knot complements known to have hidden symmetries: the complements of the two dodecahedral knots of Aitchison and Rubinstein, and the figure eight knot complement (see [AR], [NR1]). Using SnapPea one can see that both dodecahedral knots are amphichiral. Thus, by [CGLS,Cor 4] they cannot admit a lens space surgery. Additionally, it is well known that the figure eight knot complement does not admit a lens space surgery (see [Ta] for example). Acknowledgments First, I would like to thank Alan Reid for raising the questions that lead to this paper and thoughtfully guiding this work from its formative stages to completion. Figure 1 . 1The Berge manifold is the complement of this link. The Berge manifold admits several surgery slopes of interest. First if we perform Dehn surgery along the (1, 0) slope of the unknotted cusp of the Berge manifold, Figure 2 . 2The fundamental domain for Γ together with the involution w second case, Q M ∼ = H 3 /P GL(2, O 3 ) and the [P GL(2, O 3 ) : π orb 1 (Q T )] = 8. If π orb 1 (Q T ) ⊂ P SL(2, O 3 ), [P SL(2, O 3 ) : π orb 1 (Q T )] = 4. Using GAP, there is a unique index 4 subgroup G of P SL(2, O 3 ). However, G has finite abelianization, and therefore cannot be the orbifold group of Q T . Lemma 3 . 1 . 31The orbifolds β n,m are covered by three knot complements. Further more, the degrees of the corresponding covering maps are distinct. Figure 3 . 3The decomposition of a surgered β n,m along a torus By [Be, Cor 2.9], T 1 admits three Dehn surgeries that result in a solid torus. Note, [P · Ω L : Ω L ] = d < ∞. Let Q T = H 3 /P · Ω L . Q T has two cusps: a torus cusp, corresponding to the cusp created by geometric convergence from Dehn surgery, and a rigid cusp, corresponding to the cusp with peripheral group P .However by Lemma 2.2, such a limiting Q L cannot exist. Hence, at most finitely many of the β n,m have hidden symmetries. ( 2 ) 2covered by 3 knot complements, (3) does not admit hidden symmetries, and (4) O has non-empty singular locus, then O − K ∼ = β n,m for some pair (n, m). Proof. Let γ be the singular locus of O. Denote \|O\| the underlying space of O. By [BBW, Thm 1.2] and the assumptions, we know that \|O\| is a lens space, γ is a non-empty subset of the cores of a genus 1 Heegaard splitting of \|O\|, and if S 3 − K covers O − K then it does so cyclically and corresponds to a finite cyclic filling of O − K. Finally, denote M = O − γ − K First assume γ has one component. Each of the three knot complements coveringO − K will correspond to a S 1 × D 2 filling on knotted cusp of M . Again, we appeal to the fact that there is a a unique family of knots in solid tori that admits 3 nontrivial S 1 × D 2 fillings (see[Be, Cor 9.1]). Hence, M is obtained by performing (1, m) surgery on the unknotted cusp of the Berge manifold then drilling out the core of the surgered torus. Gluing back in the neighborhood of the fixed point set of γ gives us β n,m for some n, m. I also would like to thank Cameron Gordon for showing me the family of knot complements with three lens space surgeries β 1,m . Third, I would like to thank Genevieve Walsh and Steve Boyer for a number of enlightening conversations and their suggestions on early versions of this paper. Finally, I would like to thank my fellow graduate students for a number of helpful conversations. Lemma 2.2. The Berge manifold does not cover an orbifold with a torus cusp and a rigid cusp.Proof of 2.2. Assume Q T is an orbifold with a torus cusp and a rigid cusp covered by the Berge manifold. 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González-Acuña and W.C. Whitten, Imbeddings of three-manifold groups, Mem. Amer. Math. Soc. Vol. 474 (1992). Commensurability classes of twist knots. J Hoste, P Shanahan, J. Knot Theory Ramifications. 141J. Hoste and P. Shanahan, Commensurability classes of twist knots, J. Knot Theory Rami- fications Vol. 14 no. 1,(2005) pp. 91-100. The Arithmetic of Hyperbolic 3-Manifolds. C Maclachlan, A W Reid, SpringerNew YorkC. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, Springer New York (2003). Dehn filling of the "magic" 3-manifold. B Martelli, C Petronio, Comm. Anal. Geom. 145B. Martelli and C. Petronio, Dehn filling of the "magic" 3-manifold, Comm. Anal. Geom. Vol. 14, No. 5 (2006), p. 969-1026. The cusped hyperbolic 3-orbifold of minimum volume. R Meyerhoff, Bull. Amer. Math. Soc. 13R. Meyerhoff, The cusped hyperbolic 3-orbifold of minimum volume, Bull. Amer. Math. Soc. Vol. 13 (1985) pp. 154-156. The Smith Conjecture. J. W. Morgan and H. BassAcademic PressNew YorkJ. W. Morgan and H. Bass (editors), The Smith Conjecture, Academic Press, New York (1984). Commensurability classes of (-2,3,n) pretzel knot complements. M Macasieb, T Mattman, Alg. & Geom. Top. 83M. Macasieb and T. Mattman, Commensurability classes of (-2,3,n) pretzel knot comple- ments, Alg. & Geom. Top. 8(3) (2008), pp. 1833-1853. Arithmetic of hyperbolic manifolds. W D Neuman, A W Reid, Topology '90. de Gruyter1W. D. Neuman and A. W. Reid, Arithmetic of hyperbolic manifolds, in Topology '90, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992), pp. 273-310. W D Neuman, A W Reid, Notes on Adams' small volume orbifolds. de Gruyter1Topology '90W. D. Neuman and A. W. Reid, Notes on Adams' small volume orbifolds, in Topology '90, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992), pp. 311-314. Artimeticity of knot complements. A W Reid, J. London Math. Soc. 2A. W. Reid, Artimeticity of knot complements, J. London Math. Soc. (2) (1991), pp. 171-184. Commensurability classes of 2-bridge knot complements. A W Reid, G S Walsh, Alg. & Geom. Top. 82A. W. Reid, G. S. Walsh, Commensurability classes of 2-bridge knot complements, Alg. & Geom. Top. Vol. 8 No. 2 (2008), pp. 1031-1057. Knots and Links. D Rolfsen, Publish or Perish PressBerkeleyD. Rolfsen, Knots and Links. Publish or Perish Press, Berkeley, (1976). Two-bridge knots have property P, Mem. M-O Takahashi, Amer. Math. Soc. 29M-o Takahashi, Two-bridge knots have property P, Mem. Amer. Math. Soc. 29 (1981) W Thurston, The geometry and topology of 3-manifolds. Princeton UniversityMimeographed lecture notesW. Thurston, The geometry and topology of 3-manifolds, Princeton University, 1977, Mimeographed lecture notes. Symmetry of knots and cyclic surgery. S Wang, Q Zhou, Trans. Amer. Math. Soc. 3302S. Wang and Q. Zhou, Symmetry of knots and cyclic surgery, Trans. Amer. Math. Soc. Vol. 330 No. 2 (1992), pp. 665-676. E-mail address: nhoffman@math. utexas.eduE-mail address: [email protected]	[]
[ "The Charge Form Factor of the Neutron at Low Momentum Transfer from the 2 H( e, e ′ n)p Reaction", "The Charge Form Factor of the Neutron at Low Momentum Transfer from the 2 H( e, e ′ n)p Reaction" ]	[ "E Geis \nArizona State University\n85287TempeAZ\n", "M Kohl \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "V Ziskin \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "T Akdogan \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "H Arenhövel \nInstitut für Kernphysik\nJohannes Gutenberg-Universität Mainz\nD-55099MainzGermany\n", "R Alarcon \nArizona State University\n85287TempeAZ\n", "W Bertozzi \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "E Booth \nBoston University\n02215BostonMA\n", "T Botto \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "J Calarco \nUniversity of New Hampshire\n03824DurhamNH\n", "B Clasie \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "C B Crawford \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "A Degrush \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "T W Donnelly \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "K Dow \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "M Farkhondeh \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "R Fatemi \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "O Filoti \nUniversity of New Hampshire\n03824DurhamNH\n", "W Franklin \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "H Gao \nTriangle Universities Nuclear Laboratory and Duke University\n27708-0305DurhamNC\n", "S Gilad \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "D Hasell \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "P Karpius \nUniversity of New Hampshire\n03824DurhamNH\n", "H Kolster \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "T Lee \nUniversity of New Hampshire\n03824DurhamNH\n", "A Maschinot \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "J Matthews \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "K Mcilhany \nUnited States Naval Academy\n21402AnnapolisMD\n", "N Meitanis \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "R G Milner \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "J Rapaport \nOhio University\n45701AthensOH\n", "R P Redwine \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "J Seely \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "A Shinozaki \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "S Širca \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "A Sindile \nUniversity of New Hampshire\n03824DurhamNH\n", "E Six \nArizona State University\n85287TempeAZ\n", "T Smith \nDartmouth College\n03755HanoverNH\n", "M Steadman \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "B Tonguc \nArizona State University\n85287TempeAZ\n", "C Tschalaer \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "E Tsentalovich \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "W Turchinetz \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "Y Xiao \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "W Xu \nTriangle Universities Nuclear Laboratory and Duke University\n27708-0305DurhamNC\n", "C Zhang \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "Z Zhou \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n", "T Zwart \nLaboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA\n" ]	[ "Arizona State University\n85287TempeAZ", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Institut für Kernphysik\nJohannes Gutenberg-Universität Mainz\nD-55099MainzGermany", "Arizona State University\n85287TempeAZ", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Boston University\n02215BostonMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "University of New Hampshire\n03824DurhamNH", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "University of New Hampshire\n03824DurhamNH", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Triangle Universities Nuclear Laboratory and Duke University\n27708-0305DurhamNC", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "University of New Hampshire\n03824DurhamNH", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "University of New Hampshire\n03824DurhamNH", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "United States Naval Academy\n21402AnnapolisMD", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Ohio University\n45701AthensOH", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "University of New Hampshire\n03824DurhamNH", "Arizona State University\n85287TempeAZ", "Dartmouth College\n03755HanoverNH", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Arizona State University\n85287TempeAZ", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Triangle Universities Nuclear Laboratory and Duke University\n27708-0305DurhamNC", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA", "Laboratory for Nuclear Science and Bates Linear Accelerator Center\nMassachusetts Institute of Technology\n02139CambridgeMA" ]	[]	We report new measurements of the neutron charge form factor at low momentum transfer using quasielastic electrodisintegration of the deuteron. Longitudinally polarized electrons at an energy of 850 MeV were scattered from an isotopically pure, highly polarized deuterium gas target. The scattered electrons and coincident neutrons were measured by the Bates Large Acceptance Spectrometer Toroid (BLAST) detector. The neutron form factor ratio G n E /G n M was extracted from the beam-target vector asymmetry A V ed at four-momentum transfers Q 2 = 0.14, 0.20, 0.29 and 0.42 (GeV/c) 2 .	10.1103/physrevlett.101.042501	[ "https://arxiv.org/pdf/0803.3827v2.pdf" ]	11,270,509	0803.3827	b5ea22af22c2c276557904d7ce798c4263deb31a	The Charge Form Factor of the Neutron at Low Momentum Transfer from the 2 H( e, e ′ n)p Reaction 8 Apr 2008 (Dated: April 8, 2008) E Geis Arizona State University 85287TempeAZ M Kohl Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA V Ziskin Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA T Akdogan Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA H Arenhövel Institut für Kernphysik Johannes Gutenberg-Universität Mainz D-55099MainzGermany R Alarcon Arizona State University 85287TempeAZ W Bertozzi Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA E Booth Boston University 02215BostonMA T Botto Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA J Calarco University of New Hampshire 03824DurhamNH B Clasie Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA C B Crawford Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA A Degrush Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA T W Donnelly Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA K Dow Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA M Farkhondeh Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA R Fatemi Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA O Filoti University of New Hampshire 03824DurhamNH W Franklin Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA H Gao Triangle Universities Nuclear Laboratory and Duke University 27708-0305DurhamNC S Gilad Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA D Hasell Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA P Karpius University of New Hampshire 03824DurhamNH H Kolster Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA T Lee University of New Hampshire 03824DurhamNH A Maschinot Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA J Matthews Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA K Mcilhany United States Naval Academy 21402AnnapolisMD N Meitanis Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA R G Milner Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA J Rapaport Ohio University 45701AthensOH R P Redwine Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA J Seely Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA A Shinozaki Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA S Širca Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA A Sindile University of New Hampshire 03824DurhamNH E Six Arizona State University 85287TempeAZ T Smith Dartmouth College 03755HanoverNH M Steadman Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA B Tonguc Arizona State University 85287TempeAZ C Tschalaer Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA E Tsentalovich Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA W Turchinetz Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA Y Xiao Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA W Xu Triangle Universities Nuclear Laboratory and Duke University 27708-0305DurhamNC C Zhang Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA Z Zhou Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA T Zwart Laboratory for Nuclear Science and Bates Linear Accelerator Center Massachusetts Institute of Technology 02139CambridgeMA The Charge Form Factor of the Neutron at Low Momentum Transfer from the 2 H( e, e ′ n)p Reaction 8 Apr 2008 (Dated: April 8, 2008)(The BLAST Collaboration)numbers: 1340-f1340Gp1388+e1420Dh2530Bf Keywords: Neutronform factorpolarizationinternal targetelasticdeuterium We report new measurements of the neutron charge form factor at low momentum transfer using quasielastic electrodisintegration of the deuteron. Longitudinally polarized electrons at an energy of 850 MeV were scattered from an isotopically pure, highly polarized deuterium gas target. The scattered electrons and coincident neutrons were measured by the Bates Large Acceptance Spectrometer Toroid (BLAST) detector. The neutron form factor ratio G n E /G n M was extracted from the beam-target vector asymmetry A V ed at four-momentum transfers Q 2 = 0.14, 0.20, 0.29 and 0.42 (GeV/c) 2 . The neutron is composed of charged constituents, whose net distribution is described by the charge (or electric) form factor G n E . Differences in the up and down quark distributions produce a nonuniform distribution of the net charge [1]. The neutron electric form factor G n E exhibits a maximum in the region of Q 2 ≈ 0.1 − 0.5 (GeV/c) 2 ; when Fourier-transformed this corresponds to a positively charged core and a concentration of negative charge at intermediate to large distances of ≈ 1 fm, commonly associated with a meson cloud surrounding the nucleon. Models which emphasize the role of the meson cloud have been successful in explaining important aspects of nucleon structure [2, 3, 4, 5,6]. Precise knowledge of G n E is essential for the description of electromagnetic structure of nuclei, and for the interpretation of parity violating electron scattering experiments to determine the strangeness content of the nucleon. Further, it is anticipated that exact ab initio QCD calculations of G n E using lattice techniques will eventually be possible [7]. In the absence of a free neutron target, determinations of G n E at finite Q 2 are typically carried out using quasielastic electron scattering from deuterium or 3 He targets. Despite the small value of G n E compared to the neutron magnetic form factor G n M , it can be obtained with high precision from double-polarization observables based on the interference of G n E with G n M . With the availability of high-duty-factor polarized electron beams over the last decade, experiments have employed recoil polarimeters, and targets of polarized 2 H and 3 He to perform precision measurements of G n E using polarization techniques with inherently small systematic uncertainties [8]. The slope of G n E (Q 2 ) at Q 2 = 0, which defines the square of the neutron charge radius, is determined precisely by the scattering of thermal neutrons from atomic electrons [9]. This paper reports on new measurements of G n E /G n M at low Q 2 in the vicinity of the maximum of G n E , using a longitudinally polarized electron beam incident on a vector-polarized 2 H target internal to the South Hall Ring at the MIT-Bates Linear Accelerator Center. The BLAST detector was used to detect quasielastically scattered electrons in coincidence with recoil neutrons over a range of Q 2 between 0.10 and 0.55 (GeV/c) 2 . The differential cross section for the 2 H(e, e ′ n) reaction with polarized beam and target can be written [10,11,12] d 3 σ/(dΩ e dΩ pq dω) = σ unp (1 + Σ + P e ∆) (1) with Σ = 3 2 P z A V d + 1 2 P zz A T d ∆ = A e + 3 2 P z A V ed + 1 2 P zz A T ed ,(2) where σ unp is the unpolarized differential cross section, P z = n + −n − and P zz = n + +n − −2n 0 are the vector and tensor polarizations of the deuteron target defined by the relative populations n m of the three deuteron magnetic substates with respect to the deuteron orientation axis, m = +1, 0, −1, respectively, and P e is the longitudinal polarization of the electron beam. With BLAST, all of the polarization observables A i in Eq. (2) have been measured for the first time with precision in a single experiment. The beam-target vector polarization observable A V ed is particularly sensitive to the neutron form factor ratio G n E /G n M [12]. In the Plane Wave Born Approximation (PWBA) and with the deuteron in a pure S-state, the asymmetry A V ed can be written analogously to elastic scattering from the free neutron as A V ed = a G n M 2 cos θ * + b G n E G n M sin θ * cos φ * c G n E 2 + G n M 2 ≈ a cos θ * + b G n E G n M sin θ * cos φ * ,(3) where θ * and φ * are the target spin orientation angles with respect to the momentum transfer vector and a, b, and c are known kinematic factors. This asymmetry has the largest sensitivity to G n E when the momentum transfer vector is perpendicular to the target polarization, i.e. θ * = 90 • . However, there are sizable corrections to the asymmetry in Eq. (3), mainly at low Q 2 where they are dominated by final state interactions (FSI). The relative contributions of meson exchange currents (MEC), isobar configurations (IC) and relativistic corrections (RC) become more significant as the momentum transfer increases (see Fig. 1). Extracting G n E must be done by comparison with theoretical asymmetries that include these effects. The effects of FSI can be monitored with the other polarization observables in Eq. (2). The asymmetries A e , A V d , and A T ed all vanish in the Born approximation due to parity and time reversal conservation and remain very small (below 1%) even in the presence of FSI. This permits these observables to be used to identify any false asymmetries in the experiment. FSI gives a sizable contribution to the target tensor asymmetry A T d , which is insensitive to G n E and otherwise close to zero in the quasifree limit. Figure 1 displays a Monte Carlo simulation of the reaction mechanism effects on the asymmetries A V ed (upper panel) and A T d (lower panel) as a function of Q 2 along with the measured values. The calculations use the standard dipole form factor G D = (1 + Q 2 /0.71) −2 for G p E , G p M /µ p , and G n M /µ n , and 1.91τ /(1 + 5.6τ ) G D for G n E [13], where µ p = 2.79, µ n = −1.91, and τ = Q 2 /(4m 2 n ). The good agreement of the measured tensor asymmetry A T d with the full model supports the calculations of FSI for a reliable extraction of G n E from the beam-target vector asymmetry A V ed at the percent level. On the other hand, the corrections at low Q 2 to A V ed measured in the 2 H( e, e ′ p)n reaction in quasifree kinematics are negligible [12], which allows for a precise determination of the product of beam and target polarizations P e P z along with the proton form factor ratio G p E /G p M in this reaction channel [14]. The BLAST experiment was designed to carry out spin-dependent electron scattering from hydrogen [15] and light nuclei. Details on the experimental setup can be found in [16]. The internal target consisted of an atomic beam source (ABS) combined with an open-ended storage cell through which the stored electron beam passed continuously [17]. The ABS produced polarized monoatomic deuterium gas in the storage cell with nuclear vector (V +: m=1; V −: m=−1) and tensor (T −: m=0) polarization states. In addition, the helicity h of the electron beam was flipped every injection cycle. Linear combinations of the six charge-normalized yields Y hm define all five polarization observables in Eq. (2). The experimental value of the beam-vector polarization observable A V ed can be written as A V ed = 3 2 1 P e P z Y ++ + Y −− − Y +− − Y −+ Y tot ,(4) where Y tot is the total yield obtained by summing up all six combinations hm. A modest magnetic holding field was applied to define the polarization angle θ d within the horizontal plane and to minimize the depolarization of target atoms. The variation of θ d was carefully mapped over the extent of the target cell. The average value of θ d was determined along with the tensor polarization P zz by comparing the simultaneously measured tensor asymmetries in elastic scattering from tensor-polarized deuterium [18] with those expected at low Q 2 based on a parameterization of previous data [19]. The BLAST detector is a toroidal spectrometer (8 sectors) with the horizontal sectors instrumented with wire chambers, aerogelČerenkov counters, thin plastic timing scintillators, and thick plastic scintillator walls for neutron detection. With the target polarization vector pointing into the left sector, the neutron detection efficiency was augmented in the right sector covering the kinematic region most sensitive to the neutron form factor ratio, as indicated by the sin θ * term in Eq. (3). The detection of neutrons in the left sector was primarily used to independently verify the determination of P e P z from the 2 H( e, e ′ p)n reaction. The selection of (e, e ′ n) events is very clean; the number of proton tracks misidentified as neutrons is negligible, due to the highly efficient charged particle veto provided by the thin scintillator bars and the large-volume drift chambers in front of the neutron detectors. A set of cuts applied on the time correlation between the charged and the neutral track, and on kinematic constraints for the electrodisintegration process, was employed to identify the quasielastic (e, e ′ n) events. The background from scattering off the aluminum target cell walls, measured with a hydrogen (empty) target, is less than 4% (3%) of the normalized yield obtained with deuterium. The corrected asymmetries were compared to Monte Carlo simulations based on the deuteron electrodisintegration model [11], for which events were generated according to the unpolarized cross section and weighted event-by-event with the spin-dependent terms in Eq. (2). The acceptance-averaged asymmetry A V ed was simulated for different values of G n E /G n M and compared to the experimental values. In order to extract the best value of the form factor ratio for each Q 2 bin, a χ 2 minimization was performed independently with respect to the missing momentum of the reaction and the angle of the neutron in the hadronic center-of-mass system. Both extractions produced consistent results. The data reported here were acquired in two separate runs in 2004 and 2005, corresponding to a target polarization angle of 31.64 • ± 0.43 • and 46.32 • ± 0.45 • , respectively. With a total accumulated beam charge of 451 kC (503 kC) in the first (second) data set, final samples of 268,914 (205,252) coincident electron-neutron events were collected. The average product of beam and target polarization determined from the 2 H( e, e ′ p) reaction was P e P z = 0.5796±0.0034(stat)±0.0034(sys) in the first and 0.5149±0.0043(stat)±0.0054(sys) in the second data set [14]. In comparison, the polarization product determined from 2 H( e, e ′ n)p with neutrons detected in the left sector of BLAST corresponding to θ * ≈ 0 • , was found to be 0.587 ± 0.019(stat) and 0.481 ± 0.026(stat) consistent with the above (e, e ′ p) results. The two data sets were treated as separate experiments producing two consistent results for the form factor ratio, which were combined for a final result. The data were divided into four Q 2 bins to determine G n E /G n M with a comparable statistical significance (see Table I). The systematic error of G n E /G n M is dominated by the uncertainty of the target spin angle θ d . Other systematic uncertainties include that of the beam-target polarization product P e P z , the accuracy of kinematic reconstruction, as well as the dependency on software cuts. The systematic uncertainties were evaluated individually for each Q 2 bin and data set by combining the errors from each source, taking covariances into account; the correlated and uncorrelated error categories of the two measurements were then combined for a resulting systematic error of each bin. False asymmetries were studied with the observables A V d and A T ed and found to be consistent with zero. Radiative corrections to the asymmetries calculated in a PWBA formalism using the code MASCARAD [20] are <1% and therefore also neglected. The uncertainties of the reaction mechanism and FSI corrections, which are small compared to the experimental errors, are not included in the systematic error. The world's data on G n E from double-polarization experiments [8] are displayed in Fig. 2 along with the results of this work. All of the polarization data were experimentally determined as electric to magnetic form factor ratios. We used parameterization [21] for G n M , which is in good agreement with recent measurements [22], to determine G n E from BLAST and to adjust the previously published values. The data from a variety of experiments are consistent and remove the large model uncertainty of previous G n E extractions from elastic electron-deuteron scattering [23]. The new distribution is also in agreement with G n E extracted from the deuteron quadrupole form factor [24]. The measured distribution of G n E can be parameterized as a function of Q 2 based on the sum of two dipoles, , 2), shown as the BLAST fit in Fig. 2 (blue line) with a one-sigma error band. With G n E (0) = 0 and the slope at Q 2 = 0 constrained by r 2 n = (−0.1148 ± 0.0035) fm 2 [9], one parameter is fixed, resulting in a 1 = −a 2 = 0.095 ± 0.018, b 1 = 2.77 ± 0.83, b 2 = 0.339 ± 0.046 and cov(a 1 , b 1 ) = −0.014, cov(a 1 , b 2 ) = 0.0008, cov(b 1 , b 2 ) = −0.036 with Q 2 in units of (GeV/c) 2 . The parameterization [25] (magenta dash-dotted line) is based on the form introduced in [21] with an additional bump structure around 0.2 − 0.4 (GeV/c) 2 . Also shown are recent results based on vector meson dominance (VMD) and dispersion relations (red short-dashed [4] and green long-dashed lines [5]), and the prediction of a light-front cloudy bag model with relativistic constituent quarks [6] (cyan dotted line). i a i /(1 + Q 2 /b i ) 2 (i=1 The new data from BLAST do not show a bump structure at low Q 2 as previously suggested [21,25]. The BLAST data are in excellent agreement with VMD based models [4,5] and also agree with the meson-cloud calculation [6]. The improved precision of the data at low Q 2 provides strong constraints on the theoretical understanding of the nucleon's meson cloud. We thank the staff of the MIT-Bates Linear Accelerator Center for delivering high quality electron beam and for their technical support, and A. Bernstein for suggesting the form of the BLAST fit. This work has been supported in part by the US Department of Energy and National Science Foundation. * Reported results are based on the Ph.D. theses of E.G. and V.Z. † Corresponding author, email [email protected] The "BLAST fit" (blue solid line) is a parameterization of the data based on the sum of two dipoles shown with a one-sigma error band. The recent parameterization [25] (magenta dashdotted line) is based on the form introduced in [21]. Also shown are recent results based on vector meson dominance and dispersion relations (red short-dashed [4] and green longdashed lines [5]), and of a light-front cloudy bag model with relativistic constituent quarks [6] (cyan dotted line). FIG. 1 : 1Measured (solid blue points) and calculated beamtarget vector polarization observable A V ed (upper panel) and tensor asymmetry A T d (lower panel) for the 2 H(e, e ′ n)p reaction at 850 MeV, a target orientation of θ d = 31.6 • into the left sector of BLAST, and with neutrons detected in the right sector. The colored curves are Monte Carlo simulations based on the deuteron electrodisintegration model of Ref.[11] (dotted magenta = PWBA, short-dashed green = PWBA+FSI, solid red = PWBA+FSI+MEC+IC+RC) using standard parameterizations for the nucleon form factors (see text). In addition, the corresponding curves for G n E ≡ 0 (dash-dotted red) and for elastic scattering from the free neutron (dashed black line) are shown. 1171 ± 0.0182 ± 0.0052 TABLE I: Results for the extracted neutron form factor ratio µnG n E /G n M (µn = G n M (0) = −1.91) with statistical and systematic errors, respectively. [ 1 ] 1A. Thomas and W. 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[ "Influence of optical nonlinearities on plasma waves in graphene", "Influence of optical nonlinearities on plasma waves in graphene" ]	[ "Sergey A Mikhailov \nInstitute of Physics\nUniversity of Augsburg\nD-86135AugsburgGermany\n" ]	[ "Institute of Physics\nUniversity of Augsburg\nD-86135AugsburgGermany" ]	[]	A theory of the nonlinear plasma waves in graphene is developed in the nonperturbative regime. The influence of strong electric fields on the position and linewidth of plasma resonances in the farinfrared transmission experiments, as well as on the wavelength and the propagation length in the scanning near-field optical microscopy experiments is studied. The theory shows that the fields of order of a few to a few tens of kV/cm should lead to a red shift and broadening of plasma resonances in the first type and to a reduction of the wavelength and the propagation length in the second type of experiments.	10.1021/acsphotonics.7b00468	[ "https://arxiv.org/pdf/1705.08999v2.pdf" ]	119,413,307	1705.08999	87bd947db2e0b2461ddfb16ea63595bfa11bec8b	Influence of optical nonlinearities on plasma waves in graphene 18 Jul 2017 Sergey A Mikhailov Institute of Physics University of Augsburg D-86135AugsburgGermany Influence of optical nonlinearities on plasma waves in graphene 18 Jul 2017 A theory of the nonlinear plasma waves in graphene is developed in the nonperturbative regime. The influence of strong electric fields on the position and linewidth of plasma resonances in the farinfrared transmission experiments, as well as on the wavelength and the propagation length in the scanning near-field optical microscopy experiments is studied. The theory shows that the fields of order of a few to a few tens of kV/cm should lead to a red shift and broadening of plasma resonances in the first type and to a reduction of the wavelength and the propagation length in the second type of experiments. I. INTRODUCTION The field of graphene plasmonics attracted much interest in recent years 1 . Theoretically the spectrum of twodimensional (2D) plasmons in doped graphene was calculated in Refs. 2,3 . In the long-wavelength limit q ≪ k F the 2D plasmon frequency ω is related to the 2D plasmon wavevector q, electron density n s , Fermi energy E F and the effective scattering time τ by the dispersion relation ω ω + 1 τ = 2e 2 E F 2 κ 0 q = 2e 2 √ πn s v F q κ 0 ,(1) where k F is the Fermi wavevector, v F ≈ 10 8 cm/s is the Fermi velocity and κ 0 is the dielectric constant of the surrounding medium. Experimentally, the 2D plasmons in graphene have been observed using the far-infrared (FIR) transmission spectroscopy in a system of narrow graphene stripes 4 and using the scanning near-field optical microscopy (SNOM) in Refs. 5-7 ; other experimental techniques have also been used, see Ref. 8 . In the first method 4 the plasmon wavevector q in (1) is given by the stripe width W , q ≃ π/W , and one observes a transmission resonance with the position ω ′ p and the linewidth ω ′′ p determined by the real and imaginary parts of the frequency ω calculated from Eq. (1). In the second technique 5-7 the 2D plasmon frequency is determined by the frequency of the incident radiation and the wavelength λ p = 2π/q ′ and the propagation length L p = 1/q ′′ of the plasmon are given by the real and imaginary parts of q calculated from the dispersion equation (1). Another topic actively developing in graphene optics nowadays is the nonlinear electrodynamic response of graphene. It was predicted in 2007 (Ref. 9 ) that, due to the linear energy dispersion of graphene quasi-particles this material should demonstrate a strongly nonlinear electrodynamic response. Shortly after that it was confirmed by both experimental and further theoretical studies that the nonlinear parameters of graphene are much larger than in many other materials indeed (the strong nonlinearity of graphene was disputed in Ref. 10 ; a detailed analysis of this paper can be found in a recent work 11 ). Experimentally the higher harmonics generation [12][13][14][15] , four-wave mixing 16,17 , Kerr effect [18][19][20] have been measured by different methods. Theoretically the nonlinear electrodynamic response was studied both within the quasiclassical 21,22 and quantum approaches [23][24][25][26] , and different aspects of the nonlinear graphene response have been analyzed [27][28][29][30][31][32] . In view of the great interest to the two topics outlined above a question arises how the nonlinear properties of graphene influence the plasma waves in this material. The opposite question -how plasmons influence the strength of the nonlinear effects in graphene -has been discussed in Refs. 28,33,34 . Here we address the question how the frequency ω ′ p and the linewidth ω ′′ p of graphene plasmons in the first type of experiments and the wavelength λ p and the propagation length L p in the second type of experiments are modified if the intensity of the plasmon electric field is so strong that the nonlinear effects become essential. II. THEORY In the linear electrodynamics the spectrum (1) of plasma waves in 2D electron systems is calculated from the dispersion relation ǫ lin (q, ω) ≡ 1 + 2πiσ(ω)q ωκ 0 = 0 (2) where σ(ω) is the linear conductivity of the 2D layer and ǫ lin (q, ω) is the effective dielectric function of graphene in the linear approximation. If graphene is doped and the plasmon frequency satisfies the condition ω E F (equivalent to q k F ) the inter-band contribution to the linear conductivity of graphene (see Ref. 35 ) can be neglected and σ(ω) in (2) is given by the intra-band conductivity σ(ω) ≃ σ intra (ω) = e 2 E F π 2 i ω + i/τ = σ 0 1 − iωτ ,(3) where σ 0 is the static conductivity of graphene. The Drude formula (3) together with the dispersion equation (2) give the 2D plasmon spectrum (1). The linear response approach is valid when the plasmon field is not very strong, i.e. when the field parameter F ω = eE 0 k F ω(4) is small as compared to unity, F ω ≪ 1, see Refs. 9,36 . A nonperturbative quasiclassical theory which gives a general relation between the current and the field at arbitrary values of F ω has been recently developed in Ref. 36 . It was shown there that in the nonlinear regime the linear conductivity σ(ω) should be replaced by the function σ ω,ω (ωτ, F τ ) = σ 0 S 1 (ωτ, F τ ),(5) where F τ = ωτ F ω = eE 0 τ k F (6) is a frequency independent field parameter which is convenient to use analyzing the frequency dependencies of the nonlinear response, S 1 (ωτ, F τ ) = ∞ 0 e −ξ sin(ωτ ξ/2) ωτ /2 B 1 F τ sin(ωτ ξ/2) ωτ /2 e iωτ ξ/2 dξ (7) is a complex function of ωτ and F τ , B 1 (a) = 4 π π/2 0 sin 2 x 1 + (a sin x) 2 2 F 1 1 4 , 3 4 , 2; 2a sin x 1 + (a sin x) 2 2 dx(8) and 2 F 1 (a, b, c; x) is the hypergeometric function (for details see Ref. 36 ). The dispersion equation of 2D plasmons in graphene in the nonlinear regime then reads ǫ nonlin (q, ω) ≡ 1 + 2πi ωκ 0 e 2 π E F τ S 1 (ωτ, F τ )q = 0.(9) Now we can analyze the obtained results. III. RESULTS AND DISCUSSION A. Nonlinearity in a FIR transmission experiment In the FIR transmission (absorption) experiment, in the linear regime, the absorption coefficient is proportional to A lin (ω) ∝ σ ′ (ω)/\|ǫ lin (q, ω)\| 2 where q ≃ π/W is fixed. In the nonlinear regime we get A nonlin (ω) ∝ σ ′ ω,ω (ωτ, F τ )/\|ǫ nonlin (q, ω)\| 2 which gives A nonlin (ω) ∝ S ′ 1 (ωτ, F τ ) 1 + i(ω p τ ) 2 S1(ωτ,Fτ ) ωτ 2 ,(10) where is the (linear-regime) plasma frequency, see Eq. ω 2 p = 2e 2 E F 2 κ 0 q(11) (1). Figure 1 shows the nonlinear absorption coefficient (10) as a function of frequency ω/ω p and the field parameter F τ at ω p τ = 10. If F τ → 0 the absorption spectrum has a standard Drude shape with the quality factor of order of 10. When the field parameter F τ grows but remains smaller than ≃ ω p τ the influence of the nonlinear effects is not essential: the resonance frequency experiences a red shift and the resonance becomes broader but these changes are not large. If F τ approaches the value ω p τ = 10 and exceeds it, the resonance frequency decreases dramatically, its linewidth grows and becomes comparable with the frequency. The boundary between the linear and nonlinear regimes is thus determined by the condition F p ≃ 1, where F p = F τ ω p τ = eE 0 k F ω p = eE 0 v F /ω p E F(12) (compare with (4)). The parameter F p determines how much energy electrons obtain from the external electric field during one period of plasma oscillations as compared to their average (Fermi) energy. The nonlinear regime is realized when F p 1. Figure 2 illustrates the density dependence of the plasma frequency (11) and of the "nonlinear" electric field determined by the condition F p = 1 (i.e. the field required to observe the nonlinear effects), for parameters of Ref. 4 . One sees that dependent on the electron density n s and the stripe width W the plasma frequency lies in the range from ≃ 1 to a few THz and the electric field at which the nonlinear effects can be observed is of the order of a few to tens kV/cm. B. Nonlinearity in a SNOM experiment In a SNOM experiment the frequency ω is a fixed real value and the wave-vector q is a complex function. In order to analyze its dependence on the frequency and field parameters we write it in the form q k F = v F κ 0 2e 2 E F τ 2 Q,(13) and plot the real and imaginary parts of the dimensionless wave-vector in Figure 3. At F τ → 0 the real and imaginary parts of Q depend on ωτ quadratically and linearly, respectively, Q ′ = (ωτ ) 2 , Q ′′ = ωτ . When F τ grows both Q ′ and Q ′′ increase, i.e., the wavelength and the propagation length become shorter in the nonlinear regime. At F τ > ωτ the real part of Q grows linearly with ωτ in a broad range ωτ 1, Fig. 3(a). The imaginary part of Q first linearly grows with ωτ and then saturates at approximately Q ′′ ≃ 2F τ , Fig. 3(b). The 2D plasmon wavelength λ p and the propagation length L p can be obtained from (13) and written as Q = iωτ S 1 (ωτ, F τ ) ,(14)λ p = 2πA \|S 1 (ωτ, F τ )\| 2 ωτ S ′′ 1 (ωτ, F τ ) ,(15)L p = A \|S 1 (ωτ, F τ )\| 2 ωτ S ′ 1 (ωτ, F τ ) ,(16) where A = 2e 2 E F τ 2 κ 0 2 ≈ 777.8 µm × n s [10 12 cm −2 ](τ [ps]) 2 κ 0(17) is a prefactor with the dimensionality of length. The frequency dependencies of λ p /A and L p /A are shown in Figure 3(c,d). The nonlinearity effect is seen, again, at ωτ F τ , i.e., at F ω 1. Both λ p and L p decrease under the action of the strong electric field with the propagation length being affected stronger. It is interesting that in a certain interval of ωτ (1 ωτ F τ ) the length L p becomes almost frequency independent, Figure 3(d). The absolute values of λ p and L p can be estimated from Figure 3(c,d) and Eq. (17). For example, if n s = 10 12 cm −2 , τ = 1 ps and κ 0 = 3.9, the length A is about 200 µm. Then, if the plasmon frequency is 2 THz we get λ p = 7.96 µm and L p ≈ 15.92 µm at F τ → 0 (L p /λ p = 2) and λ p = 1.02 µm and L p = 1.13 µm at F τ = 80 (L p /λ p = 1.1). At higher frequencies both λ p and L p are smaller (e.g. at 10 THz and F τ → 0 the lengths are λ p = 0.32 µm and L p ≈ 3.2 µm, L p /λ p = 10), but the influence of the strong electric field is similar: at F τ = 80 λ p = 0.176 µm and L p = 1.32 µm (L p /λ p = 7.5). IV. SUMMARY We have theoretically studied the influence of the nonlinear effects on the spectrum of 2D plasmons in graphene. In the FIR transmission experiments the strong external electric field is predicted to lead to a substantial red shift and to a broadening of the plasmon resonance. In the SNOM experiments the nonlinearity is shown to result in the reduction of both the wavelength and the propagation length of 2D plasmons as compared to the linear regime. The characteristic electric fields needed for observation of the nonlinear effects are determined by the condition F ω = eE 0 / k F ω 1 meaning that the nonlinearity in the 2D plasmon spectrum is more important at low frequencies and in samples with low charge carrier density. The absolute values of the electric field causing the nonlinear effects in the 2D plasmon spectrum lie in the range from a few to a few tens of kV/cm. p /A F τ =0 F τ =5 F τ =10 F τ =20 F τ =80p /A F τ =0 F τ =5 F τ =10 F τ =20 F τ =80 (d) FIG. 1 : 1The nonlinear 2D plasmon absorption as a function of frequency ω/ωp (horizontal axis) and electric field parameter Fτ (vertical axis) at ωpτ = 10. FIG. 2 : 2(a) The 2D plasmon frequency determined by Eq. (11) and (b) the nonlinear electric field determined by the condition Fp = 1 as a function of the electron density ns at several values of the stripe width W . It is assumed that q = π/W and κ0 = 4. 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[ "High speed spatially multimode Λ-type atomic memory with arbitrary frequency detuning", "High speed spatially multimode Λ-type atomic memory with arbitrary frequency detuning" ]	[ "T Golubeva \nSt. Petersburg State University\n198504St. Petersburg\n\nStary Petershof\nul. Ul'yanovskaya, 1Russia\n", "Yu Golubev ", "O Mishina \nSt. Petersburg State University\n198504St. Petersburg\n\nStary Petershof\nul. Ul'yanovskaya, 1Russia\n\nLaboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance\n\nTheoretische Physik\nUniversität des Saarlandes\nD-66123SaarbrückenGermany\n", "A Bramati \nLaboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance\n", "J Laurat \nLaboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance\n", "E Giacobino \nLaboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance\n" ]	[ "St. Petersburg State University\n198504St. Petersburg", "Stary Petershof\nul. Ul'yanovskaya, 1Russia", "St. Petersburg State University\n198504St. Petersburg", "Stary Petershof\nul. Ul'yanovskaya, 1Russia", "Laboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance", "Theoretische Physik\nUniversität des Saarlandes\nD-66123SaarbrückenGermany", "Laboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance", "Laboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance", "Laboratoire Kastler Brossel\nUniversité Pierre et Marie Curie\nEcole Normale Supérieure\nCNRS\nCase 74, 4 place Jussieu75252, Cedex 05ParisFrance" ]	[]	We present a general model for an atomic memory using ultra-short pulses of light, which allows both spatial and temporal multimode storage. The process involves the storage of a faint quantum light pulse into the spin coherence of the ground state of Λ-type 3-level atoms, in the presence of a strong driving pulse. Our model gives a full description of the evolution of the field and of the atomic coherence in space and time throughout the writing and the read-out processes. It is valid for any frequency detuning, from the resonant case to the Raman case, and allows a detailed optimization of the memory efficiency.	10.1140/epjd/e2012-20723-3	[ "https://arxiv.org/pdf/1112.4852v1.pdf" ]	119,177,574	1112.4852	0240fed97b2aecc2fa2d658e28ad100721be5992	High speed spatially multimode Λ-type atomic memory with arbitrary frequency detuning December 22, 2011 T Golubeva St. Petersburg State University 198504St. Petersburg Stary Petershof ul. Ul'yanovskaya, 1Russia Yu Golubev O Mishina St. Petersburg State University 198504St. Petersburg Stary Petershof ul. Ul'yanovskaya, 1Russia Laboratoire Kastler Brossel Université Pierre et Marie Curie Ecole Normale Supérieure CNRS Case 74, 4 place Jussieu75252, Cedex 05ParisFrance Theoretische Physik Universität des Saarlandes D-66123SaarbrückenGermany A Bramati Laboratoire Kastler Brossel Université Pierre et Marie Curie Ecole Normale Supérieure CNRS Case 74, 4 place Jussieu75252, Cedex 05ParisFrance J Laurat Laboratoire Kastler Brossel Université Pierre et Marie Curie Ecole Normale Supérieure CNRS Case 74, 4 place Jussieu75252, Cedex 05ParisFrance E Giacobino Laboratoire Kastler Brossel Université Pierre et Marie Curie Ecole Normale Supérieure CNRS Case 74, 4 place Jussieu75252, Cedex 05ParisFrance High speed spatially multimode Λ-type atomic memory with arbitrary frequency detuning December 22, 2011EPJ manuscript No. (will be inserted by the editor) We present a general model for an atomic memory using ultra-short pulses of light, which allows both spatial and temporal multimode storage. The process involves the storage of a faint quantum light pulse into the spin coherence of the ground state of Λ-type 3-level atoms, in the presence of a strong driving pulse. Our model gives a full description of the evolution of the field and of the atomic coherence in space and time throughout the writing and the read-out processes. It is valid for any frequency detuning, from the resonant case to the Raman case, and allows a detailed optimization of the memory efficiency. Introduction For quantum telecommunications and for quantum information processing, memory registers able to store quantum information without measuring it are essential devices. A quantum memory relies on an efficient coupling between light and matter, allowing reversible mapping of quantum photonic information in and out of the material system. In the past years, several protocols have been developed theoretically and experimentally [1,2]. Storage and retrieval of some of the basic states of light for quantum communication such as a polarization q-bit [3], squeezed light [4,5] and entangled photons [6,7] or faint coherent pulses at the level of one to few photons have been realized [8,9,10,11,12]. Recent experiments have achieved large efficiencies [10,11]. However, the processing speed and the available bandwidth of the memories remain a challenge for quantum memories. The first quantum memory registers proposed more than a decade ago [13] involve the transfer of quantum information from light to atoms (writing) and back from atoms to light (retrieval), using electromagnetically induced transparency (EIT) in atomic three-level transitions, and this process implies a limited bandwidth. The storage protocol relies on a strong control field, generating EIT for the weak field that carries the quantum signal to be stored. The group velocity for the signal field is strongly reduced and the signal pulse is compressed by several orders of magnitude. A signal pulse can thus be contained inside the atomic medium, and before it propagates outside the medium, the control is switched off. The quantum variables of the signal field are then converted from a purely photonic state to a collective spin coherence. For read-out, the control field is turned on again and the medium emits a weak pulse, carrying the quantum information contained in the original pulse. While in principle this allows direct mapping of the quantum state of light into long lived coherences in the atomic ground state, the bandwidth of the stored signal is strongly limited by the transparency window associated to EIT. Various methods have been proposed to achieve broadband memories and escape the limitations linked to EIT. Very interesting protocols are based on the implementation of controlled broadening, such as CRIB (controlled reversible inhomogeneous broadening) [14], using photon echo-type reversal [15] and AFC (atomic frequency comb) where an absorbing comb structure is created in the medium [16]. These techniques have been successfully applied for echo-type light storage in rare-earth doped crystals [17] and atomic vapours [18]. These methods allow broad bandwidth but they imply writing times which are still rather long (of the order of microseconds). In the spatial domain, an interesting phenomena, quantum holography, has been proposed to implement 3D-memories [19,20]. An alternative method is based on the use of a broadband, ultrafast control and signal field pulses for the writing process. It relies on two different atomic transitions sharing the same excited state. The control field and the signal field contribute to a two-photon process coupling two ground states. In this case the signal pulse is converted into an atomic coherence between ground and excited states and then into a ground state coherence by the control pulse. However, since the interaction times are very short, it does not allow for the buildup of EIT. This method has been proposed [21,22] and demonstrated experimentally in far off-resonance conditions [23,12]. In this paper, we present a detailed theoretical model for an ultrafast memory without adiabatic approximation and valid for arbitrary frequency detuning. We show that this protocol holds the promise for a quantum memory with high efficiency, fast operation and broad bandwidth together with spatial multimode capacity. The problem of the achievable efficiency in this case was treated in Refs. [21,22] using a numerical optimization procedure based on the search of the optimal pulse shape for the signal or driving field in the limit of adiabatic elimination of the excited state. In Ref. [24] the shaping of the driving pulse is based on the analysis of the Lagrange function, avoiding the adiabatic approximation in the optimization procedure. A very good efficiency can be obtained even for short pulse durations. In Ref. [25] a different optimization technique based on the minimization of the losses has been used in the resonant case without adiabatic approximation, in the limit of very short pulses (shorter than the excited state decay time). In the present work we extend this technique to the case of arbitrary frequency detuning. We show that the memory efficiency can be very good even for large detunings as long as the experimental parameters are properly optimized. Moreover, we explore the transition region between resonant and adiabatic regimes and we demonstrate that our technique allows finding optimal parameters for storage. The article is organized as follows. In Section 2 we present the model system and we write the main equations ruling it. In Section 3 we give the method for solving the equations for the writing and read-out processes in the semi-classical limit. In Section 4, we study the evolution of the atomic coherence and of the signal field during the writing process and we calculate the losses. In Section 5, we study the read-out process and the efficiency of the memory as a whole. Model system In this paper, we consider an ensemble of three-level atoms in a Λ-configuration ( Fig. 1) that will be used to store temporal and spatial multimode quantum fields. The atoms interact with two electromagnetic fields, a signal field E s and a driving field E d , that connect the two atomic ground states to the excited state. The driving field is a strong, classical field propagating as plane wave, while the signal field is a weak quantum field with a transverse structure. The signal and driving field are very short pulses that are assumed to be much shorter than the excited state lifetime γ −1 , so that we can neglect the spontaneous emission during the writing process. In the dipole approximation the light matter interaction Hamiltonian is given bŷ V = − jd j (t)Ê(t, r j ), E(t, r j ) =Ê s (t, r j ) +Ê d (t, r j ).(1) Hered j (t) is the electric dipole operator of the j-th atom located at r j . In the paraxial and quasi-resonant approximations the Hamiltonian can be rewritten in the form V = dz d 2 ρ i g â(z, ρ, t)σ 31 (z, ρ, t) e ik s z − i∆t −â † (z, ρ, t)σ 13 (z, ρ, t) e −ik s z + i∆t +i Ω(t)σ 32 (z, ρ, t) e ik d z − i∆t −Ω * (t)σ 23 (z, ρ, t) e −ik d z + i∆t .(2) Here k s and k d are the wave vectors of the signal and driving fields, and ρ = ρ(x, y). The one-photon detunings of the signal and driving fields are assumed to be equal, and equal to ∆ so that the two-photon resonance between levels 1 and 2 is fulfilled ∆ = ω s − ω 13 = ω d − ω 23 .(3) The spatial coordinates z and ρ describe the longitudinal and transverse propagation of the signal field. The normalized amplitude of the signal fieldâ(z, ρ, t) is written asÊ s (r, t) = −i ω s 2ε 0 c e −iωst+ikszâ (z, ρ, t) + h.c.,(4) whereâ(z, ρ, t) is the annihilation operator for the signal field. Under propagation in free space we have the following commutation relations [26] â(z, ρ, t),â † (z, ρ , t ) = δ 2 (ρ − ρ )δ(t − t ),(5)â(z, ρ, t),â † (z , ρ , t) = = c 1 − i k s ∂ ∂z − c 2k 2 s ∆ ⊥ δ 3 (r − r ). (6) The amplitudeâ(z, ρ, t) is normalized so that the mean value â † (z, ρ, t)â(z, ρ, t) is the photon number per second per unit area. The symbol ∆ ⊥ represents the transverse Laplacian with respect to ρ ∆ ⊥ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 .(7) We define the intensity of the driving field through the Rabi frequency Ω. For the sake of simplicity we consider this value as real Ω = Ω * . The driving field is a classical plane monochromatic wave propagating along the z-axis. The coupling constant between atom and signal field is g = ω s 2 0 c 1/2 d 31 .(8) where d 31 is the electric dipole element on the transition \|1 → \|3 . We define the collective coherences and population as sums of over all the atomŝ σ ik (r, t) = jσ j ik (t) δ 3 (r − r j ),(9)N i (r, t) = jσ j ii (t) δ 3 (r − r j ).(10) These quantities fulfill the commutation relations [σ ik (r, t),σ ik (r , t)] = N i (r, t) −N k (r, t) δ 3 (r − r ),(11) In this basis one can derive a full system of Heisenberg equations for the collective operators namely the field amplitudeâ(r, t), the collective atomic coherencesσ ik (r, t) and the collective atomic populationsN i (r, t). Taking into account the Hamiltonian (2) and the commutation relations (6), (11), the complete system of the equations reads ∂ ∂t + c ∂ ∂z − ic 2k s ∆ ⊥ â = −cgσ 13 ,(12)∂ ∂tσ 13 = −i∆σ 13 + Ωσ 12 + gâ(N 1 −N 3 ),(13)∂ ∂tσ 12 = −Ωσ 13 − gâσ 32 ,(14)∂ ∂tσ 32 = i∆σ 32 − Ω(N 3 −N 2 ) + gâ †σ 12 ,(15)∂ ∂tN 1 = −gâσ 31 − gâ †σ 13 ,(16)∂ ∂tN 2 = −Ω (σ 32 −σ 23 ) ,(17)∂ ∂tN 3 = − ∂ ∂tN 1 − ∂ ∂tN 2 .(18) To derive the equations (12)- (18) we have performed the substitutionsσ 13 → e ik s z − i∆tσ 13 , σ 23 → e ik d z − i∆tσ 23 ,(19)σ 12 → e −i(k d − k s )zσ 12 . We have omitted the terms related to spontaneous relaxation \|3 → \|1 since we have assumed that the relaxation rate γ is small enough for the spontaneous emission to be negligible during the short time duration of the pulses. The equations can be written in a simplified way with a few approximations given below. According to equations (10) the collective atomic operators have sharp spatial distributions due to the deltalocalization of the atoms. However, due to the collective effect of atoms localized along a field trajectory, we can average Eqs. (12)-(18) over the positions of the atoms. We will also replace the operatorN 1 −N 3 in Eq. (13) by the number giving the mean atomic density N . We will assume that in the beginning of the process most atoms are in state \|1 . During the memory processes (writing and read-out) the population of the state \|1 stays close to its initial value, because the photon number in the signal pulse is much smaller than the initial atomic number. In Eq. (14), we can neglect the second term on the right hand side since gâ is much smaller than Ω (we have assumed \|Ω\| 2 g 2 â †â ). Furthermore,σ 32 σ 13 because the populations N 2 and N 3 are much smaller than N . Now we can write a simplified system of partial differential equations describing the evolution of the system as 1 c ∂ ∂t + ∂ ∂z − i 2k s 2 ⊥ â(z, ρ, t) = −gσ 13 (z, ρ, t),(20) ∂ ∂tσ 13 (z, ρ, t) = −i∆σ 13 (z, ρ, t) +gNâ(z, ρ, t) + Ωσ 12 (z, ρ, t),(21)∂ ∂tσ 12 (z, ρ, t) = −Ωσ 13 (z, ρ, t).(22) In a previous publication [25], similar equations were derived for the case of a resonant excitation (∆ = 0). Here and below we omit the averaging over the atomic localizations. Let us renormalize the coherencesσ 13 andσ 12 σ 12 (z, ρ, t)/ √ N =b(z, ρ, t),(23)σ 13 (z, ρ, t)/ √ N =ĉ(z, ρ, t)(24) so that they obey to the bosonic commutation relations: b (r, t),b † (r , t) = ĉ(r, t),ĉ † (r , t) = δ 3 (r − r ). (25) Here again we have taken into account the fact thatN 1 − N 2,3 → N . In the Fourier domain relative to the transverse coordinates ρ the equations read ∂ ∂zâ (z, t; q) = −g √ Nĉ(z, t; q),(26)∂ ∂tĉ (z, t; q) = −i∆ĉ(z, t; q) +g √ Nâ(z, t; q) + Ωb(z, t; q),(27)∂ ∂tb (z, t; q) = −Ωĉ(z, t; q),(28) where we have introduced the transverse wavevector q and we have made the changeŝ a(z, t; q) →â(z, t; q)e −iq 2 z/(2k s ) , (29) b(z, t; q) →b(z, t; q)e −iq 2 z/(2k s ) ,(30)c(z, t; q) →ĉ(z, t; q)e −iq 2 z/(2k s ) .(31) From the system (26)-(28) one can obtain a conservation equation ∂â †â ∂z + ∂b †b ∂t + ∂ĉ †ĉ ∂t = 0.(32) This equation means that the input photons of the weak quantum field are converted into excitations of the atomic coherencesσ 13 andσ 12 . The aim is to store the information carried by the signal field in the ground state coherenceσ 12 , so that an excitation of the state \|3 is undesirable. It is possible to reduce this loss channel by an appropriate choice of the driving field. If the driving field power is high enough for the Rabi oscillation on the transition \|2 → \|3 to be more effective than a spontaneous emission (Ω γ) and if the pulse duration is short enough, so that the atoms undergoing a Rabi oscillation have no time to go back to the state \|3 , then the third term in Eq. (32) should be negligible. This will be studied in the optimization of the memory process. We neglect the time delay linked to the pulse propagation in the atomic medium. This means that, if we have long enough pulses, such that L/c T (L is the thickness of the medium and T is the pulse duration), we can neglect the time interval between the time at which the front part of the pulse enters the medium and the time at which the front part leaves it. Formally this means we can neglect the time derivative in Eq. (20). For simplicity we will assume that the driving pulse has a rectangular time distribution (in the equations Ω(t) = const for 0 < t < T ). Writing and read-out processes in the semi-classical limit The main aim of this paper is the determination of the memory efficiency. As it is well known the semiclassical description is sufficient for this and we can use the initial conditions : b(0, z; q) = c(0, z; q) = 0 for writing and a in (t; q) = c(0, z; q) = 0 for read-out. Here and below we omit the operator notation for the variables. The detailed resolution of the system of partial differential equations (26)-(28) can be found in Apps. A,B. Using the general solutions (51)-(53) one can obtain the semi-classical ones for the writing process for both the field amplitude a W (t, z; q) and the atomic coherence b(t, z; q) in the form a W (t,z; q) = T W 0 dt a in (t , q)G aa (t −t ,z), (33) b W (t,z; q) = −p T W 0 dt a in (t , q)G ab (t −t ,z),(34) where we have introduced the dimensionless timet and longitudinal spatial coordinatez according tõ t = Ω t,T W = ΩT W , z = 2g 2 N Ω z,L = 2g 2 N Ω L (35) where T W , is the common duration of signal and driving pulses for writing and L is the thickness of the memory cell. We have defined an effective interaction coefficient p given by p = g √ N Ω .(36) The kernels G aa (t, z) and G ab (t, z) are time convolutions G aa (t,z) = t 0 dt f (t ,z; r)f * (t −t ,z; −r), (37) G ab (t,z) = t 0 dt f 0 (t ,z; r)f * 0 (t −t ,z; −r),(38) where the functions f and f 0 are expressed via the n-th Bessel function of the first kind denoted by J n : f (t,z; r) = δ(t) − e −irt e −i 1 + r 2t × (1 + r)z 4t J 1 (1 + r)zt Θ(t),(39)f 0 (t,z; r) = e −irt e −i 1 + r 2t J 0 (1 + r)zt Θ(t).(40) Here the frequency detuning is given by the dimensionless parameter r = ∆/(2Ω) and Θ(t) is the step function Θ(t) = 1 for 0 <t <T W and equals zero otherwise. In order to estimate the efficiency of the memory, we calculate the field amplitude at the output of the cell. This amplitude is different in the case of forward and backward retrieval. The corresponding expressions read a R f or (t,L; q) = (41) = 1 2 T W 0 dt a in (t , q) L 0 dzG ab (t,z)G ba (t ,L −z), G ba (t,z) = G ab (t,z), and a R back (t,L; q) = (42) = 1 2 T W 0 dt a in (t , q) L 0 dzG ab (t,z)G ba (t ,z). The last two formulas are correct only in the approximation where diffraction is neglected. Indeed, we have not taken into consideration the diffraction phenomenon described by equation (29)-(31). As was demonstrated in [25] this effect does not introduce any significant corrections in the case of the forward retrieval but restricts the mode number for the backward retrieval. In the following we will use an optimization procedure based on the choice of the optimal relation between the thickness of the memory cellL and the durationT W of the signal and driving pulses. We will compare this optimization approach with the approach developed in Ref. [22] based on the optimization of the driving pulse shape in the case of short pulses. Discussion of the writing process In this section, we study the writing process, i.e. the conversion of the signal field into atomic coherence. Let us start with a simple calculation of this process at the input of the memory cell. For this we solve Eq. (34) atz = 0. In this case the atomic polarization b W (t, 0; q) corresponding to the "written" information reads b W (t, 0) = b W (t, 0; q)/(−2pa in (q)) = = 1 2 1 − e −irt cos(t r 2 + 1) + ir √ r 2 + 1 sin(t r 2 + 1) . (43) For the sake of the simplicity we have taken a in (t, q) = const(t). The normalized atomic polarization ranges from 0 to 1. In the limit of small or large detuning, we have a simple behaviour 2a shows the Rabi oscillation for a normalized detuning r = 0 while Fig. 2f shows a periodical modulation with a period of 4πr for r = 10. When r increases, one can see in Fig.2b,c the beat between two close frequencies. For larger r (Fig.2d,e) the curves show a slow oscillation (that corresponds to the term with frequency √ r 2 + 1 − r in (43)), modulated by a fast oscillation (at frequency √ r 2 + 1+r), with a modulation depth decreasing with increasing r. r 1 : \|b W (t, 0)\| 2 = sin 4t 2 ,(44)r 1 : \|b W (t, 0)\| 2 = sin 2t 4r (45) Fig. Following the variation of \|b W (t, 0)\| 2 as a function of r allows to get a first view of the behaviour of the system when the detuning is varied. It can be seen from Fig. 2 that even for a small detuning, r = 0.1, a significant distortion of the coherence profile at the input of the medium takes place as compared to the resonant case. On the other hand for r = 2 the high frequency modulation of the slow oscillations is rather small and the excitation can be considered as close to the off-resonant case, where the well-known solutions in the Raman limit can be used. A detailed study of the coherence distribution for allz, as given below, will allow to better characterize the interaction regime, between resonant and Raman. When shifting into the medium, the behavior of the coherence is much more complicated since it looses its simple harmonic character. Figure 3 shows a displacement of the maximum of the coherence along the medium. Let us follow the dependence of \|b W (t,z)\| 2 onz for a given value of t =T W = π given in Fig. 4. Comparing the curves for different values of the detuning, we see that the value of coherence atz = 0 significantly decreases with increasing detuning. From these curves it could be concluded that the larger r the less information is written in the cell. However, we will show that it is not true. Let us introduce the quantity n ef f that characterizes the proportion of signal photons converted into coherenceb W during the writing process: n ef f (T W ,L) = 1 T W 1 2 L 0 \|b W (T W ,z)\| 2 dz.(46) Here, 1/T W before the integral comes from the input pulse energy, and the factor 1/2 comes from the previously introduced dimensionless variables. The integral gives the normalized population N 2 in the medium with lengthL during the interaction timeT W . Since the transition of an atom to the level \|2 > in our model corresponds to the coherent scattering of a photon from the signal wave, this is also the number of signal photons that was recorded in the atomic coherence. We are interested in the percentage of input photons recorded in such a way. The calculated values of n ef f are presented Fig. 4a-c. We see that the number of recorded photons in the first two panels are almost identical (there is even a small increase of n ef f for r = 0.5 as compared with the resonant value), while for r = 2 the value n ef f decreases by a factor of less than 2. We can also follow the dependence of n ef f on r for a given medium length and for different durations of the signal pulse. Fig. 5 shows that the proportion of recorded photons decreases when the detuning increases, but this decrease depends on the duration of the writing process. In particular, we see that the writing efficiency depends weakly on the detuning forT W = π/4 but it stays quite low. With increasing pulse duration (T W = π) a plateau appears for detuning range 0 to 0.7. This range is reduced forT W = 2π, but a second plateau appears for r ∈ [1, 1.8]. While Figs. 3, 4 and 5 give a detailed behaviour of the efficiency of the writing process depending on the detuning, on the time duration of the pulse and on the length of the medium, it appears that the optimization of the efficiency is non trivial and requires a specific procedure. This will be studied in the next section. Estimation of the writing losses In Ref. [25] an algorithm of memory optimization, based on the minimization of leakage is described. Leakage is defined as L(T W ,L) = T W 0 \|a W (t,L)\| 2 dt T W 0 \|a in (t)\| 2 dt × 100%,(47) It characterizes the proportion of signal photons going out of the cell during the writing time. Such an estimation of the losses is justified when the leakage is the main origin of losses. However, as will be shown below, there is a range ofT W in which the population of the upper energy level is large enough to cause significant losses. We will show that for a high-speed memory the three-level atomic system can not be reduced to a two-level scheme. Note that this situation is specific of the case of simultaneous interaction of the signal and control fields with matter, and does not happen in memory protocols based on an echo [15]. We have already introduced the value n ef f , which is the proportion of signal photons that have been recorded. Then, the value of the total losses (as a percentage of the number of photons in the input signal pulse) can be expressed as follows : Fig. 6 shows the losses associated with leakage L (blue curves, dotted lines) and the total losses of photons L c , (red curves, full lines) as a function of the duration of the writing for a given medium length, and three different values of r. L c (T W ,L) = (1 − n ef f (T W ,L)) · 100%.(48) First, let us note that these curves are not monotonous and exhibit one or several minima. This means that for a given lengthL one can find the pulse duration which is recorded in the atomic medium with minimal losses. Significant difference between the curves corresponding to leakage and to total losses come from the role of the upper level in the interaction of such pulses with the atomic medium. However, in the region of minimum losses, the distance between the two curves is small. The optimization of the memory based on leakage allows to define a range of valuesT W , for which an efficient writing is expected. However, the curve giving the total losses is a more precise tool to determine the optimum ratio betweenT W andL. As can be seen from the plots, above some value of T W the two curves coincide, i.e. all the system losses are associated with leakage only and level of \|3 is not populated at the end of the process. The larger detuning r the smaller the valueT W for which this happens. We can also follow the dependence of the losses on the lengthL for a given value ofT W as shown in Fig. 7. These curves have a monotonous variation, and show that the efficiency increases with increasing medium length. When the pulse duration increases, the curves of L and L c get closer to each other, in agreement with the analysis of Fig. 6. A specific feature of these curves is their saturation for large values ofL. The presence of a plateau on the leakage plots is actually due to an approximation made in the model; we have neglected the time intervals associated with the propagation of the pulse wavefronts inside the medium, and we assumedt = 0 is the time at which the wavefront reaches the the cell output, whilet =T W is the time at which the end part of the pulse arrives at the entrance of the cell. This means that we always have some leakage in the initial time independently of the length of the medium. The difference between the levels of the plateau for L and L c characterizes the losses due to the non zero population of level \|3 . One can see that this value is constant for large enough lengthL. This value saturates because of the depletion of the signal field, so that further increase of the medium length cannot change the populations N 2 and N 3 . The difference between L and L c depends strongly on the pulse duration. In particular if the pulse is too short, many atoms are left in the upper state. It can also be seen in Figs. 7c that when the detuning increases, the saturation occurs at larger valuesL, and that a high efficiency can be reached as well if the atomic medium is long enough. Various optimizations procedures have been proposed in the limit when the excited state can be adiabatically eliminated [21,22]. In particular, the optimization used in reference [22] allows to get the maximum available efficiency for long enough durations of the writing process, but breaks down when the duration of a writing process T W gets smaller than the excited state decay time γ −1 divided by the optical depth of the medium d. In Ref. [24] the numerical optimization procedure relying on the shaping of the driving pulse was extended to the non adiabatic case, which allowed to reach better storage efficiency for short pulses. The latter technique was developed for the resonant case, yielding optimal memory efficiencies that are very close to the ones presented here. The applicability of our optimization method to various detunings shows that such a memory can be also very efficient in the off-resonant regime, bringing more flexibility for experimental realizations. Validity limits of resonant and Raman approximations The solutions that we have presented are valid for detunings ranging from zero to large values that correspond to the case of Raman interaction, where the system is effectively reduced to a two-level system. General formulas covering the full range of detunings allow a comparison with the limit cases of resonant and Raman interactions. We can identify the largest detuning for which the resonant approximation is still valid, yielding the same storage efficiency. On the other hand, we can estimate for which value of ∆ a simplified Raman description can be used without yielding appreciable errors in the memory efficiency. In Fig. 2b one can see a significant distortion of the temporal profile of the atomic ground state coherence at the input of the medium for r = 0.1 as compared to the resonant case. However this detuning does not affect the writing efficiency in a significant way. Figure 8a shows the dependence of the total losses on time for a given length in two cases: for exact resonance (blue curve, dotted line) and with a detuning r = 0.1 (red curve, full line). The curves coincide to within 1.5% over the full range ofT W . Thus, despite the local differences in the field-atom interaction in these two cases, the presence of a small detuning does not actually change the properties of memory cell as a whole. However when the detuning increases, the difference between the curves increases (at r = 0.2 it reaches 4.5%, see Fig. 8b), but in the range of interest forT W , that is the one that allows minimization of the losses, the curves are still close to each other (they agree within 1.5%). Further increase of the detuning distorts the profile of the losses further, and the value ofT W that provides minimum losses is shifted (see Fig. 8c,d). Let us now turn to the case of large detuning. One can see in Fig. 9 that for r = 3 the profile of the total losses calculated with the general formulas (33)-(34) coincides well with the profile for the same quantity, calculated in the Raman approximation with r 1 (the difference is about 2.5%, and it is less than 1% at the minimum). For a smaller detuning (for r = 2) the difference between the curves increases up to 7% (about 3% at the minimum): the calculation made in the Raman case underestimates the minimum losses and shifts toward lower values of T W . Thus, we can conclude that for r = 3 and higher the Raman approach is applicable with good accuracy, but the general solutions should used for lower values of the detuning. For large enough detunings where the adiabatic limit is valid as well as our model, we can compare the results further. Our model predicts a storage efficiency below the maximal available efficiency reached by the method proposed in reference [22]. In the third plot of Fig. 6 the efficiency is 65% for T dγ =T WL /2 = 30 and d = 2400. For the same parameters the adiabatic optimization method converges to the maximal efficiency which is close to 100%. This is due to the fact that we do not elaborate shaping of a control pulse, used in reference [22]. Thus our method, even without control pulse shaping, is quite powerful in a non-adiabatic limit. Otherwise, the numerical adiabatic optimization of the control pulse profile should be used to reach the maximal storage efficiency for the long pulses. Fig. 9. Comparison of the general calculations (blue curves, dotted lines) and calculations in the Raman limit (red curves, full lines). Writing process : relative total losses (in percent of the input field intensity) at the output of the medium as a function ofT W forL = 10 for (a)r = 2 and (b) r = 3. Discussion of the read-out process In Ref. [25] the optimization of the retrieval efficiency was based on the choice of the pulse duration providing the minimum leakage for a given medium length. Here we will look for a minimization of the total losses. Moreover, as was shown in Ref. [22,25], backward retrieval provides significantly larger efficiency than forward retrieval, therefore we will study the cases of forward and backward retrieval and compare them. Using the result of Fig. 6a, we choose a value for input signal duration ofT W = 5.5 that provide minimum total losses for the writing process. For this value ofT W we calculate the intensity of the retrieval field (normalized to the intensity of the input signal) as a function of the reading time (see Fig. 10a). The plot obtained from optimization based on leakage [25] is shown in Fig. 10b, in order to compare the results. The retrieval efficiency is defined by: E(q) = T R 0 \|a R (t,z, q)\| 2 dt T W 0 \|a in (t, q)\| 2 dt × 100%(49) In the case of optimization based on total losses, we find that the retrieval efficiency is equal to 88% atT R = 2T W . In view of the writing efficiency (L c = 11.2%), we see that using a reading timeT R = 2T W we can restore almost all the information written in the medium. When the leakage-based optimization is used, we find E = 84% atT R = 3T W . This comparison clearly demonstrates the benefits of the optimization based on the total losses. As for the temporal profiles of retrieval field, it is obvious that in both cases they are very different from the input signal profile, which is a usual result in most memory processes. Let us now consider the result of the forward and backward retrieval for the non-resonant case(see Fig. 11 for r = 0.5). The efficiency is much lower for forward retrieval than for backward retrieval one: atT R = 10 (i.e.T R ≈ 3.3T W ) the efficiency of forward process is E = 58%, while the efficiency of backward process is E = 85.6%. Moreover, in the latter case less than 2% of the available photons remain in atomic ensemble (since the writing losses were 12.6%). Comparison with the resonant case shows that the presence of detuning slows down the read-out. Finally, we can show that even for a large detuning, the total efficiency can be large, at the condition that the medium lengthL is large enough. In Fig. 12 the red (oscillating) curve corresponds to the calculated read-out signal intensity for backward retrieval forL = 100, r = 2, and for a pulse duration of the input fieldT W = 4π, which ensures a minimum in the total writing losses (writing efficiency is n ef f = 95.6%). The oscillation period is equal to 4πr, similar to the modulation observed in Fig. 2. If we calculate the intensity of the retrieved signal in the Raman approximation, the overall shape of the curve remains the same, but the oscillations disappear (blue curve, dotted line in Fig. 12). The retrieval efficiency calculated with these curves differ by 1.4% (78.6% for the exact calculation and 77.2% for the calculation in Raman approximation), so that the calculation in Raman limit can be a quite good estimation for memory efficiency at r = 2. Note, that here like for the curves in Fig. 2, the magnitude of the oscillations will decrease with increasing detuning. In addition to high speed operation, our model includes transverse coordinates, as can be seen from Eq. (2), and thus allows the treatment of a variety of multimode fields, in the same way as in Ref. [25]. The storage of a multimode signal field is performed using a single mode signal field, and the information is written in the atomic ensemble as a hologram. The quantum hologram process together with a Raman memory was proposed for the first time by Sokolov et. al. in Ref. [20] and it was shown that the memory capacity is limited by diffraction, but with the limitation depending on the direction of readout. Under forward readout the maximum number of the modes or picksels N which can be stored in principle is given by the square of the Fresnel number N ∼ F 2 N , where F N = S/(λL). This expresses the condition that the output pixel size should not exceed the transverse size of the memory cell. This rather loose limitation comes from the fact that most diffraction effects are compensated between writing and readout with similar geometries. A similar compensation does not take place for the backward readout and as a result the number of stored modes N in this case can not exceed the Fresnel number. Conclusion In this article, we have presented a model for a high speed quantum memory based on a three-level medium in the Λ-configuration. This model relies on a full calculation of the fields and of the atomic coherences as a function of time and space for arbitrary frequency one-photon detuning, while keeping the two-photon resonance. It allows to examine all the situations between interactions resonant with the excited state and non resonant interaction, which corresponds to a Raman transition. Our model allows to identify the conditions in which the interaction can be treated as a Raman transition. This corresponds to a normalized detuning r = ∆/(2Ω) > 2. For larger detunings, we have shown that the Raman model gives accurate results. In the near-resonant case, the interaction of the threelevel atomic system with the weak signal field and the strong driving field turns out to be more complicated than in the Raman case. As a matter of fact in contrast to the Raman process, where the upper atomic state is practi-cally not involved in the process, all three levels are populated, and this leads ultimately to additional undesirable losses. We have shown that it is possible to choose the parameters to make these losses very small. By controlling the Rabi oscillation one can preferentially populate the lower state \|2 and depopulate state \|3 . When the frequency detuning increases, the upper state population naturally decreases, which contributes to the reduction of losses. Moreover, for large frequency detunings, the interaction between the atomic medium and the fields is weaker, and a larger medium depth is necessary to reduce the leakage. An important question is how to optimize the memory process to obtain the highest possible quantum efficiency. In the literature two approaches are mainly discussed. The first one is based on the choice of the optimal time-shape for the signal pulse, using for this the eigenfunctions of the integral operators specific of the considered memory [21]. It was demonstrated that if the functions possess some symmetry, the efficiency can be improved. In our case with very short pulses, there is no such symmetry and this optimization turns out to be impossible. Another optimization method was proposed by Gorshkov et. al. in Ref. [22], based on a study of the driving pulse shape. However, the search procedure for the optimal shape in this article is based on the adiabatic or Raman approximations, which are not generally applicable in the case of very short pulses. Non-adiabatic pulse shape optimization was also developed for the resonant case [24] extending the validity of the method. In our model, the optimization is based on the minimization of the losses, which come from the leakage and from the population of the upper atomic level, and allows identifying the optimal combination of signal pulse duration and optical depth. We demonstrate that high memory efficiencies can be achieved by this method for very short pulses whatever the value of the detuning. Acknowledgements The study was performed within the framework of the Russian-French Cooperation Program "Lasers and Advanced Optical Information Technologies", of the European Project HIDEAS (grant No. 221906). O.M. acknowledges the support of Ile de France programme IFRAF. The study was also supported by RFBR (grant No. 08-02-92504). A General solutions for the main equations Eqs. (26)-(28) can be solved in the general form by using the Laplace domain. Some details of the formal procedures are discussed in App. B and now we start with the solutions themselves in the explicit form. For our analysis in this article we do not need full information about the solutions, nevertheless one can find it below. Here and everywhere in article under consideration of the solution we use dimensionless co-ordinatest andz given bỹ t = Ωt,z = 2g 2 N Ω z. (50) − 1 − r 2 2 [f 0 (r) * f * 0 (−r)](t,z).(66) In these formulas the time dependent factors F (t) read F 1 (t) = cos 1 + r 2t + ir √ 1 + r 2 sin 1 + r 2t e −irt ,(67)F 2 (t) = 1 √ 1 + r 2 sin 1 + r 2t e −irt ,(68) −ĉ(z, 0; q) + (s + i∆)ĉ s (z; q) = g √ Nâ s (z; q) + Ωb s (z; q), −b(z, 0; q) + sb s (z; q) = −Ωĉ s (z; q). From this we can write the closed differential equation for the field amplitudeâ s (z; q) in the form dâ s (z; q) dz = −Γ sâs (z; q) − g √ Nα s (z; q). Here the coefficient Γ s determines a rate of escape of the amplitude along the z-axis and reads Γ s = g 2 N 2 µ s + iµΩ + ν s − iνΩ ,(74) where the following notations are introduced Ω = Ω 1 + r 2 , µ = 1 + r, ν = 1 − r. (75) The inhomogeneous term on the right in Eq. (73) is determined by the initial conditions for the medium state and given bŷ α s (z; q) = 1 s(s + i∆) + Ω 2 Ωb(0, z; q) + sĉ(0, z; q) . where the termsα(t, z; q) andβ(z, t; q) are expressed via the initial conditions for the medium coherences in the formα (t, z; q) = F 3 (t)ĉ(0, z; q) + F 2 (t)b(0, z; q), (83) β(z, t; q) = −F 2 (t)ĉ(0, z; q) + F 1 (t)b(0, z; q), (84) and the kernel D(t, z) is simply proportional to G aa (see (56)) and is given by D(t, z) = ΩG aa (t,z). After some simple transformations one can obtain the solution in the form (51)-(53). At the same time for the numerical computation it is possible to use Eqs. (80)-(82). Writing process : relative losses due to leakage (blue curves, dotted lines) and relative total losses (red curves, full lines) (in percent of the input field intensity) at the output of the medium as a function ofL and (a) r = 0, (b) r = 0.5 and (c) r = 2 forTW = π/2 (first row) andTW = π (second row). Fig. 1 . 1Three level atomic system interacting with driving field Ω and signal field a. Fig. 12 . 12Reading process : field intensity \|a R (t,L)\| 2 at the output of the medium forL = 100,TW = 4π and r = 2 for backward retrieval calculated with exact formulas (red curve, full line) and within the Raman approximation (blue curve, dotted line). s + i∆) + Ω 2â s (z; q) +α s (z; q), (78) b s (z; q) = 1 s b (0, z; q) − Ωĉ s (z; q) .(79)After the inverse Laplace transformation in Eqs. (77)-(79) one can obtain all the solutions in the forma(t, z; q) = t 0 dt â in (t − t ; q) D(t , z) α(t − t , z − z ; q) D(t , z ), 3 (t − t )â(t , z; q) +α(t, z; q), 2 (t − t )â(t , z; q) +β(z, t; q),(82) Fig. 2 . 2Normalized coherence atz = 0 as a function of time for (a) r = 0,(b) r = 0.1, (c) r = 0.2, (d) r = 1, (e) r = 2, (f) r = 10. Fig. 3 . 3Distribution of the coherence \|b W (t,z)\| 2 in time and space for (a)r = 0, (b) r = 1 and (c) r = 2. Fig. 4 . 4Normalized distributions of the atomic coherence inside the medium at timet = π for (a)r = 0, (b) r = 0.5, (c)r = 2. Fig. 5 . 5Fraction of signal photons (normalized to the energy of input signal pulse) that have been converted to atomic coherence b W during writing as a function of the detuning parameter r forL = 10 and (a)T W = π/4, (b)T W = π/2, (c)T W = π, (d) T W = 2π. Fig. 6 . 6Writing process : relative losses associated with leakage (blue curves, dotted lines) and relative total losses (red curves, full lines) (in percent of the input field intensity) at the output of the medium as a function ofTW forL = 10 for (a)r = 0, (b) r = 0.5, (c)r = 2. Fig. 7 . 7Fig. 7. Writing process : relative losses due to leakage (blue curves, dotted lines) and relative total losses (red curves, full lines) (in percent of the input field intensity) at the output of the medium as a function ofL and (a) r = 0, (b) r = 0.5 and (c) r = 2 forTW = π/2 (first row) andTW = π (second row). Fig. 8 . 8Comparison of relative total losses (in percent of the input field intensity) at the output of the medium as a function of TW for resonant (blue curves, dotted lines) and detuned (red curves, full lines) cases;L = 10, Fig. 10 . 10Reading process : field intensity \|a R (T R ,L)\| 2 at the output of the medium forL = 10 and r = 0 for backward retrieval with two optimization techniques: (a) total loss minimization (T W = 5.5) and (b) leakage minimization (T W = 4.2). Fig. 11 . 11Reading process : field intensity \|a R (t,L)\| 2 at the output of the medium forL = 10,TW = 3 and r = 0.5 for (a) forward and (b) backward propagating retrieval. Let us write the general solutions in the form under the arbitrary initial and boundary conditionŝ a(t,z; q) = t 0 dt â in (t −t ; q)G aa (t ,z)where kernels G ik (t,z) are bilinear combinations of the expressions depending on the n-th Bessel functions of the first kind denoted by J nThe kernels in Eq. 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[ "A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic", "A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic" ]	[ "Barbara Zaparoli Cunha \nLaboratory of Tribology and Dynamics of Systems\nEcole Centrale Lyon\nEcullyFrance\n\nCompredict GmbH\nDarmstadtGermany\n", "Christophe Droz \nUniv. Gustave Eiffel\nInria\n\nCOSYS/SII\nI4S teamRennesFrance\n", "Abdel-Malek Zine \nInstitut Camille Jordan\nEcole Centrale Lyon\nEcullyFrance\n", "Stéphane Foulard \nCompredict GmbH\nDarmstadtGermany\n", "Mohamed Ichchou \nLaboratory of Tribology and Dynamics of Systems\nEcole Centrale Lyon\nEcullyFrance\n" ]	[ "Laboratory of Tribology and Dynamics of Systems\nEcole Centrale Lyon\nEcullyFrance", "Compredict GmbH\nDarmstadtGermany", "Univ. Gustave Eiffel\nInria", "COSYS/SII\nI4S teamRennesFrance", "Institut Camille Jordan\nEcole Centrale Lyon\nEcullyFrance", "Compredict GmbH\nDarmstadtGermany", "Laboratory of Tribology and Dynamics of Systems\nEcole Centrale Lyon\nEcullyFrance" ]	[]	The use of Machine Learning (ML) has rapidly spread across several fields, having encountered many applications in Structural Dynamics and Vibroacoustic (SD&V). The increasing capabilities of ML to unveil insights from data, driven by unprecedented data availability, algorithms advances and computational power, enhance decision making, uncertainty handling, patterns recognition and real-time assessments. Three main applications in SD&V have taken advantage of these benefits. In Structural Health Monitoring, ML detection and prognosis lead to safe operation and optimized maintenance schedules. System identification and control design are leveraged by ML techniques in Active Noise Control and Active Vibration Control. Finally, the so-called ML-based surrogate models provide fast alternatives to costly simulations, enabling robust and optimized product design. Despite the many works in the area, they have not been reviewed and analyzed. Therefore, to keep track and understand this ongoing integration of fields, this paper presents a survey of ML applications in SD&V analyses, shedding light on the current state of implementation and emerging opportunities. The main methodologies, advantages, limitations, and recommendations based on scientific knowledge were identified for each of the three applications. Moreover, the paper considers the role of Digital Twins and Physics Guided ML to overcome current challenges and power future research progress. As a result, the survey provides a broad overview of the present landscape of ML applied in SD&V and guides the reader to an advanced understanding of progress and prospects in the field.	10.48550/arxiv.2204.06362	[ "https://arxiv.org/pdf/2204.06362v1.pdf" ]	248,157,275	2204.06362	1f7332a02baa06a1c60f9262192da684a013c091	A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic Barbara Zaparoli Cunha Laboratory of Tribology and Dynamics of Systems Ecole Centrale Lyon EcullyFrance Compredict GmbH DarmstadtGermany Christophe Droz Univ. Gustave Eiffel Inria COSYS/SII I4S teamRennesFrance Abdel-Malek Zine Institut Camille Jordan Ecole Centrale Lyon EcullyFrance Stéphane Foulard Compredict GmbH DarmstadtGermany Mohamed Ichchou Laboratory of Tribology and Dynamics of Systems Ecole Centrale Lyon EcullyFrance A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic Machine LearningStructural Health MonitoringSurrogate ModelActive Vibration ControlActive Noise ControlDigital-TwinPhysics Guided Machine Learning The use of Machine Learning (ML) has rapidly spread across several fields, having encountered many applications in Structural Dynamics and Vibroacoustic (SD&V). The increasing capabilities of ML to unveil insights from data, driven by unprecedented data availability, algorithms advances and computational power, enhance decision making, uncertainty handling, patterns recognition and real-time assessments. Three main applications in SD&V have taken advantage of these benefits. In Structural Health Monitoring, ML detection and prognosis lead to safe operation and optimized maintenance schedules. System identification and control design are leveraged by ML techniques in Active Noise Control and Active Vibration Control. Finally, the so-called ML-based surrogate models provide fast alternatives to costly simulations, enabling robust and optimized product design. Despite the many works in the area, they have not been reviewed and analyzed. Therefore, to keep track and understand this ongoing integration of fields, this paper presents a survey of ML applications in SD&V analyses, shedding light on the current state of implementation and emerging opportunities. The main methodologies, advantages, limitations, and recommendations based on scientific knowledge were identified for each of the three applications. Moreover, the paper considers the role of Digital Twins and Physics Guided ML to overcome current challenges and power future research progress. As a result, the survey provides a broad overview of the present landscape of ML applied in SD&V and guides the reader to an advanced understanding of progress and prospects in the field. Introduction In the current Information Era, unprecedented amount of information is produced, stored, and transformed into actionable knowledge [1]. However, such a large amount of data requires processing and translation abilities beyond human capacity. Machine Learning (ML) algorithms have been a key part of the Big-Data revolution, as they play the role of automatically processing these copious amounts of data to extract patterns and make inferences and predictions based on them. In other terms, digitalization and connectivity provide the data, and ML translates it into meaningful information. Besides the availability of data, ML progress is powered by constant developments in computing resources and algorithm improvements. Currently, ML is widely present in our daily life, such as in health-care decision making [2], autonomous vehicles [3], economic forecasts [4], detection of fake-news [5], suggestions for consumption of content and goods [6,7], mastering games [8], image classification and generation [9,10], translations and speech recognition [11] and other subjects. ML methods are also permeating the natural sciences [12], not only by overcoming traditional data-driven methods but also by powering or even replacing firstprinciple models. The use of ML in scientific fields such as biology [13], chemistry [14,15], physics [16][17][18][19] and material science [20,21] is well developed. The range of ML applications in these domains includes identify-ing behaviors from measured data, speeding up analyses time, merging data-and domain-based knowledge, finding new materials and components, modeling systems, and discovering governing equations. Given this trend, much has been debated about the pros and cons of using ML in physical science and how it can power research progress in engineering domains such as fluids dynamics [22], seismology [23,24], thermal transport [25,26] and energy systems [27]. Recently, many relevant works in structural dynamics and vibroacoustic have used ML in three major application areas: Structural Health Monitoring (SHM) using vibration and noise signals [23,, Active Noise and Vibration Control [23, and vibroacoustic Project Design [84,. SHM benefits from the ML advantages of extracting relevant features from big data to detect and classify failures efficiently and make lifetime predictions. In Active Control, ML stands out for identifying light models of the system, since the mechanistic models are currently unknown, incomplete, or high-dimensional. Besides that, a miscellaneous of approaches uses ML to model and optimize the controller design. In Vibroacoustic Design, ML-based surrogates result in fast simulations that enable an optimized and robust design for Noise, Harshness and Vibration (NVH). The ML workflow in these applications should consider the characteristics of the vibration or sound signals under analysis. As supported by the numerous results cited throughout this article, there are many benefits to employing ML in SD&V problems. However, drawbacks, misuses, and difficulties can also be spotted and show the potential for further advancement in the field. The lack of interpretability and physical basis are the aspects that raise more apprehension in the use of ML in SD&V and other physical sciences. Furthermore, although the wave behavior of SD&V systems encloses frequency information which is well explored in SHM, it also leads to non-monotonic and rough functions behaviors, raising challenges to ML models. Currently, implementations in the industry are limited by the lack of substantial amounts of labeled data required in Deep Learning or by the cost of ML simulations in real-time applications. Another issue still open to debate is reasoning about when the use of ML is justifiable and brings gains in time and precision with an adequate level of confidence. The present paper discuss these issues alongside references and approaches that tried to tackle them, indicating viable solutions. Therefore, this work focuses on doing an original and extensive review of the main contributions and on the emerging opportunities of ML applied in structural dy-namics and vibroacoustic. The review provides a comprehensive state-of-the-use and guidelines of ML applications in SHM, Active Control, and Product Design and raises the strengths and weaknesses of ML in each of these fields. The present implementation scenario of each application is presented alongside reasoning about method choices and discussion on the identified research gaps. It is also remarked how the suitability of an ML method depends on factors such as dimensionality of the problem, nature of the data, management of uncertainties and nonlinearity of the system. In-depth theory on ML and vibroacoustic are not part of the scope of this review. In that way, the present work intends to guide vibroacoustic engineers willing to explore ML techniques by providing the current background and the future opportunities of the research field merging ML with SD&V. At first, Section 2 provides the basis of the main ML methods employed in SD&V literature. Section 3 analyzes the ML workflow in SHM, especially how to prepare vibration and noise signals to increase the ML capabilities to identify patterns, and also introduces relevant works in damage detection, diagnosis, and prognosis. Section 4 reviews the applications of ML in Active Control of noise and vibration, including system identification, reduced-order models, sensor and actuator placement, and controller design. Section 5 focus on the use of surrogate models of SD&V simulations to improve Project Design, with attention to their use in uncertainty propagation, sensitivity analysis, and optimization. Section 6 addresses the trends and perspectives in the field by analyzing how integration, physics guidance, and other aspects can leverage the ML impact and applicability in SD&V research and discussing upcoming opportunities from the integration of these fields. Overview of Machine Learning Methods Machine Learning is an Artificial Intelligent (AI) algorithm which makes an inference from data and experience without the use of explicit programming. In a simplified way, for a dataset X containing some sets of inputs x, a model m(ω, x) with parameters ω is defined to represent the relationships and patterns of the dataset and an assessment criterion called cost function C(X, m(ω, x)) is defined to quantify how well the model represent the dataset. After an optimization that minimizes the cost function in relation to the model parameters ω, the optimal model m(ω opt , x) is used to infer the relationship of unseen data. Therefore, three key elements describe an ML model: representation, which is the chosen way to model data relations through m(ω, x) and establish a hypothesis space of all possible models considered, e.g., decision trees, neural networks, hyperplane representations; evaluation, which determines how to access the model performance employing a cost function C(X, m(ω, x)), e.g., accuracy, squared error, K-L divergence; and optimization, that is the search method used to minimize the cost function, e.g.: gradient descent, greedy search, quadratic programming [182]. Previously to train an ML model, it is crucial to conduct correctly the dataset sampling, as well as the data preprocessing stages to make the data suitable and more meaningful to the training stage. During the training stage, the optimization algorithm searches for the configuration that better represents the training dataset according to the criteria of the cost function. In other words, the ML model is learning with data. Once the model is trained, it can make predictions in new data. The goal of the ML model is to generalize well for new unseen data. In other words, the goal is to have a prediction model of the true hidden distribution and not a fitting model of a sample of this true distribution (the training set). Therefore, minimizing the cost function during training does not guarantee an adequate predictive model, and the final assessment of the ML model must be based on its prediction performance in the new/unseen data from the test dataset. This leads to a crucial concept in ML, the biasvariance trade-off, which stands that the generalization error is a combination of bias, variance, and irreducible errors in supervised learning, as illustrated in Figure 1a. The bias error measures the level of incorrect hypotheses in the model and decreases with model complexity. The variance error measures the variability of model predictions and typically increases with model complexity. Therefore, a high-bias model oversimplifies the problem, leading to bad predictions in both training and test dataset (underfitting), while a low-bias model performs well in the training dataset but might lead to high-variance error (overfitting). Figure 1-a illustrates this trade-off for a given number of training points, and Figure 1-a shows how complex models with low bias become viable with increasing database size. Techniques used to control overfitting play a leading role in ML algorithms, setting a good balance between bias and variance. A key aspect is the hyperparameters selection, which is done manually by experts reasoning, or automatically via algorithms of search and optimization [183][184][185][186][187][188]. During hyperparameters selection, the generalization error estimative must use sam-ples kept out of the training dataset, which is usually done through cross-validation or by separating a validation dataset. Regularization techniques are also an important strategy to fight overfitting. The regularization technique varies depending on the ML method, but the motivation is to penalize model complexity and increase robustness to ill-posed problems. The most common regularization strategy is to add a regularization term in the loss function to account for model complexity. This ML overview mentions the fundamentals which guide most ML methods. The reader can refer to the rich ML bibliography for in-depth theory and methodology, as in the classic ML textbooks by Bishop [189], Friedman et al. [190] and Goodfellow et al. [183]. For an introduction in ML, the authors recommend the article "A high-bias, low-variance introduction to Machine Learning for physicists" from Mehta et al. [17]. This article presents a brief and comprehensible explanation of the main ML concepts, along with tutorials and Jupyter notebooks of popular ML algorithms. The article of Domingos [182] is also a reference for providing valuable expertise in implementing successful ML algorithms. Usually, the ML methods are classified according to the learning approach as: • Supervised Learning: use labeled output as ground truth during training. It is a regression model when the output has continuous values or a classification if the output are categories or discrete values. • Unsupervised Learning: no labeled output and no correct answer is provided. The algorithm searches for underlying patterns in the data to assume its structures, through clustering or association. • Reinforcement Learning: the algorithm interacts with an environment, earning points to reinforce successful decisions. Hereafter, the most relevant ML methods applied in recent literature on structural dynamics and acoustics are outlined. Diversification was an important criterion in the choice of the methods, to enclose different applications and ML categories. In Section 2.1 addresses Neural Networks and Deep Learning methods, since they are certainly the largest field of ML and due to their flexibility, they have supervised, unsupervised and reinforced learning algorithms and handle various data structures. Section 2.2 explores Decision Tree based Figure 1: Bias-variance tradeoff on Machine Learning models: (a) Illustration of how bias and variance vary in opposite ways with respect to ML model complexity and how they sum up to the generalization error. The optimal ML algorithm minimizes the generalization error and, avoiding underfitting and overfitting; (b) Illustration of how the error in the training dataset E i is smaller than the true generalization error E g and how the prediction accuracy improves with more samples in the dataset; (c) Example of a model that underfits the data, a model that overfits, and a model with appropriate bias-variance tradeoff. b-) c-) a-) methods, examples of classical ML algorithms which are extremely popular in supervised learning, in special due to their interpretability and their capabilities when applied with ensemble methods. Section 2.3 is dedicated to Gaussian Processes models, popular due to their probabilistic outputs and mainly used in supervised learning with small database. The K-means algorithm is presented in Section 2.4, as a common example of unsupervised learning. Finally, a brief discussion about reinforcement learning and its class of algorithms called Q-learning is presented in Section 2.5. Certainly, ML is much richer in methods and details than those presented here, but the purpose of this section is just to provide a broad overview of ML methods and to distinguish their advantages and applications in vibroacoustic. Neural Networks Neural Networks (NN) or Artificial Neural Networks are a set of ML algorithms inspired in the human brain which can approximate any function, as stated in the Universal Approximation Theorem [191,192]. Moreover, NN is flexible and modular, so its architecture can be adapted to different purposes in supervised, unsupervised, and reinforced learning. It is by stacking multiple layers in an NN that Deep Learning (DL) models are created with improved capabilities of extracting features and learning complex data representations. No wonder, NN is a popular algorithm in all fields of ML, having continually shown impressive results in all kinds of real-life problems. The most common and general-purpose NN are the Fully Connected Feed-forward Neural Network or Multilayer Perceptron (MLP) [193], which will be used as a starting point to introduce other NN architectures. As shown in Figure 2 and as the name illustrates, a NN is a network of artificial neural units. Each neural unit is defined by a nonlinear activation function, as ReLU and sigmoid, which fires an output based on the weighted sum of inputs added to a bias. The outputs from one layer are the inputs for the next one, in a feed-forward procedure. The last layer outputs are the NN predictions, used to evaluate the NN performance. For example, in a supervised problem, the mean squared error between NN predictions and the true values is often used as the NN loss function. The learning procedure consists of finding the set of weights ω and biases b which minimize the loss. The key to making this optimization viable in big NNs is the Backpropagation algorithm, whose rediscovery in the mid-1980s led to the boom in the popularization of NN [194,195]. Backpropagation allows to efficiently compute the gradient of the loss function with respect to the weights and biases, thanks to the Automatic Differentiation (AD) capabilities of NN [196]. Thus, backpropagation enables gradient-based optimization algorithms to be effectively used during training. To reduce memory requirements and speed up the training, the dataset might be divided in batches, so that the NN parameters are updated evaluating NN performance at each batch. One training epoch has passed when all batches in the dataset have been used to update the NN. The trained NN is a system of algebraic equations which can read- ily predict new outputs. a-) Neural Unit b-) Feed-Forward Neural Network NNs account with regularization techniques as weight regularization (L1 and L2 regularization), Dropout, Early Stopping, Data Augmentation, Soft Weight-Sharing, Batch Normalization and Entropy Regularization [17,183,189,194]. Besides that, the hyperparameters selection also plays a significant role in NN generalization, since it tunes the architecture aspects and regularization parameters. Basic reasoning for hyperparameters tuning in NN is already well-known [183], but automatized approaches are becoming increasingly popular [197]. In most cases, the use of a validation set to monitor the NN generalization is a standard procedure. The number of nodes in the first NN layer is equal to the input dimension and the number of nodes in the last layer is equal to the number of output values (regression problem) or classes (classification problem) of the analysis. Usually, in the output layer, a linear activation function is used in regression problems with unbounded output, a softmax activation function is used for classification problems with multiple mutually exclusive categories and sigmoid activation function when you have not mutually exclusive categories. The number of hidden layers, the nodes in each layer and the activation functions in the hidden layers are hyperparameters to be tuned. A crucial point in NN performance is data scaling, which makes the data range like the best range for the operation of the activation function. LeCun et al. [198] further discusses practical recommendations for the creation of NN models. MLP is the most common ML method in the applications covered in this article, due to its versatility and easy implementation. However, there are NN architectures that are appropriate to specific problem structures. Further comments are presented about Convolution Neural Networks and Recurrent Neural Network, for supervised learning of images and time series, respectively; Auto-encoders for unsupervised learning; and Deep Reinforcement Learning (Section 2.5). Other important NN architectures that are not discussed here are Boltzmann machines, deep belief networks, and generative adversarial networks. Although the literature on NN is extremely dense and expands fast, there are several references which manage to cover the topic in a didactic way. The book by Nielsen [192] contains comprehensive explanations of the NN main elements, while the classic book by Goodfellow et al. [183] has equally good NN introductions but also covers more detailed and advanced aspects. Mehta et al. [17] presents a gentle and summarized NN guide along with Python code. Implementations guides are available along with dedicated libraries for NN in Matlab [199] and in Python, with highlights to Keras [194] and PyTorch [200] libraries. The CNN performs automatic spatial feature extraction from images by successively applying feature filters that create feature maps (Convolutional layer) and compressing these maps (Pooling layer). Based on the final feature maps, a fully-connected NN does a prediction, which can be a classification or regression. Convolutional Neural Networks Convolutional Neural Networks (CNN or ConvNet) are NN designed to capture spatial patterns from multiple arrays input by exploiting the local connectivity and translational invariance characteristics from data. To put it more simply, the CNN architecture considers that the points in the same region are closely related and that the identified patterns can be found translated in the space. Therefore, it is reasonable to use CNN in problems where the input order and location matter. A very well-known example is the successful application of CNN to image processing problems, as evidenced by the numerous times it has been used to gain Ima-geNet challenge [207]. CNN can also be handy to analyze time-series as a 1D array or even by transforming the time-series into images, usually by applying timefrequency transformations. Convolutional Layer is a key element of CNN to explore the spatial patterns. It consists of applying a kernel filter to all local regions by sliding it throughout the image, what is called convolving. The kernel filter is just a matrix of weights associated with a feature. The sum of the element-wise multiplication of the filter matrix with the local region values is the resultant value that identifies how strongly the feature is detected in this region. Therefore, as it slides through the image, a feature map is created, as illustrated in Figure 3. Each convolutional layer can contain multiple filters, each of which results in a feature map which is stacked to each other along the depth dimension to form the layer output. The filtered output will also go through some nonlinear activation functions, being ReLU the most common one. The convolving procedure considers the local connections into each region defined by the size of the filters. Besides that, as the filter has shared weights for the entire image at each depth slice, the same feature will be detected at distinct locations, which contributes to the translational invariance properties in the network structure. In this way, the convolutional layer makes use of these two important characteristics of spatial signals. In the sequence, the Pooling Layer subsample the feature map shrinking the image stack, usually by calculating the maximum value for non-overlapping patches of the feature map (max-pooling) ( Figure 3). The output has a pattern like the input, but with smaller dimensions. This stage collaborates to reduce computational load and number of parameters, also avoiding overfitting. However, the pooling layer also plays a significant role in the identification of spatial patterns, since it will semantically merge similar features and add invariance to small scaling, shifts and distortions [195]. In that way, the first layers in a Deep CNN are responsible for detecting simple features, such as edges and corners in images and as the CNN depth increase, higher-order features and complex patterns can be detected. In the end, Fully Connected Layers perform the final classification or regression task. The backpropagation algorithm is normally used during optimization to adjust the weights from the kernel filters (automatically defining the prominent features to be detected) and from the fully connected layers. Comprehensive explanations on CNN methods are available in the main textbooks of Deep Learning [17,183,189,192,194] and visual and iterative guide is provided by [208]. Recent progress was covered in [209]. Transfer Learning is also an important topic in CNN [17,204] since features learned from other datasets can be useful to new problems, therefore acclaimed CNN models trained with big datasets, such as LeNet, ZF Net, AlexNet, VGGNet, GoogLeNet and ResNet, can be used as a start point to initialize new CNNs, reducing the need for data. Recurrent Neural Networks Recurrent Neural Networks (RNN) is a group of NN designed to handle dynamical systems, that is, systems changing over time. While ordinary NN do not use sequential information to do predictions, RNNs can use the time history and context. Because of this, they are extensively used in natural language processing, like chatbots, machine translations, speech recognition and generation, in Computer Vision applications and forecasting problems in the most diverse fields [210], as in economy, climate change, weather and diseases prognosis. Naturally, they are also applied to analyze dynamic systems in mechanics, for example, in the prediction of the system response in function of time [211], or to forecast the remaining useful life of a component [212]. When creating an RNN model, the input is configured as a 2D tensor containing the time steps in one dimension and the input features in the other. To have memory ability, RNNs are constructed in loops over the time steps. For each time step, the correspondent input features are provided alongside the current state of the problem to obtain the current output and this output is used as the state of the next time-step, configuring the loop [194]. Therefore, the state information will carry historical information through the RNN by assuming that outputs of different time steps are dependent on each other. This looping behavior can also be interpreted as a chain of NN which pass information to the follower [213], as illustrated in Figure 4. However, these sequential NN share the same weights, which are trained by a process called backpropagation through time. Usually, RNN suffers from exploding or vanishing gradients over time, but advanced techniques are employed to overcome the issue [195]. Another frequent problem is that over time steps the level of accumulated historical information increases and the RNN cannot define which is the valuable information to keep and suffers from a long-term dependency problem. Long short-term memory (LSTM) network [214] addresses this problem by using a memory unit cell with forget, input and output gates which explicitly defines what are the relevant information to keep and to output, as illustrated in Figure 4, showing more effective results than ordinary RNN [195]. The reader can look for a good introductory explanation about LSTM functioning provided by Olah [213]. There are many configurations of RNN, to deal with different dimensions of inputs and outputs, or bidirectional RNN to address cases where previous outputs in the network might be dependent on "future" results, or other configurations to deal with the longterm dependency problem, as the Gated recurrent units (GRUs). Generative RNN has also been extensively used, in special for text and speech generation [194]. More details on RNN theory and its architectures are available in the textbook by Graves [215], in Chapter 10 by Goodfellow et al. [183], Chapter 13 by Bishop [189] as well as in [194,195]. Recent advances in RNN are reviewed by Salehinejad et al. [216], as new RNN employing the attention mechanism [217]. Auto-encoders Dimensionality reduction are techniques which reduce data dimension while preserving the critical information on it. Due to the curse of dimensionality, this is the desired tool which allows reducing processing and storage requirements, especially in problems involving high-dimension data. As an example, dimensionality reduction is often used to create Reduced order models (ROMs) to accelerate complex and large numerical models of dynamical and control systems [218,219]. Dimensionality reduction methods are also extensively used as a data preprocessing stage previously to another ML method, since they automatically perform feature extraction or/and selection, improving the predictions of classic classification and regression algorithms. Several unsupervised ML methods perform dimensionality reduction [220], the principal ones being Autoencoders, Boltzmann Machines and Principal component analysis. For sake of brevity, the authors are going to cover only Auto-encoders in this article, which have a straightforward concept and are preeminent to nonlinear dimensionality reduction [221]. Autoencoders (AEs) are merely Neural Networks that receive high-dimensional data and encode it to a latent space representation by compressing the information through the NN with decreasing layer size until a bottleneck. The compressed information is then used to reconstruct the original input in a self-supervised learning, where the input itself is also used as the target output, as illustrated in Figure 5. Therefore, the data in the Inputs Outputs Latent Variables Encoder Decoder Decoder Bottleneck Figure 5: Illustration of an Autoencoder. The encoder stage compresses the information in the latent variables and the decoder stages decompress them. The autoencoder outputs should match the inputs, therefore it is a self-supervised learning. latent space must contain the more relevant information of the original data to be able to reconstruct it. Once the AE is trained, the encoder part is used as a dimension reduction tool. An advanced version of an AE is the Variational autoencoder (VAE) which compresses the data into the mean and variance of a statistical distribution. During training a random sample of this distribution is selected to reconstruct the original data, which forces robustness and meaningful representations in the latent space and leads to better performance than ordinary AEs [194]. Chollet [194] and Goodfellow et al. [183] presents detailed AE and VAE methodology and emphasize the prominence of using VAE decoders as generative models. Decision Tree Ensemble Methods Decision Trees are statistical learning algorithms that apply recursive partitions of the space to perform classification or regression and that are the basic element of more complex and popular models, like Random Forest and Gradient Boosting [222]. There are different Decision Tree algorithms, CART (Classification and Regression Trees) being the most famous. The CART algorithm performs successively binary partitions starting from a root node. The partitions are based on the impurity criterion and aim to gather samples from the same category or with approximate value in the final nodes of the tree, also called leaves. Random variables are tested as partition criteria, and the split is performed with the one that minimizes impurity. As finding the optimal of each node would be expensive, the algorithm implementation is greed, using heuristics methods. The scikit-learn Python library provides an easy implementation of Decision Trees, as well as good documentation on the methodology, applied [223]. Decision Trees are interpretable, fast, handle heterogeneous data and outliers, exempt data scaling, implicitly perform feature selection, and are nonparametric, allowing models with nonlinear complex relations [224]. However, Decision Trees are prone to overfitting, leading to high-variance and greedy solutions [17]. Although pruning is a regularization technique commonly applied to reduce this overfitting problem [189], it is through ensemble methods that Decision Tree models achieve good generalization. Ensemble methods are a combination of individual predictors to yield a better predictor. Any ML method can be used to create ensemble methods, but ensembles of Decision Trees are popular since Decision Tree randomized structure and low-cost training make them a suitable predictor. Bagging and Boosting are reviewed here, the two most popular ensemble methods. In Bagging each predictor collaborates with a vote in the final prediction, and the predictors are constructed independently in a parallelized framework, reducing mainly the variance, as is the case of Random Forest. In Boosting the predictors are connected and sequentially organized, each one trying to improve the prediction of the previous, reducing mainly the bias, as is the case of Gradient Boosting. Louppe [224], Chapter 8 by Mehta et al. [17] and Chapter 14 by Bishop [189] present the theory of ensemble methods and its close relation with the biasvariance tradeoff. Random Forest Random Forest (RF) is an ensemble method based on Decision Trees and Bagging. To improve the performance of the individual tree, RF grows several estimators or a forest of trees and makes the final prediction based on the average result of the individual trees, for regression problems, or based on the majority vote, for classification problems, reducing the prediction variability. Each Decision Tree should be different to ensure that the final RF generalizes better than individual Trees. Because of this, the Bagging or bootstrap aggregation procedure is applied, meaning that each Decision Tree is constructed based on random sampling with replacement of the dataset and random selection of subset features. A straightforward implementation of RG is possible with scikit-learn [223]. In comparison with NNs [225], RFs are cheaper during training and prediction, demand fewer data, and are more robust to missing data and hyperparameters choice, besides maintaining the advantages of a single Decision Tree. Another convenience of RF is that they intrinsically perform a sensitivity analysis, called outof-bag (OOB) based sensitivity-index and are more interpretable than NNs [151,226]. The sensitivity analy-sis loses accuracy if the prediction is inaccurate or if the problem has highly correlated features. Both Random Forest and NN can model nonlinear complex relations, but NN usually does it better if big data is available. Besides that, RF has the big drawback of not extrapolating. Gradient Boosting Trees Gradient Boosting (GB) trees are also an ensemble of decision trees based on the Boosting technique. Boosting algorithms rely on the idea that weak predictors can be sequentially added to each other to result in a stronger predictor [227]. Thus, a sequence of decision trees results in a stronger predictor, which also counts with a differentiable loss function, allowing a gradientbased optimization of the Decision Tree parameters. This concept is the base for creating GB methods. In a regression problem, the first tree predicts the mean of the data, while the next tree tries to predict the residuals of the prediction of the first tree while minimizing the loss function, and so forth. Therefore, the sum of the prediction of all the trees in the sequence will minimize the error and the problem complexity, in a process called Additive Training. There are two main powerful methodologies to implement GB trees: XGBoost (eXtreme Gradient Boosting) [228] and Ad-aBoost (Adaptive Boosting) [229], which also count with handy Python libraries and tutorials [223,230] GB methods usually outperform RF [194], with the downside that the GB algorithm cannot be fully parallelized and does not extrapolate well. Currently, GB methods such as XGBoost and AdaBoost are, alongside NN, the most used ML method in the competitions on the Kaggle website, which is used as an indicator of ML methods relevance [194]. Gaussian Process Gaussian Processes (GPs) in ML are algorithms that use Bayesian inference to update the GP modeling of the prior and posterior distribution until it matches the available data [231]. GP regressors are also commonly referred to as Krigging. The GP is a stochastic process that assumes a joint Gaussian distribution over all variables and, thus, a distribution over functions: f GP (t) ∼ GP(m(t), k(t, t )) m(t) = E[ f GP (t)] k(t, t ) = E[ f GP (t) − m)( f GP (t ) − m(t )](1) where m is the mean function and k is the covariance function or kernel, which models the correlation between the variables (t, t ). Now, considering that X and Y are in input and output vectors, respectively, of the training points and assuming a zero mean GP with a matrix of covariance given by K(X, X), the posterior distribution for any set of function inputs X * is inferred as: Figure 7 illustrates the process of updating the prior with input data and evaluation of the posterior distribution. Usually, a maximum likelihood estimator approach is applied to optimize the kernel hyperparameters. In other words, the searching process looks for the best fit of the output data to the input data evaluated by the marginal likelihood of data. A detailed explanation, also including the formulation for GP with non-zero mean and inference considering a Gaussian white noise in the observation's outputs, can be found in the works by Beckers [231] and Bachstein [206]. The book by Rasmussen [232] is also a classic reference for Gaussian Processes. Implementation is also widely supported in many libraries in python, like scikit-learn [223], GPy, GPflow, SMT [233] and GPyTorch. f GP (x * )\|[ f GP (X) = Y] ∼ N(m c , k c ) m c = K(x * , X) · K(X, X) · Y k c = K(x * , x * ) + K(x * , X) · K(X, X) −1 · K(X, x * ) (2) GP belongs to the class of probabilistic surrogates that predict a distribution of the outputs, which makes it possible to access the prediction variance, often used as a level of trust in the prediction. The variance information is used in informed decision-making and in adaptive sampling techniques that evaluate where to sample new training points. Abdar et al. [234] and Bachstein [206] do an extensive review of several probabilistic ML methods besides GP, such as Bayesian Neural Networks, Monte Carlo Drop-out, Deep Ensembles, Dropout Ensembles, Quantile Regression, etc. Although several recent techniques try to overcome the problem, GP typically does not scale well with big data and therefore is commonly applied when the input dimension is small and when a limited amount of data is available, such as to construct surrogate models, where GP probabilistic properties are useful. Bayesian Optimization is a field that makes extensive use of the GP as a surrogate model in the optimization of expensive functions [235,236]. K-means Clustering is an unsupervised learning method that aims to group data into similar groups, or clusters, based on some similitude or distance measurement [17]. Clustering is extensively used to explore data and discover its structure when little or no previous knowledge is available, being especially relevant in Data Mining. Many clustering methods have already been developed based on different principles and are appropriate for a wide variety of cases, like K-means, supportvector clustering, hierarchical clustering, mixture models, density-based. Xu and Wunsch [237] present a survey on several clustering methods, while the implementation of several of these methods is provided by Scikitlearn libraries [223] alongside documentation with tutorials and a summary of the appropriate use cases [238]. The K-means algorithm is one of the simplest and most used clustering algorithms. K-means is a centroidbased method that divides data into a pre-defined number of k disjoint clusters, minimizing the Euclidean distance between each cluster sample and the cluster centroid, also known as cluster moment of inertia. According to Mehta et al. [17], this can be interpreted as the minimization of the variance within each cluster. The algorithm consists of random initialization of the centroids after which the algorithm iterates through a loop that first assigns each sample to its nearest centroid and then updates the centroids as the mean value of the samples assigned to it until the change in the position of the centroid is smaller than a defined threshold. K-means is very efficient and scales well for big data, however, it performs poorly for irregular and elongated clusters, it is sensitive to initialization and outliers, and it requires a pre-defined number of clusters. Reinforcement Learning Reinforcement Learning (RL) is a class of ML algorithms in which an agent interacts with an environment and learns from the success and errors of these experiences. The agent actions A transform the environment state S , which generates a reward ( Figure 8). The goal Figure 8: Reinforcement Learning framework: the agent performs an action A t in an interactive environment, resulting in a change from state S t to state S t+1 and in a reward R t+1 . The information is used to update the value function (or Q-function in the case of Q-learning), which learns the relation between an action performed in a given state with the reward in the long term. In that way, the agent learns the actions that optimize the system rewards. Action A t State S t Reward R t Agent Update Q-function Environment S t+1 R t+1 of RL algorithms is to find the optimal sequence of actions that maximize the Quality function (or value function), which is a function that models the expected longterm cumulative reward [239]. The classical book by Sutton and Barto [239] explains the methodology behind RL, and the work by Li [240] presents an overview of Deep RL aspects, applications, and relevant references. The Q-learning algorithm developed by Watkins [241] is a popular RL algorithm used here to illustrate the RL workflow. Q-learning is an off-policy algorithm which uses temporal-difference learning (bootstrap approach) to learn the optimal sequence of actions that maximize the Q-function based on the Bellman equation. Therefore, although a partly random policy is used to select the actions to update the optimal Q-function, the learned optimal action-value function is independent of a policy that determines the agent actions [239,242]. A Qtable commonly represents the Q-function with discrete variables, whereas the use of Deep-NN to approximate the Q-function shows notorious results in Deep RL with discrete and continuous action spaces [243][244][245]. Recent outcomes with RL have drawn attention to how these algorithms might be a key part of the future of artificial intelligence, with notable autonomous machines playing games, controlling robots, recognizing, and classifying images, generating human language content, proposing financial investments, suggesting purchases, among many other applications [240]. In mechanical engineering fields, as the vibroacoustic field, the use of RL to develop adaptive controlling systems is a developing research area, as further discussed in Section 4.2. Structural Health Monitoring by Machine Learning SHM deals with the detection, diagnosis, and prognosis of recipient failures, as well as the prediction of the Remaining Useful Life (RUL) of engineering structures based on its measurements. The benefits of SHM are manifold and well known to structural reliability and integrity management, including avoiding catastrophic failures, optimizing service time with scheduled maintenance, planning missions to minimize wear, tracking the cause of failures, and reusing healthy components. Data-driven statistical methods are suitable for SHM as they incorporate information of the true operating conditions and the uncertainties involved, while physical-based models usually fail to model the real conditions of the system, often unknown and commonly involve costly simulations, making them unfeasible for some online SHM applications [246]. The growing amount of data available after the Big Data revolution and the fast advance of ML as a powerful data analysis method paved the way for ML-based SHM to become a major area of research. Currently, SHM is the most developed application of ML in engineering and many successful works and methodologies exist in the literature, most including vibroacoustic-related problems. The book by Farrar and Worden [28] presents an in-depth analysis of all aspects of SHM related to ML, including main applications, data collection and processing, and ML algorithms. More recent surveys on SHM driven by ML were carried out by Khan and Yairi [29], Azimi et al. [30], Toh and Park [31], Lin et al. [32], Bao and Li [33], Malekloo et al. [34], Lei et al. [247] and Zhao et al. [35], most of them focusing on Deep-learning models. The main applications are in Rotating machinery [36,37], Civil Engineering [38][39][40][41], Earthquake engineering [23] and Aerospace Structures [42][43][44][45]. The ML methodologies applied in SHM are closely related to the characteristics of the data. Inspection images, vibration signals, or acoustic signals are the three most commonly used data sources. In the review by Azimi et al. [30], 44 references that use images datasets in SHM are summarized, most of them using CNN to take advantage of its image-processing capabilities. However, in vibroacoustic applications, it is usually more convenient to use time-domain signals, which can be continuously monitored, as vibration and noise signals. Avci et al. [39] presents a review of vibration-based health monitoring in civil structures. It is often necessary to transform vibration and noise signals to reveal their relevant characteristics, as they might be represented best in the frequency or wavelet domain. Because of this, Data Transformation is an important pre-processing step in SHM and much literature dedicates to investigating it. Analyses in the Frequency domain are usually applied for stationary signals and can be obtained with Fast Fourier Transform (FFT) or bispectrum analysis [46][47][48]. Time-frequency or Wavelet domain analyses are convenient for nonstationary signals, and their methods and aspects are reviewed by Taha et al. [49]. Many new references can also be listed employing different Wavelet transform methods, as Discrete Wavelet Transform (DWT) [50], Wavelet Packet Transform (WPT) for more noise reduction and adaptive resolution [51][52][53][54], Morlet Wavelet [55], Short Term Fourier Transform [56,57], Hilbert-Huang transform [57] and Empirical model decomposition [58]. Vibration and noise signals are often transformed into multiple arrays or images to train a CNN, taking advantage of the CNN ability to extract features from big data. However, the transformation method from signal to image also plays a key role in these cases. For data in the time-domain, the methods include reshaping the time-series as matrices [47,59], Omnidirectional regeneration [60], Gramian Angular Displacement Field and Markov Transition Field [47]. Many works used the time-series signals with 1D CNN [66][67][68][69][70][71][72]. In frequency domain, the images are created using Dempster-Shafer theory [61], 3D image method [46], or combining multiple sensor data [62,63]. Wavelet analysis can also create spectrograms to analyses in time-frequency domain [47,64,65]. Reshaping statistical features into a matrix was also used to create images [73]. As data in SHM applications are usually highdimensional, it is also crucial to perform Feature Extraction and Selection to reveal and filter the most pertinent information. The classical approach has handcrafted feature extraction followed by order reduction methods and, lastly, the ML model. However, Deep Learning (DL) has been increasingly used in SHM as its depth enables the extraction of relevant features automatically, ending the need for handcrafted methods Khan and Yairi [29]. In this way, even raw data can be input in DL methods due to its ability to learn high complex and nonlinear patterns. On the other hand, DL usually demands more data. The different procedures of feature selection and extraction in both Classical ML and Deep Learning are illustrated in Figure 9. Classical ML methods often use handcrafted features in cases with little data availability [47,68,74,75,83]. The extracted features might be in time-domain (e.g., Root Mean Square, Skewness and Kurtosis), in the frequency domain (e.g., Power Bandwidth, Harmonics, and spectral skewness) [47,[248][249][250][251] and in the time-frequency domain, where Wavelet Packet Energy-Entropy is commonly used to extract features [52,[252][253][254]. Other feature extraction methods used in vibroacoustic problems are Multi-Domain Statistical Feature [68,251,[255][256][257], Compressed sensing techniques [52,65] and Histogram of Oriented Gradients for vibration images [60,258,259]. After the feature extraction, feature selection is performed to dimension reduction, currently done by an Unsupervised ML method. Varanis and Pederiva [252] compares some feature selection methods and concludes that Linear Discriminant Analysis (LDA) is suitable for non-stationary cases, while Principal component analysis (PCA) is convenient for stationary signals and independent component analysis (ICA) for problems with combined faults. Malekloo et al. [34] also reviews several supervised and unsupervised feature selection methods. The classical ML approach in SHM is addressed by Worden and Manson [260] for damage detection, localization, and assessment problems. The use of DL algorithms in SHM occurs mainly in two distinct configurations: an Unsupervised DL algorithm to perform dimension reduction, usually Autoencoders (AE) or Boltzmann Machines (BM), followed by a simple ML classifier or regressor, which can be either a supervised algorithm (as a decision tree) [54,74,80,93,261] or an unsupervised algorithm (as k-means) [79,94]; or a DL algorithm which performs all stages, that is, feature extraction and selection and the final prediction, as the case of CNN or LSTM [47,76,77]. Malekloo et al. [34] provides a complete review of all processes that should be considered in an ML-based SHM analysis, from the excitation source, going to data acquisition, data normalization, data cleaning, data compression, feature extraction and selection, data fusion, and the prediction. Hereafter, some ML-based SHM works using vibroacoustic signals are presented to exemplify applications in damage detection and diagnosis -including damage location, extent, and typeand in the prognosis of RUL and mission planning. Failure Detection and Diagnosis The first level of complexity of SHM problems is failure detection, that is, identifying if a signal is healthy or not. Unsupervised learning algorithms are suitable for these applications, once the failure can be detected just by identifying anomalies and outliers in the signal, without the need to have labeled data, which is currently the case in real applications. Rizzo et al. [78] show an example of unsupervised learning crack detection using hand-crafted extracted features by discrete wavelet transform and outlier analysis. The article also exemplifies how hand-crafted features can lead to good accuracy, but also how laborious it can be to select the best features to be used. Automatic feature extraction by unsupervised deep learning approach is implemented by Reddy et al. [79], that used a deep auto-encoder to extract features and reconstruct the signal from unlabeled datasets with raw and heterogeneous data (coming from different sensors modalities). A threshold in the error of the reconstruction signal was used to identify unhealthy signals. The next complexity level in SHM is fault diagnosis, which includes the prediction of damage location, extent, and class [34]. For example, in the work of Reddy et al. [79], after detecting the fault, a clustering unsupervised method is applied to classify the fault type. However, supervised methods are more commonly used when the diagnosis of fault location, class and extent is required. Sun et al. [80] used a Sparse Autoencoder to extract features from the vibration signals of an induction motor, followed by a Dropout NN to classify the fault. Gecgel et al. [47] compared several approaches to identify and classify gear tooth crack based on simulated-based vibration signals. Various levels of tooth profile error were also considered, and the noise was artificially added to the signals to augment the robustness. Gecgel et al. [47] performed the classi-cal approach of feature extraction followed by classical ML methods and compared it with Deep-Learning approaches without feature extraction, as CNN and LSTM. The DL methods overcome the classical approach for all ML algorithms, while among the DL methods, CNN beat LSTM. However, Gecgel et al. [47] shows that the CNN accuracy level is highly dependent on the preprocessing method used to encode the vibration signals into images. As indicated by these examples, vibration signals are the most used inputs in ML-based SHM problems, and their many uses are reviewed by Toh and Park [31]. MLbased methods can also use acoustic emission signals to detect and diagnose failures. The reader can consult the recent work by Suwansin and Phasukkit [81] or the review by Muir et al. [82] to have more details and references, being that the latter focus on diagnosis of composites structures. Acoustic measurements can also be used as nonintrusive sensors to failure diagnosis, as investigated by Janssen and Arteaga [46]. The signals from a microphone array placed close to a vibrating plate were used to localize a failure in the plate, which was experimentally represented by a disturbance mass attached to the plate. Janssen and Arteaga [46] investigate processing and data augmentation methods, as well as the number of microphones needed for an acceptable accuracy in the location prediction. The psycho-acoustic features extracted from acoustic signals also demonstrated to be meaningful inputs for failure detection in gears [83], which can also be indicative of NVH performance. Li et al. [54] implemented a method to merge acoustic emission and vibration signals by extracting the signal features through separately Deep Boltzmann Machines and merging them with a Random Forest, showing improved accuracy in the classification of gearbox failures in comparison to other approaches. Hybrid models integrating physics-based and MLbased models are also found in the literature for fault detection and diagnosis. Abbiati et al. [84] implemented a Hybrid Model to detect Euler buckling failure in a beam using Kriging meta-models and active learning to assess structural reliability. The Hybrid model proposed by Ritto and Rochinha [85] configures a Digital Twin due to its bi-directional connection, which allows the model to be calibrated with data from the physical twin (real asset) and the digital twin predictions can be used to update the physical twin operation parameters and control strategy. The methodology is demonstrated in a bar structure, which is trained offline using displacement synthetic data to detect damage and identify its severity and location. The study also analyses the ro-bustness of the DT to different damage intensities, noise and uncertainties levels, the number of sensors and the excitation frequency and location. Prognosis and Mission Planning More than detection and diagnostics, the potential of the SHM relies on prognosis. Predictive analyses allow us to estimate the RUL and therefore, it can be used to optimize the maintenance schedule, minimize g downtime, and lead to money savings more safety operation conditions. Si et al. [86] and Jardine et al. [87] provide a review of the Data-Driven methods to evaluate RUL and apply Condition-Based Monitoring. When degradation historical data is available, the DT can predict the RUL just based on online-sensor data and in the component operation conditions. As an example, an LSTM-based model with an attention mechanism was implemented by Muneer et al. [88] to predict RUL curves with uncertainties using a turbofan benchmark dataset, outperforming CNN, RNN, and GRU models. Zhao and Yuan [89] implemented a CNN which detects and classifies faults in the outer race, inner race, and the cage of a bearing and, once the fault is detected, the DT predicts its RUL in real-time with an online adaptive delay correction method to obtain an improved accuracy result. Other examples include a semi-supervised approach using VAE and RNN [90], an LSTM with dimension reduction methods for multisensor data of machining tools [91], a hybrid model with NN and GPR to predict fatigue failure time with adaptive confidence interval [92], or even just an exponential fit [262]. Similarity models can also be used when only runto-failure data from other similar components are available. In this case, the RUL is estimated based on a curve with a similar profile, as implemented in MathWorks [263] and Liao et al. [93]. Finally, if no degradation or failure data is available, the failure can be estimated with relation to an established threshold in some of the monitored data, just by predicting its future state. Booyse et al. [94] shows the possibility of using a dataset only with healthy vibration data to detect damage, indicate damage severity, and identify failure mode. To illustrate the methods, analyses were performed in simulated datasets of a gearbox with tooth damage and of an Aero-Propulsion System with compressor or fan failures, as well as an experimental dataset with bearing failures. An order tracking preprocessing was applied to normalize the data with respect to rotational speed, as well as time-synchronous average over the recording period. Generative Adversarial Network (GAN) and VAE were used as the ML unsupervised algorithms, being that GAN presented the best performance. Recently, DT models have been proposed in SHM to analyze "what if?" scenarios, perform risk assessments and optimize the asset operation to maximize its life. Karve et al. [95] implements a DT which performs failure diagnosis and prognosis, as well as optimize the operational parameters of the system, resulting in damagetolerant planning which minimizes fatigue crack growth while still ensuring that the mission aim is achieved, all considering both aleatory and epistemic errors. Different methods of fusion Physics-based and data-based models are applied in each stage of the DT and experimental results are obtained to validate the methodology. Kapteyn et al. [96] also developed a data-driven DT to mission planning based on both component and model libraries applied to a fixed-wing unmanned aerial vehicle. All assets share the same physics models, and the analyses are efficiently scaled to the entire system because of the component-based approach, which uses the Static-Condensation Reduced-Basis-Element method. In addition, a library of physical models is used, so that the more appropriate model is applied according to the scenario detected. The physics models use model order reduction to speed up the computation and enable fast predictions. Optimal trees are applied as interpretable classifiers to select the best physical model and update the DT based on the sensor data. Therefore, informed by the level of damage estimated by the DT, the system decides what maneuvers to do, avoiding structural failure. Stender et al. [97] approaches the acoustic brake squeal problem in two steps: brake NVH assessment and brake squeal prediction. For the NVH assessment, a Short-time Fourier transform creates 2D data representations, which pass through random modification to augment the data and avoid overfitting before training a CNN. The trained CNN identifies the class of brake noise and indicates when and in which frequency it occurs with accuracy. In the second task, the problem parameters over time are the inputs of an LSTM algorithm that predicts when the squeal will occur. However, this methodology showed poor performance when the brake configuration under analysis differentiates from the brake used during LSTM training. Active Control powered by Machine Learning Active Control is the area of study which aims to model dynamic systems and design control mechanisms to guide the system behavior to the desired state. Active Vibration Control (AVC) of flexible structures plays a significant role in the safety and ergonomics of vehicles, aircraft, machines, and buildings as in manufacturing accuracy [264][265][266][267]. Active Noise Control (ANC) or noise-canceling is based on destructive interference and is a subject of longstanding research [268,269]. The growing importance of user comfort and product development focused on ergonomic and NVH performance [270] increase the efforts to control vibration and noise. The need for AVC and ANC is greater in the low-frequency ranges, where the application of passive control is limited [271]. Active Control and ML are deeply correlated. Both are data-based science, which development relates to the popularization of sensors and IoT devices, improvements in signal processing, and growth of computational memory and power. In addition, ML methods can be applied in several elements of a control system, as analyzed in the book by Brunton and Kutz [272]. The Least Mean Square (LMS) filter, used in control to estimate the state and controller parameters, is a basic linear ML algorithm. More advanced ML algorithms, in special NN, are also widely used, in special to model and control nonlinear systems in which linear control theory might fail. When possible, linear control methods should still be prioritized, because of their smaller response time and the well-developed control algorithms suited to them. Back to the 1990s, many NN applications in active control have already been identified with three usual configurations [273][274][275]: NNbased Model Predictive Control (MPC), in which an NN black-box models the forward dynamics of the system [276]; as an NN-based model-free controller [277]; and a model reference control, where NN models the plant and optimizes the controller parameters [278]. The first and third, use NN in the system modeling stage, as analyzed in 4.1, while the second and third configurations use NN to learn the optimal control design, as discussed in 4.2. Brunton's series of videos, named "Data-Driven Control with Machine Learning" [279], covers overall aspects of ML applied in Active control. In a recent article, Brunton et al. [22] also reviewed the ML methods applied to fluid dynamics control. Kim et al. [280] reviews the ML methods applied in the control of soft robots, with a focus on soft sensors and actuators. This paper explores the methods which have been applied to the vibroacoustic domain, in special ML-based methods supporting dynamic system modeling, using system identification and reduced-order models, intelligent placement of sensors and actuators, and adaptive control algorithm. Within this topic, it is included the control of buildings vibration under seismic activity, also reviewed by Xie et al. [23]. An extensive, but not an exhaustive number of applications of ML in ANC and AVC is illustrated in Figure 10. Dynamic System Modeling with Machine Learning A big part of Active control theory relies on modelbased control techniques in which control needs to have a mathematical model of the physical system, as in model predictive control and linear optimal control [272]. However, in practical situations, there are two main obstacles: • The physical model of the system is unknown, or the model parameters which fit the system equation are unknown. In this case, it is necessary to apply system identification techniques. • The physical model is known, but its complexity is unfeasible for real-time control applications. Here, it is necessary to apply model reduction techniques. System Identification System identification (SI) are techniques that use the measured data of a system to model the relationship between the input and output of the system. This description coincides with the definition of ML models of inferring model from data [272,281]. Therefore, according to Duriez et al. [282], classical system identification methods -as eigensystem realization algorithm, Kalman filters, and linear parameter varying -can be considered an early form of ML. From this, it can be inferred that modern ML methods should be used for more complex and nonlinear SI. Chiuso and Pillonetto [283] and Pillonetto et al. [284] did a review of SI methods based on ML, especially the so-called kernel-based methods, highlighting their continuous structure selection capabilities about classical SI methods. A comparison of the online learning performance of adaptive filters in an ANC application, showed the superiority of kernel-based models, as Kernel-LMS and Kernel Affine Projection Algorithms, over classical LMS and NN algorithms [285,286]. However, NNs have been similarly applied to classical SI methods since they are nonlinear autoregressive exogenous (NARX) models [281]. The sequential dynamic structure of RNNs is also suitable for SI of dynamics systems [287]. citetljung2020deep analyzes the similarity of Deep Learning and SI concepts and shows the workflow and the results of implementing an LSTM in the identification of a nonlinear state-space model. [98], Xu and Fei [99] and buildings [100]. NN black-boxes are implemented in AVC of vehicle suspension Vidya and Dharmana Another advantage of NNs over classical SI models is that, besides being able to model nonlinear dynamics, they can be trained to predict the model output some steps ahead. This method is convenient in Model Predictive Control (MPC), which uses a prediction of the system response based on a model of the plant to optimize the control signal over a finite-time horizon in relation to the control cost function in a feedforward configuration. Jamil et al. [100] uses an NN Predictive Control applied to a tall building, combining the good aspects of pole-placement and Neuro-fuzzy control. Although state-space representations can be obtained from ML black-box models [100], they may also be constructed based on first-principle knowledge of the system dynamics, as with Kalman filters. MLbased state-space models which use both data and firstprinciples are found in the literature [288][289][290], but the authors did not find applications in vibroacoustic. The use of Koopman theory in dynamic mode decomposition (DMD) to describe a nonlinear system on a linear basis has also been leveraged by ML methods, especially in fluid dynamics [22,291,292], allowing the use of advanced linear control methods. Recently, Saito and Kuno [101] investigated the application of Data-driven DMD in structural dynamics problems. The sparse identification of nonlinear dynamics (SINDy) method proposed by Brunton et al. [293] to discover governing equation using ML and sparsity techniques have also been used to identify the structural dynamics equations of a geometrically nonlinear system [294] and a oscillator [295], but it was not implemented in AVC. Reduced Order Models and Sensors/Actuators Placement Reduced Order Models (ROM) are also essential to construct efficient real-time control, since they use lower-rank representations of the system without losing valuable information about its dynamics, balancing accuracy, and efficiency. In that way, ROMs lead to reduced response time and memory requirements, key aspects in active control. Once more, ML methods play a significant role, since there is an intrinsic relation between ML and many ROM. The first scenario in which ROMs are applied in control is when there is a high dimensional numerical model of the system, as a Finite Element Model, which is computationally expensive for real-time applications. Therefore, ROMs or meta-models are used to speed up the simulation of the system prediction in model-based control. Feedback control might require further reduction in the space-state representation. Besides the MLbased surrogate models presented in section 5, Component mode synthesis (CMS) are used for linear ROM of structures from FEM models, while ROM based on Proper Orthogonal Decomposition (POD) can be obtained directly from measured data [22,109]. POD, also known as Principal component analysis (PCA), is also an ML method, once it is equivalent to a symmetric autoencoder with a linear activation function, as demonstrated by Baldi and Hornik [296]. The POD/PAD applies a coordinate transformation from the physical coordinates to an orthonormal basis formed by the system eigenvectors. By selecting only the main modal contributions, or first principal components, the system model is represented on a reduced basis, which is very convenient to model acoustic and structural dynamic problems and has for long been used in spacestate vibroacoustic control [102][103][104][105][106]. This modal basis representation also provides useful information on the controllability, observability, and stability of the system, which are key factors to determine the optimal placement of sensors and actuators and therefore PCA has also been used for this purpose [103,107]. The location of sensors and actuators is a crucial aspect in active control design since it influences the control efficiency, cost, and stability [267]. Real-time predictive control applies other tools combining ML methods with ROM. The following examples explore these techniques, which could be used in online control. Liu et al. [108] developed an automatic FEM model updating by using CMS together with Kriging. Simpson et al. [109] used autoencoder to get nonlinear ROM (or Nonlinear normal modes) of a frame structure with hysteresis and used it alongside an LSTM model to predict the system dynamics in near real-time. A Deep Learning-based ROM of structural dynamic systems with inertia and geometric nonlinearities is implemented and benchmarked with simulations of a Doubly clamped beam resonator and a MEMS Micromirror [110]. Using cluster-based ROM, already explored in fluid control [297] and static structural mechanics [218], could have potential use in the vibroacoustic domain. ML driven Control Design Another application of ML methods is in the control design, that is, in optimizing the control signal or control laws regarding the cost function that quantifies the control performance. While in the last section ML computes the forward output of the system, the following references use ML to learn effective control laws. ML-based controllers are mainly used to handle systems with nonlinearities, epecially using NN methods, as evidenced in the survey on Nonlinear ANC by Lu et al. [111]. Several configurations use ML to support the control design, such as Model Reference Control, inverse-dynamics control, Machine Learning Control, neuro-fuzzy control, and Reinforcement Control. In NN-based Model Reference Control, two NNs form the control system: one NN plant model predicts the system response and the other NN defines the controller parameters which are optimized to minimize the error between system response and the reference signal. Vidya and Dharmana [98] implemented a Model Reference Control of a vehicle suspension using an NN reference model and an RNN controller, claiming that it leads to better adaptivity and stability. The drawback of NN reference control is that it uses dynamic backpropagation in the optimization, which is computationally expensive [273]. Adaptive NN controllers are popular in noise and vibration control with diverse methodologies. One of them is the NN-based inverse dynamics control, which consists of training an NN with the inverse system dynamics and use it to determine the controller parameters, as in a regressor-based control. De Abreu et al. [112] implemented a direct inverse NN control of a vibratory system by training an NN as the inverse model of the plant, such that the NN receives the current state and the desired state and outputs the actuator signal. Similarly, Ariza-Zambrano and Serpa [113] applied direct inverse NN control to a beam cantilever, in which the NN was trained both with a full state FEM model and with a reduced model to account for dynamic uncertainties in practical scenarios, showing more stable results than H-infinity control. Nerves and Krishnan [114] uses NN direct controller to control wind-induced vibrations in a building-TMD system, by considering the plant as the output layer of the NN, as in a feedback linearization control. Bani-Hani [115] uses NN to model both a direct forecasting model and an inverse model also applied in the control of wind-induced vibrations. Model-free NN controllers have also been used in a different configuration of ANC, as a nonlinear alternative to the commonly used adaptive Filtered-X LMS (FxLMS) algorithm. Park [116] tested different configurations of NN as the adaptive controller in a feedback configuration for different ANC datasets. CNN was the one that performed the best, followed by MLP and RNN, all of them with better performance than typical LMS-based controllers. For the case of a feedforward noise control system with a nonlinear primary path, Zhang et al. [117] also obtained better performance with LSTM based controller than with FxLMS. Zhang and Wang [118] implement Deep-ANC in a feedforward configuration in which a Convolutional Recurrent Network (CNR) is used to estimate the optimal control signal-to-noise cancellation. The supervised training of the CNR uses the reference signal as input and the ideal anti-noise as the target, both in their spectrogram format. Besides that, the CNR predicts the canceling signal with some frames in advance, to compensate for the ANC delay. The approach improved noise canceling in noise-only and noisy speech scenarios in comparison to typical ANC. Other examples of NN applications are found in the review by Lu et al. [111] on ANC for nonlinear systems. Heuristic algorithms, such as genetic algorithm (GA) and particle swarm optimization (PSO), can search for an arbitrary optimal control law in Machine Learning Control (MLC). According to Hansen et al. [271], MLC can optimally adapt the weights of any nonlinear filter structure, including an NN. As MLC does not rely on a fixed structure of the controller, nor on a model of the system, it gives more flexibility to the optimization, with the downside that it adapts slowly, preventing its online application to a transient system. Chapter 2 of Duriez's book [282] has a gentle introduction to MLC. [119] was a pioneer to apply MLC in ANC and AVC and was followed by many others in acoustics [120][121][122][123][124][125][126] and vibration [127][128][129][130][131] control. Neuro-fuzzy control systems, especially using adaptive network-based fuzzy inference system (ANFIS), have been widely applied in ANC [132][133][134][135] and AVC [136,137], to name a few works. Neuro-fuzzy systems usually use expert knowledge to set initial fuzzy rules in an NN-like structure, then the neuro-fuzzy parameters can be adapted during a training process to fit measured data. The resultant Neuro-fuzzy systems combine advantages of using interpretable explicit rules from fuzzy rules with the learning capabilities of NN. Finally, noteworthy results are being achieved by exploring Reinforcement Learning Control (RLC) [138]. As explained in Section 2.5, in RL the agent (the controller) can interact with the environment (the dynamic system) and its actions will affect the output of the system and, therefore, the Quality function quantifying the long-term performance of the control, which the algorithm optimizes. In this way, RL interactively learns information about the system and the controller behavior altogether, in a similar fashion to human learning. Detailed RLC explanation and references examples are presented in [138,298]. RLC has gained prominence in applications such as autonomous car control and robot control [299] but has also shown applicability to control acoustic and structural dynamic systems, special in problems with high uncertainty and stochastic behavior [138]. Latifi et al. [139] shows a successful example in which they applied RLC to manipulate an acoustic field by controlling a centrally-actuated vibrating plate (Chladni plate) and guided a particle towards a target location on the plate surface. Implemented ANC using Q-learning algorithm also had satisfactory results, showing the capabilities of RLC to adapt, as when the secondary path of the noise changes suddenly [140]. Qiu et al. [141] carried out bending and torsional vibration control via an RL algorithm virtually trained with a validated FEM model and transferred to an experimental setup where it shows better performance than PD control. The vibration control of a rotating machine was also performed through RLC using pad actuators [142]. Gulde et al. [143] implemented a method to compensate vibrations in an industrial machine tool using RLC. Eshkevari et al. [144] and Gao et al. [145] also achieved good controllability of flexible buildings structures through RLC. Although RLC shows potential as a real-time decision-making control for complex and uncertain scenarios, it demands considerable training time and expensive computational resources and, therefore, its use may be superfluous to applications already mastered with simpler solutions. Vibroacoustic Product Design by Physics-Driven Surrogates Physics-Driven Surrogates or Meta-models are simpler and cheaper replicas of a high-fidelity simulation constructed based on the information from some inputoutput points of the true simulation. They have long been used as practical and efficient tools for decision making and risk management in the early stages of product development, once they make it workable to carry out domain exploration, uncertainty propagation analysis, sensitivity analysis, and optimizations, studies in which many evaluations of the same function are necessary. In the article "Modelling for Digital Twins -Potential Role of Surrogate Models", Bárkányi et al. [300] coins the term Physics-driven surrogate models (PDSM) for surrogate generated from a large amount of data got from a high-fidelity physical model and enumerates several advantages and applications of DT with surrogate models. First, after training, PDSM is much faster than traditional first-principle simulations. In addition, despite being a "black-box" algorithm, it is guided by the physics of the supporting data. Unlike a datadriven model, PDSM is less susceptible to different bias sources, and its uncertainties can be related to the physical model, so they can be estimated and bounded. As for drawbacks of the PDSM, Bárkányi et al. [300] mentions the lack of interpretability, inability to extrapolate the prediction to unseen scenarios, and the difficulty of assimilating long-term historical data. However, some built-in or implemented methods can increase the interpretability of ML, and Adaptive Sampling can deal with extrapolation when the prediction uncertainty is high. Hybrid methods to embed physical knowledge in the ML can also alleviate these issues. Despite being a well-known tool, its use in structural dynamics and acoustic applications is still not well established, as in other physical domains. The reason might be that many vibration and acoustic analyzes present a harsh and discontinuous behavior, especially close to the system resonances, which hinders the ML generalization ability [146,301]. One of the main assumptions of ML is that data is locally smooth [17,182]. Therefore, the ML surrogates smooth the system response and underestimate sharp peaks and valleys, regions of interest in vibroacoustic analysis. Tsokaktsidis et al. [147] uses an NN surrogate to predict the acceleration response of a source-receiver structure as a function of the excitation and geometry, showing overall good agreement, but with some inaccuracies in the peak amplitudes. After going through data reduction techniques, data from a beam acceleration response was clustered and used to train NNs aiming to replace FE models Birky et al. [148], but some prediction curves still present high percentage errors. Approaches as Adaptive Sampling [302], Physics Guided ML methods [303], global surrogates with local refinements, Domain-decomposition methods [301] and Low-bias ML are suitable for dealing with model-ing of rough functions. For example, a Physics-guided convolution neural network, with embedded physical constraints, was used to predict building response under earthquakes excitations [149]. The following sections review the workflow for the construction of the surrogates and their applications in acoustics and structural dynamics analyses. Applications in uncertainty propagation and optimization problems stand out since there is a cross-fertilization of research of these domains with surrogate models [300,301,[304][305][306], especially because of the benefits of surrogates when performing several evaluations of the same function and because of their intrinsic statistical characteristics. Besides that, ML may also improve similitude techniques to scale models and prototypes during product design, especially for complex structures with incomplete or distorted data [150]. Workflow of Surrogate Construction and Related Methods As the name implies, the Physics-driven surrogate mimics the behavior of a true function f (x) = y. For this, the surrogate models use statistical methods to map the relationship between a sample of inputs X and the corresponding outputs of f (X) = Y, which are called support points. In this way, the surrogate generates a new approximate functionf (x) ≈ f (x), which can generalize the observed behavior of the true function and then predict the outputs for a new set of inputsỹ =f (x) with low computational cost and accuracy lost, providing a compromise between computational cost and fidelity. Several statistical methods are used as surrogates, such as Polynomial Chaos [307,308], Response Surface Model (RSM), Polynomial Function, Radial Basis Function (RBF) [309], Low-rank tensor approximations [305], spectral expansions [301], however, this The basic steps to build a surrogate model are schematized in Figure 11. The first step is to generate the support points. As the approximate function is created based on their information, it is of major importance to generate an informative set of support points, according to the Design of Experiments (DOE) theory. Methods as Latin Hypercube Sampling or Quasi-Monte Carlo are used, since they have good spacefilling properties, providing information in the entire design domain, including the interaction between parameters, with an affordable amount of sampling points [306]. Adaptive Sampling automatically defines regions where to add support points to update the surrogate model, as reviewed by Liu et al. [302]. Adaptive Sampling comprises training a surrogate with a sparse initial database and defining new sampling points based on so-called Acquisition Functions to query and update the surrogate. In that way, Adaptive Sampling increases the surrogate accuracy near points of interest. This procedure continues until it reaches a stopping criterion, as illustrated in Figure 12, showing the entire workflow of surrogate construction with Adaptive Sampling. The Acquisition Function, also called Infill Sampling Criteria, accounts for the surrogate mean and variance when choosing enrichment points that should present a compromise between the exploration of new regions (global search), where the surrogate variance is high and the exploitation of promising regions (local search), where the surrogate prediction is of interest, as near the optimal. The surrogate should use a probabilistic ML method once the prediction variance is necessary. Comparisons of Acquisition Functions are presented by Chaiyotha and Krityakierne [310] in constrained optimization and by Emmerich et al. [311] in multi-objective optimization. Figure 12 illustrates a probabilistic surrogate model prediction with the respective Acquisition Function, which determines new observation points. Adaptive Sampling improves the accuracy of the surrogate, especially in regions of interest, minimizing the number of points that must be evaluated with the true function to achieve the required performance and are widely used in optimization problems, as also illustrated in Figure 12 and further discussed in Section 5.3. Aspects such as dimensionality, data format, presence of outliers, non-linearity aspects of the model and the need for variance information should be considered when choosing the ML method of the surrogate. Although the surrogate can be a classifier, the target is more often composed of continuous values which predict the physical system response. In front of this, most of the supervised regression methods are appropriate to construct surrogates, like Support Vector Machines [178], Gaussian Process Regressor or Kriging [312,313], Neural Networks [304], Random Forest, Gradient Boosting, etc. As surrogate models rarely involve high dimensions inputs, since the ML inputs are the parameters of the physical problem, the choice of Deep Learning methods is not always as advantageous as in SHM applications. However, the limited number of samples and input dimensions make Kriging-based surrogates viable to use and, because of their probabilistic properties, they become a common choice. Marelli et al. [301] mentions different classes of surrogate models which perform better in different scenarios. Localized surrogates generate predictions that rely on the proximity of the support points and are good at interpolating. Global surrogates have better extrapolation capabilities but usually have lower local accuracy. Global approximations with local refinements or domain-decomposition-based methods are suitable for functions with highly localized behavior in some specific regions of the input space. Uncertainty Quantification with Surrogates Uncertainties are an inherent part of every phenomenon and computational analysis. Their study improves comprehension of the phenomenon and enables an adequate level of reliability. The main steps of an Uncertainty Quantification (UQ) analysis are uncertainty propagation and sensitivity analysis, being that both demand several evaluations of the system model and are used with surrogates ( Figure 13). Sensitivity analysis enables us to understand the variability of the outputs by the uncertainties in the inputs and their combinations. The influence of the inputs can be studied separately by performing a local sensitivity analysis (LSA), or simultaneously with a global sensitivity analysis (GSA), which allows capturing effects of the interaction between inputs [151]. Sensitivity analysis improves understanding of system behavior and the interpretability of surrogate models and allows the choice of the most influential inputs for the system response to perform further optimization on a lower-dimensional problem, without major loss of accuracy. Sensitivity techniques can be based on multiple solutions for the forward model, so they benefit from surrogate models. Besides that, surrogate models have intrinsic properties to evaluate sensitivity indexes, as they come as a by-product of trained surrogates such as polynomial chaos expansion, low-rank tensor, and random-forest [305] or are evaluated with minor effort in Neural Network [314] and Kriging. Cheng et al. [315] presents an overview of global sensitivity analysis evaluated with surrogate models with their performance comparison. Focusing on improving surrogate interpretability, Pizarroso et al. [316] lists several methods to analyze input-output relationships in ML-based surrogates. Gradient interpretability is one method studied for this purpose [317][318][319]. Chai et al. [151] constructed a surrogate for the Sound Transmission Loss analyses of sandwich panels using Random Forest and, as a by-product, got the out-ofbag (OBB) based global sensitivity analysis method. Although the RF surrogate presents bias and smoothing effects, Chai et al. [151] showed an overall good agreement of the sensitivity indexes obtained by the Fourier amplitude sensitivity test (FAST) and OOBbased method and highlighted that the latter can be more easily interpreted. Abbiati et al. [152] show a framework to do global sensitivity analysis in hybrid surrogates, which merge physical and numerical substructures, showing an application in a structural dynamic problem modeled by polynomial chaos expansion surrogates. With a similar method, Abbiati et al. [84] creates a hybrid model for buckling failure reliability analysis using a GP classifier, obtaining a failure surface prediction with good accuracy against experimental and analytical references. With Uncertainty Propagation analysis, the input uncertainties propagate through the model to quantify statistical moments and probability density function (PDF) of the system response, as well as the failure probability [305]. Spectral stochastic methods as Polynomial Chaos Expansion (PCE) and direct simulation methods as Monte Carlo Method (MCM) are used to propagate the uncertainties [156]. Nobari et al. [153] used MCM to quantify the uncertainties of the squeal instability analysis by using a surrogate model of the Complex Eigenvalue Analysis constructed with polynomial regressor and a GP. The model not only provided the PDF of the instability modes, but it also allowed an in-depth knowledge of the effects of each variable on the response, since the global result given by the surrogate diverged from the local sensitivity result obtained with FEM. Hurtado and Alvarez [154] implemented surrogates based on MLP and RBF networks with MCM for analyzing the probability of failure of structures and showed that RBF performed better for the cases under static load, but MLP is better in nonlinear dynamic analysis, where similar inputs may lead to distinct outputs. Wang et al. [155] uses the automatic differentiation property of NN to evaluate the first and second-order derivatives of the surrogate model, to get the response bounds by applying the subinterval method. Cicirello et al. [146] calls attention to the nonmonotonic behavior of vibroacoustic problems, which makes Vertex Methods inappropriate for performing uncertainty propagation of this system. Statistical methods which involve several evaluations of the system are then required. To tackle this problem, Cicirello et al. [146] use Adaptive Sampling techniques with different Activation Functions to build GP surrogates that predicted the upper and lower bounds of the system. This leads to a decrease in the number of evaluations of the true function during uncertainty propagation. The upper and lower bounds estimations consider the GP variance to lead to more conservative predictions. The proposed method has good accuracy and is faster than a Sub-Interval method, but it struggles when applied to more complex analysis with higher dimensionality. Chapter 8 of [156] reviews traditional uncertainty quantification methods applied to structural dynamics and vibroacoustic problems. Surrogates may support reliability analysis and risk assessment, in which a failure threshold is established. For nonlinear and time-varying analysis, as often met in vibroacoustic, Extremum Response Surface Method (ERSM) as a surrogate can be handy, since just the extreme responses of the system are considered [157]. Kriging surrogate with moving extremum framework used to model the extreme structural dynamic responses in an interval of time was implemented by Lu et al. [158,159] to evaluate the reliability and sensitivity analysis of turbine and compressor blisks deformation under dynamic loads. Guo et al. [160] uses active learning Kriging to improve computation efficiency in the reliability analysis of resonance fault of pipelines excited by fluid-structure interactions. Guo et al. [161] used Ac-tive Sampling with Kriging to do a sensitivity analysis which quantifies the effects of different variables and the contribution of each failure mode to the system reliability, as for different resonances modes. The surrogate model implemented by You et al. [162] used Random Forest and Stacking methods to predict the probability of failure of tuned mass damper-based structures under random excitation. The surrogate was trained to replace the entire Monte Carlo analysis and not just the computational model, so each call of the surrogate outputs the probability of failure based on the input distributions. Specific strategies have been developed for different applications and complexities added to the models. For instance, Bhattacharyya et al. [320] combines Kriging with NARX applied to UQ in dynamical systems since time-domain UQ with Monte Carlo is expensive even when using surrogate models. For high-dimensional problems in which the surrogates may struggle, Tripathy and Bilionis [321] proposes a deep neural network, which comprises an encoder followed by an MLP, to reduce the problem dimension and then apply the uncertainty propagation methods. Luo and Kareem [322] proposes a Deep convolutional neural network approach for dealing with uncertainty quantification in high-dimensional problems, with no dimension reduction. Chaudhuri et al. [323] tackled the complex problem of addressing uncertainty propagation in feedbackcoupled multidisciplinary systems using Kriging surrogates and Adaptive Sampling. Böttcher et al. [304] used polymorphic uncertainty propagation method in an eigenvalue problem modeled by Kriging surrogate and Adaptive Sampling. Optimization with Surrogate Models Surrogate models are suitable for domain exploration and optimization, being that optimization and multiobjective optimization are the most common links to surrogates in the literature review performed by [300]. Once again, this is due to the surrogate ability to speed up the simulation, allowing to perform several analyses with affordable cost. Surrogate-based optimization of the vehicle mass subjected to NVH and crashworthiness constraints was performed using an RSM [163] and RBF [164] as surrogate models. The acoustic optimization of an electric motor was tackled by [165] through local surrogates replacing FEM, being that the performance of several ML methods was evaluated, namely Linear regression, Decision Tree, SVM, and GP. Zhang et al. [166] used RBFbased surrogate to replace the modal and vibroacoustic coupling simulation of the volute case of a centrifugal fan in the optimization of the thickness parameters of the geometry to decrease the radiated sound power and the total mass. Transmission Loss (TL) optimization using Gaussian Process surrogates was performed for intake systems [167] and for meta-material properties [168]. [169] carried out Bandgap optimization of meta-materials supported by RBF surrogate. A surrogate model based on quadratic polynomial regression also leveraged the aerodynamic and acoustic optimization of a fuel cell vehicle fan [325]. Park and Papadimitriou [170] used GP surrogate together with Dynamic substructuring to perform NVH Optimization of a vehicle, while [171] used a surrogatebased in Elman Network, like an RNN, to minimize the vehicle sound pressure while constraining mass, sideimpact intrusion, and first-order global modal. Instead of using the vehicle parameters, Tsokaktsidis et al. [172] used time-domain acceleration at the component level as input of an NN surrogate which predicts the sound pressure level in the passenger cabin. Parametric optimization of the kinematic hardpoints of a vehicle suspension aiming to decrease road noise was performed through an NN surrogate model replacing costly FE analysis [173]. The optimization approach combines criteria that set an FRF curve as an up-limit and as a matching target, aiming to control both amplitudes in a specific frequency and frequency shift. The NN-driven optimization achieved good time-saving and allowed to increase the problem dimensionality in comparison with previous polynomial meta-models constructed for the same problem. Despite that, inaccuracies in the predicted curve are also visible [174]. Inac-curacies do not invalidate surrogates, once they are used in the early stages of the design development and later high-fidelity models and experiments must validate the final results. Intelligent space exploration methods such as Adaptive Sampling are applied in surrogate-based optimizations as it may be expensive to obtain enough supporting points in the entire domain to build an accurate surrogate. The exploration from Adaptive Sampling allows avoiding local minima, while the exploitation guarantees a good accuracy of the surrogate near the predicted optima. Optimizations using probabilistic surrogates being updated according to the new points selected by the Acquisition Function are called Bayesian optimization and represent a vast field of research. Back in 1998, Jones et al. [236] published "Efficient Global Optimization (EGO) of Expensive Black-Box Functions", one of the first studies applying optimization with GPR surrogates. citetFrean gives an in-depth explanation of Bayesian optimization, while Dwight et al. [306] provides a tutorial with codes for its implementation. Besides that, several toolboxes implement Bayesian Optimization using GP [233,326,327], although other probabilistic ML methods are suitable. Mohanasundaram et al. [175] used the EGO approach in the multi-objective optimization of a disc-pad shape under squeal noise criteria modeled by Kriging, after the previous performance of a variance-based sensitivity analysis. Du et al. [176] applied an Adaptive Hierarchical Kriging model to optimize the modal characteristics of an engine. Adaptive Sampling was applied in the optimization of a mechanical low-cutting Metafilter modeled by RBF surrogate [177]. Another use of surrogate models applies to Reliability-Based Design Optimization (RBDO), once both reliability analysis and optimization analysis require several evaluations. Moustapha and Sudret [178] presents a complete survey on surrogate-assisted RBDO with detailed implementation details and several approaches to tackle the reliability analysis. Fei et al. [179] performed an RBDO of turbine blade radial deformation under dynamic loads using an Extreme Support Vector Machine surrogate and Importance Degree Model. Zhang et al. [180] used a fuzzy multi-extremum response surface method to perform an RBDO of fatigue and creep failures of a turbine bladed disk, showing accuracy as in MCM but in a fraction of the time. A reliability EGO approach was implemented to optimize friction-TMD (tuned mass damper) controlled structures modeled by Kriging [181]. Polynomial-Chaos-based Kriging was used as a surrogate to speed up the dynamic simulations in the [324]. Figure (a) shows the optimization path, including the enrichment points selected during adaptive learning. Figure (b) shows how the accuracy criterion decreases with optimization iterations to guarantee the surrogate accuracy near the optimal point. The blue points are admissible; green points are the successive best points; the red points are unfeasible points; and cyan points are those around which enrichment has been done during optimization. RBDO of a Nonlinear Energy Sink, a passive control device to mitigate vibration [328]. An example of RBDO applied to the buckling analysis of a column under pressure aiming to minimize its cross area and keep the probability of failure under 5% is shown in Figure 14. Using adaptive sampling with adaptive accuracy criterion led to minimized computational cost with maximized accuracy. In his Thesis, Moustapha [324] also applied this methodology to perform RBDO of the crash analysis of a lightweight vehicle. Besides adaptive sampling, global approximations with local refinements and domain-decompositionbased methods are other techniques to improve the surrogates in the region of interest, as regions near the optimal or with highly localized behavior [301]. Importance sampling is also performed to decrease the number of evaluation points in uncertainty propagation analysis [329]. Neural Network-based surrogate model could benefit from the automatic differentiation (AD) [196] properties of NN to perform optimization by using the derivative of the output function. A computational packet with AD implemented is shown by Bouhlel et al. [330], as well as an example of optimization implementation using AD. However, AD is difficult to implement for complex NN architectures. On Future Trends and Perspectives Digital transformation is already a reality and has been changing the way to solve several problems, including mechanical problems traditionally solved solely by physical models. The various works referenced in this article illustrate how this transformation is taking place and bringing advantages to acoustic and structural dynamic field. Despite the current progress, much should be done to scale DT implementation and to take full advantage of them. Integration is a cornerstone on this path and two main discussions field are raising in this direction: Digital Twins, which approach the concept of integrating all levels of simulation and information of an asset through its life-cycle; and Physics Guided Machine Learning, in which physics knowledge is embedded into datadriven methods to support learning of consistent representations. Next, these and other topics are discussed to evaluate future paths in ML research applied to vibroacoustic problems through the observation of recent results of ML in other physic domains and through the observation of the current gaps in the field. Digital Twin The Digital Twin is a time-evolving highly fidelity replica of a product/process, with a bidirectional connection of information. The concept was first addressed in 2003 by Grieves in a presentation on Product Lifecycle Management (PLM) [331], but it only spread with the famous article by Glaessgen and Stargel [332]. This article envisioned the great potential of DT as an integrated multiphysics and multiscale simulation of the real system using the best physical models and data available, to create a virtual copy able to continuously forecast the system health and create plans to mitigate the damage or improve performance while accounting for the system associated uncertainties. However, the concept of DT is still loose and broad and is constantly evolving while DT enablers are under ongoing development and DT applications are spreading to many sectors. In view of this, several works focused on reviewing the characteristics and achievements of the DT [85,303,[333][334][335][336][337][338][339][340][341][342][343][344][345]. According to Gardner et al. [313], the Digital Twin is built from components from four main categories: simulations which model the physics of the system; the knowledge from experts and previous experiences about the product and the environment variables; the available data of the physical twin; and the connectivity which links the other elements and gives DT the ability to evolve with information. These components and their interconnections are the building blocks for creating a Digital Twin, as illustrated in Figure 15. As pointed out by many authors [331,333,338,340,344,[346][347][348], the DT must evolve throughout the life of the product. During the product development, where DT is called Digital Prototype [333], surrogate models are used to explore the design space, leading to optimized and robust design. During the usage phase, monitoring the product and its environment assists in the early detection of failures, optimization of control strategies and mission planning. Data can also be processed and merged to generate virtual sensors, leading to more informative operation without extra hardware [349,350]. Finally, component life estimation is used to optimize scheduled maintenance and to support end-of- Figure 17: Three-storey structure used by Gardner et al. [313] to construct an operational DT (a) and predicted acceleration response of the third floorÿ 3 when column and bumper are in contact (nonlinear response) for different stages of the DT implementation (b-e). Adapted from [313]. life decisions on disposal, reuse and market value [351]. A representation of the Digital Twin components, advantages and uses throughout the life cycle is shown in Figure 16. The information collected throughout the product life cycle also helps in the design of the next generation, as it is possible to evaluate the components that were over-or under-designed. Moreover, DT could make it possible to investigate the causality of the observed phenomena by exploring sensitivity features in Section 5.2. Therefore, a complete DT must store and manage the product data, as well as integrate data-driven and high-fidelity simulations, both for an individual product (Digital Instance) and an assembly of them (Digital Aggregate) [333]. In summary, the DT aims to avoid wasting valuable data and information. A complete DT does not exist yet, and its implementation might take decades of further development, as predicted by Glaessgen and Stargel [332]. However, integrating several key elements has led to the development of incomplete DTs that are noteworthy. One example aforementioned is the DT applied to SHM developed by Karve et al. [95], which merges data and physical-driven methods to perform a mission-planning that minimizes damage while accounting for uncertainties, meaning that the DT is time-evolving and has a bi-directional connection. The DT of an aircraft implemented by Kapteyn et al. [96] points to an interesting route. The DT identifies the current damage scenario through a classification method and selects the proper surrogate model, making an in-formed decision to replan the maneuvers. Aivaliotis et al. [350] presents a methodology of DT implementation in predictive maintenance, including physics-based modeling, virtual sensors modeling, automatic calibration of model parameters. The implemented DT is used to deliver RUL predictions, as demonstrated in the case study of an industrial welding robot. Gardner et al. [313] implemented a DT in several stages. First, measured data was used to calibrate the physical model parameters. Then the outputs of this model are used as input of a GPR, which ameliorates the output prediction to care for uncertainties and nonmodeled physics using online adaptive sampling. The methodology is demonstrated in the model of the threestorey structure, as shown in Figure 17. Although the physical model used was linear, the DT could predict the nonlinear behavior resulting from the contact between column and bumper, which occurs at specific excitations. Besides that, as the DT is trained with lagged information and can make predictions steps ahead in time, it is conveniently used in the Active Control of the structure. Physics Guided Machine Learning The big drawback of ML models, especially when applied to physical problems, is the lack of the theoretical base and interpretability, raising skepticism with ML by part of the scientific community. Indeed, ML models may lead to physically inconsistent results, may fail to generalize to unseen scenarios, and rely on the availability of big data. However, Physical-driven mod- Residual Modeling In els (PDM) rely on hypotheses and simplifications of the real boundary conditions and struggle to account for uncertainties and historical and environmental conditions. Physics Guided Machine Learning (PGML) is an incipient but fast-growing research field which suggest merging physics-driven and data-driven model to take the best of both worlds, as shown in Table 1 [303,[352][353][354][355]. Recent reviews by Willard et al. [303], Karpatne et al. [352], Rai and Sahu [356] and Wang and Yu [353] classify and describe PGML works developed in different domains in the last years. In his thesis, Stender [357] develops a data science process for mechanical vibrations explicitly considering physics aspects in all steps of the process, namely obtain, pre-process, transform, model, and explain (OPTME). Some of the ML applied in SHM, Active Control, and surrogates from the last sections might be classified as PGML. The state-of-art of PGML is described here according to the configurations in which the physical knowledge is merged with the ML algorithm, as illustrated in Figure 18. Two categories can be defined: Physics leveraged by Machine Learning, in which ML models are used to improve the results from the simplified physical models; and Machine Learning leveraged by Physics, in which physical laws and constraints are intrinsically embedded into the ML, guiding it to have physical consistent results. Willard et al. [303] presented a similar categorization and provided a table associating each PGML configuration with an objective for which it may be appropriate. One way in which ML can leverage the results of physical simulations is when the results of the latter (and possibly its inputs) are used as ML input in a Inseries hybrid model configuration. The ML is trained to correct the results of the physical model by using the output of the real system as the target [313,358]. Similarly, in Residual Modelling, ML learns to model the PDM error, therefore the ML can correct the PDM output or classify its validity, as in [354,[359][360][361][362][363]. Finally, the ML can be used just as a sub-process of the PDM to evaluate one of its parameters [95,[364][365][366][367][368]. In the configurations that physics leverage ML, the structure is case-specific since it depends on the physical equations which govern the system. The most common approach is Physics Guided Loss [149,358,[369][370][371][372][373][374], in which the loss function contains penalization terms for non-physical predictions, e.g., an unexpected non-monotonic behavior. A thorough case of Physics Guided Loss is in Physics-Informed Neural Network (PINN) [375][376][377][378][379][380][381][382][383][384][385], in which the loss function is solely composed of the residue of a partial differential equation formulated in its derivative form. The equation variables are also the NN inputs, therefore the residue (loss function) is minimized by using automatic differentiation of NNs [196] and the equation is solved with no data needed. PINN also solves inverse problems, discovering equation parameters or constitutive relationships [386][387][388][389][390][391][392][393][394][395]. Another popular approach is Physics-guided architecture, in which a physical behavior is inserted somehow in the model architecture. At some level, this is done to insert sequential behavior in RNN, for example. Advantages of PGML provided by the datadriven model Advantages of PGML provided by the physics embedded on it Improve state-of-the-art physical models by comprising unknown relations; Improve ML predictions with domain knowledge and inductive bias; Solve inverse problems and lead to better parameter identification in the physical model; Provide physically consistent models; Handle noisy input; Reduce or end need of data; Reduce model order; Increase interpretability of ML model; Estimate aleatory and epistemic errors bounds; Improve ML generalization for unseen scenarios; Mitigate instability issues in time integrators; Reduce search space of ML algorithm Provide lagged predictions to active control; Improve long term-forecasting. Discover governing equations and unknown physics; Computationally cheap to evaluate. Expected physical behavior can be embedded through constraints in weighs and biases [396] or intermediate variables [372,397]. Zhang et al. [211] used LSTM and graph-based tensor differentiator to enforce physical constraints in the architecture and loss-function of metamodels of nonlinear structural systems. Besides improving the prediction accuracy and robustness, the PGML implemented in [211] models non-observable latent nonlinear state variables, such as the hysteretic meter, and nonlinear restoring force, delivering a more interpretable surrogate. Elements of physics-guided loss and physics-guided architecture are used in Neural ordinary differential equations (NODE) and Energy-Conserving Neural Networks (ECNN). In NODEs, explicit integration steps are performed in each layer of the NN as one step evaluation of a common ODE solver [398][399][400][401][402]. In ECNN, the structure of Lagrangian and Hamiltonian equations have been embedded into the NN construction to ensure an energy conservative behavior, as reviewed by Lutter and Peters [403] and implemented in different structures in [398,[404][405][406][407][408][409][410][411][412]. Willard et al. [303] survey presents other approaches, while Ba et al. [413] merged several PGML approaches to create an NN able to generalize well to different mechanical problems. Although it is a new topic, several recent works employed PGML, underlying its potential. However, most PGML research concentrate in other fields, e.g., fluids dynamics [22,360,414], lake modeling [358,369,370,372,415], climate modelling [416,417] and material science [21,418,419]. The spread of PGML techniques to other domains is a matter of time. For example, NODE and ECNN can be used as time integration solvers for modeling structural dynamics and acoustics and the physics information can help to solve the difficulties posed by the rough and discontinuous behavior of these systems. Recently, [420] created a PGML of a structural dynamic system using an RNN encoding the equation of motion. The PGML showed superior results even for scarce and noisy data, with better generalizability and robustness compared to purely data-driven. Besides that, it allowed time-saving by applying big time-steps without facing stability issues from the purely mechanistic approach. NODE and ECNN are suitable for introducing inductive biases in dynamic systems. Examples of how this approach can improve ML performance under high nonlinearities and discontinuities are presented in recent works which applied ECNNs to leverage data efficiency in non-smooth contact dynamics problems [214,421]. Yin et al. [422] introduced the APHYNITY framework to augment physical models with data information applied to dynamics forecasting. The Residual Modeling approach considers that the final response constitutes in both physics and ML models while ensuring that the ML response contribution is minimal so that the physics explains most of the prediction as possible. In addition, ECNNs are used in the APHYNITY framework to ensure physical consistency. The work presents results for reaction-diffusion equations, wave equations, and nonlinear damped pendulum, showing better accuracy than the simplified physical-based model and the solely data-driven approach. APHYNITY also leveraged the identification of physical parameters. Thus, PGML could ameliorate ML techniques used in the applications mentioned in this paper, increasing the coherence, interpretability, and reliability of ML models in structural dynamics and acoustic. Besides that, a burgeoning discussion explores how to use PGML to unveil unknown governing equations and physics intuition based on data [293-295, 370, 423-425]. Recently, Lai et al. [426] applied NODE to learn the governing structural dynamics and experimentally showed its effectiveness in a structure equipped with a negative stiffness device. Incipient research applied ECNN to learn the dynamics of the pendulum and multi-body problems [398,404,427,428]. Further research in the area might consider dynamic system with flexible elements. Research gaps and emerging opportunities This survey identified drawbacks and difficulties in the employment of ML in SD&V that should be addressed in future works. Based on the spotted research gaps and in the observed research trends in the integration of ML with other physical sciences, some future research opportunities that arise are: • Explore the many configurations of Physics-Guided ML in vibroacoustic problems to enforce physics consistency and improve accuracy, as carried out in other physical domains by [211,403]. • Ameliorate ML interpretability with sensitivity analysis [162] and PGML [211]. • Use ML to discover governing equations [293-295, 370, 423-425] and meta-materials [21,418] in SD&V problems. • Create a Digital Twin in the vibroacoustic domain by using entire life-cycle data, integrating multiple assets information, and performing real-time decision-making. • Investigate lifelong learning [429] applied to SHM, Active Control, and Product Design. • Overcome problems due to the lack of a dedicated database in SHM through transfer learning methods, similitude methods applied to SHM signals, and crowd-sourcing database [247]. • Use trained ML models to identify and mitigate the cause of failures. • The research exploring surrogate models in vibroacoustic problems is still incipient. To overcome the difficulties because of the rough behavior of the functions, one should investigate approaches such as learning from the function derivatives [430], global surrogates with local refinements, domain-decomposition methods [301], Adaptive Sampling [302], and Physics-Guided Machine Learning methods [303]. • Present a systematic study to clarify in which scenarios a surrogate model is justifiable in SD&V applications, considering the problem dimensionality, function smoothness, number of supporting points, loss of accuracy, and time gains. • Study methods to improve robustness of ML-based control of noise and vibration, as adversarial reward learning with reinforcement learning [431], NN with provable guarantees [432], and many others presented in [433]. Conclusion and Discussion This article presents a review of the intersection between the fields of Machine Learning (ML) and Structural Dynamics and Vibroacoustic (SD&V). First, the main ML methods are revised, paving the way for a broader and more advanced understanding of ML tools applied in the field. Subsequently, the reviewed literature reveals the ability of ML to perform critical tasks in SD&V, often surpassing physics-based methods in efficiency and accuracy. ML is notable for handling nonlinear analyses, uncertainties, and expensive/unmodeled physical problems. Three major application areas were identified: Structural Health Monitoring (SHM), Active Control of noise and vibration, and vibroacoustic Product Design. The capabilities of ML in extracting and recognizing fault patterns from monitored signals in the time and frequency domain make SHM the most developed and explored of these areas. Recurrently used data transformation and preprocessing methods are presented with examples of their significance in prediction accuracy. The prominence of Deep-learning methods in SHM is noteworthy since they automatically extract relevant features and reveal more complex patterns. SHM allows early failure detection and enables Remaining Useful Life predictions. Consequently, SHM avoids catastrophic failures and might implement a preventive main-tenance schedule to optimize uptime, maximizing the use of component lifetime. In addition, ML methods locate and classify damage and plan actions to minimize risks. Therefore, ML for SHM results in operational safety and economic savings, with simpler implementations and better efficiency than model-based methods. The correspondence between the areas of ML and Active Control of Vibration and Noise is natural since both have data-based optimization as a pillar. Thus, active control theory exploits ML tools by various means. System identification and reduced-order modeling techniques use ML to model the system under control. ML methods also support the study of the optimal location of sensors and actuators. Various control system approaches and configurations use ML methods to optimize controller parameters and policies, offline or online. Usually, ML applications in Active Control are justified and show better performance than traditional methods in nonlinear and complex scenarios. Although the memory and processing costs of ML algorithms are prohibitive in many control applications, the computer science advances occurring in parallel should make them affordable in more scenarios. Replacing costly high-fidelity simulations with fast ML-based surrogate models also enhances product design. The alleviation of the computational cost enables the evaluation of more designs, allowing uncertainty propagation and optimization analyses. Moreover, the statistical ground of ML methods makes them suitable to handle uncertainties, being that many have built-in sensitivity analysis to improve the comprehension of the system physics and ML interpretability. Besides that, ML surrogates trained with adaptive sampling require fewer observation points, while ensuring better accuracy near regions of interest, being widely used in optimizations. Despite the advantages, surrogates are not used in SD&V as often as in other domains. This occurs mainly because of the difficulties found by ML when fitting functions with rough behavior, recurrent in SD&V. However, methods such as domain subdivision, adaptive sampling, local refinements, and ML guided by physics are areas of potential study to overcome this issue. The wide range of ML application possibilities in SD&V and the advantages of this integration are explicit in this article, justifying the great interest in the area. Furthermore, the advances of successful recent research, likewise the analysis of ML uses in other fields of natural science, point to a path of opportunities. Connectivity and data management are the key aspects explored in the Digital Twins concept, which aims to benefit from the data and knowledge available to improve Product Lifecycle Management. Incorporating more theoretical and expert knowledge into ML models, as studied in Physics Guided Machine Learning models, is another trend subject of research, onde it leads to more interpretable models, less need for training data, and more physically consistent predictions. The drawbacks and difficulties identified in the application of ML in SHM, Active Control and Product design also show the open discussions and room for progress. The authors' perspectives for the upcoming research fields merging ML and SD&V were compiled in a list to be used as a guideline for future research. The extensive review presented reinforces that ML can strongly collaborate for the development of SD&V projects that are safer, more stable, controllable, robust, and optimized in design and operation. In conclusion, the union of data-driven and physics-driven methods can lead to a greater understanding of phenomena involved in SD&V analyses and open the way for further developments in the field. Acknowledgements This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 860243. The author would like to acknowledge all the Institutions and Partners involved within the LIVE-I project. Figure 2 : 2In the neural unit of a NN, the weighted sum of the inputs is added to a bias and goes through a nonlinear activation function, firing the neuron output (a). Supervised training workflow of a Fully Connected Feed-Forward Neural Network with backpropagation of the errors (b). Figure 3 : 3Representation of a Convolutional Neural Network. Figure 4 : 4Recurrent Neural Network rolled and unrolled, showing how the network uses the state from the previous time step as information to the current step and so forth (a). Long Short-Term Memory Network with forget, input and output gates forming the memory unit cell that avoids vanishing gradient problem (b). Figure 6 : 6Individual decision tree partitions the space minimizing the classification/regression impurity (a). Random Forest uses Bagging to ensemble decision trees and get the final prediction by majority or average voting (b). Gradient Boosting ensembles decision trees in sequence, so that the individual prediction of one decision tree is improved by the next and so on (c). Figure 7 : 7In Gaussian Process Regression, the prior distribution (left) is defined by kernels functions and the posterior distribution (right) is updated with information from the observation points using Bayesian inference. Figure 9 : 9Structural Health Monitoring workflow: in the classic approach, feature extraction and selection are handcrafted and followed by an ML method (a); if Deep-Learning is used, feature extraction and selection are performed automatically by the ML method (b). Figure 10 : 10Applications of Active Control of vibrations and noise powered by Machine Learning (ML). (a) Active Control workflow highlighting with red shadow the process that can use ML; (b) Scheme of ML training in System Identification problems; (c) Model Predictive Control (MPC) based on ML-model; (d) Controller design using Neural Network (NN)-based Inverse Model; (e) Active control using Adaptive NN filter to define control parameters; (f) ML control using heuristic methods to optimize control structure and parameters; (g) Reinforcement Learning (RL) applied to active control. Figure 11 : 11Steps to build a surrogate model: apply DOE to define supporting points location; sample results with the high-fidelity model; preprocess the data; train the surrogate model; predict new outputs using the surrogate model. Adaptive Learning is optional and may be applied to select new supporting points used to update the surrogate model in regions of interest or uncertainty. Figure 12 : 12(left) and with (right) Adaptive Learning Illustration of the Adaptive Sampling process. The upper image show a Gaussian Process (GP) surrogate model and its Acquisition function used to select new sampling points; the lower images illustrate the initial observation points (black) and the new samples points (red) for the cases with (right) and without (left) Adaptive Sampling. work focus in ML techniques, capable of model complex and nonlinear relations. Figure 13 : 13Uncertainty Quantification framework: the Uncertainty Propagation analysis propagates the inputs uncertainties through the system model to obtain the output distribution; and the Sensitivity Analysis evaluates the input contributions to the output uncertainties. Figure 14 : 14Surrogate based RBDO of a column under compression performed by Moustapha Figure 15 :Figure 16 : 1516Main components and interconnections which are the building blocks for creating a Digital Twin. Source: Gardner et al. [313]. Robust Product (Design) Maintenance Schedule (Usage) Real-Time Control (Usage) Risk Alarms and (Usage) Disposal/Reuse Plan (Retirement) Transfer Learning (Retirement/Design) Digital Twin framework: the data from the Physical Twin is processed by Physics and Data Driven methods by the Digital Twin, which supports optimized and robust decisions throughout the product life cycle. 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[ "SUBARU AND GEMINI OBSERVATIONS OF SS 433: NEW CONSTRAINT ON THE MASS OF THE COMPACT OBJECT 1", "SUBARU AND GEMINI OBSERVATIONS OF SS 433: NEW CONSTRAINT ON THE MASS OF THE COMPACT OBJECT 1" ]	[ "K Kubota ", "Y Ueda ", "S Fabrika ", "A Medvedev ", "E A Barsukova ", "O Sholukhova ", "V P Goranskij " ]	[]	[]	We present results of optical spectroscopic observations of the mass donor star in SS 433 with Subaru and Gemini, with an aim to best constrain the mass of the compact object. Subaru/FOCAS observations were performed on 4 nights of October 6-8 and 10, 2007, covering the orbital phase of φ = 0.96 − 0.26. We first calculate cross correlation function (CCF) of these spectra with that of the reference star HD 9233 in the wavelength range of 4740-4840 Å. This region is selected to avoid "strong" absorption lines accompanied with contaminating emission components, which most probably originate from the surroundings of the donor star, such as the wind and gas stream. The same analysis is applied to archive data of Gemini/GMOS taken at φ = 0.84 − 0.30 byHillwig & Gies (2008). From the Subaru and Gemini CCF results, the amplitude of radial velocity curve of the donor star is determined to be 58.3±3.8 km s −1 with a systemic velocity of 59.2±2.5 km s −1 . Together with the radial velocity curve of the compact object, we derive the mass of the donor star and compact object to be M O =12.4±1.9 M ⊙ and M X =4.3±0.6 M ⊙ , respectively. We conclude, however, that these values should be taken as upper limits. From the analysis of the averaged absorption line profiles of strong lines (mostly ions) and weak lines (mostly neutrals) observed with Subaru, we find evidence for heating effects from the compact object. Using a simple model, we find that the true radial velocity amplitude of the donor star could be as low as 40±5 km s −1 in order to produce the observed absorption-line profiles. Taking into account the heating of the donor star may lower the derived masses to M O = 10.4 +2.3 −1.9 M ⊙ and M X = 2.5 +0.7 −0.6 M ⊙ . Our final constraint, 1.9 M ⊙ ≤ M X ≤ 4.9 M ⊙ , indicates that the compact object in SS 433 is most likely a low mass black hole, although the possibility of a massive neutron star cannot be firmly excluded.	10.1088/0004-637x/709/2/1374	[ "https://arxiv.org/pdf/0912.2797v1.pdf" ]	119,261,950	0912.2797	cdf433de6ed9ac0e94cb233976506a6bcf6507e6	SUBARU AND GEMINI OBSERVATIONS OF SS 433: NEW CONSTRAINT ON THE MASS OF THE COMPACT OBJECT 1 15 Dec 2009 K Kubota Y Ueda S Fabrika A Medvedev E A Barsukova O Sholukhova V P Goranskij SUBARU AND GEMINI OBSERVATIONS OF SS 433: NEW CONSTRAINT ON THE MASS OF THE COMPACT OBJECT 1 15 Dec 2009accepted to ApJACCEPTED TO APJ Preprint typeset using L A T E X style emulateapj v. 03/07/07Subject headings: accretionaccretion disks -stars: individual (SS 433V1343 Aquilae) -supergiants - X-rays: binaries -techniques: spectroscopic We present results of optical spectroscopic observations of the mass donor star in SS 433 with Subaru and Gemini, with an aim to best constrain the mass of the compact object. Subaru/FOCAS observations were performed on 4 nights of October 6-8 and 10, 2007, covering the orbital phase of φ = 0.96 − 0.26. We first calculate cross correlation function (CCF) of these spectra with that of the reference star HD 9233 in the wavelength range of 4740-4840 Å. This region is selected to avoid "strong" absorption lines accompanied with contaminating emission components, which most probably originate from the surroundings of the donor star, such as the wind and gas stream. The same analysis is applied to archive data of Gemini/GMOS taken at φ = 0.84 − 0.30 byHillwig & Gies (2008). From the Subaru and Gemini CCF results, the amplitude of radial velocity curve of the donor star is determined to be 58.3±3.8 km s −1 with a systemic velocity of 59.2±2.5 km s −1 . Together with the radial velocity curve of the compact object, we derive the mass of the donor star and compact object to be M O =12.4±1.9 M ⊙ and M X =4.3±0.6 M ⊙ , respectively. We conclude, however, that these values should be taken as upper limits. From the analysis of the averaged absorption line profiles of strong lines (mostly ions) and weak lines (mostly neutrals) observed with Subaru, we find evidence for heating effects from the compact object. Using a simple model, we find that the true radial velocity amplitude of the donor star could be as low as 40±5 km s −1 in order to produce the observed absorption-line profiles. Taking into account the heating of the donor star may lower the derived masses to M O = 10.4 +2.3 −1.9 M ⊙ and M X = 2.5 +0.7 −0.6 M ⊙ . Our final constraint, 1.9 M ⊙ ≤ M X ≤ 4.9 M ⊙ , indicates that the compact object in SS 433 is most likely a low mass black hole, although the possibility of a massive neutron star cannot be firmly excluded. INTRODUCTION The microquasar SS 433 is a target of great interest in modern astronomy as a unique Galactic source that shows steady relativistic (v = 0.26c) jets (for a review see, e.g., Margon 1984;Fabrika 2004). It gives us an ideal opportunity to study the formation mechanism of astrophysical jets under supercritical mass accretion onto a compact object. Although SS 433 has been studied for about 30 years since its discovery, the identification of the compact object, the most fundamental issue to understand this system, has remained unsolved. In particular, the question whether it is a neutron star or a black hole is still open. A direct way to identify the compact object in a binary system is to determine its mass function by measuring the Doppler shifts of the stars due to the orbital motion. In the case of SS 433, both the inclination angle and the orbital period have been measured with high accuracy to i = 78.8 • (Margon & Anderson 1989) and P = 13.082 days (Goranskii et al. 1998), respectively. Hence, if the radial velocity amplitudes of the compact object and the donor star (or companion star) are known, their masses can be firmly determined. Most of the optical light from SS 433 is emitted from the compact object (i.e., from the jets and the accretion disk). Hence,it is relatively easy to measure the radial velocity of the compact object, K X , utilizing emission lines that can be observed even with moderate-size telescopes (hereafter, the subscripts X and O represent the compact object and the donor star, respectively). Using He II λ4686, Fabrika & Bychkova (1990) obtained K X = 175 ± 20 km s −1 . Their analysis is based on out-of-eclipse data with minimum inclination, i.e., when the disk is oriented maximally toward us. Gies et al. (2002b) used C II λ7231, 7236 blended lines and determined K X = 162 ± 29 km s −1 . The systemic velocity of the compact object, γ X , is different between Fabrika & Bychkova (1990) and Gies et al. (2002b), however. Gies et al. (2002b) argue that this is caused by the difference of the line emitting regions in the accretion disk. The measurement of the radial velocity of the donor star, K O , is more complicated, since the signal from the donor star, detectable as absorption lines from its surface, is hidden in the high flux from the compact object. Gies et al. (2002a) detected faint absorption lines in the blue part of the optical spectra of SS 433, showing Doppler shifts expected from the orbital motion of the donor star. They found that the spectrum resembles that of an A-type evolved star, as confirmed by Cherepashchuk et al. (2005). Latest results on the radial velocity of the donor star are reported by Hillwig & Gies (2008), who used data obtained at the Kitt Peak National Observatory (previously published in Hillwig et al. (2004)) and the Gemini telescope. They derived a donor star semi-amplitude of K O = 58.2 ± 3.1 km s −1 and a systemic velocity of γ O = 73 ± 2 km s −1 . Combining their results with K X = 168 ± 18 km s −1 , the average value of Fabrika & Bychkova (1990) and Gies et al. (2002b), they conclude the mass of the donor and compact object to be M O = 12.3 ± 3.3 M ⊙ and M X = 4.3 ± 0.8 M ⊙ , respectively. In the discussion below ( § 6.1), we review in detail the history of the radial velocity determinations of the compact object and the donor star. The selection of absorption lines originating from the photosphere of the donor star is a key issue for a reliable determination of the radial velocity of the donor star. It is also important to observe SS 433 when the disk is oriented maximally towards the observer and the outflowing material does not intersect with the line of sight. Charles et al. (2004), Barnes et al. (2006), and Clark et al. (2007) suggested that the spectral type of the donor star is an A-type supergiant. However, the wavelengths of the lines they observed did not follow the expected orbital velocity curve. Barnes et al. (2006) and Clark et al. (2007) pointed out that some absorption lines originate from the mass accretion flow onto the compact object, and not from the surface of the donor star. More importantly, the heating of the donor surface by the compact object may significantly affect the accurate measurement of the radial velocity . The intensity of the absorption lines is maximum in the central phase of the accretion-disk eclipse and it rapidly decreases when the compact object moves out from the eclipse (Hillwig et al. 2004). There are at least two types of absorption lines in the SS 433 spectra, (1) absorption lines with emission components, and (2) pure absorption lines. The former ones are usually stronger, showing typical "shell-type" line profiles, where the absorption line is located between two emission components. The pure absorption lines are weaker, and they are apparently not accompanied by emission features. Hereafter, we call these two types of absorption lines "strong" and "weak", respectively. The strong absorption lines have large oscillator strengths, and are usually formed in the upper regions of a star's atmosphere. In the case of SS 433, considering the heavy massloss rate of the donor ofṀ ∼ 10 −4 M ⊙ yr −1 (Fabrika 2004), and the underlying emission components, we expect that the strong absorption lines and associated emission lines may be formed in places not directly related to the donor photosphere. Hillwig et al. (2004) used a "highly rectified continuum" (see below) to smooth out the emission components of the strong absorption lines. Generally speaking, using the strong absorption lines without a detailed modeling of each spectral feature, has to be considered as risky. The heating of the donor star by the strong UV radiation from the supercritical accretion disk is known to be important. For a disk UV luminosity of ∼ 10 40 erg s −1 , the heated surface of the donor star in SS 433 is expected to have a temperature of ∼ 20000 K (Fabrika 2004), instead of only 9500 K in the absence of heating. (Hillwig et al. 2004;Cherepashchuk et al. 2005). Such a strong heating effect can produce the emission components observed in the strong absorption lines. It can also distort the radial velocities measuring from the weak absorption lines , since they are observed in the non-heated (or partly heated) regions of the donor surface, whose configuration changes with the orbital phase. Here, we present the most recent determination of the radial velocities and mass of the donor star and the compact object. In our analysis, we consider various effects as discussed above. Firstly, we carefully select absorption lines that originate from the donor's photosphere with minimum contamination by emission components, such as those from the wind, the gas stream, and the heated surface of the donor star. For this purpose, we use high quality optical spectroscopic data obtained with Subaru FOCAS in 2007 October during four nights. Archival data taken at the Gemini telescope published by Hillwig & Gies (2008) are analyzed as well. We present our best constraints on the mass of the compact object in SS 433, based on a model that accounts for the averaged absorption line profiles with consideration of the heating effects from the compact object. OBSERVATIONS AND DATA REDUCTION Subaru Data We observed SS 433 with the FOCAS instrument (Kashikawa et al. 2002) on the Subaru telescope on October 6-8 and 10, 2007. The jet of this source is known to precess with a period of 162.15 days. This epoch was chosen to observe the system in a particular phase. Firstly, the disk was oriented maximally towards us (ψ ≈ 0, where φ is precessional phase), which prevented the gas outflow from the accretion disk to intersect with the line of sight. Secondly, the orbital phase included the eclipse of the compact object by the donor star (φ ≈ 0, where φ is orbital phase). The spectra cover the orbital phase of 0.96 ≤ φ ≤ 0.26 and precession phase of 0.02 ≤ ψ ≤ 0.04, based on the orbital light curves presented below and the precession ephemeris given by Gies et al. (2002b). FOCAS was operated with the 0 ′′ .4 slit, VPH450 Grism, and 3×1 binning for the CCD chip. The sky condition was mostly photometric, except for October 10, 2007, with a typical seeing of ≈ 1 ′′ .0. The resulting spectra cover the wavelength range of 3750-5250 Å with a dispersion of 0.37 Å pixel −1 . Per night, we took 5 to 8 frames with 11-20 minutes exposure each. As reference stars we observed HD 9233 (spectral type A4 Iab), whose spectrum is similar to the donor star in SS 433 (Hillwig et al. 2004), HD 187982 (A1 Ia) and HD 332044 (B3 Ia). All spectra are reduced by the IRAF package (Tody 1993) in the usual way. We first subtract the bias, using averaged data of 20 bias frames. Then, to correct for the individual difference of each frame, we further subtract the remaining offsets in the over-scanned region from the exposed region. For the data of SS 433 with long exposures, the cosmic ray particle traces are removed using the lacos_spec task (van Dokkum 2001). The averaged flat image is created from 13 frames taken each night, which is then corrected approximately for the wavelength dependence of the flux to give a "normalized" flat frame. Finally, we divide the object frames by the normalized flat frame. Any remaining cosmic ray traces are removed manually in this stage. Accurate wavelength calibration is a critical point for our scientific goals. We utilize a Thorium-Argon lamp with the identify task on IRAF. The accuracy is confirmed to be better than 4 km s −1 by checking the interstellar absorption feature of Ca II λ3933.66. The flux calibration is made using the standard stars, BD+28D4211 (first night) and BD+40D4032 (second-fourth night). We ignore the effects of the slit loss, as we are mainly interested in the change of the relative flux. Finally, the atmospheric extinction is corrected. For the spectra of the fourth night, when the sky condition was not photometric, we correct the flux level relative to that of the third night by using the B-band magnitudes reported in § 2.3. To achieve the best signal-to-noise ratio, we add all the individual spectra produced in this way, except for those with low statistics or poor observing conditions, to obtain one spectrum for each night. We utilize 5, 7, 5 and 7 frames for the 1st, 2nd, 3rd and 4th night, respectively. We then produce "normalized" spectra, by dividing the original spectra by a smooth continuum. Gemini Data We analyze the archival data of SS 433 observed with the GMOS instrument on the Gemini telescope on June 7-13,2006 (UT). This data was also used by Hillwig & Gies (2008). The observations cover the orbital phase of φ = 0.84 − 0.30 at the precession phase of ψ = 0.02 − 0.06, when the accretion disk is oriented maximally toward us. In the epoch of the Gemini observations, SS 433 was found to be more active than during the Subaru observations. Starting from the original frames available from the Gemini web site 6 , we reduced the data using the ESO-MIDAS package (Warmels 1992), according to standard procedures. Like for the Subaru data, we produce an averaged spectrum for each night, "normalize" by a continuum fit, and apply a heliocentric correction to the wavelengths. Table 1 summarizes the observation dates and exposures of the Subaru and Gemini data analyzed in this paper. Photometric data We obtained photometric data of SS 433 in the standard B and V bands at the 1m telescope of the Special Astrophysical Observatory (SAO RAS) on October 2-11, 2007 with the CCD detector EEV 42-40. In addition, we obtained Subaru V-band images taken just before the spectral observation. To determined the B magnitudes of SS 433 during the Subaru spectroscopic observations, we used the V data and an interpolation of the (B-V) versus V relation, which is established very well in SS 433 (Goranskii et al. 1998). The final photometric accuracy is 0.01 and 0.02 magnitudes in the V and B bands, respectively, in direct observations. For the B-band magnitudes interpolated to the Subaru observation time, we obtained an accuracy of 0.03 mag. Figure 1 shows the V and B photometric light curves of SS 433. The SS 433 brightness out of eclipse is V = 14.0, indicating that the object was in "passive state" (Fabrika 2004). The middle eclipse took place between the first and the second night of the Subaru observations. The minimum is very well shaped and regular. Using these data and all previous photometric data of SS 433 we have, we update the orbital ephemeris. The main minimum is Min I = JD 2450023.76 ± 0.2 + (13.08227 ± 0.00008) × E. The new orbital period is slightly greater than the previous one (13.08211) published by Goranskii et al. (1998). This does not mean, however, that we detect a change of the period. The orbital ephemeris satisfy the previous photometric data as a solution with a constant period. The particular photometric eclipse displayed in Figure 1 has occurred 0.375 day after the predicted time from Goranskii et al. (1998) and 0.18 day after the time predicted by our new ephemeris. Such deviations are well-known from 6 http://www4.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/gsa/ the photometric behavior of SS 433. In the following, we assume JD 2454380.335 as the peak of the eclipse to calculate the orbital phase. For the analysis of the Gemini spectra, taken in June 2006, we apply the new orbital ephemeris as presented above. Figure 2 shows the flux-calibrated spectra of SS 433 in the 3750-5250 Å range obtained with the Subaru FOCAS instrument. Apparently, the continuum fluxes were small in the first and second night, corresponding to orbital phases close to the eclipse (φ = 0.956 and 0.034), and increased as the compact object move out of the eclipse. The most prominent features in these spectra are emission lines originating from the accretion disk and the gas stream (Fabrika 2004), including H I lines (from Hβ to H11), He I (the strongest are 5048 Å, 5015 Å, 4922 Å, 4713 Å, 4471 Å), He II (4686 Å), Fe II (the strongest is 5169 Å), and the C III / N III Bowen blend (≈ 4640 Å). The broad lines produced by relativistic jet were not strong during our Subaru observations. The Hβ − line is detected close to Hγ 0 line, from λ ∼ 4400 Å to λ ∼ 4270 Å due to the jet nutation motion. SPECTRAL FEATURES FROM THE DONOR STAR A large fraction of the optical emission of SS 433 originates from the compact object, i.e., from the accretion disk and the jet bases (Fabrika 2004). When the accretion disk move out from the eclipse, absorption lines from the donor star become very weak and hence careful analysis is required to study their features. To measure the orbital motion of the donor star ( § 4), we need to determine the cross-correlation function (CCF) with the spectrum of a reference star first. For this, we define three different spectral regions that are not affected by prominent emission lines from compact object; Region 1 (4490-4630 Å), Region 2 (4740-4840 Å), and Region 3 (4950-4990 Å). The normalized spectra of Region 1 and Regions 2-3, taken during first night, are plotted in Figures 3 and 4, respectively. Region 1 contains many strong absorption lines of Fe II and Ti II surrounded by emission components. By contrast, Region 2 is practically void of strong absorption lines and contains the Cr II λ4824 line with a weak emission component. Region 3, the narrowest one, contains the Fe I λ4957 line with a weak emission component. To make the absorption features clearly visible, we further divide the "normalized spectra" by a continuum function modeled by Legendre polynomials of order ≈15 in each region. We call the resultant spectra "highly rectified spectra". Figures 5 and 6 show the highly rectified spectra of SS 433 in Region 1 and Region 2-3, respectively, together with the normalized spectrum of the standard star HD 9233. HD 9233, spectral type A4 Iab, is known to show an absorption line spectrum similar to the donor star of SS 433 (Hillwig et al. 2004). For an easy comparison, all the spectra have been shifted into the rest frame by correcting for its radial velocity as determined by the CCF analysis in the next section. It is known that even during the eclipse the surroundings of the compact object (probably the accretion disk wind) contribute significantly to the total brightness of the system. The spectrum of HD 9233 is scaled to match the flux of the SS 433 spectra by multiplying with a factor of 0.36 (Hillwig et al. 2004). The deep absorption lines at 4500 Å, 4760 Å, 4780 Åand 4980 Å are due to interstellar absorptions (Hobbs et al. 2008). Apart from these, the spectra of HD 9233 and SS 433 contain the same set of absorption lines. These absorption features in the SS 433 spectra become deeper as the donor star hides the compact object (Hillwig et al. 2004), providing evidence that they originate from the donor star. The Gemini spectra observed for 7 nights are analyzed with the same procedure. Figure 7 shows the resulting highly rectified spectra in Regions 2 and 3 obtained with the Gemini GMOS. For comparison, we also plot the Subaru HD 9233 spectrum in the same figure. The wavelengths are corrected for its radial velocity, except for that of the seventh night where the absorption-line features are found to be extremely faint. We do not analyze the Gemini spectra of Region 1, since the strong absorption lines in the region appear as pure emission lines. This is probably due to the higher activity during the Gemini observations than during the Subaru observations. This make it impossible to use them for our CCF analysis. THE RADIAL VELOCITY OF THE DONOR STAR Cross Correlation Function Analysis We derive the radial velocity of the donor star by crosscorrelating the spectra of SS 433 with those of HD 9233 in each spectral region (Region 1, 2 or 3). Thereby, we assume the heating effects by the compact object at the surface of the donor star are negligible (see discussion in § 4.3). We further ignore wavelengths with strong interstellar absorption. We pay special attention to the determination of the radial velocity of HD 9233, which is known to be a radial velocity variable star (Hillwig & Gies 2008). By measuring the Doppler shifts of 13 non-blended absorption lines (Chentsov & Sarkisyan 2007) from the A4 Iab supergiant, we derive a systemic velocity of HD 9233 of γ HD 9233 = −44.2 ± 1.3 km s −1 . To verify our analysis, we also measure the radial velocity of another reference star HD 187982 (Type A1 Ia), which was observed on October 7, 2007 (i.e., during the second night of our Subaru observations). Our result is in good agreement with the literature value (Wilson 1953) within the error range. Figure 8 shows the radial velocity curve of the donor star in SS 433 from the CCF analysis of the Subaru spectra. The error bars in each point correspond to the "standard error" of the CCF (Fernie 1989). The amplitudes of the radial velocity curve are different between the three spectral regions used in our analysis, with Region 2 showing the largest amplitude and Region 1 showing the smallest one. In Figure 9, we display the CCF results for the Gemini spectra of SS 433 together with the Subaru spectrum of HD 9233. We confirm that the amplitude of the radial velocity curve derived from the Gemini spectra shows the same tendency as in the Subaru case (i.e., the amplitude derived from Region 2 is greater than the on from Region 3). Table 2 summarized our results on the radial velocities. We have demonstrated that the selection of absorption lines does affect the estimate of the radial velocity within our simple analysis. The differences in the amplitudes are related to the strength of the spectral features; the absorption lines in Region 1 are the deepest and have underlying emission components, while those in Region 2 are the weakest and mainly do not show emission components. We interpret that the strong absorption lines are more significantly affected by the emission from the wind, the gas stream, and the heated surface of the donor star, which decrease the amplitude of the radial velocity curve. From Region 1, we obtain K O = 24 ± 9 km s −1 with a systemic velocity of γ O = 52 ± 6 km s −1 . This value for K O is even smaller than the result from Hillwig et al. (2004), K O = 45 ± 6 km s −1 , who studied the same spectral region. This is probably due to different conditions of the sur-roundings between the two epochs of observations. In this context, the selection of "weak" lines is important to determine correctly the motion of the donor star, as for as the region responsible for the production of the absorption lines is constant over the orbital phase. Under this assumption, we can estimate the amplitude of the radial velocity curve by fitting the velocities with a Keplerian solution. Figure 10 shows the Subaru and Gemini results obtained from the CCF analysis of Region 2 together with the best-fit curve. We obtain a semi-amplitude of the radial velocity of K O = 58.3 ± 3.8 km s −1 and a systemic velocity of γ O = 59.2 ± 2.5 km s −1 . This value of K O is consistent with the result of Hillwig & Gies (2008) within the error bars. We note that Figure 10 may indicate a distortion of the donor's radial velocity curve in the orbital phases 0.0-0.15. Average Absorption Lines Profiles The high quality of the Subaru spectra allows us to study individual absorption lines. In the case of CCF analysis ( § 4.1), one compares two stars (i.e., of SS 433 and reference stars) with almost identical spectra. Hence, the complex blending and crowding of the absorption lines is not critical for the study. The analysis of the individual lines, however, depends strongly on such line blending effects and requires knowledge of the "laboratory" wavelengths of the blends. We carefully check the whole spectrum of SS 433 for the first and second night, when the system was in maximum eclipse. First, we select two groups of absorption lines: "strong" lines, which clearly show emission components, and "weak" lines with pure absorption line profiles. For the line identification, laboratory wavelengths and relative line strengths, we refer to the Atomic Spectra Database 7 and to the Atomic Line List 8 . We estimate the effective wavelengths of the blends by weighing the wavelengths of the individual lines according to their line strengths. We test each line or line blend to have the same radial velocity in the given Subaru night, allowing a difference of up to 20 km s −1 . The set of strong lines as well as the set of weak lines are both formed from lines that have same radial velocities within the first and the second night. The data of the other nights are not considered at the line selection. Obvious or resolved line blends are not included in the two line groups. Finally, we add the line profiles for each group in the normalized spectra in the radial-velocity space to create the strong and weak average line profiles. In the averaging procedure, we apply a weight of unity for a single line and a smaller weight for obvious (but non-resolved) blends. The line blending may distort the line profiles. The averaging procedure minimizes this distortion, because the line blending is only occasional, and because obvious and strong blends are excluded from the set of lines. To derive the average line profiles, we do not use the "highly rectified" spectra. Instead, we apply a linear continuum rectification to the final average line profiles in order to subtract the continuum levels near the lines. Both the "strong" and the "weak" absorption lines are considerably fainter than the strong emission lines in the SS 433 spectrum. Hence, in case where an absorption line is located near the wing of a strong and broad emission line, its local continuum becomes 7 National Institute of Standards and Technology; http://physics.nist.gov/PhysRefData/ASD/ 8 Department of Physics and Astronomy, University of Kentucky; http://www.pa.uky.edu/ ∼ peter/atomic/ not flat. Applying a linear function for the continuum rectification of the average spectra makes it possible to obtain a flat normalized continuum for the average line profiles. Using our B-band photometry results (Figure 1), we scale the average line profiles to a relative intensity unit, where the contribution from the donor star can be directly compared for the four nights. In the scaling process, we multiply the average profiles by coefficients depending on the system brightness and adopt the first Subaru night coefficient as 1.0. We . We find that nearly all strong absorption lines with emission components are generated by ions, while most of the weak, pure absorption lines are produced by neutral atoms. This implies that the ion absorption lines are partial formed in a more extended gas envelope of the donor star (i.e., the donor's wind). This wind could be a low velocity wind, since the donor overfills its Roche lobe. The emission components in the strong absorption lines may be partly formed in the gas stream (probably the mass flow, see below), which is best observed in hydrogen and He I emission lines (Crampton & Hutchings 1981;Fabrika et al. 1997;Fabrika 2004). The average strong and weak line profiles are shown in the right hand panels of Figures 11 and 12, respectively. The strong lines show clear evolution of their emission components over the four nights of observations. The absorption components shift with time to positive velocities with an amplitude of ≈ 40 km s −1 . The weak lines show the same systematic evolution with orbital phase, although its orbital shift in radial velocity is notably larger than that of the strong lines. The final signal-to-noise ratio in the average line profiles is very high. We detect several features in the line profiles that change from night to night in ways that are difficult to interpret. They would be due to line-emitting and -absorbing regions in the system having a complex structure. Additionally, emission components could partially fill the absorption profiles of the weak lines as well. In the following, we study the main features only, such as the line positions and intensities. We measure the absorption line positions for the average weak and strong line profiles and compare them with the CCF results. Note that the CCF analysis and the study of the average absorption line profiles are independent. We find that the weak absorption lines show the same behavior as the lines in the CCF analysis of Region 2, which is free from strong lines. The total radial velocity amplitude measured between the first and 4th spectrum is ≈ 63 km s −1 , which is very close to ≈ 67 km s −1 derived from the CCF analysis (Table 2). The absolute velocities are about the same as well, with the difference between the two methods being within 5 km s −1 . We further compare the behavior of the strong absorption lines with the CCF results of Region 1, which mainly contains strong lines with emission components. The difference is 10 km s −1 for the total radial velocity amplitude, although the systemic velocity is larger by 15 km s −1 for the average strong line profile than for the CCF analysis. Note that for the CCF analysis, we used the "highly rectified" spectra, where smoothing of the emission components may produce a systematic shift in the absorption line position. In the following, we study the average line profiles using a simple model of a close binary by taking into account the heating effects from the compact object. Figure 13 sketches the binary system with its main components. Since the UV luminosity of the compact object is as high as L UV ∼ 10 40 erg s −1 (Fabrika 2004), the compact object can heat the donor surface up to ∼ 20000 K during the 13day binary period. Although the detailed of geometry of the system is unknown, we know that the size of the optical continuum source is about the size of the donor-star or slightly exceed it, because optical eclipses are never total (about half of the continuum light remains present always). The donor star has an extended and dense envelope, which is probably due to a strong, low-velocity wind. Studies of X-ray eclipses in SS 433 (Filippova et al. 2006) revealed that the radius of the envelope that is opaque to X-rays exceeds the donor radius (or its Roche lobe radius) by 10-20 %. This proves the existence of a gas envelope ("coat") around the donor star, which can produce the emission components observed in the strong absorption line profiles. This gas envelope is also sketched in the figure. Study with a Simple Model Including Heating Effects by the Compact Object Model Description We construct a simple toy model of the system to study the heating effect on the absorption lines. Heating effects increase the observed radial velocity amplitude of absorption lines Cherepashchuk et al. 2005), because, due to the spin of the donor star, the non-heated side moves with a larger velocity than that of the center of mass. In out model, we consider three different regions of the donor surface ( Figure 13): Region I, which is not heated and produces absorption lines only, Region II, which is heated and produces emission lines only, and Region III, which is overheated and therefore does not produce any emission or absorption lines of elements with low ionization potentials (like, i.e., Ti II or Fe II). We assume that the system is synchronized and the donor star is a sphere with a volume equal to that of the Roche lobe for a given mass ratio q = M X /M O . The orbital inclination of the system is adopted as 79 • (Fabrika 2004). The semiamplitude of the compact object is set to K X = 160 km s −1 , as derived late in this paper ($ 5). The semi-amplitude of the donor star K O (or the mass ratio q) is taken as free parameter in this model. The donor surface is divided into 100 grid cells both in longitude and latitude, with each grid cell producing a Gaussian absorption line profile in the non-heated region or an emission line profile in the heated region. A Gaussian line width of FWHM = 5 km s −1 is adopted. For absorption lines formed at the donor's surface which is not exposed to the UV source (Region I in Figure 13), the Gaussian line intensity is normalized to unity. For emission lines formed in the heated region of the surface (Region II in Figure 13), we calculate the normalization depending on the heating parameters (see below). We adopt a quadratic limb-darkening law (Kallrath & Milone 1999) with x = y = 1 for absorption lines only. Using this limb-darkening law, we can easily fit the observed absorption lines. Note, however, that determining the correct limb-darkening law for this supergiant with heavy mass loss and heating is a truly complex task. In the case of emission lines, we do not account for a limb-darkening effects, since one expects an inverse temperature gradient in the emission region. The gas envelope is modeled in a similar way than the donor surface, with the difference that it produces emission lines in the heated region (Region II), but no emission or absorption lines in the other two regions. The radial extent is set to 10 % of the donor radius. It has 10 individual segments in the radial direction. We assume that the emitting gas in the envelope rotates with the same velocity as the donor surface and moves in radial direction with the escape velocity V esc . The UV source is spherical and has the same size as the donor star in the model (Figure 13). Each point of the donor surface sees all the points of the extended source visible from it. We calculate the angle of incidence of the UV radiation in each point of the donor surface. Thus, we specify only geometrical properties of the source. We suggest that the donor's regions exposed to the UV radiation of the source produce emission lines. This is expected if the temperature gradient in the donor's atmosphere is inverse. Antokhina et al. (2005) confirmed this behavior in their X-ray heating model of a low-mass X-ray binary. UV radiation is subject to strong extinction and may therefore not penetrate deeply into the donor atmosphere. Considering the high luminosity (L UV ∼ 10 40 erg/s) of the accretion disk in SS 433, however, we suggest that the UV radiation can indeed produce the inverse temperature gradient in the atmosphere and that the gas in the donor's wind may be ionized down to the photosphere. We introduce two empirical parameters, the heating coefficients γ phot and γ env , which define the intensities of the emission lines formed in the heated regions of the donor surface and in the envelope, respectively. From the relative fluxes of the incident radiation in each point exposed to the UV source, we calculate the expected emission-line components. We then determine the values of γ phot and γ env by comparing the line profiles between the model and the data observed in all four nights. The values are independent for the strong and the weak lines (in particular, γ env = 0 for the weak lines). The obtained heating coefficients are relative ones only, and cannot be used to estimate the heating effects physically. To produce a reasonable agreement between the observed and the modeled line profiles, γ phot has to be 2-3 times larger for the weak lines than for the strong lines. The weak lines do not require an additional emission component formed in the envelope. In case of the strong lines, this component (γ env ) is necessary, since it produces emission line wings which are notably broader than the photospheric line profiles. By varying other parameters of the model, such as the mass ratio, we can infer the required amount of heating that is necessary to account for the observed line profiles. Finally, we integrate the line profiles from individual regions of the donor surface and of the envelope which are visible from the observer during the orbital phases of the Subaru observations (indicated in Figure 13). The final line profiles are convolved with the instrumental response, which is derived from single line measurements in the comparison-lamp spectra. The normalization of the final absorption lines is determined from the data of the first night only. These normalization coefficients are kept constant for all four nights of the observations. The final line profiles are notably broader (FWHM ∼ 100 km s −1 ) than our Gaussian lines formed in the individual surface cells and they do not depend on the adopted line width of the individual lines (when it is less than ∼ 20 km/s). This simple approach is sufficient for the following study, because we do not compare in detail the observed and calculated line profiles, but investigate the main features of the heating effects only. Several effects are not taken into account here. In reality, heating by the UV radiation is a complex process. For example, the UV photons may not reach the donor surface because of strong absorption in the wind. The UV absorption can heat the gas deeply down to the surface, however. We further assume isotropy of the UV source, which is probably are oversimplification. For instance, the thick outer rim of the disk (Filippova et al. 2006) may cast an extended shadow. This effect is most important in the fourth Subaru night, since, at precession phases ψ ∼ 0, the donor star crosses the disk plane at φ ≈0.25 and 0.75. The assumption of isotropic UV radiation will lead to an overestimate of the heating effects in this case. Since the environment of the compact object (i.e., the wind, the jet bases and the disk structure) are basically unknown, a complex modeling of the heating is problematic. Thus, in this paper, we illustrate how the heating distorts the radial velocities of the absorption lines in SS 433 for a better understanding of the principal difference between the strong and weak lines. The gas stream is a strong source of hydrogen and He I emission lines. Crampton & Hutchings (1981) showed that the radial velocity curves have the largest redshifts at orbital phases ∼ 0 close to the inferior conjunction of the donor star; their orbital phases lag behind the accretiondisk phase by 0.2-0.25. Later studies of the hydrogen and He I emission lines (Kopylov et al. 1989;Fabrika et al. 1997;Goranskii et al. 1997) revealed that Hβ and He I radial velocities show the largest redshifts at orbital phases 0.9-0.95 and 0.85-0.9, respectively. These allows also detected a partial eclipse in the hydrogen emission lines at orbital phases 0.1-0.2. This indicates that the He I and hydrogen emission lines in the SS 433 spectra are formed in the gas stream onto the accretion disk. Goranskii et al. (1997) and Gies et al. (2002b) also discussed whether the behavior of the hydrogen and He I radial velocity curves may result from an evacuation of the disk wind surrounding the donor star, which leads to anisotropic wind and the observed radial velocity curves. This cannot explain, however, both the partial eclipses and the differences between the hydrogen and He I radial velocity curves. If hydrogen and He I lines are formed in the anisotropic wind, their radial velocity amplitudes have to be greater than that of the accretion disk (traced by He II line), since the accretion disk powers the wind. This does not agree with the observed radial velocity amplitudes. We therefore conclude that the hydrogen and He I emission lines are formed in the gas stream, although a fraction of this emission may be formed in the disk wind as well. In any case, the emission region must be extended. A probable location of the gas stream region is shown in Figure 13. It is noteworthy that a fraction of the emission of the strong absorption lines (Figures 11) may be formed at the same location as the hydrogen and He I lines. During all for nights, we detected an additional red emission line component in the strong and even in the weak absorption lines, which we fail to reproduced with our model. We ascribe this component to the gas stream, which contributes stronger to the red emission in the strong lines than in the week lines. We do not model any probable eclipses of the gas stream (Kopylov et al. 1989;Fabrika et al. 1997;Goranskii et al. 1997), which may change the intensity of the red emission components. To model this additional red emission, we decide to follow the radial velocity curves of hydrogen and He I emission lines by Fabrika et al. (1997), where the lines show the largest redshifts (100-150 km s −1 ) at the orbital phases ∼0.85-0.95 and the velocity decreases gradually with the orbital phase. Finally, the gas stream is modeled to produce Gaussian emission lines, whose parameters are tuned to reproduce the observed line profiles. Comparison with the Data The left hand panels of Figures 11 and 12 show the best-fit models of the strong and weak line profiles, respectively, to be compared with the observed ones in the right panels. These line profiles are modeled using a radial velocity amplitude of the compact object of K X = 160 km s −1 and the real radial velocity amplitude of the donor star K O = 40 km s −1 (i.e., q = 0.25). The relative heating coefficient γ phot for the weak lines is twice as large than that of the strong lines, and the wind velocity is set to V esc = 260 km s −1 (for the strong lines only). We see that our model can reproduce the overall features of the observations. For the strong absorption lines, the emission components evolve in agreement with the idea that the gas envelope, which rotates with the same velocity as the donor star, is heated by the compact object. For the weak absorption lines, emission lines originating from the heated donor surface are required, while those from the envelope are not. These emission lines are seen as a low intensity emission bump near the absorption line (as for the strong lines). This emission component alters the position and intensity of the absorption lines. The absorption lines move in accordance with the orbital phase, and their velocity amplitudes are smaller for the strong lines than for the weak lines. The absorption line intensities generally decrease with orbital phase because of the heating (note that the observed spectra are scaled from the photometric data in order to keep the non-illuminated continuum radiation of the donor constant). Naturally, the observed absorption line profiles are more complex than those produced by our simple model. For example, the weak absorption line profile of the first night (solid line in Figure 12) shows either an additional absorption in its red wing or an additional emission component. Such a feature is not produced in our model, since we do not take into account a possible absorption of the continuum radiation from the compact object in the donor's wind. This effect might be important during the first and second night (see Figure 13). Additional absorption features may be present for the second and third night in the blue wing of the strong and weak absorption lines (Figures 11, 12). During these orbital phases, the speculated wind from the donor, which is seen projected onto the strong continuum source, is directed toward us because of the stellar rotation. This feature may also produce the distortion of radial velocity curve observed in Figure 10. We conclude that the model reproduces both the intensity and the radial velocity variations of the emission components and the absorption lines. The modeled line profiles are influenced mostly by the following parameters; the real velocity amplitude of the donor star K O , and the heating efficiency coefficients γ phot and γ env . As mentioned above, the best-fit value is K O ≈ 40 km s −1 (i.e., q ≈ 0.25 with K X ≈ 160 km s −1 ). This is 18 km s −1 or 30 % less than that measured in our CCF analysis (58.3±3.8 km s −1 ). It is impossible to produce the average line profiles in our model for K O < 35 km s −1 . Further, for velocity amplitude of the compact object between 170 and 150 km s −1 , we obtain values of K O between 35 and 45 km s −1 . We thus conclude that the donor's real radial velocity amplitude is K O = 40 ± 5 km s −1 , based on our simple model. From the set of models reproducing the observed averaged line profiles, we find that the size of the overheated region (Region III in Figure 13), modeled as a cone with origin at angle of the donor, has a half-opening angle of ≈15-20 • for the strong lines and ≈20-25 • for the weak lines, respectively. The wind velocity of the donor is V esc ∼ 260 km s −1 . The mass flow, which produces parts of the red emission components in the absorption lines, has a radial velocity decreasing from 160 km s −1 in the first night to 90 km s −1 in the last night for the strong absorption lines, and from 100 km s −1 to 80 km s −1 for the weak also lines, respectively. The emission line components formed in the gas stream are broad with a FWHM of 140-200 km s −1 . The intensity of this component is two times weaker in the weak absorption lines than in the strong absorption lines. The origin and formation of the gas stream (mainly observed in the hydrogen and He I emission lines) were discussed in previous papers (Crampton & Hutchings 1981;Fabrika et al. 1997;Fabrika 2004). It is important to note that the introduction of the red emission components in the modeled line profiles is necessary, although its formation remains unclear. All these parameters do not change, however, the modeled absorption line profiles so strongly as the real radial velocity amplitude of the donor and heating efficiency coefficients do. Finally, in Figure 14, we present the radial velocity curves derived from the observed average line profiles, simply based on the position of the absorption line minimum. For comparison, the values from our best-fit models (K O = 40 km s −1 and K X = 160 km s −1 ) are displayed as well. Clearly, the heating model reproduces well the observed radial velocities. Note that these velocities are apparent ones and differ from the real radial velocities of the donor star considered in the model. RADIAL VELOCITY OF THE COMPACT OBJECT To constrain the radial velocity of the compact object, we analyze the He II λ4686 emission line, the brightest line known to originate from the compact object. Figure 15 shows the line corresponding profiles. The flux level is normalized to that of the first night (φ=0.956), calibrated using the B-band magnitudes (Figure 1). Although the line profile is complex, it is obvious that the line center moves from the red to the blue with increasing orbital phase. To determine the radial velocity of the He II line, we calculate the center of gravity above a certain threshold in order to discard the broad wings. The line wings are stronger in the red than in the blue. We estimate the error of the so-derived radial velocities by changing the flux thresholds (upper and lower) used in the calculation of the center of gravity. The results are summarized in Table 3. In Figure 16, we show the radial velocities of the compact object as measured from the He II line. A large velocity of ≈ 150 km s −1 is required to fit them with a Keplerian velocity curve. This is unlikely and probably due to the fact that the He II line was significantly affected by the eclipse during the first three nights, as observed in previous studies (Fabrika & Bychkova 1990). Indeed, the effects of the eclipse are clearly seen in Figure 15, where the He II line profile changed notably across the eclipse of the line emitting region by the donor star. If we use the data of the fourth night only, together with a fixed systemic velocity of γ O = 59.2 km s −1 , the same value of γ x for the donor star, we obtain K X =159±7 km s −1 . This value is consistent with previous results. Fabrika & Bychkova (1990) reported K X = 175 ± 20 km s −1 , using the He II λ4686 line observed in the precession phase of 0.9 ≤ ψ ≤ 0.1, but outside of the eclipse. Fabrika et al. (1997) constructed precessional and orbital radial velocity curves of the He II line using additional spectral data. They found K X = 176 ± 15 km s −1 for the same precessional phase of 0.9 ≤ ψ ≤ 0.1, while Gies et al. (2002b) used C II λ7231, 7236 blended lines and derived K X = 162 ± 29 km s −1 . For consistency, we adopt K X =168±10 km s −1 in this paper, the average between these three studies and our own estimate of the K X . Note that Hillwig et al. (2004) and Hillwig & Gies (2008) adopted the same value, 168±18 km s −1 , as the average between two studies, Gies et al. (2002b) and Fabrika & Bychkova (1990). 6. DISCUSSION Review of the Dynamical Mass Determination of SS 433 In this subsection, we review recent work on the dynamical determination of the mass function of SS 433 from measurements of K X and K O , which we compare with our results. 1. Gies et al. (2002b) interpreted that an absorption feature in the strong emission line of He I λ6678 may originate from the donor star, using the spectra taken at the KPNO 0.9 m telescope. (Fabrika & Bychkova 1990), they determined M X ≈ 18 M ⊙ and M O ≈24 M ⊙ . They noted that this radial velocity semi-amplitude is probably an upper limit, because the heating of the donor star increases the observed amplitude. 5. Barnes et al. (2006) observed SS 433 with the Calar Alto Observatory 3.5 m telescope, and the Observatory del Roque de Los Muchachos 2.5 m and 4.2 m telescope. They cross-correlated the SS 433 spectra in the 4500-4630 Å range with those of HD 9233, using the radial velocity given in Hillwig et al. (2004). They obtained K O =69±4 km s −1 and γ O =−53±3 km s −1 . But, their observations were performed when the accretion disk was close to an edge-on orientation, where the outflowing material produces strong absorption lines. Note that they assumed a velocity for HD 9233 of −34 km s −1 (Hillwig et al. 2004). Since this star is velocity variable, its velocity might have been different at the time of the observations, and the systemic velocity γ 0 has an additional factor of uncertainty. 6. Hillwig & Gies (2008) 6.2. Constraints on the Mass of the Compact Object Once the amplitudes of the radial velocity curves of both the donor star and the compact object are known, one can deduce the mass of each component. By fitting the radial velocity curve of the donor star obtained from the CCF analysis with a Keplerian curve (i.e., without consideration for the heating effects), we derived K O = 58.3 ± 3.8 km s −1 , which is consistent with the value obtained by Hillwig & Gies (2008). This is not surprising, since we use the same Gemini data for our analysis in addition to the Subaru data, although we restrict the wavelength range to Region 2 only, in order to avoid systematic effects from emission components in the absorption line profiles. We adopt the amplitude of the radial velocity of the compact object of K X =168±10 km s −1 and conclude the mass of the donor star and compact object to be M O = 12.4 ± 1.9 M ⊙ and M X = 4.3 ± 0.6 M ⊙ , respectively. The corresponding mass ratio is q = 0.35. Again, these values should be taken as upper limits only if we consider the heating effect, as discussed in Section 4.3. By taking into account the heating effects, we derive lower limits on the masses, since we assumed a real radial velocity amplitude of the donor star of K O = 40 ± 5 km s −1 . This leads to lower masses of M O = 10.4 +2.3 −1.9 M ⊙ and M X = 2.5 +0.7 −0.6 M ⊙ . We thus conclude that the compact object in SS 433 is most likely a low mass black hole. However, the possibility of a massive neutron star cannot be firmly ruled out at present, given the fact that a neutron stars could have masses of up to 3 M ⊙ as inferred from both theory (Lattimer & Prakash 2007) and observations (Freire et al. 2008). 7. CONCLUSIONS 1. To study the radial velocity curve of the mass donor star in SS 433, we obtained high quality optical spectra with Subaru/FOCAS, covering the orbital phase of φ = 0.96 − 0.26. We combine these observations with the Gemini data reported by Hillwig & Gies (2008) to analyze the largest set of the best quality spectra observed right now from this source. This allows us to study in detail the behavior of the "weak" absorption lines from the donor surface, which are least affected by the emission components from the surroundings of the donor star. 2. We demonstrate that the selection of the spectral region is critical for the cross correlation function (CCF) analysis. We adopt the 4740-4840 Å range (Region 2) for this study, where only "weak" absorption lines from the surface of the donor star are present. If we instead use the 4490-4630 Å range (Region 1), which contains many "strong" absorption lines associated with emission components, we obtain a significantly smaller velocity amplitude than Region 2. 3. From the Subaru and Gemini CCF results (Region 2), we determine the amplitude of the radial velocity curve of the donor star to be 58.3±3.8 km s −1 . Together with the radial velocity of the compact object, 168±10 km s −1 , we derive masses of the donor star and the compact object of M O = 12.4 ± 1.9 M ⊙ and M X = 4.3 ± 0.6 M ⊙ , respectively. These values should be taken as upper limits, because of the heating of the donor star by the compact object. 4. We calculated average absorption line profiles for the strong and weak lines separately, each line using 8 individual lines. The position of the line centers of the average absorption line agree within our CCF results. Prominent emission components are observed in the strong lines, indicating that the heating effects are important for a proper interpretation. 5. We construct a simple model where we take into account the UV heating effects on the donor star surface and on its envelope, and where also consider the emission from the gas stream. The model reproduces well the emission components and absorption lines in the average line profiles, both in intensity and radial velocity variations. These results indicate that the heating could have a significant impact on the estimate of the real ra-dial velocity of the donor star, which may be as low as K O = 40 ± 5 km s −1 . We then estimate the masses of the components as M O = 10.4 +2.3 −1.9 M ⊙ and M X = 2.5 +0.7 −0.6 M ⊙ . 6. The final constraints for the compact object mass are 1.9 M ⊙ ≤ M X ≤ 4.9 M ⊙ , where the lower and upper limits are inferred from the modeling of the average absorption line profiles and from the CCF analysis, respectively. We conclude that the compact object in SS 433 is most likely a low mass black hole, although the possibility of a massive neutron star cannot be firmly excluded. We thank the observatory staff of the Subaru telescope, in particular the support astronomer Dr. Takashi Hattori, for their help on our observation run, the scheduling, and for useful advice on the reduction of the FOCAS data. We also thank Dr. E. Chentsov for the very useful comments, and the referee, Prof. Douglas Gies, for his careful reading of the manuscript that helped to improve the clarity of the paper. We are grateful to Dr. Dominikus Heinzeller, who gave extensive suggestions to the manuscript. KK and YU greatly appreciate the warm hospitality of staffs in the SAO during their visits in 2007 and 2008. This work was partly supported by the Grant-in-Aid for JSPS Fellows for young researchers (KK), by the Grantin-Aid for the Global COE Program "The Next Generation of Physics, Spun from Universality and Emergence" from from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and by Russian RFBR grants 07-02-00909, 07-02-00630, and 09-02-00163. 5.-Rest frame spectra of SS 433 (highly rectified) and HD 9233 (normalized) in Region 1 obtained with Subaru FOCAS. The SS 433 spectra are shifted by offsets of 0.4, 0.3, 0.2, and 0.1 for the first, second, third, and fourth night, respectively. Numbers on the right side indicate the corresponding orbital phases φ. The flux level of HD 9233 spectrum is reduced by a factor of 0.36. The absorption feature near 4501.79 Å is of interstellar origin. 13.-Sketch of the binary system SS 433 (not to scale). The system consists of a mass donor star with an extended gas envelope, a UV heating source (surrounding the compact object), and a gas stream. The orbital motion and rotation of the donor are shown. We distinguish between three different regions of the donor surface, (I) the non-heated region from which we observe absorption lines only, (II) the heated region from which we observe emission lines only, and (III) the overheated region which does not emit any spectral lines. The extended envelope (i.e., the wind) produces emission lines only in Region II and in those parts of Region I that are exposed to the UV source. The orbital phases seen by the observer in the four Subaru nights are indicated. km s −1 as determined from the CCF analysis for the donor star. The dotted, sinusoidal curve represents the best-fit obtained using the data of the fourth night (K X =159±7 km s −1 ). FIG. 1 .FIGFIG. 3 .FIG. 4 . 134-Optical photometry of SS 433 during the Subaru observations. Crosses show data points interpolated to the time of the spectral observations. Variations larger than 0.01 in V and 0.02 in B represent real photometric activity in SS 433. . 2.-Optical spectra of SS 433 observed with Subaru FOCAS covering the 3750-5250 Å range. From top to bottom, the curves represent the spectra taken on the fourth (φ = 0.262), third (φ = 0.115), first (φ = 0.956), and second night (φ = 0.034). -"Normalized" spectrum of Region 1 taken during the first night of the Subaru observations. It contains several strong absorption lines with emission components (cf. the highly rectified spectrum of Region 1 inFigure 5). The strong absorption line at λ4501 is a diffuse interstellar band (DIB) blended with a Ti II line. -"Normalized" spectrum of Regions 2 and 3 taken on the first night of the Subaru observations. FIGFIG . 8.-Radial velocity curve of the donor star in SS 433 obtained by the CCF analysis of the Subaru data. squares indicate the results from Region 1. 9.-Radial velocity curve of the donor star in SS 433 obtained by the CCF analysis of the Gemini data. Filled circles and triangles indicate the results from Region 2 and from Region 3, respectively. FIG. 10.-Radial velocity curve of the donor star in SS 433 complied from both Subaru and Gemini results of the CCF analysis in Region 2. Filled circles and triangles represent Subaru and Gemini results, respectively. The best-fit Keplerian solution is displayed by the solid curve. Dashed lines correspond to the 1σ error. . 11.-Observed average absorption-line profiles for the "strong" lines with emission components (right) and its best-fit model (left). From the first to the fourth Subaru night, the profiles are denoted by solid, dashed, dash-dotted and dotted lines. FIG. 12.-Observed average absorption-line profiles for the "weak" lines without notable emission components (right) and its best-fit model (left). From the first to the fourth Subaru night, the profiles are denoted by solid, dashed, dash-dotted and dotted lines.FIG. FIG . 14.-Absorption-line radial velocity curves derived from the observed average line profiles (filled symbols) and those measured from the best-fit models using the same method as for the observed ones (open symbols). Circles represent the strong lines, and squares the weak lines, respectively. K O = 40 km s −1 and K X = 160 km s −1 are assumed. The heating model well reproduces the observed radial velocities. . 15.-Line profiles of He II λ4686. Numbers shown in the upper right corner indicate the orbital phases. FIG. 16.-Radial velocity curve of the compact object measured from the He II line. The horizontal line corresponds to the systemic velocity of γ O =59.2 include 8 individual lines in the average strong line profile (Mg II λ4481.21, Ti II+Fe II λ4549.63, Ti I+Fe II λ4555.49, Fe II λ4576.39, Ti II+Fe II λ4583.41, P II+Cr II λ4823.84, Si II λ5041.03, Fe I λ5226.86) and 8 individual lines in the average weak line profile (Cr I λ4161.42, Fe I λ4271.76, Ti II λ4290.22, Ti I λ4325.13, Fe I λ4528.87, λ4983.85, λ5125.11, Mg I λ5183.60) FIG. 6.-Same as Figure 5, but for Regions 2 and 3. The absorption features near 4762.61 Å, 4780.02 Å, and 4963.88 Å are of interstellar origin. FIG. 7.-Highly rectified spectra of SS 433 obtained the Gemini GMOS in Region 2 and 3, together with the normalized spectrum of HD 9233 obtained with Subaru FOCAS. The absorption features near 4762.61 Å, 4780.02 Å, and 4963.88 Å are of interstellar origin.0.9 1 1.1 1.2 1.3 1.4 4750 4800 Normalized Intensity Wavelength (Angstrom) C I 4771.720 Ti II 4798.535 Ti II 4805.105 C I 4812.84 Cr II 4824.13 4960 4980 HD9233 0.262 0.115 0.034 0.956 Fe I 4957.603 0.9 1 1.1 1.2 1.3 1.4 4750 4800 Normalized Intensity Wavelength (Angstrom) C I 4771.720 Ti II 4798.535 Ti II 4805.105 C I 4812.84 Cr II 4824.13 4960 4980 HD9233 0.300 0.223 0.145 0.069 Fe I 4957.603 0.988 0.918 0.838 TABLE 1 1 . E A Antokhina, A M Cherepashchuk, V V Shimanskii, Astronomy Reports. 49109Antokhina, E. A., Cherepashchuk, A. M., & Shimanskii, V. V. 2005, Astronomy Reports, 49, 109 . A D Barnes, J Casares, P A Charles, J S Clark, R Cornelisse, C Knigge, D Steeghs, MNRAS. 365296Barnes, A. D., Casares, J., Charles, P. A., Clark, J. 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[ "The Hyperfine Splitting in Charmonium: Lattice Computations using the Wilson and Clover Fermion Actions UKQCD Collaboration", "The Hyperfine Splitting in Charmonium: Lattice Computations using the Wilson and Clover Fermion Actions UKQCD Collaboration" ]	[ "C R Allton \nPhysics Department\nDepartment of Physics\nThe University\nSO9 5NHSouthamptonUK\n", "C T Sachrajda \nPhysics Department\nDepartment of Physics\nThe University\nSO9 5NHSouthamptonUK\n", "S P Booth \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "K C Bowler \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "D S Henty \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "R D Kenway \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "B J Pendleton \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "D G Richards \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "J N Simone \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "A D Simpson \nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n", "C Michael \nDAMTP\nUniversity of Liverpool\nL69 3BXLiverpoolUK\n", "P W Stephenson \nDAMTP\nUniversity of Liverpool\nL69 3BXLiverpoolUK\n" ]	[ "Physics Department\nDepartment of Physics\nThe University\nSO9 5NHSouthamptonUK", "Physics Department\nDepartment of Physics\nThe University\nSO9 5NHSouthamptonUK", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "The University of Edinburgh\nEH9 3JZEdinburghScotland", "DAMTP\nUniversity of Liverpool\nL69 3BXLiverpoolUK", "DAMTP\nUniversity of Liverpool\nL69 3BXLiverpoolUK" ]	[]	We compute the hyperfine splitting m J/ψ − m ηc on the lattice, using both the Wilson and O(a)-improved (clover) actions for quenched quarks. The computations are performed on a 24 3 ×48 lattice at β = 6.2, using the same set of 18 gluon configurations for both fermion actions. We find that the splitting is 1.83 + 13− 15 times larger with the clover action than with the Wilson action, demonstrating the sensitivity of the spinsplitting to the magnetic moment term which is present in the clover action. However, even with the clover action the result is less than half of the physical mass-splitting. We also compute the decay constants f ηc and f −1 J/ψ , both of which are considerably larger when computed using the clover action than with the Wilson action. For example for the ratio f −1 J/ψ /f −1 ρ we find 0.32 + 1 − 2 with the Wilson action and 0.48 ± 3 with the clover action (the physical value is 0.44(2)).	10.1016/0370-2693(92)91195-f	[ "https://export.arxiv.org/pdf/hep-lat/9208018v1.pdf" ]	15,823,104	hep-lat/9208018	4ebc0a87593c78eed6de185fb5c7e9c1640802b5	The Hyperfine Splitting in Charmonium: Lattice Computations using the Wilson and Clover Fermion Actions UKQCD Collaboration 21 Aug 1992 C R Allton Physics Department Department of Physics The University SO9 5NHSouthamptonUK C T Sachrajda Physics Department Department of Physics The University SO9 5NHSouthamptonUK S P Booth The University of Edinburgh EH9 3JZEdinburghScotland K C Bowler The University of Edinburgh EH9 3JZEdinburghScotland D S Henty The University of Edinburgh EH9 3JZEdinburghScotland R D Kenway The University of Edinburgh EH9 3JZEdinburghScotland B J Pendleton The University of Edinburgh EH9 3JZEdinburghScotland D G Richards The University of Edinburgh EH9 3JZEdinburghScotland J N Simone The University of Edinburgh EH9 3JZEdinburghScotland A D Simpson The University of Edinburgh EH9 3JZEdinburghScotland C Michael DAMTP University of Liverpool L69 3BXLiverpoolUK P W Stephenson DAMTP University of Liverpool L69 3BXLiverpoolUK The Hyperfine Splitting in Charmonium: Lattice Computations using the Wilson and Clover Fermion Actions UKQCD Collaboration 21 Aug 1992Southampton Preprint: SHEP 91/92-27 Edinburgh Preprint: 92/510 We compute the hyperfine splitting m J/ψ − m ηc on the lattice, using both the Wilson and O(a)-improved (clover) actions for quenched quarks. The computations are performed on a 24 3 ×48 lattice at β = 6.2, using the same set of 18 gluon configurations for both fermion actions. We find that the splitting is 1.83 + 13− 15 times larger with the clover action than with the Wilson action, demonstrating the sensitivity of the spinsplitting to the magnetic moment term which is present in the clover action. However, even with the clover action the result is less than half of the physical mass-splitting. We also compute the decay constants f ηc and f −1 J/ψ , both of which are considerably larger when computed using the clover action than with the Wilson action. For example for the ratio f −1 J/ψ /f −1 ρ we find 0.32 + 1 − 2 with the Wilson action and 0.48 ± 3 with the clover action (the physical value is 0.44(2)). The Hyperfine Splitting Lattice computations of the vector-pseudoscalar mass-splittings m D * − m D and m J/ψ − m ηc , using the standard Wilson action for the quarks in the quenched approximation, give results which are much too small [1,2]. At β = 6.2, for which the inverse lattice spacing (a −1 ) is approximately 2.7 GeV, the discrepancy is about a factor of 2 for m D * − m D and a factor of about 4 for m J/ψ − m ηc . In this letter we compute the hyperfine splitting m J/ψ − m ηc using both the Wilson fermion action, S W F = a 4 x 1 a q(x)q(x) + κ µ q(x)(γ µ − r)U µ (x)q(x +μ) −q(x +μ)(γ µ + r)U † µ (x)q(x)(1) and the nearest-neighbour O(a)-improved (or "clover") fermion action [3], S C F = S W F − irg 0 κ a 2 a 4 x,µ,νq (x)F µν (x)σ µν q(x)(2) with the same set of 18 gluon configurations in each case. The computations are performed on a 24 3 × 48 lattice at β = 6.2 with r = 1. These 18 configurations have been used earlier in our study of light quark spectroscopy and meson decay constants, the results and computational details can be found in ref. [4,5]. For the quantities studied in [4] the results obtained with the two actions were broadly compatible. However, here we show that the hyperfine splitting in charmonium is almost a factor of 2 larger with the clover action than with the Wilson action (see eq.(3) below), demonstrating the sensitivity of this quantity to the magnetic moment term in eq. (2). We also present the results for the decay constants of the η c and J/ψ mesons. Qualitatively similar results for the hyperfine splitting were obtained by the Fermilab group [6], who have performed simulations using a fermion action which is similar to that in equation 2 but with a factor of 1.4 multiplying the second term. This factor is their estimate of the effects of higher order perturbative corrections, and was obtained using a mean field theory calculation [7]. These authors also find a larger value of the hyperfine splitting with their action than with the Wilson action. However this comparison is obtained from computations on lattices of different size. Below we will compare our results with those of ref. [6]. The errors presented in this letter were obtained using the bootstrap procedure, described in detail in ref. [4,5]. The main result of this letter comes from the entries in the third column of table 1, from which it is clear that the hyperfine splittings are very different for the two actions. We stress that the results were obtained using the same gluon configurations and with the same analysis techniques. It is therefore likely that a number of the systematic errors would cancel in the ratio, for which we find: (m J/ψ − m ηc ) clover (m J/ψ − m ηc ) Wilson = 1.83 + 13 − 15(3) In fig. 1 we plot the values of m 2 V − m 2 P (where V and P represent vector and pseudoscalar respectively) as a function of the square of the pseudoscalar mass. We include not only the values for charmonium obtained from table 1, but also those for mesons composed of light quarks for three different light quark masses [4]. We see that for small masses the two actions give similar results, but as the mass is increased a gap gradually opens, with the clover action giving a larger value for the hyperfine splitting. It is interesting to note that the suggestion that this quantity might be susceptible to lattice artefacts has been made previously by the APE collaboration in a comparative study [8] Wilson and staggered fermion actions, albeit at stronger coupling and lighter quark masses than the present work. The result for the hyperfine splitting of charmonium obtained with the clover action is still much smaller than the experimental value. Taking a −1 = 2.73 GeV, the values in table 1 correspond to: m J/ψ − m ηc = 28 + 2 − 2 MeV Wilson Action (4) m J/ψ − m ηc = 52 + 3 − 4 MeV Clover Action(5) to be compared to the experimental value of 117(2) MeV. The errors quoted in (4) and (5) are statistical only, and the reader should bear in mind the uncertainty in the value of the lattice spacing. The corresponding values found by El-Khadra et al. [6] are: 51(3) MeV at β = 5.7, 62(4) MeV at β = 5.9 and 68(5) MeV at β = 6.1. Extrapolating linearly in a 2 to the continuum limit, these authors quote: m J/ψ − m ηc = 73 ± 10 MeV(6) where the error includes an estimate of the systematic uncertainty. The lattice spacing in this work was determined from the 1P-1S mass splitting (a quantity which is considerably less sensitive to the form of the action [6]). Thus it appears that there is a difference of about 20 MeV due to the different action used in ref. [6]. Decay Constants The (dimensionless) decay constant of the J/ψ-meson is defined by: 0\|c(0)γ µ c(0)\|J/ψ = ǫ µ m 2 J/ψ f J/ψ (7) where ǫ µ is the polarisation vector of the J/ψ. The measured value of the decay constant is 1/f J/ψ = 0.124 (5). In our computations we take for the lattice vector current, the local operator Z W Vc (0)γ µ c(0) when using the Wilson action, and the "improved" current Z C Vc (x)(1 + ra 2 γ· ← D )γ µ (1 − ra 2 γ· → D )c(x)(8) when using the clover action. Z W V and Z C V are the renormalisation constants (which ensure that the currents are correctly normalised), and can be evaluated in perturbation theory. We obtain the results 1 Z W V 1 f J/ψ = 0.152 + 5 − 5 Wilson Action (9) 1 Z C V 1 f J/ψ = 0.179 + 8 − 7 Clover Action The Z V 's have been calculated to one-loop order, Z W V =0.83 and Z C V =0.90 if the lattice bare coupling constant g 2 0 is used as the expansion parameter of perturbation theory, whereas Z W V =0.71 and Z C V =0.83 if an "effective" coupling g 2 ef f = 1.75g 2 0 is used (following suggestions in ref. [9]). For example using the effective coupling we find 1/f J/ψ = 0.108 + 4 − 4 using the From eqs.(9)-(13) we note that both the quantities 1/Z V f −1 J/ψ and 1/Z A f ηc are larger when obtained using the clover action than the Wilson action. This is opposite to the results for the corresponding quantities for the light mesons π and ρ [4]. Thus the difference obtained for the decay constants with the two actions is amplified significantly in the ratios f −1 J/ψ f −1 ρ and fη c fπ . Using the chiral extrapolations for f π and f −1 ρ , we find f −1 J/ψ f −1 ρ = 0.32 + 1 − 2 Wilson Action (14) f −1 J/ψ f −1 ρ = 0.48 + 3 − 3 Clover Action (15) where the physical value of this ratio is 0.44 (2), and f ηc f π = 2.3 + 5 − 3 Wilson Action (16) f ηc f π = 4.0 + 12 − 9 Clover Action (17) In these ratios the dependence on the renormalisation constants cancels. The differences between the results for the two actions in eqs.(14)-(17) indicate significant errors due to the finiteness of the lattice spacing (at least, presumably, for the Wilson action) in the decay constants for the charmonium system. Conclusions In this letter we have shown that the value of the hyperfine splitting in charmonium in lattice simulations is very sensitive to the fermion action which is used, and in particular to the magnetic moment term in the improved action. Using the same gluon configurations, we have found that at β = 6.2, the ratio of the splittings for the clover and Wilson actions is about 1.8 (see eq.(3) ). Even using the clover action, the value we obtain for the hyperfine splitting is only about one half of the physical value. The clover action is a "tree-level improved action", i.e. there are no errors of O(a), but the leading remaining errors due to the finiteness of the lattice spacing are of O(α s a). Presumably at least some of the discrepancy between the value in eq.(5) and the physical one of 117(2) MeV is due to these remaining O(α s a) errors. El-Khadra et al. have tried to reduce these by performing a mean field calculation to estimate the effects of the higher-order perturbative terms on the magnetic moment term in the action [6,7]. Their result for the hyperfine splitting, of about 73 MeV, although larger than that in eq.(5), is still considerably smaller than the physical value. In view of the sensitivity of the splitting to the magnetic moment term in the action, it is likely that at least part of the discrepancy is still due to the finiteness of the lattice spacing. Unfortunately it is not possible at present to determine how much of the discrepency is due to the inadequacy of the mean field calculation, and how much to other systematic errors (such as quenching). We are forced to accept that the hyperfine splitting is currently not amenable to an accurate lattice determination. Nevertheless it is reassuring to find that one of the few quantities for which lattice computations with the Wilson fermion action give a result which disagrees significantly with experiment, is unusually sensitive to known systematic errors. For the decay constants of charmonium we also found significant differences between the values obtained using the two actions. For the ratio f −1 J/ψ /f −1 ρ (see eqs. (14) and (15) ) we find a result which is about 75% of the physical one with the Wilson action (consistent with the results in ref. [2]), and a result which is consistent with the physical one with the clover action. For the decay constant of the η c (see eqs. (16) and (17) ) we find a larger result with the clover than with the Wilson action (note however that the error for the clover action is large, this error is dominated by the uncertainty in the extrapolated value of f π ). It will be very interesting to repeat this study for heavy-light mesons. In particular simulations with Wilson fermions indicate a substantial violation of the scaling law f P √ M P ≃ constant (up to logarithmic corrections) [2,10] for the decay constants of heavy-light pseudoscalar mesons P . This would imply a value of f B of about 200 MeV (one which is consistent with the simulations using the static approximation [11,12]), larger than many earlier expectations. It is now important to check whether these results will be stable under the reduction of the errors of O(a) achieved by the use of the clover action. For this study we took κ = 0.1350 for the Wilson action and κ = 0.1290 for the clover action. These values were chosen so that the pseudoscalar meson masses are almost equal for both actions, and correspond approximately to the physical mass of the η c . In [4] 1 1, and using this value we see that the masses of the mesons are within a few percent of their physical values. The masses were obtained by fitting the correlation functions in the time range t = 14 − 20, and the mass differences were obtained by fitting the ratio of the vector and pseudoscalar propagators over the same range. The fits were performed taking the correlations between the values at different time slices into account, and the values of χ 2 /d.o.f were acceptable. of 1 1The values of the inverse lattice spacing obtained by comparing the lattice values of the masses of the light hadrons to the physical ones lie within 15% of the result from the string tension. We take this as an indication of the size of the systematic uncertainty. Figure 1 : 1m 2 V − m 2P versus m 2 P , in lattice units, for the Wilson and clover actions. Wilson action, and 1/f J/ψ = 0.149 + 7 − 6 with the clover action. Hence the value of 1/f J/ψ is about 30-40% larger with the clover action than the Wilson action (although the uncertainty in the values of the renormalisation constants should be borne in mind).The decay constant of the η c has not been measured, however we present the lattice results in order to compare the values obtained with the two actions. The decay constant is defined by \| 0\|c(0)γ µ γ 5 c(0)\|η c (p) \| ≡ f ηc p µ(11) (with such a normalisation f π ≃ 132 MeV). For the lattice axial current we take the same operators as in the vector case with γ µ → γ µ γ 5 . The corresponding results are: when the bare (effective) coupling is used as the expansion parameter. Thus the values of f ηc are also about 30-40% larger when determined using the clover action than those obtained with the Wilson action. Table 1 : 1Masses (in lattice units) of the pseudoscalar and vector heavy-heavy mesons.present our results for the masses of the vector and pseudoscalar mesons (in lattice units), and for their difference. Setting the scale from the string tension gives a −1 = 2.73(5) GeV . M Bochicchio, Nucl. Phys. 372403M. Bochicchio et al., Nucl. Phys. B372 (1992) 403 . A Abada, Nucl. Phys. 376172A. Abada et al., Nucl. Phys. B376 (1992) 172 . B Sheikholeslami, R Wohlert, Nucl. Phys. 259572B. Sheikholeslami and R. Wohlert, Nucl. Phys. 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[ "MACHINE LEARNING FAIRNESS NOTIONS: BRIDGING THE GAP WITH REAL-WORLD APPLICATIONS", "MACHINE LEARNING FAIRNESS NOTIONS: BRIDGING THE GAP WITH REAL-WORLD APPLICATIONS" ]	[ "Karima Makhlouf [email protected] ", "Sami Zhioua [email protected] ", "Catuscia Palamidessi [email protected] ", "\nHigher Colleges of Technology Dubai\nInria, Ecole Polytechnique\nUniversité du Québec à Montréal Québec\nCanada, United Arab Emirates\n", "\nIPP Paris\nFrance\n" ]	[ "Higher Colleges of Technology Dubai\nInria, Ecole Polytechnique\nUniversité du Québec à Montréal Québec\nCanada, United Arab Emirates", "IPP Paris\nFrance" ]	[]	Fairness emerged as an important requirement to guarantee that Machine Learning (ML) predictive systems do not discriminate against specific individuals or entire sub-populations, in particular, minorities. Given the inherent subjectivity of viewing the concept of fairness, several notions of fairness have been introduced in the literature. This paper is a survey that illustrates the subtleties between fairness notions through a large number of examples and scenarios. In addition, unlike other surveys in the literature, it addresses the question of "which notion of fairness is most suited to a given real-world scenario and why?". Our attempt to answer this question consists in (1) identifying the set of fairness-related characteristics of the real-world scenario at hand, (2) analyzing the behavior of each fairness notion, and then (3) fitting these two elements to recommend the most suitable fairness notion in every specific setup. The results are summarized in a decision diagram that can be used by practitioners and policy makers to navigate the relatively large catalogue of ML fairness notions.	10.1016/j.ipm.2021.102642	[ "https://arxiv.org/pdf/2006.16745v5.pdf" ]	236,253,472	2006.16745	c015ebe85c0cb28edf624198e8da90e2ea0c04a7	MACHINE LEARNING FAIRNESS NOTIONS: BRIDGING THE GAP WITH REAL-WORLD APPLICATIONS 7 Jun 2022 Karima Makhlouf [email protected] Sami Zhioua [email protected] Catuscia Palamidessi [email protected] Higher Colleges of Technology Dubai Inria, Ecole Polytechnique Université du Québec à Montréal Québec Canada, United Arab Emirates IPP Paris France MACHINE LEARNING FAIRNESS NOTIONS: BRIDGING THE GAP WITH REAL-WORLD APPLICATIONS 7 Jun 2022Fairness · Machine learning · Discrimination · Survey · Systemization of Knowledge (SoK) Fairness emerged as an important requirement to guarantee that Machine Learning (ML) predictive systems do not discriminate against specific individuals or entire sub-populations, in particular, minorities. Given the inherent subjectivity of viewing the concept of fairness, several notions of fairness have been introduced in the literature. This paper is a survey that illustrates the subtleties between fairness notions through a large number of examples and scenarios. In addition, unlike other surveys in the literature, it addresses the question of "which notion of fairness is most suited to a given real-world scenario and why?". Our attempt to answer this question consists in (1) identifying the set of fairness-related characteristics of the real-world scenario at hand, (2) analyzing the behavior of each fairness notion, and then (3) fitting these two elements to recommend the most suitable fairness notion in every specific setup. The results are summarized in a decision diagram that can be used by practitioners and policy makers to navigate the relatively large catalogue of ML fairness notions. Introduction Decisions in several domains are increasingly taken by "machines". These machines try to take the best decisions based on relevant historical data and using Machine Learning (ML) algorithms. Overall, ML-based decision-making (MLDM) 1 is beneficial as it allows to take into consideration orders of magnitude more factors than humans do and hence outputting decisions that are more informed and less subjective. However, in their quest to maximize efficiency, ML algorithms can systemize discrimination against a specific group of population, typically, minorities. As an example, consider the automated candidates selection system of St. George Hospital Medical School [65,74]. The aim of the system was to help screening for the most promising candidates for medical studies. The automated system was built using records of manual screenings from previous years. During those manual screening years, applications with grammatical mistakes and misspellings were rejected by human evaluators as they indicate a poor level of English. As non-native English speakers are more likely to send applications with grammatical and misspelling mistakes than native English speakers do, the automated screening system built on that historical data ended up correlating race, birthplace, and address with a lower likelihood of acceptance. Later, while the overall English level of non-native speakers improved, the race and ethnicity bias persisted in the system to the extent that an excellent candidate may be rejected simply for her birthplace or address. Given that MLDM can have a significant impact in the lives and safety of human beings, it is no surprise that social and political organization are becoming very concerned with the possible consequences of biased MLDM, and the related issue of lack of explanation and interpretability of ML-based decisions. The European Union has been quite active in this respect: already in the General Data Protection Regulation (GDPR) there were directives concerning Automated Decision Making: for instance, Article 22 states that "The data subject shall have the right not to be subject to a decision based solely on automated processing." Other initiatives include the European Union's Ethics Guidelines for Trustworthy AI (April 2019), and OECD's Council Recommendation on Artificial Intelligence (May 2019). In the scientific community, the issue of fairness in machine learning has become one of the most popular topics in recent years. The number of publications and conferences in this field has literally exploded, and a huge number of different notions of fairness have been proposed, leading sometimes to possible confusion. This paper, like other surveys in the literature (cf. Section 2), attempts to classify and systematize these notions. The characteristic of our work, however, consists in our point of view, which is that the very reason for having different fairness notions is how suitable each one of them is for specific real-world scenarios. We feel that none of the existing surveys has addressed this aspect specifically. Discussion about the suitability (and sometimes the applicability) of the fairness notions is very limited and scattered through several papers [71,37,100,58,21,7]. In this survey paper we show that each MLDM system can be different based on a set of criteria such as: whether the ground-truth exists, difference in base-rates between sub-groups, the cost of misclassification, the existence of a government regulation that needs to be enforced, etc. We then revisit exhaustively the list of fairness notions and discuss the suitability and applicability of each one of them based on the list of criteria. Another set of results from the literature which is particularly related to the applicability problem we are addressing in this paper is the tensions that exist between some definitions of fairness. Several papers in the literature provide formal proofs of the impossibility to satisfy several fairness definitions simultaneously [71,58,7,18,35]. These results are revisited and summarized as they are related to the applicability of fairness notions. The results of this survey are finally summarized in a decision diagram that hopefully can help researchers, practitioners, and policy makers to identify the subtleties of the MLDM system at hand and to choose the most appropriate fairness notion to use, or at least rule out notions that can lead to wrong fairness/discrimination result. The paper is organized as follows. Section 3 lists notable real-world MLDMs where fairness is critical. Section 4 identifies a set of fairness-related characteristics of MLDMs that will be used in the subsequent sections to recommend and/or discourage the use of fairness notions. Fairness notions are listed and described in the longest section of the survey, Section 5. Section 6 discusses relaxations of the strict definitions of fairness notions. Section 7 describes classification and tensions that exist between some fairness notions. The decision diagram is provided and discussed in Section 8. Related Work and Scope With the increasing fairness concerns in the field of automated decision making and machine learning, several survey papers have been published in the literature in the few previous years. This section revisits these survey papers and highlights how this proposed survey deviates from them. In 2015, Zliobaite compiled a survey about fairness notions that have been introduced previously [107]. He classified fairness notions into four categories, namely, statistical tests, absolute measures, conditional measures, and structural measures. Statistical tests indicate only the presence or absence of discrimination. Absolute and conditional measures quantify the extent of discrimination with the difference that conditional measures consider legitimate explanations for the discrimination. These three categories correspond to the group fairness notions in this survey. Structural measures correspond to individual fairness notions 2 . Most of the fairness notions listed by Zliobaite are variants of the group fairness notions in this survey. For instance, difference of means test (Section 4.1.2 in [107]) is a variant of balance for positive class (Section 5.7 in this paper). Although, he dedicated one category for individual notions (structural measures), Zliobaite did not mention important notions, in particular fairness through awareness. Regarding the applicability of notions, the only criterion considered was the type of variables (e.g. binary, categorical, numerical, etc.). The survey of Berk et al. [10] listed only group fairness notions that are defined using the confusion matrix. Similar to this survey, they used simple examples based on the confusion matrix to highlight relationships between the fairness notions. The applicability aspect has not been addressed as the paper focused only on criminal risk assessment use case. The survey of Verma and Rubin [93] described a list of fairness notions similar to the list in this survey. To illustrate how each notion can be computed in real scenarios, they used a loan granting real use case (German credit dataset [5]). Rather than using a benchmark dataset, this survey uses a smaller and fictitious use case (job hiring) which allows to illustrate better the subtle differences between the fairness notions. For instance, counterfactual fairness is more intuitively described using a small job hiring example than the loan granting benchmark dataset. Verma and Rubin did not address the applicability aspect in their survey. Gajane and Pechenizkiy [37] focused on formalizing only notable fairness notions (e.g. statistical parity, equality of opportunity, individual fairness, etc.) and discussed their implications on distributive justice from the social sciences literature. In addition, they described two additional fairness notions that are studied extensively in the social sciences literature, namely, equality of resources and equality of capability. These notions, however, do not come with a mathematical formalization. This survey is more exhaustive as it analyzes a much larger number of fairness notions. However, being focused on the implication on distributive justice, Gajane and Pechenizkiy's survey addresses the suitability of the discussed fairness notions in real world domains. Mehrabi et al. [69] considered a more general scope for their survey: in addition to briefly listing 10 definitions of fairness notions (Section 4.2), they surveyed different sources of bias and different types of discrimination, they listed methods to implement fairness categorized into pre-processing, in-processing, and post-processing, and they discussed potential directions for contributions in the field. This survey is more focused on fairness notions which are described in more depth. A more recent survey by Mitchell et al. [71] presents an exhaustive list of fairness notions in both categories (group and individual) and summarizes most of the incompatibility results in the literature. Although Mitchell et al. discuss a "catalogue" of choices and assumptions in the context of fairness, the aim of these choices and assumptions is different from the criteria defined in this survey (Section 4). The assumptions and choices discussed in Section 2 in [71] address the question of how social goals are abstracted and formulated into a prediction (ML) problem. In particular, how the choice of the prediction goal, the choice of the population, and the choice of the decision space can have an impact on the degree of fairness of the prediction. Whereas the choices and criteria discussed in this survey (Section 4) are used to help identify the most suitable fairness notion to apply in a given scenario. Other surveys include the one by Friedler et al. [36] which considered only group fairness notions and focused on surveying algorithms to implement fairness. Overall most of existing review papers do not address all flavors of fairness notions in the same survey. In particular, most of them focus on statistical and group fairness notions. Causality based fairness notions, however, are not covered in several surveys while it is the most reliable category of notions in the disparate treatment legal framework. However, the main contribution of this survey is the focus on the applicability of fairness notions and the identification of fairness-related criteria to help select the most suitable notion to use given a scenario at hand. Brief discussions about the suitability of specific fairness notions can be found in few papers. For instance, Zafar et al. [100] mentioned some application scenarios for statistical parity and equalized odds. Kleinberg et al. [58] discussed the applicability of calibration and balance notions. Through a discussion about the cost of unfair decision on society, Corbett-Davies et al. [21] analyzed the impact of using statistical parity, predictive equality, and conditional statistical parity on public safety (criminal risk assessment). Gajane and Pechenizkiy [37] discuss the suitability of notable fairness notions (statistical parity, individual fairness, etc.) from the distributive justice point of view. Unlike the scattered discussions about the applicability of fairness notions found in the literature, this survey provides a complete reference to systemize the selection procedure of fairness notions. A short version of this paper was presented in BIAS 2020 workshop at ECMLPKDD 2020 [67]. Fairness in machine learning can be categorized according to two dimensions, namely, the task and the type of learning. For the first dimension, there are two tasks in fairness-aware machine learning: discrimination discovery (or assessment) and discrimination removal (or prevention). Discrimination discovery task focuses on assessing and measuring bias in datasets or in predictions made by the MLDM. Discrimination removal focuses on preventing discrimination by manipulating datasets (pre-processing), adjusting the MLDM (in-processing) or modifying predictions (post-processing). For the second dimension, fairness can be investigated for different learning types including fairness in classification, fairness in regression [50,2], fairness in ranking [15], fairness in reinforcement learning [45], etc. This survey focuses on the task of discrimination discovery (assessing fairness) in "pure prediction" [57] classification problems with a single decision making task (not sequential) and where decisions do not impact outcomes [22]. Real-world scenarios with critical fairness requirements As the paper is focusing on the applicability of fairness notions, we provide here a list of notable real-world MLDMs where fairness is critical. In each of these scenarios, failure to address the fairness requirement will lead to unacceptable biased decisions against individuals and/or sub-populations. These scenarios will be used to provide concrete examples of situations where certain fairness notions are more suitable than others. Job hiring: MLDMs in hiring are increasingly used by employers to automatically screen candidates for job openings 3 . Commercial candidate screening MLDMs include XING 4 , Evolv [62], Entelo, Xor, EngageTalent, GoHire and SyRI 5 . Typically, the input data used by the MLDM include: affiliation, education level, job experience, IQ score, age, gender, marital status, address, etc. The MLDM outputs a decision and/or a score indicating how suitable/promising the application is for the job opening. A biased MLDM leads to rejecting a candidate because of a trait that she cannot control (gender, race, sexual orientation, etc.). Such unfairness causes a prejudice on the candidate but also can be damaging for the employer as excellent candidates might be missed. Granting loans: Since decades, statistical and MLDM systems are used to assess loan applications and determine which of them are approved and with which repayment plan and annual percentage rate (APR). The assessment proceeds by predicting the risk that the applicant will default on her repayment plan. Loan Granting MLDMs currently in use include: FICO, Equifax, Lenddo, Experian, TransUnion, etc. The common input data used for loan granting include: credit history, purpose of the loan, loan amount requested, employment status, income, marital status, gender, age, address, housing status and credit score. An unfair loan granting MLDM will either deny a deserving applicant a requested loan, or give her an exorbitant APR, which on the long run will create a vicious cycle as the candidate will be very likely to default on her payments. College admission: Given the large number of admission applications, several colleges are now resorting to MLDMs to reduce processing time and cut costs 6 . Existing college admission MLDMs include GRADE [95], IBM Watson 7 , Kira Talent 8 . Typically, the candidates' features used include: the institutions previously attended, SAT scores, extra-curricular activities, GPAs, test scores, interview score, etc. The predicted outcome can be a simple decision (admit/reject) or a score indicating the candidate's potential performance in the requested field of study [35]. Unfair college admission MLDMs may discriminate against a certain ethnic group (e.g. African-American [82]) which could lead, in the long term, to economic inequalities and corrupting the role of higher education in society as a whole. For instance, in 2020 Ofqual, the UK Office of Qualifications and Examinations Regulation, used a MLDM to assess students for university admission decisions. Nearly 40% of students ended up receiving exam scores downgraded from their teachers' predictions, threatening to cost them their university spots. Analysis of the algorithm revealed that it had disproportionately hurt students from working-class and disadvantaged communities and inflated the scores of students from private schools [42]. Criminal risk assessment: There is an increasing adoption of MLDMs that predict risk scores based on historical data with the objective to guide human judges in their decisions. The most common use case is to predict whether a defendant will re-offend (or recidivate). Examples of risk assessment MLDMs include COMPAS [19], PSA [66], SAVRY [70], predPol [78]. Predicting risk and recidivism requires input information such as: number of arrests, type of crime, address, employment status, marital status, income, age, housing status, etc. Unfair risk assessment MLDMs, as revealed by the highly publicized 2016 proPublica article [4], may result in biased treatment of individuals based solely on their race. In extreme cases, it may lead to wrongful imprisonments for innocent people, contributing to the cycle of violation and crime. 3 In 2014, the automated job screening systems market was estimated at $500 million annual business and was growing at a rate of 10 to 15% per year [96] 4 A job platform similar to LinkedIn. It was found that this platform ranked less qualified male candidates higher than more qualified female candidates [60]. 5 System Riscico Indicatie, or SyRI for short, is a risk profiling system being deployed in the Netherlands by the Department of Social Affairs and Employment with the intention of identifying individuals who are at a high risk of committing fraud in relation to employment and other matters like social security and taxes. Its use raised a lot of controversy, and its case was brought to the Court of the Hague, that concluded on the 5th of February 2020 that the Government's use of SyRI violates the European Convention on Human Rights. To a very large extent, the Court's judgment was based on the lack of transparency in the algorithm at the heart of the system. 6 While the final acceptance decision is taken by humans, MLDMs are typically used as a first filter to "clean-up" the list from clear rejection cases. 7 A platform that uses natural language processing and personality traits in order to help students find the suitable and right college for them. 8 A Canadian startup that sells a cloud-based admissions assessment platform to over 300 schools. Teachers evaluation and promotion: MLDMs are increasingly used by decision makers to decide which teachers to retain after a probationary period [16] and which tenured teachers to promote. An example of such MLDM is IMPACT [81]. Teacher evaluation MLDMs take as input teacher related features (age, education level, experience, surveys, classroom observations), students related features (test scores, sociodemographics, surveys), and principals related features (surveys about the school and teachers), to predict whether teachers are retained. A biased teacher evaluation MLDM may lead to a systematic unfair low evaluation for teachers in poor neighborhoods, which, very often, happen to be teachers belonging to minority groups [80]. On the long term, this may lead to a significant drop in students' performance and the compromise of overall school reputation [74]. Child maltreatment prediction: The objective of the MLDM in child maltreatment prediction is to estimate the likelihood of substantiated maltreatment (neglect, physical abuse, sexual abuse, or emotional maltreatment) among children. The system generates risk scores, which would then trigger a targeted early intervention in order to prevent children maltreatment. PRM (predictive risk model) [91] has been developed to estimate the likelihood of substantiated maltreatment among children enrolled in New Zealand's public benefit system. In Finland, the government uses a ML-based system called "Kela" to administer benefits and to identify risk factors indicating that a child might need welfare services. In the US, the Allegheny County uses AFST (Allegheny Family Screening Tool) [32] to improve decision-making in child welfare system. The features considered in this type of MLDM include both contemporaneous and historical information for children and caregivers. An unfair MLDM may use a proxy variable to predict decisions based on the community rather than which child gets harmed. For example, a major cause of unfairness in AFST is the rate of referral calls; the community calls the child abuse hotline to report non-white families at a much higher rate than it does to report white families [32]. On the long term, this creates a vicious cycle as families which have been reported will be the subject of more scrutiny and more requirements to satisfy, and eventually, will be more likely to fail short of these requirements and hence confirm the prediction of the system. Health care: Since decades, ML algorithms are able to process anonymized electronic health records and flag potential emergencies, to which clinicians are invited to respond promptly. Examples of features that might be used in disease (chronic conditions) prediction include vital signs, blood test, socio-demographics, education, health insurance, home ownership, age, race, address. The outcome of the MLDM is typically an estimated likelihood of getting a disease. A biased disease prediction MLDM can misclassify individuals in certain sub-populations in a disproportionately higher rate than the dominant population. For instance, diabetic patients have known differences in associated complications across ethnicities [88]. Obemeyer et al. [73] give another example of an MLDM that predicts the health care spending for individuals in the coming years (useful information for insurance companies). They observe that the MLDM is biased against African-Americans because it uses the cost of health services in the previous year to predict the spending in the coming years. As African-Americans were spending less on health services than whites in the previous year, they were predicted to be spending less in the coming years. Hence, for the same prediction score, African-Americans were found to be sicker (more health issues) than whites. Consequently, white patients were benefiting more from additional help programs than African-Americans. More generally, because different sub-populations might have different characteristics, a single model to predict complications is unlikely to be best-suited for specific groups in the population even if they are equally represented in the training data [90]. Failure to predict disease likelihood in a timely manner may, in extreme cases, have an impact on people's lives. Online recommendation: Recommender systems are among the most widespread MLDMs in the market, with many services to assist users in finding products or information that are of potential interest [46]. Such systems find applications in various online platforms such as Amazon, Youtube, Netflix, LinkedIn, etc. An unfair recommender MLDM can amplify gender bias in the data. For example, a recommender MLDM called STEM, which aims to deliver advertisements promoting jobs in Science, Technology, Engineering, and Math fields, is deemed unfair as it has been shown that less women compared to men saw the advertisements due to gender imbalance [61]. Datta et al. [25] found that changing the gender bit in Google Ad Setting [41] resulted in a significant difference in the type of job ads received: men received much more ads about high paying jobs and career coaching services towards high paying jobs compared to women. Facial analysis: Automated facial analysis systems are used to identify perpetrators from security video footage, to detect melanoma (skin cancer) from face images [31], to detect emotions [26,33,89], and to even determine individual's characteristics such as IQ, propensity towards terrorist crime, etc. based on their face images [97]. The possible applications of Facial Analysis are innumerable. For instance, in France, FRT (Facial Recognition Tool) has been used on an experimental basis at various schools, with the aim of making access more fluid and secure for pupils. Furthermore, the government announced in 2020 that it would start to use an FRT system called "Alicem" in order to create a digital identification system by which its citizens could access governmental online services. Both of these, however, have sparked a lot of controversy leading to an announcement that the French government would be reviewing the use of FRT. Indeed, these devices are particularly intrusive and present major risks of invasion of the privacy and individual freedoms. Worse yet, a flawed MLDM may lead to biased outcomes such as wrongfully accusing individuals from specific ethnic groups (e.g. Asians, dark skin populations) for crimes (based on security video footage) at a much higher rate than the rest of the population. For instance, African-Americans have been reported to be more likely to be stopped and investigated by law enforcement due to a flawed face recognition system [40]. An investigation of three commercial face-based gender classification systems found that the error rate for dark-skinned females can be as high as 34.7% while for light-skinned males the maximum error rate is 0.8% [14]. Others: Other MLDMs with fairness concerns include: insurance policy prediction [85], income prediction [69], [105,30,83,3], and university ranking [68,74]. For a survey of the various kinds of MLDMs used in European countries, and a description of the debates and legal actions they have triggered, we recommend the excellent report by Robin Allen QC and Dee Masters [79] for the European Network of Equality Bodies. Fairness notion selection criteria In order to systemize the procedure for selecting the most suitable fairness notion for a specific MLDM system, we identify a set of criteria that can be used as as roadmap. For each criterion, we check whether it holds in the problem at hand or not. Telling whether a criterion is satisfied or not does not typically require an expertise in the problem domain. This section presents a list of 13 selection criteria. These criteria are derived mainly from three sources. First, the types of bias. For instance, the unreliable outcome criterion is a manifestation of a historical bias. Second, the mathematical formulation of the fairness notions themselves. For instance, the emphasis on precision vs recall criterion reflects a fundamental difference in the mathematical formulations of two families of notions, namely, predictive parity and equal opportunity. Third, the existing anti-discrimination legislation. The last two criteria are inspired by the current legislation. We note here that in some cases, these criteria can, not only indicate if a fairness notion is suitable, but whether it is "acceptable" to use in the first place. Ground truth availability: A ground truth value is the true and correct observed outcome corresponding to given sample in the data. It should be distinguished from an inferred subjective outcome in historical data which is decided by a human. An example of a scenario where ground truth is available is when predicting whether an individual has a disease. The ground truth value is observed by submitting the individual to a blood test 9 for example. An example of a scenario where ground truth is not available is predicting whether a job applicant is hired. The outcome in the training data is inferred by a human decision maker which is often a subjective decision, no matter how hard she is trying to be objective. It is important to mention here that the availability of the ground truth depends on how the outcome is defined. Consider, for example, college admission scenario. If the outcome in the training data is defined as whether the applicant is admitted or rejected, ground truth is not available. If, however, the outcome is defined as whether the applicant will ultimately graduate from college with a high GPA, ground truth is available as it can be observed after a couple of years. Base rate is the same across groups: The base rate is the proportion of positive outcome in a population (Table 1). A positive outcome is the goal of the prediction (e.g. a candidate to college is admitted, a child is maltreated, an individual is granted a loan, etc.). Note that the positive outcome can be desirable (e.g. hiring, admission) or undesirable (e.g. firing, high criminal risk). The base rate can be the same or differs across sub-populations. For example, the base rates for diabetes disease occurrence for men and women is typically the same. But, for another disease such as prostate cancer, the base rates are different between men and women 10 . (Un)reliable outcome: In scenarios where ground truth is not available, the outcome (label) in the data is typically inferred by humans. The outcome in the training data in that case can or cannot be reliable as it can encode human bias. The reliability of the outcome depends on the data collection procedure and how rigorous the data has been checked. Scenarios such as job hiring and college admission may be more prone to the unreliable outcome problem than recommender system for example. A "one-size-fit-all" MLDM model in disease prediction that does not take into consideration the ethnic group of the individual may result in unreliable outcome as well. Presence of explanatory variables: An explanatory variable 11 is correlated with the sensitive attribute (e.g. race) in a legitimate way. Any discrimination that can be explained using that variable is considered legitimate and is acceptable. 9 Assuming the blood test is flawless. 10 While male prostate cancer is the second most common cancer in men, female prostate cancer is rare [28]. 11 Referred also as a resolving variable. For instance, if all the discrepancy between male and female job hiring rates is explained by their education levels, the discrimination can be deemed legitimate and acceptable. Emphasis on precision vs recall: Precision (the complement of target population error [27]) is defined as the fraction of positive instances among the predicted positive instances. In other words, if the system predicts an instance as positive, how precise that prediction is. Recall (the complement of model error [27]) is defined as the fraction of the total number of positive instances that are correctly predicted positive. In other words, how many of the positive instances the system is able to identify. There is always a trade-off between precision and recall (increasing one will lead, very often, to decreasing the other). Depending on the scenario at hand, the fairness of the MLDM may be more sensitive to one on the expense of the other. For example, granting loans to the maximum number of deserving applicants contributes more to fairness than making sure that an applicant who has been granted a loan really deserves it 12 . When firing employees, however, the opposite is true: fairness is more sensitive to wrongly firing an employee, rather than, firing the maximum number of under-performing employees. Emphasis on false positive vs false negative: Fairness can be more sensitive to false positive misclassification (type I error) rather than false negative misclassification (type II error), or the opposite. For example, in criminal risk assessment scenario, it is commonly accepted that incarcerating an innocent person (false positive) is more serious than letting a guilty person escape (false negative). Cost of misclassification: Depending on the scenario at hand, the cost of misclassification can be significant (e.g. incarcerating an individual, firing an employee, rejecting a college application, etc.) or mild and without consequential impact (e.g. useless product recommendation, misleading income prediction, offensive online translation, abusive results in online autocomplete, etc.) Prediction threshold is fixed or floating: Decisions in MLDM are typically made based on predicted real-valued score. In the case of binary outcome, the score is turned into a binary value such as {0, 1} by thresholding 13 . In some scenarios, it is desirable to interpret the real-value score as probability of being accepted (predicted positive). The threshold used as a cutoff point where positive decisions are demarcated from negative decisions can be fixed or floating. A fixed threshold is set carefully and tends to be valid for different datasets and use cases. For instance, in recidivism risk assessment, high risk threshold is typically fixed. A floating threshold can be selected and fine-tuned arbitrarily by practitioners to accommodate a changing context. Acceptance score in loan granting scenarios is an example of a floating threshold as it can move up or down depending on the economic context. When the threshold is floating in a given application, assessing fairness should be done using a suitable fairness notion (e.g. calibration) otherwise, the result of the assessment may be misleading for specific threshold values. Likelihood of intersectionality: Intersectionality theory [24] focuses on a specific type of bias due to the combination of sensitive factors. An individual might not be discriminated based on race only or based on gender only, but she might be discriminated because of a combination of both. Black women are particularly prone to this type of discrimination. Likelihood of masking: Masking is a form of intentional discrimination that allows decision makers with prejudicial views to mask their intentions [8]. Masking is typically achieved by exploiting how fairness notions are defined. For example, if the fairness notion requires equal number of candidates to be accepted from two ethnic groups, the MLDM can be designed to carefully select candidates from the first group (satisfying strict requirements) while selecting randomly from the second group just to "make the numbers". Sources of Bias: Bias in the MLDM outcome can arise from several possible sources at any stage in the data generation and machine learning pipeline. Framing sources of bias necessitates deep understanding of the application at hand and, typically, can only be identified after a "post-mortem" analysis of the predicted outcome. However, in some real-world scenarios, one or more sources of bias may be more likely than others. In such cases, the suspected source of bias can be used as a criterion to select the most appropriate notion for fairness assessment. Sources of bias can be grouped broadly into six categories: historical, representation, measurement, aggregation, evaluation, and deployment [90]. Historical bias arises when the data reliably collected from the world leads to outcomes which are unwanted and socially unfavorable. For example, while data reliable collected indicates that only 5% of Fortune 500 CEOs are women [102], the resulting outcome of a prediction system based on this data is typically not wanted 14 . Representation bias arises when some non-protected populations are under-represented in the training data. Measurement bias arises when the features or label values are not measured accurately. For example, Street Bump is an application used in 12 It is important to mention here that from the loan granting organization's point of view, the opposite is true. That is, it is more important to make sure that an applicant who has been granted a loan really deserves it and will not default in payments because the interest payments resulting from a loan are relatively small compared to the loan amount that could be lost. Our aim here is fairness, while the loan granting organization's goal is benefit. 13 The threshold is defined by the decision makers depending on the context of interest. 14 For this reason, Google has changed their image search result for CEO to return a higher proportion of women. Boston city to detect when residents drive over potholes thanks to the accelerometers built into smartphones [23]. Collecting data using this application introduces a measurement bias due to the disparity in the distribution of smartphones according to the different districts in the city, which are often correlated with race or level of income. Aggregation bias arises when sub-populations are aggregated together while a single model is unlikely to fit all sub-populations. For instance, the genetic risk scores derived largely on European populations have been shown to generally perform very poorly in the prediction of osteoporotic fracture and bone mineral density on non-European populations, in particular, on Chinese population [63]. Evaluation bias arises when the training data differs significantly from the testing data. For instance, several MLDMs are trained using benchmark datasets which may be very different from the target dataset. Deployment bias arises when there is a disparity between the initial purpose of an MLDM and the way it is actually used. For instance, a child maltreatment MLDM might be designed to predict the risk of child abuse after two years from the reception of a referral call, while in practice it may be used to help social agents take decisions about an intervention. This can lead to a bias since the decision has an impact on the outcome [22]. Legal Framework: Anti-discrimination regulations in several countries, in particular US, distinguish between two legal frameworks, namely disparate treatment and disparate impact [8]. In the disparate treatment framework, a decision is considered unfair if it uses (directly or indirectly) the individual's sensitive attribute information. In the disparate impact framework, a decision is unfair if it results in an outcome that is disproportionately disadvantageous (or beneficial) to individuals according to their sensitive attribute information. Zafar et al. [100] formalized another fairness criterion, namely, disparate mistreatment according to which, a decision is unfair if it results in different misclassification rates for groups of people with different sensitive attribute information. Note that this criterion is currently not supported by a legal framework. Machine learning fairness notions can be classified according to the type of fairness it is evaluating. For instance, if a plaintiff is accusing an employer for intentional discrimination, she should consider the disparate treatment legal framework, and hence a fairness notion which falls in that framework. The existence of regulations and standards: In some domains, laws and regulations might be imposed to avoid discrimination and bias. For instance, guidelines from the U.S. Equal Employment Opportunity Commission state that a difference of the probability of acceptance between two sub-populations exceeding 20% is illegal [7]. Another example might be an internal organizational policy imposing diversity among its employees. Fairness notions Let V , A, and X be three random variables representing, respectively, the total set of attributes, the sensitive attributes, and the remaining attributes describing an individual such that V = (X, A) and P (V = v i ) represents the probability of drawing an individual with a vector of values v i from the population. For simplicity, we focus on the case where A is a binary random variable where A = 0 designates the protected group, while A = 1 designates the non-protected group. Let Y represent the actual outcome andŶ represent the outcome returned by the prediction algorithm (MLDM). Without loss of generality, assume that Y andŶ are binary random variables where Y = 1 designates a positive instance, while Y = 0 a negative one. A perfect MLDM will match perfectly the actual outcome (Ŷ = Y ). Typically, the predicted outcomeŶ is derived from a score represented by a random variable S where P (S = s) is the probability that the score value is equal to s. All fairness notions presented in this section address the following question: "is the outcome/prediction of the MLDM fair towards individuals?". So fairness notion is defined as a mathematical condition that must involve eitherŶ or S along with the other random variables. As such, we are not concerned by the inner-workings of the MLDM and their fairness implications. What matters is only the score/prediction value and how fair/biased it is. Most of the proposed fairness notions are properties of the joint distribution of the above random variables (X, A, Y , Y , and S). They can also be interpreted using the confusion matrix and the related metrics (Table 1). While presenting and discussing fairness notions, whenever needed, we use the simple job hiring scenario of Table 2. Each sample in the dataset has the following attributes: education level (numerical), job experience (numerical), age (numerical), marital status (categorical), gender (binary) and a label (binary). The sensitive attribute is the applicant gender, that is, we are focusing on whether male and female applicants are treated equally. Table 2(b) presents the predicted decision (first column) and the predicted score value (second column) for each sample. The threshold value is set to 0.5. A simple and straightforward approach to address fairness problem is to ignore completely any sensitive attribute while training the MLDM system. This is called fairness through unawareness 15 . We don't treat this approach as fairness notion since, given MLDM prediction, it does not allow to tell if the MLDM is fair or not. Besides, it suffers 15 Known also as: blindness, unawareness [71], anti-classification [20], and treatment parity [64]. Table 2: A simple job hiring example. Y represents the data label indicating whether the applicant is hired (1) or rejected (0).Ŷ is the prediction which is based on the score S. A threshold of 0.5 is used. Female 1 8 2 39 single 0 Female 2 8 2 26 married 1 Female 3 12 8 32 married 1 Female 4 11 3 35 single 0 Female 5 9 5 29 married 1 Male 1 11 3 34 single 1 Male 2 8 0 48 married 0 Male 3 7 3 43 single 1 Male 4 8 2 26 married 1 Male 5 8 2 41 single 0 Male 6 12 8 30 single 1 Male 7 10 Y = 1 Y = 0 Predicted Positive TP (True Positive) FP (False Positive) PPV = T P T P +F P FDR = F P T P +F P Y = 1TN (True Negative) FOR = F N F N +T N NPV = T N F N +T N Y = 0 Type II error False Omission Rate Negative Predictive Value Success Predictive Error PV- ww TPR = T P T P +F N FPR = F P F P +T N OA = T P +T N T P +F P +T N +F N BR = T P +F N T P +F P +T N +F N True Positive Rate False Positive Rate Overall Accuracy Base Rate Sensitivity Model Error Prevalence (p) Recall ww FNR = F N T P +F N TNR = T N F P +T N False Negative Rate True Negative Rate Model Error Specificity 2 28 married 1 (b) Prediction Y S 1 0.5 0 0.1 1 0.5 0 0.2 0 0.3 1 0.8 0 0.1 0 0.1 1 0.5 1 0.5 1 0.8 0 0.3 from the basic problem of proxies. Many attributes (e.g. home address, neighborhood, attended college) might be highly correlated to the sensitive attributes (e.g. race) and act as proxies of these attributes. Consequently, in almost all situations, removing the sensitive attribute during the training process does not address the problem of fairness. Statistical parity Statistical parity [29] (a.k.a demographic parity [59], independence [6], equal acceptance rate [106], benchmarking [86], group fairness [29]) is one of the most commonly accepted notions of fairness. It requires the prediction to be statistically independent of the sensitive attribute (Ŷ ⊥ A). Thus, a classifierŶ satisfies statistical parity if: P (Ŷ \| A = 0) = P (Ŷ \| A = 1)(1) In other words, the predicted acceptance rates for both protected and unprotected groups should be equal. Using the confusion matrix (Table 1), statistical parity implies that (T P + F P )/(T P + F P + F N + T N ) should be equal for both groups. In the MLDM of Table 2, it means that one should not hire proportionally more applicants from one group than the other. The calculated predicted acceptance rate of hiring male and female applicants is 0.57 (4 out of 7) and 0.4 (2 out of 5), respectively. Thus, the MLDM of Table 2 does not satisfy statistical parity. Statistical parity is appealing in scenarios where there is a preferred decision over the other, and provided there are no other considerations relevant for the decision, in which case, the following fairness notion namely, conditional statistical parity, is more suitable. For example, being accepted to a job, not being arrested, being admitted to a college, etc. 16 . What really matters is a balance in the prediction rate among all groups. Statistical parity is suitable when the label Y is not trustworthy due to some flawed or biased measurement 17 . An example of this type of problem was observed in the recidivism risk prediction tool COMPAS [4]. Because minority groups are more controlled, and more officers are dispatched in their regions, the number of arrests (used to assess the level of crime [90]) of those minority groups is significantly higher than that of the rest of the population. Hence, for fairness purposes, in the absence of information to precisely quantify the differences in recidivism by race, the most suitable approach is to treat all sub-populations equally with respect to recidivism [47]. Statistical parity is also well adapted to contexts in which some regulations or standards are imposed. For example, a law might impose to equally hire or admit applicants from different sub-populations. The main problem of statistical parity is that it doesn't consider a potential correlation between the label Y and the sensitive attribute A. In other words, if the underlying base rates of the protected and unprotected groups are different, statistical parity will be misleading. In particular, modifying an MLDM with a perfect prediction (ŷ = y) so to satisfy statistical parity while the base rates are different will lead to loss of utility [43]. As an example, Figure 1 illustrates a scenario for hiring computer engineers where equal proportions of male/female applicants have been predicted hired (60%) thus, satisfying statistical parity. However, when considering the label and more precisely the base rates that differ in both groups (0.3 for men versus 0.4 for women), the classifier becomes discriminative against female applicants (50% of qualified female applicants are not predicted hired). More generally, when the ground truth is available and is used in the training of the MLDM, statistical parity is not recommended because, very often, it conflicts with the ground truth [100]. Another issue with this notion is its "laziness"; if we hire carefully selected applicants from male group and random applicants from female group, we can still achieve statistical parity, yet leading to negative results for the female group as its performance will tend to be worse than that of male group. This practice is an example of self-fulfilling prophecy [29] where a decision maker may simply select random members of a protected group rather than qualified ones, and hence, intentionally building a bad track record for that group. Barocas and Selbst refer to this problem as masking [8]. Masking is possible to game several fairness notions, but it is particularly easy to carry out in the case of statistical parity. Conditional statistical parity Conditional statistical parity [21], called also conditional discrimination-aware classification in [49] is a variant of statistical parity obtained by controlling on a set of legitimate attributes 18 . The legitimate attributes (we refer to them as E) among X are correlated with the sensitive attribute A and give some factual information about the label at the same time leading to a legitimate discrimination. In other words, this notion removes the illegal discrimination, allowing the disparity in decisions to be present as long as they are explainable [21]. In the hiring example, possible explanatory factors that might affect the hiring decision for an applicant could be the education level and/or the job experience. If the data is composed of many highly educated and experienced male applicants and only few highly educated and experienced women, one might justify the disparity between predicted acceptance rates between both groups and consequently, does not necessarily reflect gender discrimination. Conditional statistical parity holds if: P (Ŷ = 1 \| E = e, A = 0) = P (Ŷ = 1 \| E = e, A = 1) ∀e (2)(b) Prediction Y S 1 0.5 0 0.1 1 0.5 1 0.5 1 0.5 1 0.8 Table 3 shows two possible combinations values for E. The first combination (education level=8 and job experience=2) includes samples Female 1, Female 2, Male 4, and Male 5 for which the prediction is clearly discriminative against women as the predicted acceptance rates for men and women are 1 and 0.5, respectively. The second combination (education level=12 and job experience=8) includes Female 3 and Male 6 in which the prediction is fair (predicted acceptance rate is 1 for both applicants). Overall, the prediction is not fair as it does not hold for one combination of values of E. In practice, conditional statistical parity is suitable when there is one or several attributes that justify a possible disparate treatment between different groups in the population. Hence, choosing the legitimate attribute(s) is a very sensitive issue as it has a direct impact on the fairness of the decision-making process. More seriously, conditional statistical parity gives a decision maker a tool to game the system and realize a self-fullfilling prophecy. Therefore, it is recommended to resort to domain experts or law officers to decide what is unfair and what is tolerable to use as legitimate discrimination attribute [49]. Equalized odds Unlike the two previous notions, equalized odds [44] (separation in [6], conditional procedure accuracy equality in [10], disparate mistreatment in [100], error rate balance in [18]) considers both the predicted and the actual outcomes. Thus, the prediction is conditionally independent from the protected attribute, given the actual outcome (Ŷ ⊥ A \| Y ). In other words, equalized odds requires both sub-populations to have the same TPR and FPR (Table 1). In our example, this means that the probability of an applicant who is actually hired to be predicted hired and the probability of an applicant who is actually not hired to be incorrectly predicted hired should be both same for men and women: P (Ŷ = 1 \| Y = y, A = 0) = P (Ŷ = 1 \| Y = y, A = 1) ∀y ∈ {0, 1}(3) In the example of Table 2, the TPR for male and female groups is 0.6 and 0.33, respectively while the FPR is exactly the same (0.5) for both groups. Consequently, the equalized odds does not hold. By contrast to statistical parity, equalized odds is well-suited for scenarios where the ground truth exists such as: disease prediction or stop-and-frisk [9]. It is also suitable when the emphasis is on recall (the fraction of the total number of positive instances that are correctly predicted positive) rather than precision (making sure that a predicted positive instance is actually a positive instance). A potential problem of equalized odds is that it may not help closing the gap between the protected and unprotected groups. For example, consider a group of 20 male applicants of which 16 are qualified and another equal size group of 20 females of which only 2 are qualified. If the employer decides to hire 9 applicants and while satisfying equalized odds, 8 offers will be granted to the male group and only 1 offer will be granted to the female group. While this decision scheme looks fair on the short term, on the long term, however, it will contribute to confirm this "unfair" status-quo and perpetuate this vicious cycle 19 . Whether to consider this long term impact as a problem of equalized odds is a controversial issue as it overlaps with the different but related question of "how to address unfairness?". Note that other fairness notions, such as statistical parity, help closing the gap between the protected and unprotected groups on the long term. Because equalized odds requirement is rarely satisfied in practice, two variants can be obtained by relaxing Eq. 3. The first one is called equal opportunity [44] (false negative error rate balance in [18]) and is obtained by requiring only TPR equality among groups: P (Ŷ = 1 \| Y = 1, A = 0) = P (Ŷ = 1 \| Y = 1, A = 1)(4) In the job hiring example, this is to say that we should hire equal proportion of individuals from the qualified fraction of each group. As T P R = T P/(T P + F N ) ( Table 1) does not take into consideration F P , equal opportunity is completely insensitive to the number of false positives. This is an important criterion when considering this fairness notion in practice. More precisely, in scenarios where a disproportionate number of false positives among groups has fairness implications, equal opportunity should not be considered. The scenario in Table 4 shows an extreme case of a job hiring dataset where the male group has a large number of false positives (Male 7 − 100) while equal opportunity is satisfied. (b) Prediction Y S 1 0.5 0 0.1 0 0.3 1 0.5 0 0.2 0 0.4 1 0.8 1 . . . 1 0.7 To decide about the suitability of equal opportunity in the job hiring example, the question that should be answered by stakeholders and decision makers is "if all other things are equal, is it fair to hire disproportionately more unqualified male candidates?". For the employer, it is undesirable to have several false positives (regardless of their gender) as the company will end up with unqualified employees. For a stakeholder whose goal is to guarantee fairness between males and females, it is not very critical to have more false positives in one group, provided that these two groups have the same proportion of false negatives (a qualified candidate which is not hired). In the scenario of predicting which employees to fire, however, a false positive (firing a well-performing employee) is critical for fairness. Hence, equal opportunity should not be used as a measure of fairness. The second relaxed variant of equalized odds is called predictive equality [21] (false positive error rate balance in [18]) which requires only the FPR to be equal in both groups. In other words, predictive equality checks whether the accuracy of decisions is equal across protected and unprotected groups: P (Ŷ = 1 \| Y = 0, A = 0) = P (Ŷ = 1 \| Y = 0, A = 1)(5) In the job hiring example, predictive equality holds when the probability of an applicant with an actual weak profile for the job to be incorrectly predicted hired is the same for both men and women. Since F P R = F P/(F P + T N ) ( Table 1) is independent from F N , predictive equality is completely insensitive to false negatives. One can come up with an extreme example similar to Table 4 with a disproportionate number of false negatives but predictive equality will still be satisfied (keeping all other rates equal). Hence, in scenarios where fairness between groups is sensitive to false negatives, predictive equality should not be used. Such scenarios include hiring and admission where a false negative means a qualified candidates are rejected disproportionately among groups. Predictive equality is acceptable in criminal risk assessment scenarios as false negatives (releasing a guilty person) are less critical than false positives (incarcerating an innocent person). Predictive equality is particularly suitable to measure the fairness of face recognition systems in crime investigation where security camera footage are analyzed. Fairness between ethnic groups with distinctive face features is very sensitive to the FPR. A false positive means an innocent person is being flagged as participating in a crime. If this false identification happens at a much higher rate for a specific sub-population (e.g. dark skinned group) compared to the rest of the population, it is clearly unfair for individuals belonging to that sub-population. Looking to the problem from another perspective, choosing between equal opportunity and predictive equality depends on how the outcome/label is defined. In scenarios where the positive outcome is desirable (e.g. hiring, admission), typically fairness is more sensitive to false negatives rather than false positives, and hence equal opportunity is more suitable. In scenarios where the positive outcome is undesirable for the subjects (e.g. firing, risk assessment), typically fairness is more sensitive to false positives rather than false negatives, and hence predictive equality is more suitable. The following proposition states formally the relationship between equalized odds, equal opportunity, and predictive equality. Proposition 5.1 Satisfying equal opportunity and predictive equality is equivalent to satisfying equalized odds: Eq. 3 ⇔ Eq. 4 ∧ Eq. 5 Conditional use accuracy equality Conditional use accuracy equality [10] (called sufficiency in [6]) is achieved when all population groups have equal P P V = T P T P +F P and N P V = T N F N +T N . In other words, the probability of subjects with positive predictive value to truly belong to the positive class and the probability of subjects with negative predictive value to truly belong to the negative class should be the same: P (Y = y \|Ŷ = y, A = 0) = P (Y = y \|Ŷ = y, A = 1) ∀y ∈ {0, 1}(6) Intuitively, this definition implies equivalent accuracy for male and female applicants from both positive and negative predicted classes [93]. By contrast to equalized odds (Section 5.3), one is conditioning on the algorithm's predicted outcome not the actual outcome. In other words, this notion emphasis the precision of the MLDM system rather than its sensitivity (a trade-off discussed earlier in Section 4). The calculated PPVs for male and female applicants in our hiring example (Table 2) are 0.75 and 0.5, respectively. NPVs for male and female applicants are both equal to 0.33. Overall the dataset in Table 2 does not satisfy conditional use accuracy equality. Predictive parity [18] (called outcome test in [86]) is a relaxation of conditional use accuracy equality requiring only equal PPV among groups: P (Y = 1 \|Ŷ = 1, A = 0) = P (Y = 1 \|Ŷ = 1, A = 1)(7) In our example, this is to say that the prediction used to determine the candidate's eligibility for a particular job should reflect the candidate's actual capability of doing this job which is harmonious with the employer's benefit. Like predictive equality (Eq. 5), predictive parity is insensitive to false negatives. Hence in any scenario where fairness is sensitive to false negatives, predictive parity should not be considered sufficient. Choosing between predictive parity and equal opportunity depends on whether the scenario at hand is more sensitive to precision or recall. For precision-sensitive scenarios, typically predictive parity is more suitable while for recallsensitive scenarios, equal opportunity is more suitable. Precision-sensitive scenarios include disease prediction, child maltreatment risk assessment, and firing from jobs. Recall-sensitive scenarios include loan granting, recommendation systems, and hiring. Very often, precision-sensitive scenarios coincide with situations where the positive prediction (Ŷ = 1) entails a higher cost [100]. For example, a predicted child maltreatment case will result in placing the child in a foster house which will generally entail a higher cost compared to a negative prediction (low risk of child maltreatment) in which case the child stays with the family and typically no action is taken. Conditional use accuracy equality (Eq. 6) is "symmetric" to equalized odds (Eq. 3) with the only difference of switching Y andŶ . The same holds for equal opportunity (Eq. 4) and predictive parity (Eq. 7). However, there is no "symmetric" notion to predictive equality (Eq. 5). For completeness, we define such notion and give it the name negative predictive parity. Definition 5.1 Negative predictive parity holds iff all sub-groups have the same N P V = T N F N +T N : P (Y = 1 \|Ŷ = 0, A = 0) = P (Y = 1 \|Ŷ = 0, A = 1)(8) The following proposition states formally the relationship between conditional use accuracy equality, predictive parity, and negative predictive parity. Proposition 5.2 Satisfying predictive parity and negative predictive parity is equivalent to satisfying conditional use accuracy equality: Eq. 6 ⇔ Eq. 7 ∧ Eq. 8 Overall accuracy equality Overall accuracy equality [10] is achieved when overall accuracy for both groups is the same. Thus, true negatives and true positives are equally considered and desired. Using the confusion matrix (Table 1), this implies that (T P + T N )/(T P + F N + F P + T N ) is equal for both groups. In our example, it is to say that the probability of well-qualified applicants to be correctly accepted for the job and non-qualified applicants to be correctly rejected is the same for both male and female applicants: Gender YŶ P (Ŷ = Y \|A = 0) = P (Ŷ = Y \|A = 1)(9)F1 1 1 F2 1 0 F3 1 0 F4 0 0 F5 1 1 F6 1 1 F7 1 0 F8 1 1 Group 2 (Male) Gender YŶ M1 1 1 M2 0 1 M3 0 1 M4 0 0 M5 0 0 M6 0 0 M7 0 1 M8 1 1 OA = 0.625 PPV = 0.4 NPV = 1 Overall accuracy equality is closely related to equalized odds (Eq. 3) and to conditional use accuracy equality (Eq. 6). The main difference is that overall accuracy equality aggregates together positive class and negative class misclassifications (FP and FN). Aggregating together FP and FN (and hence TP and TN) without any distinction is very often misleading for fairness purposes. Proposition 5.3 An MLDM that satisfies equalized odds or conditional use accuracy equality always satisfies overall accuracy. Eq. 3 ∨ Eq. 6 ⇒ Eq. 9 The reverse, however, is not true. That is, an MLDM that satisfies overall accuracy does not necessarily satisfy equalized odds or conditional use accuracy equality. To prove it, consider the example in Table 5 satisfying overall accuracy equality but not conditional use accuracy equality. For the female group, there are only FN misclassifications (no FP) and more TPs than TNs, while in the male group, there are only FP misclassifications (no FN) and more TNs than TPs. But since the proportion of correct classifications is the same in both groups (5 out of 8), overall accuracy equality holds. In real-world applications, it is very uncommon that TP (or FN) and TN (or FP) are desired at the same time and without distinction. For example, overall accuracy equality is not suitable to measure fairness in child maltreatment prediction because a False Positive (misclassifying a child case which is not at risk 20 ) is less damaging than a False Negative (misclassifying a child case which is at risk 21 ). A hypothetical health care scenario where overall accuracy equality is suitable is when both types of misclassifications have the same cost/benefit. For example, an eventual health condition that yields very similar complications (1) when the treatment is administered wrongly and (2) when the treatment is not administered while it is needed. Treatment equality Treatment equality [10] is achieved when the ratio of FPs and FNs is the same for both protected and unprotected groups: F N F P (A=0) = F N F P (A=1)(10) Treatment equality is insensitive to the numbers of TPs and TNs which are important to identify bias between subpopulations in most real-world scenarios. Berk et al. [10] note that treatment equality can serve as an indicator to achieve other kinds of fairness. Table 6 shows a dataset which fails to satisfy all previous notions, yet, treatment equality is satisfied. Treatment equality can be used in real-world scenarios where only the type of rate of misclassification matters for fairness. Treatment equality can be suitable to use in case the cost (or benefit) of a FP is a fixed ratio (or reciprocal) of the cost (or benefit) of a FN. For example, one can think of a loan granting scenario where the cost of a FP (misclassifying a non-defaulter) is exactly a fraction (e.g. 1/3) of the cost of a FN (misclassifying a defaulter). Total fairness [10] is another notion which holds when all aforementioned fairness notions are satisfied simultaneously, that is, statistical parity, equalized odds, conditional use accuracy equality (hence, overall accuracy equality), and treatment equality. Total fairness is a very strong notion which is very difficult to hold in practice. Table 7 shows a scenario where total fairness holds. More generally, total fairness is satisfied in the very uncommon situation where the proportions of TPs, TNs, FPs, and FNs are the same in all groups. Total fairness can be considered in scenarios where any deviation in misclassification or acceptance rates between sub-populations is very costly 22 . Balance The predicted outcome (Ŷ ) is typically derived from a score (S) which is returned by the ML algorithm. All aforementioned fairness notions do not use the score to assess fairness. Typically, the score value is normalized to be in the interval [0, 1] which makes it possible to interpret the score as the probability to predict the sample as positive. Balance for positive class [58] focuses on the applicants who constitute positive instances and is satisfied if the average score S received by those applicants is the same for both groups. In other words, a violation of this balance means that applicants belonging to the positive class in one group might receive steadily lower predicted score than applicants belonging to the positive class in the other group: Table 8 shows a job hiring scenario where the average score for female candidates that should be hired (Y = 1) is 7.1 while it is 4.7 for male candidates. The scenario is not balanced for positive class. Note that, despite the significant difference between these two average values, for a score threshold value of 5, the scenario of Table 8 satisfies both statistical parity (Eq. 1) and equal opportunity (Eq. 4). Balance of negative class [58] is an analogous fairness notion where the focus is on the negative class: E[S \| Y = 1, A = 0)] = E[S \| Y = 1, A = 1](11)E[S \| Y = 0, A = 0] = E[S \| Y = 0, A = 1](12) The scenario in Table 8 is not balanced for the negative class either since the average scores for the negative class (Y = 0) for the female and male groups are 5.3 and 2.8, respectively. Both variants of balance can be required simultaneously (Eq. 11 and 12) which leads to a stronger notion of balance. Since no previous work reported such fairness notion, for completeness, we define it and call it overall balance. Definition 5.2 Overall balance is satisfied iff: E[S \| Y = y, A = 0] = E[S \| Y = y, A = 1] ∀y ∈ {0, 1}(13) Balance fairness notions are relevant in the criminal risk assessment scenario because a divergence in the score values of individuals from different races may indicate a difference in the type of crime that can be committed (high risk score typically means a serious crime). Balance fairness notions are also suitable in the teacher firing scenario since any discrepancy between the average evaluation scores of fired teachers in different groups is a clear indicator of bias. On the other hand, balance fairness notions can be misleading in presence of clusters of samples sharing very similar attribute values and having score values in the vicinity of the positive/negative outcome threshold. In such case, the average score of the positive/negative class can change significantly due to a slight increase/decrease of the threshold value. Calibration Calibration [18] (a.k.a. test-fairness [18], matching conditional frequencies [44]) relies on the score variable as follows. To satisfy calibration, for each predicted probability score S = s, individuals in all groups should have the same probability to actually belong to the positive class: P (Y = 1 \| S = s, A = 0) = P (Y = 1 \| S = s, A = 1) ∀s ∈ [0, 1](14) Eq. 14 is very unlikely to be satisfied in practice as the probability of two individuals having exactly the same real number score is very small. Moreover, technically, the probability that S exactly equal to s is typically 0. Therefore, in practice, the space of score values [0, 1] is binned into intervals called bins such that any two values falling in the same bin are considered equal [58,93,56]. In our job hiring example, this implies that for any score value s ∈ [0, 1], the probability of truly being hired should be the same for both male and female applicants. Eq. 14 is very similar to Eq. 7 corresponding to predictive parity. Table 9 illustrates a job hiring scenario that may or may not satisfy predictive parity depending on the score threshold to hire a candidate; for a threshold value of 0.6, PPV rate for both male and female groups is the same, 0. Table 9 does not satisfy calibration. Interestingly, calibration is not always stronger than predictive parity [39]. Table 10 shows a job hiring scenario satisfying calibration, but not predictive parity. Calibration is suitable to use in scenarios where the threshold is not fixed and is very likely to be tuned to accommodate a changing context. A first example is the acceptance score in loan granting applications which may change abruptly due to economic instability. A second example is the child maltreatment risk assessment where the threshold for intervention (withdrawing a child from his family) depends on the available seats in foster houses. Well-calibration [58] is a stronger variant of calibration. It requires that (1) calibration is satisfied, (2) the score is interpreted as the probability to truly belong to the positive class, and (3) for each score S = s, the probability to truly belong to the positive class is equal to that particular score: P (Y = 1 \| S = s, A = 0) = P (Y = 1 \| S = s, A = 1) = s ∀ s ∈ [0, 1](15) Intuitively, for a set of applicants who have a certain probability s of being hired, approximately s percent of these applicants should truly be hired. Table 11 (a) is a job hiring scenario which is calibrated (the proportion of applicants which should be hired for every score value is the same for male and female groups) but not well-calibrated (the score value does not coincide with the proportion of applicants that should be hired). Table 11 (b) is both calibrated and well-calibrated. Garg et al. [39] show that the difference between calibration and well-calibration is a simple difference in mapping. That is, "the scores of a calibrated predictor can, using a suitable transformation, be converted to scores satisfying well-calibration". Group vs individual fairness notions All the fairness notions discussed above are considered as group fairness where their common objective is to ensure that groups who differ by their sensitive attributes are treated equally. These notions, mainly based on statistical measures, generally ignore all attributes of the individuals except the sensitive attribute A. Such treatment might hide unfairness. Dwork et al. [29] stated that group fairness, despite its suitability for policies among demographic subpopulations, does not guarantee that individuals are treated fairly. This is illustrated in the simple example in Table 12. The example satisfies most of group fairness notions, including total fairness (Section 5.6). However, based on the applicants profiles, it is clear that the predictor is unfair towards applicant Female 4. The fairness notions which follow attempt to address such issues by not marginalizing over non-sensitive attributes X of an individual, therefore they are called individual fairness notions 23 . Causal discrimination Causal Discrimination [38] implies that a classifier should produce exactly the same prediction for individuals who differ only from gender while possessing identical attributes X. In our hiring example, this is to say that male and female applicants with the same attributes X should have the same predictions: X (A=0) = X (A=1) ∧ A (A=0) = A (A=1) ⇒ŷ (A=0) =ŷ (A=1)(16) In our example, this implies that male and female applicants who otherwise have the same attributes X will either both be assigned a positive prediction or both assigned a negative prediction. Considering the example of Table 2, two applicants of different genders (Female 2 and Male 4) have identical values of X yet, getting different predictions (negative for female applicant while positive for male applicant). The predictor is then unfair towards Female 2 applicant. At a first glance, causal discrimination can be seen as an extreme case of conditional statistical parity (Section 5.2) when conditioning on all non-sensitive attributes (E = X). However, conditional statistical parity is a group fairness notion which is satisfied if the proportion of individuals having the same non-sensitive attribute values and predicted accepted in both groups (e.g. male and female) is the same. This is why Eq. 2 is expressed in terms of conditional probabilities. Causal discrimination, however, consider every individual separately regardless of its contribution to sub-population proportions. To illustrate this subtlety, consider the following scenario: Causal discrimination is suitable to use in decision making scenarios where it is very common to find individuals sharing exactly the same attribute values. For example, admission decision making based mainly on test scores and categorical attributes. To apply this fairness notion on a loan granting scenario where there are only few individuals with exactly the same attribute values, Verma and Rubin [93] generated, for every applicant in the dataset, an identical individual of the opposite gender. The result of applying causal discrimination is the percentage of violations in the entire population (i.e. how many individuals are unfairly treated?). Fairness through awareness Fairness through awareness [29] (a.k.a individual fairness [37,59] Let D be a distance metric between probability distributions. Fairness through awareness is achieved iff, for any pair of individuals i and j: D(M (v i ), M (v j )) ≤ d(v i , v j )(17) For our hiring example, this implies that the distance between the distribution of outcomes of two applicants should be at most the distance between those applicants 24 . A possible relevant features to use for measuring the similarity between two applicants might be the education level and the job experience. The distance metric d between two applicants could be defined as the average of the normalized difference (the difference divided by the maximum difference in a dataset) of their education level and their job experience. More formally, let E vi and E vj be the education levels of individuals i and j, respectively, and let N E be the normalized difference between education levels, that is, N E = \|Ev i −Ev j \| mE where m E is the maximum difference in education level in the dataset. Similarly, let J vi and J vj be the job experience of individuals i and j, while N J is the normalized difference of the job experience, that is, N J = \|Jv i −Jv j \| mJ where m J is the maximum difference in job experience in the dataset. The distance metric is defined as: d(v i , v j ) = N E + N J 2 , The distance between the probability distributions over the outcomes could be the Hellinger distance [72]. Let {y 1 , y 2 , . . . , y K } be the set of possible outcomes and let P and Q two (discrete) probability distributions. The Hellinger distance between P and Q is defined as: Table 13 shows a sample from the job hiring dataset on which fairness through awareness is applied. The result of applying fairness through awareness is shown in Table 14. Each cell at the left of the shaded diagonal represents a distance between two individuals and each cell at the right of the shaded diagonal represents the distance between probability outcomes of two individuals. For instance: H(P, Q) = 1 √ 2 K k=1 P (y k ) − Q(y k ) 2d(F 1, F 2) = 0.25 While: D(M (F 1), M (F 2)) = 1 √ 2 √ 0.4 − √ 0.3 2 + √ 0.6 − √ 0.7 2 = 1 √ 2 √ 0.0081 + 0.0036 = 0.07 The cell values in bold represent the cases where fairness through awareness is not satisfied: D d. For example, 0.07 (< 0.0) implies that F 1 is discriminated compared to M 3 . Similarly, M 2 is discriminated compared to F 3 , F 2 , and M 3 . Fairness through awareness is more fine-grained than any group fairness notion presented earlier in Sections 5.1-5.8. For instance, in the example of Table 13, statistical parity is satisfied: 0.33 for both men and women. Likewise, d(v i , v j ) equalized odds ( 5.3) is satisfied as the TPR and the FPR are equal for male and female applicants (0.5 and 0, respectively). Nevertheless, Table 14 shows that when comparing each pair of individuals (regardless of their gender) cases of discrimination have been discovered. It is important to mention that, in practice, fairness through awareness introduces some challenges. For instance, it assumes that the similarity metric is known for each pair of individuals [55]. That is, a challenging aspect of this approach is the difficulty to determine what is an appropriate metric function to measure the similarity between two individuals. Typically, this requires careful human intervention from professionals with domain expertise [59]. For instance, suppose a company is intending to hire only two employees while three applicants i 1 , i 2 and i 3 are eligible for the offered job. Assume i 1 has a bachelor's degree and 1 year related work experience, i 2 has a master's degree and 1 year related work experience and i 3 has a master's degree but no related work experience ( Figure 2). Is i 1 closer to i 2 than i 3 ? If so, by how much? This is difficult to answer, especially if the company overlooked such specific cases and did not carefully define and set a suitable and fair similarity metric in order to rank applicants for job selection. Thus, fairness through awareness can not be considered suitable for domains where trustworthy and fair distance metric is not available. Causality-based fairness notions Causality-based fairness notions differ from all aforementioned statistical fairness approaches in that they are not totally based on data but consider additional knowledge about the structure of the world, in the form of a causal model. This additional knowledge helps us understand how data is generated in the first place and how changes in variables propagate in a system. Most of these fairness notions are defined in terms of non-observable quantities such as interventions (to simulate random experiments) and counterfactuals (which consider other hypothetical worlds, in addition to the actual world). A variable X is a cause of a variable Y if Y in any way relies on X for its value [77]. Causal relationships are expressed using structural equations [13] and represented by causal graphs where nodes represent variables (attributes) and edges represent causal relationships between variables. Figure 3 shows a possible causal graph for our hiring example where directed edges indicate causal relationships. Statistical parity (Section 5.1) is known also as total variation (TV) as it can be expressed by subtracting the two terms in Eq. 1 as follows: T V a1,a0 (ŷ) = P (Ŷ =ŷ \| A = a 1 ) − P (Ŷ =ŷ \| A = a 0 )(18) A T V equal zero indicates fairness according to statistical parity. As T V is purely a statistical notion, it is unable to reflect the causal relationship between A and Y , that is, it is insensitive to the mechanism by which data is generated. Total effect (T E) [76] is the causal version of T V and is defined in terms of experimental probabilities as follows : T E a1,a0 (ŷ) = P (ŷ A←a1 ) − P (ŷ A←a0 )(19) P (ŷ A←a ) = P (Ŷ =ŷ \| do(A = a)) is called an experimental probability and is expressed using intervention. An intervention, noted do(V = v), is a manipulation of the model that consists in fixing the value of a variable (or a set of variables) to a specific value. Graphically, it consists in discarding all edges incident to the vertex corresponding to variable V . Intuitively, using the job hiring example, while P (Ŷ = 1 \| A = 0) reflects the probability of hiring among female applicants, P (Ŷ A←0 = 1 = P (Ŷ = 1) \| do(A = 0)) reflects the probability of hiring if all the candidates in the population had been female. The obtained distribution P (Ŷ A←a ) can be considered as a counterfactual distribution since the intervention forces A to take a value different from the one it would take in the actual world. Such counterfactual variable is noted alsoŶ A=a orŶ a for short. T E measures the effect of the change of A from a 1 to a 0 onŶ =ŷ along all the causal paths from A toŶ . Intuitively, while T V reflects the difference in proportions ofŶ =ŷ in the current cohort, T E reflects the difference in proportions ofŶ =ŷ in the entire population. A more involved causal-based fairness notion considers the effect of a change in the sensitive attribute value (e.g. gender) on the outcome (e.g. probability of hiring) given that we already observed the outcome for that individual. This typically involves an impossible situation which requires to go back in the past and change the sensitive attribute value. Mathematically, this can be formalized using counterfactual quantities. The simplest fairness notion using counterfactuals is the effect of treatment on the treated (ETT) [76]. The effect of treatment on the treated (ETT) is defined as: ET T a1,a0 (ŷ) = P (ŷ A←a1 \| a 0 ) − P (ŷ \| a 0 )(20) P (ŷ A←a1 \| a 0 ) reads the probability ofŶ =ŷ had A been a 1 , given A had been observed to be a 0 . For instance, in the job hiring example, P (Ŷ A←1 \| A = 0) reads the probability of hiring an applicant had she been a male, given that the candidate is observed to be female. Such probability involves two worlds: an actual world where A = a 0 (the candidate is female) and a counterfactual world where for the same individual A = a 1 (the same candidate is male). Notice that P (ŷ A←a0 \| a 0 ) = P (ŷ \| a 0 ), a property called consistency [76]. Counterfactual fairness [59] is a fine-grained variant of ETT conditioned on all attributes. That is, a predictionŶ is counterfactually fair if under any assignment of values X = x, P (Ŷ A←a1 =ŷ \| X = x, A = a 0 ) = P (Ŷ A←a0 =ŷ \| X = x, A = a 0 )(21) where X is the set of all attributes excluding A. Since conditioning is done on all remaining variables X, counterfactual fairness is an individual notion. According to Eq. 21, counterfactual fairness is satisfied if the probability distribution of the outcomeŶ is the same in the actual and counterfactual worlds, for every possible individual. In the job hiring example, an MLDM is counterfactually fair if: P (Ŷ A←1 \| X = x, A = 0) = P (Ŷ A←0 \| X = x, A = 0)(22) The main problem with the applicability of TE, ETT, and counterfactual fairness is the computation of the nonobservable terms in Eqs 19, 20, and 21. These terms are either interventional (e.g. P (ŷ A←a1 )) or counterfactual (e.g. P (Ŷ A←a1 =ŷ \| X = x, A = a 0 ). In scenarios where these quantities can be expressed in terms of observable probabilities (e.g. joint probabilities, conditional probabilities, etc.), it is said that they are identifiable. Otherwise, they are unidentifiable. Typically, the identifiability of interventional and counterfactual quantities depends on the structure of the causal graph [84,76]. Alternatively, if all parameters of the causal model are known (including the latent variables distributions P (U = u)), any counterfactual is identifiable and can be computed using the three steps abduction, action, and prediction (Theorem 7.1.7 in [76]). The details of the computation of a counterfactual probability using a simple deterministic example are provided in A. A simple but important implication of Eq. 21 is that, given a causal graph, a predictorŶ is counterfactually fair if it is a function of non-descendants of the sensitive variable A. In other words, ifŶ is a function of variables that depend on A (there is a directed path between any one of those variables and A), it is not counterfactually fair. Consequently, one can tell if a predictor is counterfactually fair by simply checking the causal graph 25 . No unresolved discrimination [53] is another causal-based fairness notion which is satisfied when no directed paths from the sensitive attribute A to the predictorŶ are allowed, except via a resolving variable. A resolving variable is any variable in a causal graph that is influenced by the sensitive attribute in a manner that is accepted as nondiscriminatory (this is similar to explanatory attributes in conditional statistical parity (Section 5.2)). In the job hiring example, if we assume that the effect of A on the education level is nondiscriminatory, it implies that the differences in education level for different values of A are not considered as discrimination. Thus, a disparity in the predictions between men and women might been explained and justified by their corresponding education levels. Hence, the education level acts as a resolving variable. Figure 4 shows two similar causal graphs for our hiring example, yet differ in some of the causal relations between variables. By considering the education as a resolving variable, the graph at the left exhibits unresolved discrimination along the dashed paths: A → Experience →Ŷ and A →Ŷ . By contrast, the graph at the right does not exhibit any unresolved discrimination as the effect of A onŶ is justified by the resolved variable Education: A → Education →Ŷ . Figure 4: Two possible graphs for the hiring example. If Education is a resolving variable, the predictorŶ exhibits unresolved discrimination in the left graph (along the dashed paths), but not in the right one. No unresolved discrimination is equivalent to other fairness notions in some interesting special cases [53]. For instance, if no resolving variables exist, no unresolved discrimination is analogous to statistical parity (Section 5.1) in a causal context. A andŶ are statistically independent and no directed paths from A toŶ are allowed. Likewise, no unresolved discrimination might be equivalent to equalized odds (Section 5.3) in a causal context if the set of resolving variables is the singleton set of actual outcomes: {Y }. Compared to counterfactual fairness, no unresolved discrimination is a weaker notion. That is, a counterfactually unfair scenario may be identified as fair based on no unresolved discrimination. This can happen in case one or several variables in the causal graph are identified as resolving. A causal graph exhibits potential proxy discrimination [53] if there exists a path from the protected attribute A to the predicted outcomeŶ that is blocked by a proxy variable P x . A proxy is a descendant of A that is chosen to be labelled as a proxy because it is significantly correlated with A. Given a causal graph, a predictorŶ exhibits no proxy discrimination if following equality holds for all potential proxies P x . P (Ŷ Px←p ) = P (Ŷ Px←p′ ) ∀ p, p′(23) In other words, Eq. 23 implies that changing the value of P x should not have any impact on the prediction. In the job hiring example, the job experience can be considered as a proxy of an individual's gender. Figure 5 shows two similar causal graphs. The one at the left presents a potential proxy discrimination via the path: A → Experience →Ŷ . However, the graph at the right is free of proxy discrimination as the edge between A and its proxy P x (here Experience) has been removed along with all incoming arrows of P x (the edge between Education and Experience). Figure 5: Two possible graphs to describe proxy discrimination. If we consider Experience as a proxy of the sensitive attribute A, the graph at the left exhibits a potential proxy discrimination (along the dashed edge between A and Experience), but not in the right one. Other causal based fairness notions include direct/indirect effect [75], FACE/FACT [52], counterfactual effects [104], counterfactual error rates [103], and path-specific counterfactual fairness [17,98]. As a general rule, causality-based fairness notions can be used as long as the causal relationships between the attributes are identified and represented using a reliable and plausible causal graph. The construction of the causal graph requires typically domain-specific expertise and can be validated by existing datasets. In practice, however, causality-based fairness notions are recommended in at least two notable scenarios. The first scenario is when the legal framework of the case at hand is disparate treatment. In such framework, to win a discrimination case, the plaintiff must show that the defendant has used (directly or indirectly (via proxy)) the sensitive attribute A to take the discriminatory decision Y . In other words, she must prove that the variable A is a cause ofŶ while the causal effect of A onŶ is central to all causal-based fairness notions mentioned above. The second scenario is when there is confounding between A andŶ . That is, there is a covariate which is a common cause of A andŶ . Such scenario can lead to statistical anomalies such as Simpson's paradox [87,76] where the statistical conclusions drawn from the sub-populations differ from that from the whole population. The Berkeley admission case [11] is a known real-world example of such statistical anomaly. In such scenarios, any statistical fairness notion which relies solely on correlation between variables, will fail to detect bias. Hence, causality-based fairness notions are necessary to appropriately address the problem of fairness. Relaxation Almost all fairness notions presented so far involve a strict equality between quantities, in particular probabilities. In real scenarios, however, it is more suitable to opt for an approximate or relaxed form of fairness constraint. The need for relaxation might be due to the impossibility to apply fairness strictly on the application at hand, or merely, it is not a requirement to impose an exact constraint [54]. Fairness notion definitions can be relaxed by considering a threshold on the ratio or difference between quantities. For instance, the requirement for statistical parity (Section 5.1) can be relaxed in one of the two following ways: • By allowing the ratio between the predicted acceptance rates of protected and unprotected groups to reach the threshold of ǫ (a.k.a p% rule defined as satisfying this inequality when ǫ = p/100 [101]): P (Ŷ \| A = 0) P (Ŷ \| A = 1) ≥ 1 − ǫ ∀ ǫ ∈ [0, 1](24) For ǫ = 0.2, this condition relates to the 80% rule in disparate impact law [34,8]. • By allowing the difference between the predicted acceptance rates of different groups to reach a threshold of ǫ [29]: \| P (Ŷ \| A = 0) − P (Ŷ \| A = 1) \| ≤ ǫ ∀ ǫ ∈ [0, 1](25) A notable difference between the two types of relaxation is that the second one (Eq. 25) is insensitive to which group/individual is the victim of discrimination as the formula is using absolute value. Fairness through awareness can be relaxed using three threshold values, α 1 , α 2 , and γ as follows [99]: P P [\|M (v i ) − M (v j )\| > d(v i , v j ) + γ ] > α 2 ≤ α 1 .(26) The relaxation is allowing M (v i )−M (v j ) to exceed d(v i , v j ) by a margin of γ, but the fraction of individuals differing from them by γ should not exceed α 2 . If the fraction exceeds α 2 , the individual is said to be α 2 -discriminated against. To allow for more flexibility in the application of fairness notions, other relaxations can be considered. For instance, Eq. 2 of conditional statistical parity (Section 5.2) can be modified by relaxing the strict equality E = e as follows: P (Ŷ = 1 \| e − ǫ ≤ E ≤ e + ǫ, A = 0) = P (Ŷ = 1 \| e − ǫ ≤ E ≤ e + ǫ, A = 1)(27) Classification and tensions Group fairness notions fall into three classes defined in terms of the properties of joint distributions, namely, independence, separation, and sufficiency [7]. These properties are used in the literature to prove the existing of tensions between fairness notions, that is, it is impossible to satisfy all fairness notions simultaneously except in extreme, degenerate, and dump scenarios. Besides, the applicability of most of fairness notions can be ameliorated by relaxing their strict definitions. Classification Group fairness (a.k.a statistical fairness) notions can be characterized by the properties of the joint distribution of the sensitive attribute A, the label Y , and the classifierŶ (or score S). This means that we can write them as some statement involving properties of these three random variables resulting in the three following fairness criteria [6,7]: Independence Independence means that the sensitive feature A is statistically independent of the classifierŶ (or the score S).Ŷ ⊥ A (or S ⊥ A) (28) In the case of binary classification, independence is equivalent to statistical parity as defined in Section 5.1, Eq. 1. Conditioning on explanatory variables (E) yields a variant of independence as follows. Conditional independenceŶ ⊥ A \| E (or S ⊥ A \| E)(29) This class includes conditional statistical parity defined in Section 5.2, Eq. 2. Separation Separation denotes a class of fairness notions satisfying, at different degrees, conditional independence between the predictionŶ and the sensitive attribute A given the actual outcome Y . Y ⊥ A \| Y (or S ⊥ A \| Y )(30) In the case whereŶ is a binary classifier, the formulation of separation is equivalent to that of the equalized odds (Eq. 3). Equal opportunity (Eq. 4), predictive equality (Eq. 5), balance for positive class (Eq. 11), and balance for negative class (Eq. 12) are all relaxations of separation. Some incompatibility results do hold for separation, but do not hold for the relaxations. More on this in the next section (Section 7.2). Sufficiency Sufficiency is a class of fairness notions satisfying, at different degrees, conditional independence between the target variable Y and the sensitive attribute A given the predictionŶ . Y ⊥ A \|Ŷ (or Y ⊥ A \| S)(31) In the case of binary classification, strict sufficieny corresponds to conditional use accuracy equality (Eq. 6). Using the score S, calibration (Eq. 14), and well-calibration (Eq. 15) can be considered as sufficiency [18]. Relaxation of sufficiency yields to predictive parity (Eq. 7) which also does not satisfy exactly the same incompatibility result as sufficiency (Section 7.2). Table 15 lists all fairness notions along with their classification. Machine learning fairness notions: Bridging the gap with real-world applications Statistical parity [29] P (Ŷ \| A = 0) = P (Ŷ \| A = 1) Independence (equivalent or relaxed ⋆ ) Conditional statistical parity [21] P (Ŷ = 1 \| E = e, A = 0) = P (Ŷ = 1 \| E = e, A = 1) ⋆ Equalized odds [44] P (Ŷ = 1 \| Y = y, A = 0) = P (Ŷ = 1 \| Y = y, A = 1) ∀y ∈ {0, 1} Equal opportunity P (Ŷ = 1 \| Y = 1, A = 0) = P (Ŷ = 1 \| Y = 1, A = 1) ⋆ Separation (equivalent or relaxed ⋆ ) Predictive equality [21] P (Ŷ = 1 \| Y = 0, A = 0) = P (Ŷ = 1 \| Y = 0, A = 1) ⋆ Balance for positive class [58] E[S \| Y = 1, A = 0)] = E[S \| Y = 1, A = 1] ⋆ Balance for negative class E[S \| Y = 0, A = 0] = E[S \| Y = 0, A = 1] ⋆ Overall balance * E[S \| Y = y, A = 0] = E[S \| Y = y, A = 1] ∀y ∈ {0, 1} Group Conditional use acc. equality Total effect [76] T Ea 1 ,a 0 (ŷ) = P (ŷA←a 1 ) − P (ŷA←a 0 ) Effect of treatment on treated ET Ta 1 ,a 0 (ŷ) = P (ŷA←a 1 \| a0) − P (ŷ \| a0) No unresolved discrimination [53] − Causality No proxy discrimination P (Ŷ \| do(Px = p)) = P (Ŷ \| do(Px = p′)) ∀Px and ∀ p, p′ Tensions It has been proved that there are incompatibilities between fairness notions. That is, it is not always possible for an MLDM to satisfy specific fairness notions simultaneously [6,7,18,100,71]. In presence of such incompatibilities, the MLDM should make a trade-off to satisfy some notions on the expense of others or partially satisfy all of them. Incompatibility 26 results are well summarized by Mitchell et al. [71] as follows: Statistical parity (independence) versus conditional use accuracy equality (sufficiency) Independence and sufficiency are incompatible, except when both groups (protected and non-protected) have equal base rates orŶ and Y are independent. Note, however, thatŶ and Y should not be independent since otherwise the predictor is completely useless. More formally, Y ⊥ A AND Y ⊥ A \|Ŷ ⇒ Y ⊥ A ORŶ ⊥ Y (independence) (strict sufficiency) (equal base rates) (useless predictor) It is important to mention here that this result does not hold for the relaxation of sufficiency, in particular, predictive parity. Hence, it is possible for the output of an MLDM to satisfy statistical parity and predictive parity between two groups having different base rates. Such example needs to satisfy the following constraints, assuming two groups a and b: T Pa+F Pa T Pa+F Pa+F Na+T Na = T P b +F P b T P b +F P b +F N b +T N b (independence) T Pa T Pa+F Pa = T P b T P b +F P b (predictive parity) T Pa+F Na T Pa+F Pa+F Na+T Na = T P b +F N b T P b +F P b +F N b +T N b (different base rates) An example scenario satisfying the above constrains is the following: P P V a = 0.4 baserate a = 0.43 T P a = 9 F P a = 6 F N a = 4 T N a = 11 T P b = 12 F P b = 8 F N b = 2 T N b = 18 P P V b = 0.4 baserate b = 0.35 Statistical parity (independence) versus equalized odds (separation) Similar to the previous result, independence and separation are mutually exclusive unless base rates are equal or the predictorŶ is independent from the actual label Y [7]. As mentioned earlier, dependence betweenŶ and Y is a weak assumption as any useful predictor should satisfy it. More formally, Y ⊥ A ANDŶ ⊥ A \| Y ⇒ Y ⊥ A ORŶ ⊥ Y (independence) (strict separation) (equal base rates) (useless predictor) Considering a relaxation of equalized odds, that is, equal opportunity or predictive equality, breaks the incompatibility between independence and separation. An MLDM whose output satisfies independence and equal opportunity, but with different base rates between groups should satisfy the following constraints: T Pa+F Pa T Pa+F Pa+F Na+T Na = T P b +F P b T P b +F P b +F N b +T N b (independence) T Pa T Pa+F Na = T P b T P b +F N b (equal opportunity) T Pa+F Na T Pa+F Pa+F Na+T Na = T P b +F N b T P b +F P b +F N b +T N b (different base rates) An example scenario satisfying the above constrains is the following: T P R a = 0.6 baserate a = 0.55 T P a = 9 F P a = 3 F N a = 2 T N a = 6 26 The term impossibility is commonly used as well. T P b = 12 F P b = 6 F N b = 8 T N b = 4 T P R b = 0.6 baserate b = 0.71 Equalized odds (separation) vs conditional use accuracy equality (sufficiency) Separation and sufficiency are mutually exclusive, except in the case where groups have equal base rates. More formally: Y ⊥ A \| Y AND Y ⊥ A \|Ŷ ⇒ Y ⊥ A (strict separation) (strict sufficiency) (equal base rates) Both separation and sufficiency have relaxations. Considering only one relaxation will only drop the incompatibility for extreme and degenerate cases. For example, predictive parity (relaxed version of sufficiency) is still incompatible with separation (equalized odds), except in the following three extreme cases [18]: • both groups have equal base rates. • both groups have F P R = 0 and P P V = 1. • both groups have F P R = 0 and F N R = 1. The incompatibility disappears completely when considering relaxed versions of both separation and sufficiency. For example, the following scenario satisfies equal opportunity (relaxed version of separation) and predictive parity (relaxed version of sufficiency) while base rates are different in both groups: T P R a = 0.4 P P V a = 0.75 baserate a = 0.6 T P a = 9 F P a = 6 F N a = 3 T N a = 2 T P b = 12 F P b = 8 F N b = 4 T N b = 8 T P R b = 0.4 P P V b = 0.75 baserate b = 0.5 Group vs individual fairness Compared to individual fairness notions, the main concern for group fairness notions is that they are only suited to a limited number of coarse-grained, predetermined protected groups based on some sensitive attribute (e.g. gender, race, etc.). Hence group fairness notions are not suitable in presence of intersectionality [24] where individuals are often disadvantaged by multiple sources of discrimination: their race, class, gender, religion, and other inner traits. Typically, statistical fairness can only be applied across a small number of coarsely defined groups, and hence failing to identify discrimination on structured subgroups (e.g. single women) known also as "fairness gerrymandering" [51]. A simple alternative might be to apply statistical fairness across every possible combination of protected attributes. There are at least two problems to this approach. First, this can lead to an impossible statistical problem with the large number of sub-groups which may lead in turn to overfitting. Second, groups which are not (yet) defined in anti-discrimination law may exist and may need protection [94]. Another issue with group fairness notions is their susceptibility to masking. Most of group fairness notions can be gamed by adding arbitrarily selected samples to satisfy the fairness notion formula, that is, to just "make up the numbers". Compared to group fairness notions, individual fairness notions have the drawback that they can result in "unjust disparities in outcomes between groups" [12]. For illustration, consider the example in Table 16 where fairness through awareness is satisfied (Eq. 17) whereas statistical parity Eq. (1) is not. Fairness through awareness is satisfied since for every pair of candidates, the distance between the probability distributions on the outcomes (M ()) is smaller than the distance between the pair of candidates. On the other hand, if the hiring threshold is 0.6, only one female candidate (F 2) will be hired as she has a probability of acceptance P (Ŷ = 1) = 0.8 > 0.6 whereas all male candidates will be hired. Another important issue for similarity-based individual fairness (e.g. fairness through awareness) is the difficulty to obtain a similarity value between every pair of individuals. For example, even with the assumption that the similarity can be quantified between all individuals in the training data, it might be challenging to generalize to new individuals [12]. Several researchers assume that both group and individual fairness are prominent, yet, conflicting and suggest approaches to minimize the trade-offs between these notions [12]. For instance, [35] define two different worldviews, WYSIWYG and WAE. The WYSIWYG (What you see is what you get) worldview assumes that the unobserved (construct) space and observed space are essentially the same while the WAE (we're all equal) worldview implies that there are no innate differences between groups of individuals based on certain potentially discriminatory characteristics. These two worldviews highlight the tension between group and individual fairness. For instance, in the job hiring example, the WYSIWYG might be the assumption that attributes like education level and job experience (which belong to the observed space) correlate well with the applicant's seriousness or hardworking (properties of the construct space). This is to say that there is some way to combine these two spaces to correctly compare true applicant aptitude for the job. On the other hand, the WAE claims that all groups will have almost the same distribution in the construct space of inherent abilities (here, seriousness and hardworking), chosen as important inputs to the decision d(v i , v j ) making process. The idea is that any difference in the groups' performance (e.g., academic achievement or education level) is due to factors outside their individual control (e.g., the quality of their neighborhood school) and should not be taken into account in the decision making process. Thus, the choice between fairness notions must be based on an explicit choice in worldviews. Diagram and discussion With the large number of fairness notions and the subtle resemblance between MLDM scenarios, deciding about which fairness notion to use is not a trivial task. More importantly, selecting and using a fairness notion in a scenario inappropriately may detect unfairness in an otherwise fair scenario, or the opposite, i.e., fail to identify unfairness in an unfair scenario. One of the objectives of this survey is to systemize the selection procedure of fairness notions. This is achieved by identifying a set of fairness-related characteristics (Section 4) of the scenario at hand and then use them to recommend the most suitable fairness notion for that specific scenario. The proposed systemized selection procedure is illustrated in the decision diagram of Figure 6. The diagram is called "decision diagram" and not "decision tree" for the following reason. In typical decision trees, every leaf corresponds to a single decision, which is a fairness notion that should be used. However, the diagram in Figure 6 is designed such that every node indicates which notions are recommended, which notions to be avoided, and which notions must not be used. In addition, if a notion is not mentioned along the path, it means, it can be safely used. The diagram is composed of four types of nodes: • Decision node (diamond): based on fairness-related characteristics (Section 4). • Recommended node (rectangle): a leaf node indicating that the fairness notion is suitable to be used given all fairness-related characteristics in the path to that node. • Warning node (triangle): indicates that the fairness notion(s) is/are not recommended in all the branch in the right of the node. This node can appear in the middle of the edge between two decision nodes. • Must-not node (circle): the fairness notion must not be used. To illustrate how the diagram should be interpreted, consider the recommended node predictive parity (node 38). According to the diagram, predictive parity is recommended in the scenario where the legal framework is disparate impact (decision node 1), intersectionality and/or masking are unlikely (decision node 2), there is no evidence that representation bias is likely (decision node 2), standards do not exist (decision node 4), ground-truth is available or outcome Y is reliable (decision node 11), historical and measurement bias are unlikely (decision node 11), fairness is more sensitive to precision rather than recall (decision node 22), the prediction threshold is typically fixed (decision node 24) and the emphasis is on false positives rather than false negatives (decision node 28). In that particular scenario, equal opportunity must not be used (must-not node 45) because fairness in this scenario is particularly sensitive to false positives, while equal opportunity is completely insensitive to false positives. Similarly, negative predictive parity must not be used (must-not node 46) as fairness is sensitive to precision rather than recall. The warning node 17 along the same path indicates that statistical parity is not suitable in this scenario. Finally, any fairness notion for which there is no a warning node or a must-not node along the path of the scenario can be used in this scenario. For instance, all individual fairness notions can be used. As concrete example of situations where predictive parity (node 38) is recommended, consider the following. In situations when the outcome is influenced by the decision, some statistical quantities (e.g. FN, TN, etc.) are unlikely to be observed, and hence, any fairness notion that is defined in terms of those quantities is not suitable to use. For example, in real-world cases of loan-granting, a loan application which is predicted to be defaulting, will not be approved. Consequently, both negative statistics (true negative (TN) and false negative (FN)) will not be typically observed. Hence, fairness notions such as equalized odds and equality of opportunity cannot be used as they are defined in terms of TN and FN. In such cases, predictive parity (node 38) is recommended. Node 1: Assessing fairness is very often performed in the context of a legal case where a plaintiff is filing a claim against a party that is using an MLDM. According to real-world legislation, in particular, the American antidiscrimination law, this can fall into one the two legal frameworks, namely, disparate impact and disparate treatment. If the plaintiff is filing the claim under the disparate impact framework, she can prove the liability of the defendant by using an observational group or individual fairness notion as the goal is to show that the practices and policies used by the defendant are facially neutral but have a disproportionately adverse impact on the protected class [8]. If, however, the plaintiff is filing a claim under the disparate treatment framework, observational fairness notions are often not enough to prove the liability of the defendant as the goal is to show that the defendant has used the sensitive attribute to take the discriminatory decision. The recommended fairness notions in that case are causality-based (recommended node 3) since all of them are expressed in terms of the causal effect of the sensitive attribute on the prediction. Node 2: As explained above, any unintentional type of bias can also be "orchestrated" intentionally by decision makers with prejudicial views. For instance, decision makers can purposefully bias the data collection step to ensure that the MLDM remains less favorable to protected classes. To reliably assess the bias in presence of such masking attempts, all group fairness notions should be avoided as they are defined in terms of statistics about the different sub-populations and hence can more easily be gamed by prejudicial decision makers. Intersectionality is similar to masking as both lead to a discrimination which is difficult to detect using statistical measures and consequently requires more finegrained measures. Therefore individual fairness notions are recommended in presence of both criteria (nodes 9 and 18). Nodes 2, 3, and 11: In case one or more sources of bias are suspected ahead of time (before assessing fairness), the information can help warn against the use of some fairness notions. If representation bias is likely, the performance (accuracy) of the MLDM on under-represented categories will often be worse. Such disparity in performance between groups may lead to unreliable fairness assessment in case a group fairness notion is used, in particular disparate mistreatment notions (grayed section of the diagram). In such case, individual fairness notions can assess fairness more reliably provided that measurement bias is not likely (node 2). A suspicion of historical or measurement bias means that the features (X) and/or the label (Y) are not reliable. All group fairness notions using the label Y (disparate mistreatment) as well as individual notions are not recommended in that case. Statistical parity is recommended in such situation. Finally, in presence of either aggregation, evaluation, or deployment bias, causality-based fairness notions are recommended. The reason is that the interventional and counterfactual quantities used in the definitions of these notions go beyond mere correlations and hence allow to assess fairness more reliably in presence of such bias. For instance, Coston et al. [22] propose counterfactual formulations of fairness metrics to properly account for the effect of intervention (decision) on the outcome. Such effect is a type of deployment bias. Node 3: As discussed in Section 5.12, there are several notions that use causal reasoning to assess fairness. Counterfactual fairness is suitable in case a fine-grained assessment is required as the equality of Eq. 21 conditions on all features (X). Counterfactual fairness, however, requires strong assumptions to be applicable in real scenarios (the availability of the full causal model including the latent variables distributions). Total effect (TE), effect of treatment on treated (ETT), and no proxy discrimination (nodes 13, 14 and 10), on the other hand, require a weaker assumption to be applicable, namely, the identifiability of the causal quantities used in their definitions. No proxy discrimination is recommended in presence of potential proxies, however, the identification of proxy variables requires a domain expertise of the application at hand. Finally, in case there are variables in the causal graph which are correlated with the sensitive attribute but in a manner that is accepted as nondiscriminatory, no unresolved discrimination is recommended while the remaining causal based fairness notions should be avoided. No unresolved discrimination is easier to apply in practice as it only needs the availability of the causal graph. Node 4: To reduce inequality and historical discrimination against sub-populations, in particular, minorities, some states and organizations resort to equality standards and regulations such as the laws enforced by the US Equal Employment Opportunity Commission [1]. In presence of such standards, to be deemed fair, an MLDM should satisfy such standards. Consequently, all what matters for fairness assessment is the proportion of positive prediction across all groups which corresponds to statistical parity. Node 17: If no standards/regulations exist (node 4) and either the ground truth exists or the outcome label Y is available (node 11), statistical parity is not recommended (node 17) as it can lead to misleading results such as detecting unfairness in an otherwise fair scenario or failing to identify fairness in an unfair scenario. For instance, in stop-andfrisk real world scenario applied in New York city starting 1990 [9] 27 , the ground truth is available as by frisking an individual, a police officer can know with certainty the presence or no of illegal substance. In such case, one or several disparate mistreatment notions (nodes 30-41) are more suitable to assess fairness. Nodes 22-47: The bulk of Figure 6 is dedicated for disparate mistreatment fairness notions and the criteria leading to each one of them. These notions define fairness in terms of the disparity of misclassification rates among the different groups in the population. Based on their definitions, selecting the most suitable notion to use depends on four citeria, namely, whether the emphasis is on precision or recall (node 22), whether the threshold is fixed or floating (nodes 23 and 24), whether the emphasis is on false negatives or false positives (nodes 26 and 28), and finally whether the emphasis is on the positive or negative class (node 27). As some notions focus only on either FP or FN (nodes 31, 32, 38, and 39), any notion that is insensitive to either FP or FN must not be used (nodes 42 -47). The diagram may be misleading if it is interpreted very categorically. This occurs when a user of the diagram navigates it and ends up using the recommended fairness notion without considering other important elements specific to the scenario at hand. The diagram can be misleading also when it is not clear which branch to take in a decision node. For example, the question in decision node 22 (emphasis on precision or recall?) is difficult to answer categorically in several scenarios. The decision nodes 4, 21, 12, and even 2, are typically easier to navigate, but can be challenging to settle in a number of scenarios. Moreover, in presence of measurement bias, the values of some features and even the outcome label may not be reliable which can make the diagram navigation more challenging. A potential solution would be to label one of the branches as default (to be followed when the answer is not clear), but this can, often result in a suboptimal decision. In summary, the diagram should be considered as guide and should never be used to supersede important elements specific to the scenario at hand. Table 17 states explicitly the relationship between every selection criterion and every fairness notion. The table uses four symbols, namely, recommended (✓), warning ( ), must-not (✗), and insensitive (−). Insensitive means that the choice of the fairness notion is independent of the selection criterion. Conclusion With the increasingly large number of fairness notions considered in the relatively new field of fairness in ML, selecting a suitable notion for a given MLDM (machine learning decision making) becomes a non-trivial task. There are two contributing factors. First, the boundaries between the defined notions are increasingly fuzzy. Second, applying inappropriately a fairness notion may report discrimination in an otherwise fair scenario, or vice versa, fail to identify discrimination in an unfair scenario. This survey tries to address this problem by identifying fairness-related characteristics of the scenario at hand and then use them to recommend and/or discourage the use of specific fairness notions. The main contribution of this survey is to systemize the selection process based on a decision diagram. Navigating the diagram will result in recommending and/or discouraging the use of fairness notions. One of the main objectives of this survey is to bridge the gap between the real-world use case scenarios of automated (and generally unintentional) discrimination and the mostly technical tackling of the problem in the literature. Hence, the survey can be of particular interest to civil right activists, civil right associations, anti-discrimination law enforcement agencies, and practitioners in fields where automated decision making systems are increasingly used. More generally, in real-scenarios, there are still two important obstacles to address the unfairness problem in automated decision systems. First, the victims of such systems are, very often, members of minority groups with limited influence in the public sphere. Second, automated decision systems are geared towards efficiency (typically money) and to optimize profit, they are designed to sacrifice the outliers as tolerable collateral damage. After all, the system is benefiting most of the population (employers finding ideal candidates, banks giving loans to minimum risk borrowers, a society with recidivists locked in prisons, etc.). To illustrate how counterfactual quantities are computed, consider the simplified deterministic version of the hiring example in Figure 7. For simplicity, the hiring score variable S depends on the observable variable JE representing job experience and the exogenous variable U h representing how hard working the candidate is. The variable JE in turn depends on the observable sensitive variable A representing the gender (male or female) and the exogenous variable U s representing the seriousness of the candidate. The causal graph in Figure 7 is represented by the two following equations: JE = a.A + c.U s (32) S = b.JE + d.U h(33) For simplicity of the illustration, assume that both U (U s and U h ) variables are independent and all the parameters of the model (Eq. 32 and 33) are known. Assume that the values of the coefficients are given as follows: The second step consists in setting the sensitive attribute A John to the opposite gender (0) and updating all equations of the model. This consists in replacing the variable A in Eq. 32 by 0. a The third step consists in the prediction, that is computingŜ A←0 in the counterfactual world. This requires the computation of JE John A←0 , that is, the job experience of John in a world where John is a female. Hence, the hiring score of John had he was female isŜ John A←0 = 0.48 which is considered a violation of counterfactual fairness as the predicted hiring score of John in the original world isŜ John = 0.55. Consider now a female candidate Marie (A Marie = 0), with the a job education level JE Marie = 0.61 and a predicted scoreŜ Marie = 0.65. The question to investigate is now: what would Marie's hiring score have been had she was male? This boils down to computingŜ Marie A←1 and comparing it withŜ Marie = 0.65. Applying the three-steps process: 28 To keep the computation simple, all variable values are normalized between 0 and 1. 29 Since this example is deterministic, every individual is characterized by a unique assignment for exogenous variables Us and U h . In typical (non-deterministic) scenarios, every individual is assigned a probability distribution over the exogenous variables. Figure 1 : 1F i and M i (i ∈ [1 − 10]) designate female and male applicants, respectively. The grey shaded circles indicate applicants who belong to the positive class while white circles indicate applicants belonging to the negative class. The dotted vertical line is the prediction boundary. Thus, applicants at the right of this line are predicted hired while applicants at the left are predicted not hired. 6 , 6while for a threshold value of 0.75, PPV for female group is 0.66 but for male it is 1.0. However, the calibration score (P (Y = 1 \| S = s, A = a) a ∈ {0, 1}, s ∈ [0, 1]) for every value of s is as follows: satisfied for score values 0.4 and 0.85, but not satisfied for score values 0.7 and 0.8. Overall, the scenario of Conditional statistical parity with E = X (conditioning on all non-sensitive attributes) is satisfied as the proportion of males and females having the exact same attribute values and predicted accepted is the same (0.5). However, at the individual level, causal discrimination is not satisfied as there are two violations: Female 1 vs Male 1 and Female 2 vs Male 2. The two violations compensated each others and as a result conditional statistical parity is satisfied. ) is a generalization of causal discrimination which implies that similar individuals should have similar predictions. Let i and j be two individuals represented by their attributes values vectors v i and v j . Let d(v i , v j ) represent the similarity distance between individuals i and j. Let M (v i ) represent the probability distribution over the outcomes of the prediction. For example, if the outcome is binary (0 or 1), M (v i ) might be [0.2, 0.8] which means that for individual i, P (Ŷ = 0) = 0.2 and P (Ŷ = 1) = 0.8. Figure 2 : 2An example showing the difficulty of selecting a distance metric in fairness through awareness Figure 3 : 3A possible causal graph for the hiring example. [ 10 ] 1 101P (Y = y \|Ŷ = y, A = 0) = P (Y = y \|Ŷ = y, A = 1) ∀y ∈ {0, 1} Predictive parity [18] P (Y = 1 \|Ŷ = 1, A = 0) = P (Y = 1 \|Ŷ = 1, A = 1) ⋆ . Sufficiency (equivalent or relaxed ⋆ ) Negative predictive parity * P (Y = 1 \|Ŷ = 0, A = 0) = P (Y = 1 \|Ŷ = 0, A = 1) ⋆ Calibration [18] P (Y = 1 \| S = s, A = 0) = P (Y = 1 \| S = s, A = 1) ∀s ∈ [0, 1] Well-calibration [58] P (Y = 1 \| S = s, A = 0) = P (Y = 1 \| S = s, A = 1) = s ∀ s ∈ [0, 1] Overall accuracy equality P (Ŷ = Y \|A = 0) = P (Ŷ = Y \|A = ] P (ŶA←a(U ) = y \| X = x, A = a) = P (Ŷ A←a ′(U ) = y \| X = x, A = a) Individual Causal discrimination [38] X(A=0) = X(A=1) ∧ A(A=0) = A(A=1) ⇒ŷ(A=0) =ŷ(A=1) Similarity Metric Fairness through awareness [29] D(M (vi), M (vj )) ≤ d(vi, vj ) Given this causal model, consider a candidate John who is male (A John = 1), with the normalized 28 job education level JE John = 0.6 and a predicted scoreŜ John = 0.55. Assessing the fairness of the hiring score prediction with respect to gender is achieved through answering the following question: what would John's hiring score have been had he was of opposite gender (female)? This corresponds to the hiring score of John in the counterfactual world where John is a female (Ŝ John A←0 ). To compute this quantity, the three-steps process above is used, namely, abduction, action, and prediction.The abduction step consists in using the evidence (A John = 1, JE John = 0.6,Ŝ John = 0.55) to identify the specific characteristics of John, namely, his level of seriousness and hard working (U s and U h ) 29 as follows: A←1 = 0.72 >Ŝ Marie = 0.65 is another violation for counterfactual fairness. Table 1 : 1Metrics based on confusion matrix.Actual Positive Actual Negative Table 3 : 3Application of conditional statistical parity by controlling on education level and job experience.(a) Dataset Gender Education Level Job Expe- rience Age Marital Status Y Female 1 8 2 39 single 0 Female 2 8 2 26 married 1 Female 3 12 8 32 married 1 Male 4 8 2 26 married 1 Male 5 8 2 41 single 0 Male 6 12 8 30 single 1 Table 4 : 4An extreme job hiring scenario satisfying equal opportunity. All Male 7 − 100 samples are false positives (label Y is 0 and predictionŶ is 1).(a) Dataset Gender Education Level Job Expe- rience Age Marital Status Y Female 1 8 2 39 single 1 Female 2 8 2 26 married 0 Female 3 12 8 32 married 1 Male 4 8 2 26 married 1 Male 5 8 2 41 single 0 Male 6 12 8 30 single 1 Male 7 10 5 32 married 0 . . . . . . . . . . . . . . . 0 Male 100 8 10 27 single 0 Table 5 : 5A job hiring scenario satisfying overall accuracy but not conditional use accuracy equality.OA = 0.625 PPV = 1 NPV = 0.25 Group 1 (Female) Table 6 : 6A job hiring scenario satisfying treatment equality but not satisfying all of the previous notions.TPR = 0.33 FPR = 0.8 PPV = 0.2 NPV = 0.33 OA = 0.25 FN/FP = 0.5 Group 1 (Female) Gender YŶ F1 1 1 F2 0 0 F3 0 1 F4 0 1 F5 0 1 F6 0 1 F7 1 0 F8 1 0 Group 2 (Male) Gender YŶ M1 1 1 M2 1 1 M3 1 1 M4 1 1 M5 0 0 M6 0 1 M7 0 1 M8 1 0 TPR = 0.8 FPR = 0.66 PPV = 0.66 NPV = 0.5 OA = 0.625 FN/FP = 0.5 Table 7 : 7A job hiring scenario satisfying total fairness.TPR = 0.5 FPR = 0.66 PPV = 0.33 NPV = 0.5 OA = 0.4 FN/FP = 0.5 Group 1 (Female) Gender YŶ F1 1 1 F2 0 0 F3 0 1 F4 0 1 F5 1 0 Group 2 (Male) Gender YŶ M1 1 1 M2 1 1 M3 0 0 M4 0 0 M5 0 1 M6 0 1 M7 0 1 M8 0 1 M9 1 0 M10 1 0 TPR = 0.5 FPR = 0.66 PPV = 0.33 NPV = 0.5 OA = 0.4 FN/FP = 0.5 Table 8 : 8A job hiring scenario satisfying statistical parity and equal opportunity (for a score threshold value of 5) but neither balance for positive class nor balance for negative class.(a) Group 1 (Female) Gender Y S F1 1 9 F2 1 8 F3 0 8 F4 1 4.5 F5 0 4.5 F6 0 3.5 (b) Group 2 (Male) Gender Y S M1 1 6.2 M2 1 6 M3 0 5.5 M4 0 1 M5 1 2 M6 0 2 Table 9 : 9A job hiring scenario satisfying predictive parity (for any threshold smaller than 0.7 or larger than 0.8) but not calibration. (a) Group 1 (Female) Gender Y S F1 1 0.85 F2 1 0.8 F3 0 0.8 F4 1 0.7 F5 0 0.7 F6 0 0.4 F7 1 0.4 F8 0 0.4 (b) Group 2 (Male) Gender Y S M1 1 0.85 M2 1 0.8 M3 1 0.8 M4 0 0.7 M5 0 0.7 M6 1 0.4 M7 0 0.4 M8 0 0.4 Table 10 : 10A job hiring scenario satisfying calibration but not predictive parity (for any threshold).(a) Group 1 (Female) Gender Y S F1 1 0.8 F2 1 0.8 F3 1 0.7 F4 1 0.7 F5 0 0.7 F6 0 0.7 F7 0 0.3 F8 0 0.3 (b) Group 2 (Male) Gender Y S M1 1 0.8 M2 1 0.8 M3 1 0.7 M4 0 0.7 M5 0 0.3 M6 0 0.3 Table 11 : 11Calibration vs well-calibration. Table 12 : 12A simple job hiring example satisfying most of group fairness notions, but unfair towards Female 4 applicant.Gender Education Level Job Expe- rience Age Marital Status YŶ Female 1 8 2 39 single 0 1 Female 2 8 2 26 married 0 1 Female 3 6 1 32 married 0 0 Female 4 12 8 35 single 1 0 Female 5 9 10 29 married 1 1 Male 1 7 3 34 single 0 1 Male 2 8 0 28 married 1 0 Male 3 11 8 43 single 1 1 Male 4 7 1 26 married 0 0 Male 5 8 2 41 single 0 1 TPR = 0.5 FPR = 0.66 PPV = 0.33 OA = 0.4 TPR = 0.5 FPR = 0.66 PPV = 0.33 OA = 0.4 Table 13 : 13Job hiring sample used to apply fairness through awareness.(a) Dataset Gender Education Level Job Expe- rience Age Marital Status Label Female 1 12 2 39 single 1 Female 2 12 1 26 married 0 Female 3 13 1 32 married 1 Male 1 13 1 26 married 1 Male 2 12 1 41 single 0 Male 3 12 2 30 single 1 (b) Prediction Y S 0 0.4 0 0.3 1 0.9 0 0.2 0 0.2 1 0.7 Table 14: Application of fairness through awareness. Each cell at the left of the shaded table's diagonal represents a distance between a pair of applicants. Those at the right represent the distance between probability distributions. Values in bold imply cases where D > d, meaning fairness through awareness is not satisfied. F1 F2 F3 M1 M2 M3 D(M (vi), M (vj )) F1 0.07 0.26 0.16 0.16 0.07 F2 0.25 0.18 0.08 0.08 0.29 F3 0.75 0.5 0.1 0.54 0.18 M1 0.75 0.5 0.0 0.0 0.08 M2 0.25 0.0 0.5 0.5 0.37 M3 0.0 0.25 0.75 0.75 0.25 Table 15 : 15Classification of fairness notions. ( * notion newly defined in this paper)Fairness Notion Ref. Formulation Table 16 : 16A job hiring scenario satisfying fairness through awareness (Eq. 17) but not statistical parity (Eq. 1) for a threshold of 0.6. The second row (M ()) indicates the probability distribution on the outcomes. For example, for the first female applicant F 1, P (Ŷ = 1) = 0.58 and P (Ŷ = 0) = 0.42. Each cell at the left of the shaded table's diagonal represents a distance between a pair of applicants. Those at the right represent the distance between probability distributions on the outcomes.F1 F2 F3 M1 M2 M3 M() [0.58, 0.42] [0.8, 0.2] [0.55, 0.45] [0.65, 0.35] [0.81, 0.19] [0.61, 0.39] F1 0.17 0.021 0.051 0.18 0.02 D(M (vi), M (vj )) F2 0.21 0.19 0.11 0.008 0.15 F3 0.06 0.22 0.07 0.20 0.04 M1 0.1 0.15 0.1 0.12 0.029 M2 0.2 0.01 0.3 0.15 0.15 M3 0.05 0.17 0.08 0.05 0.17 Table 17 : 17Correspondence between Fairness notions and the selection criteria: C1: disparate impact , C2: disparate treatment , C3: intersectionality/masking, C4: historical bias, C5: representational bias, C6: measurement bias, C7: aggregation/evaluation/deployment bias, C8: standards, C9: ground truth available, C10: y not reliable, C11: explanatory variables, C12: precision, C13: recall, C14: FP, C15: FN, C16: causal graph available, C17: threshold floating.Notation: ✓: recommended, : warning, ✗: must not, −: insensitive.Legal Frame Suspected source of bias Emphasis on Emphasis on Fairness notion Criterion C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 Statistical parity We focus on automated decision-making system supported by ML algorithms. In the rest of the paper we refer to such systems as MLDM. Zliobaite does not use group vs individual notions, but indirect and direct discrimination. This might not be the case in other scenarios such as disease prediction, child maltreatment, where imposing a parity of positive predictions is meaningless.17 This is also known as differential measurement error[92].18 Called explanatory attributes in[49]. If the job is a well-paid, male group tends to have a better living condition and affords better education for their kids, and thus enable them to be qualified for such well-paid jobs when they grow up. The gap between the groups will tend to increase over time. Results in a useless intervention, because the child is not at risk anyway.21 Results in a failure to anticipate a child maltreatment. The cost can be financial, ethical, reputation, etc. The term individual fairness is used in some papers to refer to fairness through awareness (Section 5.11). In this paper, the term individual fairness refers to fairness notions which cannot be considered as group fairness notions. Reducing all difference between two applicants/instances to a single distance value is often not easy to do in practice. Kusner et al. [59] identify some exceptions, but guaranteeing that they will not happen in general. Machine learning fairness notions: Bridging the gap with real-world applicationsFigure 6: Fairness notions applicability decision diagram. Assuming the absence of measurement bias. . Prediction: compute the outcome (Ŷ ) value using the updated probability P (U \| X = x, A = a 0 ) and structural functions. AcknowledgementThe work of Catuscia Palamidessi was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme. 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[ "Inferring the parameters of a Markov process from snapshots of the steady state", "Inferring the parameters of a Markov process from snapshots of the steady state" ]	[ "Simon L Dettmer \nInstitute for Theoretical Physics\nUniversity of Cologne\nZülpicher Straße 7750937CologneGermany\n", "Johannes Berg \nInstitute for Theoretical Physics\nUniversity of Cologne\nZülpicher Straße 7750937CologneGermany\n" ]	[ "Institute for Theoretical Physics\nUniversity of Cologne\nZülpicher Straße 7750937CologneGermany", "Institute for Theoretical Physics\nUniversity of Cologne\nZülpicher Straße 7750937CologneGermany" ]	[]	We seek to infer the parameters of an ergodic Markov process from samples taken independently from the steady state. Our focus is on non-equilibrium processes, where the steady state is not described by the Boltzmann measure, but is generally unknown and hard to compute, which prevents the application of established equilibrium inference methods. We propose a quantity we call propagator likelihood, which takes on the role of the likelihood in equilibrium processes. This propagator likelihood is based on fictitious transitions between those configurations of the system which occur in the samples. The propagator likelihood can be derived by minimising the relative entropy between the empirical distribution and a distribution generated by propagating the empirical distribution forward in time. Maximising the propagator likelihood leads to an efficient reconstruction of the parameters of the underlying model in different systems, both with discrete configurations and with continuous configurations. We apply the method to non-equilibrium models from statistical physics and theoretical biology, including the asymmetric simple exclusion process (ASEP), the kinetic Ising model, and replicator dynamics.	10.1088/1742-5468/aaa8ea	[ "https://arxiv.org/pdf/1707.04114v3.pdf" ]	88,515,584	1707.04114	74832b72c8673b274ecefff65fe81403a00973ac	Inferring the parameters of a Markov process from snapshots of the steady state 14 Dec 2017 Simon L Dettmer Institute for Theoretical Physics University of Cologne Zülpicher Straße 7750937CologneGermany Johannes Berg Institute for Theoretical Physics University of Cologne Zülpicher Straße 7750937CologneGermany Inferring the parameters of a Markov process from snapshots of the steady state 14 Dec 2017numbers: 0250Ga0230Zz0250Tt8975-k 7550Lk0570Ln Keywords: stochastic inferenceMarkov processnon-equilibrium steady stateIsing modelneural networksreplicator dynamicsasymmetric exclusion process We seek to infer the parameters of an ergodic Markov process from samples taken independently from the steady state. Our focus is on non-equilibrium processes, where the steady state is not described by the Boltzmann measure, but is generally unknown and hard to compute, which prevents the application of established equilibrium inference methods. We propose a quantity we call propagator likelihood, which takes on the role of the likelihood in equilibrium processes. This propagator likelihood is based on fictitious transitions between those configurations of the system which occur in the samples. The propagator likelihood can be derived by minimising the relative entropy between the empirical distribution and a distribution generated by propagating the empirical distribution forward in time. Maximising the propagator likelihood leads to an efficient reconstruction of the parameters of the underlying model in different systems, both with discrete configurations and with continuous configurations. We apply the method to non-equilibrium models from statistical physics and theoretical biology, including the asymmetric simple exclusion process (ASEP), the kinetic Ising model, and replicator dynamics. I. INTRODUCTION The problem of inferring the parameters of a stochastic model from data is ubiquitous in the natural and social sciences, and engineering. Many systems, like gene regulatory networks, electric power grids, virus populations, or financial markets have a complex dynamics which is often modelled by stochastic processes. Such stochastic processes are characterised by potentially many free parameters, which need to be estimated from data. For a review in the context of the inverse Ising problem, see [1]. Here, we ask how to infer the parameters characterising a non-equilibrium stochastic process. We consider a system with configurations x in some configuration space and time-homogeneous transition probabilities between configurations. Our focus is on time-homogeneous Markov processes, which are fully defined by instantaneous transition rates. These rates are parametrized by a model with parameters denoted Θ. Configurations can be discrete or continuous, and also time can be discrete or continuous. For the concrete example of a colloidal particle undergoing Brownian motion, the configurations x are positions in space and the model parameter specifies the diffusion constant of the particle. We restrict ourselves to ergodic processes, so for any initial state the system eventually settles into a unique steady state characterised by the steady-state probability distribution p Θ (x). Our aim is to infer the underlying parameters Θ true from M samples x µ , with µ = 1, . . . , M , drawn independently from the steady state distribution. Parameter inference hinges on the description of the empirical data by a model. For a model whose steady * [email protected], and [email protected] state p Θ (x) is known explicitly, the maximum-likelihood estimate Θ inf = argmax Θ M µ=1 p Θ (x µ )(1) provides an estimate of the model parameters which becomes exact in the limit of a large number of samples. However, for non-equilibrium models, the steady state p Θ (x) is hard to compute and generally unknown. This is a major difference to equilibrium models and prevents the use of established inference methods. In some cases, time series data is available and one can use the empirically observed transitions between configurations to compute the likelihood of the observed time series. This likelihood can be computed directly from the transition probabilities specified by the model; the underlying model parameters are then estimated as the parameters that maximise the likelihood of the time series [2,3]. Inference from time series can be performed even more efficiently using mean-field approximations [2,4,5]. However, for many systems, classical as well as quantum, time series data is not available. An extreme case is whole-genome single-cell gene expression profiling, where cells are destroyed by the measurement process. In such cases, we have only independent samples from which to infer the model parameters. To this end, we use the transition rates between configurations and their dependence on the model parameters to construct a quantity we call the propagator likelihood. We show how this likelihood can be used to infer the model parameters from independent samples taken from the steady state. This article is organised as follows: First, we introduce the propagator likelihood through an intuitive argument and then offer a systematic derivation based on relative entropies. Second, we apply the propagator likelihood to pedagogical examples with both discrete and continuous configurations, specifically the asymmetric simple exclusion process (ASEP) and the Ornstein-Uhlenbeck process. Finally, we address the more challenging problem of inferring the parameters of two prominent models from statistical physics and theoretical biology: the kinetic Ising model and replicator dynamics. II. THE PROPAGATOR LIKELIHOOD Suppose we knew the functional dependence of the steady-state distribution p Θ (x) on the model parameters Θ. Then a standard approach would be to maximise the (log-) likelihood of the samples L(Θ) = 1 M M µ=1 log p Θ (x µ ) = xp (x) log p Θ (x) ,(2) where the set of sampled configurations characterises the empirical distributionp(x) with probability mass functionp (x) = 1 M M µ=1 δ x µ ,x ,(3) and δ x µ ,x denotes a Kronecker-δ. However, in non-equilibrium systems we frequently do not know the steady-state distribution. Non-equilibrium systems lack detailed balance, so the steady state is not described by the Boltzmann distribution and lacks a simple characterisation. Our solution to this inference problem is based on exploiting one elementary fact: since the distribution p Θ is stationary, it remains unchanged if we propagate it forward in time by an arbitrary time interval. Thus, we can replace the steadystate distribution p Θ (x) in the log-likelihood function (2) with the same distribution propagated forward in time y p Θ (x, τ \|y, 0)p Θ (y). The propagator p Θ (x, τ \|y, 0) is the conditional probability of observing the system in configuration x at time t = τ , given it was in configuration y at time t = 0. By replacing the unknown steadystate distribution p Θ (y) with the empirical distribution p(y), we arrive at the propagator likelihood PL(Θ; τ ) = xp (x) log y p Θ (x, τ \|y, 0)p(y) = 1 M M µ=1 log 1 M M ν=1 p Θ (x µ , τ \|x ν , 0) . (4) In this way, we have shifted the parameter-dependence from the (unknown) steady-state distribution p Θ (x) to the (known) propagator p Θ (x, τ \|y, 0). For models with continuous configurations, p Θ (x µ , τ \|x ν , 0) is the transition probability density. The propagator likelihood (4) has a straightforward probabilistic interpretation: 1 M M ν=1 p Θ (x, τ \|x ν , 0) is a probability distri-bution over x, conditional on the sampled configurations {x ν }. The propagator likelihood is the corresponding log-likelihood of this probability distribution, evaluated for M independent draws of the empirically observed configurations and rescaled by M . In the limit τ → ∞, the propagator likelihood (4) approaches the log-likelihood (2), since lim τ →∞ p Θ (x, τ \|y, 0) ≡ p Θ (x). However, the complexity of calculating the propagator increases with τ . In principle, the propagation time interval τ is arbitrary: it parameterizes different measures of how close a given empirical probability distribution is to being stationary under a particular set of model parameters. Different choices of τ will be discussed in sections IV A and V A. Maximizing the propagator likelihood does not involve sampling the probability distribution at different times, but seeks model parameters that would leave the empirical distribution invariant, if one did propagate it forward in time. Correspondingly, although rates of transitions between configurations x ν and x µ appear in (4), these transitions are entirely fictitious: the empirical configurations {x ν } are sampled independently from the non-equilibrium steady state. In the following, we will assume that the parameters maximizing the propagator likelihood are unique. One might want to prove this for the particular model used by analytically calculating the propagator likelihood and checking its convexity. (3), shown on the left. We use the transition probabilities pΘ(x, τ \|y, 0) to propagatep forward in time by an arbitrary interval τ to generate a new distribution qΘ,τ (see Eq.(5)), shown on the right. The functional form of the propagator is thought to be known, but it is parametrized by a set of unknown parameters Θ. Demanding stationarity of the empirical distribution, we can estimate the underlying parameters Θ true by finding the parameters Θ inf that minimise the distance betweenp and qΘ,τ as measured with relative entropy. This is equivalent to maximising the propagator likelihood (see main text). A. Minimising relative entropy A second interpretation of the propagator likelihood can be found by rephrasing parameter inference from a steady state as finding a set of parameters Θ such that the propagator p Θ (x, τ \|y, 0) is compatible with the empirical distributionp being stationary (see Fig. 1). Demanding stationarity corresponds to requiring thatp is in some sense close to a distribution q Θ,τ generated by propagating the empirical distribution for an arbitrary time interval τ , q Θ,τ (x) = y p Θ (x, τ \|y, 0)p(y) .(5) To quantify this notion of closeness for discrete configurations, we use the relative entropy or Kullback-Leibler divergence [6] D(p q Θ,τ ) = xp (x) logp (x) q Θ,τ (x) .(6) Inserting the probability mass function q Θ,τ (x) defined by (5) into the relative entropy, we find that the relative entropy can be written as the negative sum of the Shannon entropy of the empirical distribution, S(p) = − xp (x) logp(x) and the propagator likelihood (4): D(p q Θ,τ ) = −S(p) − PL(Θ; τ ) .(7) The first term depends only on the sampled configurations and is independent of the model parameters; thus minimising the relative entropy with respect to Θ is equivalent to maximising the propagator likelihood. Furthermore, due to the positivity of relative entropy, the propagator likelihood is bounded from above by the negative Shannon entropy, and this bound will be saturated only for a model that makes the empirical distribution exactly stationary. The propagator likelihood (4) thus emerges from a variational approach aiming to find the model parameters that are most consistent with the sampled distribution being the steady state. A similar argument can be made also for models with continuous configurations x ∈ R d . We consider the differential relative entropy D = dxp s (x) log(p s (x)/q Θ,τ,s (x)), which can be computed by estimating the probability density of the steady state from the samples via a Gaussian mixture modelp s (x) = 1 M M µ=1 exp(−(x − x µ ) 2 /2s 2 )/(2πs 2 ) d/2 . Here, s > 0 is the width of the Gaussians in the mixture model, and q Θ,τ,s (x) = dy p Θ (x, τ \|y, 0)p s (y) denotes the time-propagated density estimate. Minimising this estimate of the differential relative entropy is then equivalent to maximising a quantity that converges to the propagator likelihood for s → 0. It is easy to show that the maximum propagator likelihood estimate Θ inf converges to the underlying parameters Θ true in the limit of large sample sizes: for M → ∞, the empirical distributionp(y) converges to the steady-state distribution p Θ true (y). Hence, the propagator likelihood converges to PL(Θ; τ, M = ∞) = x p Θ true (x) ln y p Θ (x, τ \|y, 0)p Θ true (y). According to (7), this function has its maximum over Θ where the relative entropy between the underlying distribution p Θ true (x) and its propagated version y p Θ (x, τ \|y, 0)p Θ true (y) is minimal. This minimum is realised for Θ = Θ true , since the relative entropy is non-negative and the steady-state by definition remains unchanged when propagated with the parameter value Θ = Θ true . The propagator likelihood for a simple twoconfiguration system. The inset shows the single-step dynamics of the system with configurations 0 and 1, controlled by the hopping probability r ∈ (0, 1). In the main figure, the solid lines show the propagator likelihood for varying propagation time intervals τ . The dashed line shows the log-likelihood (2), which corresponds to an infinite propagation time interval. The maximum likelihood estimate of the hopping probability, r inf = 1−p(0) p(0) , is marked on the top axis and coincides with the maximum for all propagator likelihoods with an uneven number of time steps τ (see the main text for the case of even time steps). To illustrate the propagator likelihood with a toy example, we consider a system with only two configurations, denoted by 0 and 1 (see inset of Fig. 2). At each time step, if the system is in configuration 1, it moves to configuration 0. If it is in configuration 0, it moves to configuration 1 with probability r ∈ (0, 1) or remains in configuration 0 with probability 1 − r. The steady-state distribution is easily computed, giving p r (0) = 1/(1 + r) and p r (1) = 1 − p r (0) = r/(1 + r). We are now given samples {x µ } M µ=1 ∈ {0, 1} M taken independently from the steady state and want to infer the model parameter r. The empirical distribution is given by the frequencies of the two configurations,p(0) = 1 M M µ=1 δ 0,x µ andp(1) = 1 −p(0). Since we know the steady state for this particular model, we can infer r from the relationship p(0) = 1/(1 + r), yielding r inf = (1 − p(0))/p(0). For comparison, we also use the propagator likelihood (4) with the single-step propagator p r (x, τ = 1\|y, 0) = δ y,1 δ x,0 + δ y,0 (rδ x,1 + (1 − r)δ x,0 ), giving PL(r; 1) =p 0 log((1 − r)p 0 0→0 +p 1 1→0 ) +p 1 log( rp 0 0→1 ) =p 0 log(1 − rp 0 ) + (1 −p 0 ) log(rp 0 ) . (8) Maximising the propagator likelihood analytically with respect to r by setting dPL dr (r inf ) = 0, we recover the same result as obtained above by analysing the known steadystate distribution. Indeed, for uneven propagation time intervals, the propagator likelihood shows a unique maximum at the same point where the likelihood has its maximum, r inf = 1−p0 p0 . Also, the propagator likelihood approaches the log-likelihood for increasing τ , as expected. For even propagation time intervals, however, a second (global) maximum occurs at the boundary r = 1: since the choice r = 1 makes the two configurations simply exchange their probabilities in each step, the Markov chain loses its ergodicity and becomes periodic. In this case, any distribution is stationary over an even number of time steps. While stationarity with respect to a single time step is both necessary and sufficient to define the steady state, stationarity with respect to longer propagation time intervals is necessary but not sufficient. Hence, spurious maxima of the likelihood can appear when using longer propagation time interval. B. Continuous time: the asymmetric simple exclusion process (ASEP) Markov processes with discrete configurations in continuous time are characterised by instantaneous transition rates between distinct configurations W Θ (x\|y) = lim τ →0 p Θ (x, τ \|y, 0)/τ , (x = y). The system hops away from configuration y at a random time that is exponentially distributed with parameter −W Θ (y\|y) ≡ x =y W Θ (x\|y). For the purpose of inferring the model parameters, it is convenient to map the continuous-time process onto a discrete-time process with the same steady state. This can be achieved by choosing the single-step transition matrix p Θ (x, τ = 1\|y, 0) = δ x,y + λW Θ (x\|y) .(9) The parameter λ affects the overall rate at which transitions occur. Choosing 0 < λ < [max y {−W (y\|y)}] −1 ensures a well-defined stochastic matrix. Since the steadystate distribution p Θ (y) itself is not associated with a time scale, the choice of λ is in principle arbitrary. As an example of a model with continuous time, we consider the asymmetric simple exclusion process (ASEP) on a ring with asynchronous updates (see inset of Fig. 3). The ASEP is a simple model of a driven lattice gas and has been applied to traffic flow, surface growth, and directed paths in random media [7][8][9]. The steady-state distribution in 1D can be calculated analytically in terms of matrix products [8,9]. In higher dimensions, however, there is no such systematic approach and, to the best of our knowledge, the steadystate distribution is unknown. ¤ inf ¥ 2 ¦ 1 § 4 3F IG. 3. Reconstruction of hopping rates in the asymmetric simple exclusion process (ASEP). The inset schematically shows the dynamics: K particles move on a periodic onedimensional lattice with N > K lattice sites, see text. In the main figure, we plot the relative mobilitiesμ inf i inferred using the propagator likelihood versus the underlying relative mobilitiesμ true i = µ true i / j µ true j that were used to generate the data. We simulated K = 10 particles hopping on a lattice with N = 15 sites and took M = 10 10 samples independently from the steady state. The underlying mobilities µi were drawn independently from a uniform distribution on the unit interval (0, 1). The model consists of K particles moving on a periodic one-dimensional lattice with N > K lattice sites. Each lattice site can be occupied by at most one particle. Particles labelled i = 1, . . . , K independently attempt to jump one step in the clockwise direction at a rate µ i , which is called the intrinsic mobility or hopping rate of a particle. The configuration of the system can be characterised by the number of free lattice sites in front of each particle, n = (n 1 , . . . , n K ) ⊂ (N 0 ) K , with the restriction that the particle gaps add up to the number of free lattice sites: n 1 + n 2 + . . . + n K = N − K. For the transition n = (n 1 , . . . , n K ) → n ′ = (n ′ 1 , . . . , n ′ K ) between two distinct configurations there is a non-zero transition rate only if the configurations are connected by the jump of a single particle i, i.e. all gaps are identical except for the gap in front of particle i, which must be decreased by one, n ′ i = n i −1, and the gap behind par-ticle i, which must by increased by one n ′ i−1 = n i−1 + 1. The transition rate is then simply the hopping rate of the particle W µ (n ′ \|n) = µ i . To infer the parameters, we define a discrete-time version of the process with transition probabilities defined by (9). We choose λ such that the hopping rates add to one, λ = (µ 1 + µ 2 + . . . + µ K ) −1 in (9). The steady-state distribution is characterised by the relative hopping ratesμ i ≡ µ i / j µ j . The single-step propagator likelihood of the discrete-time process then reads PL(μ, 1) = n ′p (n ′ ) log p(n ′ ) + n Wμ(n ′ \|n)p(n) .(10) We use this result to evaluate the propagator likelihood (4) and infer the relative mobilitiesμ i . As an example, we consider a system of K = 10 particles hopping on N = 15 lattice sites. The particle mobilities µ i are independently and uniformly drawn from the interval (0, 1). We generate M = 10 10 Monte Carlo samples, recorded every 10 jumps after an initial settling time interval of 10 5 jumps to reach the steady state. We then maximise the propagator likelihood numerically using the sequential least squares programming algorithm as implemented in the SciPy library [10]. In Fig. 3 we plot the inferred relative mobilities versus the relative mobilities used to generate the samples. IV. MODELS WITH CONTINUOUS CONFIGURATIONS Markov processes with continuous configurations pose an additional challenge: Finite-time propagators are generally not known explicitly. Instead, finite-time propagators are characterised indirectly as the solution of a Fokker-Planck equation. Rather than solving a Fokker-Planck equation, which for systems with a large number of degrees of freedom is generally infeasible, we proceed by approximating the propagator for short time intervals τ via a linearisation of the corresponding Langevin equation (LE) that describes the stochastic dynamics of the model. Again, we first demonstrate this procedure using a toy model. We consider one of the simplest processes with continuous configurations, the Ornstein-Uhlenbeck process (OUP), which describes the Brownian dynamics of an overdamped particle in a quadratic potential. Note that, again, for this particular case the steady-state distribution is known exactly, so one could infer the model parameters using the standard maximum likelihood approach. We use this case to illustrate the propagator likelihood before turning to more complex models where the likelihood-based approach is not feasible. A. The Ornstein-Uhlenbeck process Consider a single particle diffusing in a onedimensional harmonic potential U (x) = b 2 x 2 with diffusion constant σ 2 . A physical realisation of this model is a colloidal particle in solution being held in place by optical tweezers and confined to a one-dimensional channel. The dynamics of the particle is modelled by the Langevin equation dx dt = −bx + σξ(t) ,(11) where the random force ξ(t) describes δ-correlated white noise interpreted in the Itô convention. As for the exclusion process, one model parameter must be eliminated by rescaling time, since the steadystate distribution is time-independent. We rescale time to be dimensionless with t ′ = tσ 2 , so that the particle has unit diffusivity. To calculate the propagator likelihood for short time intervals τ ≪ 1, we linearise the LE (11) in time x(τ ) ≈ x(0) − b σ 2 x(0)τ + τ 0 dt ′ ξ(t ′ ) .(12) Since the integrated white noise τ 0 dt ′ ξ(t ′ ) is normally distributed with mean 0 and variance τ , we obtain an approximate short-time Gaussian propagator p b/σ 2 (x, τ \|y, 0) ≈ exp −[x − x] 2 /2τ √ 2πτ ,(13) where x = y − (b/σ 2 )yτ is the most likely future position of the particle. Such a Gaussian form of the propagator emerges for any linearised LE with white noise and is not specific to the OUP. For coloured and multiplicative noise, ξ(t) → f (x(t), t)η(t), where f is some function and the random force η(t) has a finite correlation time, we can proceed similarly. In this case, the normal distribution of the integrated white noise is replaced with the appropriate distribution of the integrated coloured noise τ 0 dt ′ f (x(t ′ ), t ′ )η(t ′ ) ≈ f (x(0), 0) τ 0 dt ′ η(t ′ ). Inserting the short-time propagator (13) into the propagator likelihood (4), we perform a one-dimensional maximisation of the propagator likelihood to infer the parameter Θ = b/σ 2 . Fig. 4(a) shows the relative reconstruction error versus the dimensionless propagation time interval τ for various sample sizes, both for the short-time propagator and for the exact finite-time propagator. The non-monotonic behaviour of the error for the short-time propagator shows that the optimal choice for τ involves a trade-off: At short time intervals τ , the distances typically crossed during the interval τ are small. In this case, the sum over pairs of sampled configurations in the propagator likelihood (4) is dominated by few transitions with small steps, and, in the limit τ → 0 it is dominated by transitions of the type x µ → x µ . For this reason, the parameter inference at small values of τ is more strongly affected by sampling fluctuations than at large values of τ . At large values of τ , on the other hand, the approximation used to derive the short-time propagator (13) becomes invalid. As a result, both the optimal value of τ and the total reconstruction error decrease as the sample size is increased. The exact finite-time propagator exhibits only sampling fluctuations, so the reconstruction error decreases monotonically with τ , converging to the maximum likelihood estimate at large τ . Note that the results for the approximate and exact propagators do not converge for τ → 0, since the relative difference of the propagators converges to 0 only for the peak x = y, even though the absolute difference converges to 0 for all values of x. a. Choosing the optimal propagation time interval. The non-monotonic behaviour of the reconstruction error ǫ = \|Θ inf − Θ true \|/Θ true raises the question how to choose the optimal propagation time interval without prior knowledge of the underlying parameter Θ true . We find an answer by assuming that the error is a smooth function of the propagation time interval: we seek the minimal error by demanding 0 = ∂ǫ/∂τ = sgn(Θ inf −Θ true ) \|Θ true \| ∂Θ inf ∂τ ∼ ∂Θ inf /∂τ . The error derivative will become small only for ∂Θ inf /∂τ → 0. The latter quantity can be estimated directly from the data by repeating the inference for a set of propagation time intervals {(τ i , τ i +∆τ )} and computing the forward difference quo- tients ∂Θ inf /∂τ (τ i ) ≈ [Θ inf (τ i +∆τ )−Θ inf (τ i )]/∆τ . Since estimating the derivative from the data will incur numerical errors, we relax the condition 0 = ∂Θ inf /∂τ and demand only that \|∂Θ inf /∂τ \| is minimal. In Fig. 4(b) we show that these minima indeed coincide with the optimal choice of τ as judged from the reconstruction error shown in Fig. 4(a). V. NON-EQUILIBRIUM MODELS IN STATISTICAL PHYSICS AND THEORETICAL BIOLOGY We now turn to non-equilibrium applications where the standard maximum likelihood approach is not feasible, as the steady-state distribution is unknown. A. The kinetic Ising model The kinetic Ising model consists of a set of N binary spins s i = ±1, which interact with each other via couplings J ij and are subject to external fields h i (see inset of Fig. 5). Crucially, the couplings are not symmetric (J ij = J ji in general). A stochastic dynamics of this model is specified by the so-called Glauber dynamics [11]: In each time step, a spin i is chosen in and its value s i (t + 1) one time step later is updated according to the probability distribution p(s i (t + 1)\|s s s(t)) = exp{s i (t + 1)θ i (t)} 2 cosh(θ i (t)) , where the effective local field at time t is θ i (t) = h i + N j=1 J ij s j (t) .(15) The kinetic Ising model has been used to model gene regulatory and neural networks [12][13][14]. For a symmetric coupling matrix without selfcouplings, the Glauber dynamics (14) converges to the equilibrium state characterised by the Boltzmann distribution p B (s) = e −H(s) /Z with the well-known Ising Hamiltonian H(s) = − i s i (h i + j>i J ij s j ). For asymmetric couplings, however, Glauber dynamics (14) converges to a non-equilibrium steady state, which lacks detailed balance and is hard to characterise. In recent work we have shown how the spin couplings J ij and external fields h i can be inferred from independent samples taken from the steady state by fitting couplings and fields to match the magnetisations, two-, and three-point correlations sampled in the data [15]. Here we demonstrate that the couplings can be inferred even more accurately with the propagator likelihood (4), which uses information from the full empirical distribution. We insert the single-step propagator (14) into the propagator likelihood (4) and maximise the propagator likelihood with respect to the external fields h i and off-diagonal couplings J ij (we consider a model without self-interactions: J ii = 0). For the last step, we use the Broyden-Fletcher-Goldfarb-Shanno algorithm as implemented in the SciPy library [10], and initialise the algorithm with the naive mean-field parameter estimate as described in [15]. Fig. 5 compares the relative error of coupling reconstruction ǫ = J inf −J true 2 / J true 2 based on the single-step propagator likelihood with the corresponding reconstruction error of fitting finite spin moments up to three-point correlations. It turns out that parameter inference in the kinetic Ising model requires more samples than in the equilibrium inverse Ising problem. To achieve a relative reconstruction error of 10 −2 for an equilibrium system of N = 10 spins, the pseudolikelihood method requires of the order of 10 6 samples [16]. In the non-equilibrium model considered here, we require at least 10 8 independent samples for a similar reconstruction accuracy (see Fig. 5). Naturally, inference in the kinetic Ising model becomes significantly easier if time-correlated data is available. For example, the Gaussian mean-field theory [4] requires only on the order of 10 6 pairs of samples {s s s(t), s s s(t + 1)} to achieve a similar reconstruction accuracy for a system as large as 100 spins. The reason for this is that, in the kinetic Ising model, couplings are not uniquely determined by pairwise correlations. Instead, many different models can reproduce the same pairwise correlations. For this reason, we need information from higher order spin correlations, which require more samples to determine them accurately. The inset schematically shows a system of binary spins interacting via couplings Jij subject to external fields hi (not shown). In the main figure, we plot the relative error of couplings ǫ = J inf − J true 2/ J true 2 versus the number of independent samples used for inference, using (i) finite spin moments up to three-point correlations (•) and (ii) the single-step propagator likelihood ( ). Both methods are exact, so the relative error decreases with the sample size as ǫ ∼ M −1/2 . The propagator likelihood (which uses the full set of configurations sampled) performs only a little better than the fit to the first three moments, showing that most information required for reconstruction is already contained in the first three moments. The underlying off-diagonal couplings were drawn independently from a Gaussian distribution with mean 0 and standard deviation 1/ √ N (we excluded self-interactions, Jii = 0), the external fields were drawn independently from a Gaussian distribution with mean 0 and standard deviation 1. The system size was N = 10 spins. a. Sparse networks. We now consider a particular situation, where the parameter inference requires fewer samples: sparse coupling matrices with known topology of the couplings, so only the values of the couplings are to be reconstructed. Specifically, we look at the kinetic Ising model with sparse couplings (so most interactions are zero) and assume as prior knowledge the pairs (i, j) that have a non-zero coupling between them, i.e. J true ij = 0 or J true ji = 0, regardless of the direction of the coupling. This problem has been addressed for undirected equilibrium systems like Ising models with ferromagnetic or binary couplings [16,17]. We apply the propagator likelihood to a network of N = 10 spins, where each possible directed link J ij from spin i to spin j is non-zero with probability p = 0.2. The non-zero couplings are again drawn independently from a Gaussian distribution with mean 0 and variance 1/N . Self-interactions are excluded and the external fields h i drawn independently from a Gaussian distribution with mean 0 and variance 1. Figure 6 shows that the directed couplings can be inferred with slightly fewer samples when the topology of the couplings is known. The inset schematically shows a system of binary spins interacting via sparse couplings Jij subject to external fields hi (not shown). In the main figure, we plot the relative error of couplings ǫ = J inf − J true 2/ J true 2 versus the number of independent samples. The underlying off-diagonal couplings were chosen sparsely: they were set to zero with probability 1−p = 0.8, and with probability p = 0.2 were drawn independently from a Gaussian distribution with mean 0 and variance 1/N (we excluded self-interactions, Jii = 0 ). The external fields were drawn independently from a Gaussian distribution with mean 0 and variance 1. The system size was N = 10 spins. The couplings were inferred by maximising the singlestep propagator likelihood over the set of couplings between directly interacting spin pairs (i, j), i.e. there is at least one true non-zero coupling between the spin pair, J true ij = 0 or J true ji = 0, regardless of the direction. b. Increasing the propagation time interval. So far we have restricted ourselves to the single-step propagator (τ = 1). Can the inference be improved by increasing the propagation time interval? Intuitively, we expect that the single-step propagator cannot be improved on when all configurations have been sampled, since this implies that all transitions over longer propagation time intervals consist of single-step transitions that have already been probed by the single-step propagator likelihood: x ν τ → x µ = x1,x2,...,xτ−1 x ν τ =1 → x 1 τ =1 → x 2 . . . τ =1 → x τ −1 τ =1 → x µ . Indeed, the examples with discrete time considered so far in this article fall into this category and our numerical evidence confirms that increasing the propagation time interval does not improve the inference. If, however, the configuration space is undersampled, some of the transitions appearing in the longer-time propagator likelihood will involve intermediate configurations that are not present in the sample and therefore do not appear in the single-step propagator likelihood. In this case, we expect to find that increasing the propagation time interval improves the inference for a fixed sample size. In principle, one could even compute the loglikelihood (2) numerically by using sufficiently long propagation time intervals τ . However, the computational cost of taking the 2 N -dimensional transition matrix to a large power τ is often prohibitive. Furthermore, the matrix products needs to be computed many times in order to evaluate the likelihood and its (N 2 -dimensional) gradient over many iterations of a maximisation algorithm. In Fig. 7 we consider a kinetic Ising model where only a small fraction of system configurations appear in the sampled configurations. Increasing the propagation time interval from τ = 1 to τ = 3 improves the inference markedly. Also, we find that the reconstruction error is much smaller for the symmetric part of the coupling matrix (shown in Fig. 7(a)) than for the antisymmetric part (shown in Fig. 7(b)). This is because the symmetric part of the couplings is governed by the pairwise spin-correlations, while the antisymmetric part is dominated by higher-order spin-correlations, which require more samples for an accurate computation, see [15]. The benefit of increasing the propagation time interval is also larger for the symmetric part, suggesting that the reconstruction of the antisymmetric part of the couplings is mainly limited by the sample size and that increasing the propagation time interval even further will not lead to a more accurate reconstruction. B. The replicator model The replicator model describes a dynamics of selfreplicating entities, for instance genotypes, different animal species, RNA-molecules, or an abstract strategy in the game-theoretic problem. The replicator model has been used in population genetics, ecology, prebiotic chemistry, and sociobiology [18]. We consider a population consisting of N different species and denote by x i the fraction of species i in the total population (scaled for convenience by a factor on N so i x i = N ). The growth rate of species i, called its fitness, is denoted by f i . The population fraction change in time depends on the growth rate f i and the average growth rate of the population f t). The set of equations (16) defines the replicator model. The average fitness f enters to ensure that the fractions remain normalised such that i x i (t) = N for all times. dx i dt = x i (t)(f i (x, t) − f (x, t)) ,(16)with f (x, t) = 1 N N j=1 x j (t)f j (x, Here we consider a fitness which for each species i depends linearly on the population fractions of the other f i (x(t)) = N j =i J ij x j (t) .(17) The inter-species interactions J ij are quenched random variables with mean u (called the cooperation pressure) and standard deviation 1/ √ N . There are no selfinteractions, J ii = 0. For symmetric interactions, J ij = J ji , the fitness vector can be written as the gradient of a Lyapunov function. This implies that the system converges to an equilibrium steady state, which can be characterised by methods from statistical physics [19]. In the socio-biological context, however, there is no reason for the interactions to be symmetric, or in fact to assume deterministic dynamics. Assuming an asymmetric matrix J ij and allowing random fluctuations σξ i (t) in the reproduction of species i leads to a set of Langevin equations dx i dt = x i (t) (f i (x(t)) + σξ i (t) − λ(x, t)) ,(18) where the ξ i (t) are N independent sources of white noise interpreted in the Stratonovich convention, the parameter σ > 0 controls the overall noise strength, and the factor λ(x(t), t) = 1 N j x j (t)(f j (x(t)) + σξ j (t)) ensures normalisation, i.e. i x i (t) = N for all times. This dynamics converges to a non-equilibrium steady state. Its characteristics for typical realisations of the matrix of couplings have been studied in the limit of a large number of species using dynamical mean field theory [20]. We now turn to the problem of inferring the couplings J ij of the replicator model from a set of configurations {x µ } M µ=1 taken independently from the non-equilibrium steady state. For simplicity, we focus on the so-called cooperative regime, in which all species survive in the long-time limit, i.e. lim t→∞ x i (t) > 0 ∀i. This regime is characterised by a sufficiently large value of the cooperation pressure u [20]. Our results can be generalised to the case where species go extinct by restricting the transitions x ν → x µ considered in the propagator likelihood to those between configurations with the same set of surviving species. Again, to make time dimensionless, we rescale time t ′ = tσ 2 , resulting in a noise-term with unit magnitude. The steady state and the propagator depend only on the rescaled couplingsĴ ij ≡ J ij /σ 2 . By linearising the LE (18) for short times and eliminating x N via the normalisation constraint, For small values of τ the effects of sampling fluctuations dominate and the inferred parameter saturates as discussed in section IV A. For large τ , the error due to the linearisation of the Langevin equations is large and the inference becomes unstable, as signalled by the erratic changes in the value of the inferred parameter. The interval of reasonable propagation time intervals must lie between those two regimes and we choose a propagation time (marked by the circle) that lies in the (logarithmic) centre of this transition region (marked by the two vertical dashed lines). The other parameters show a similar behaviour and the same transition time interval, so the choice of the propagation time interval does not depend on the parameter considered. For each parameter, we take the vertical extent of the transition region as the estimation error. To illustrate the effects of sampling fluctuations, we repeated the procedure above a second time with the same model parameters but different samples (continuous line without markers). As expected, the sampling fluctuations influence mainly the inferred parameters for τ → 0, while the inference for larger values of τ is far less sensitive to the fluctuations. The system consisted of N = 3 species, the noise strength was set to σ = 0.1, and the underlying interactions J true ij were quenched random variables chosen independently from a Gaussian with mean u = 2.0 and standard deviation 1/ √ N (no selfinteractions: Jii = 0). We used an Euler discretisation of the Langevin equation (18) with time steps of length ∆t = 10 −6 /σ 2 and a total of M = 10 4 samples were taken every 10 4 steps after an initial settling time of 10 9 steps. x N = N − N −1 i=1 x i , we arrive at the Gaussian short-term propagator p(x, τ \|y, 0) ≈ 1 √ 2πτ N −1 √ DetΣ × exp    − 1 2τ N −1 i,j=1 (x i − y i − µ i τ ) Σ −1 ij (x j − y j − µ j τ )   (19) with drift 1 µ i = y i (f i (y) −f (y)) − y i N   y i − 1 N N j=1 y 2 j  (20) and covariance matrix Σ = AA T ∈ R N −1×N −1 with A ij = y i (y j /N − δ i,j ) .(21) 1 The second term in the drift arises from the difference between the Itô and Stratonovich convention in the Langevin equation. We denote byf i (y) the fitness (17) calculated with the rescaled variablesĴ ij = J ij /σ 2 , instead of the original interactions J ij , and byf (y) = 1 N j y jfj (y) its speciesweighted average. To reconstruct the rescaled interactionsĴ ij , we insert the Gaussian short-term propagator (19) into the propagator likelihood (4) and maximise it using the Broyden-Fletcher-Goldfarb-Shanno algorithm (see Fig. 8). As for the OUP, the reconstruction error depends nonmonotonically on the choice of the dimensionless propagation time interval τ , due to the trade-off between the error from linearising the LE and the error from effectively reducing the sample size by exponentially damping the propagators of most transitions. Unfortunately, the simple procedure we used for the OUP, minimising the parameter derivative \|∂Θ inf /∂τ \|, cannot easily be generalised to higher dimensions. The reason is that the derivative of the reconstruction error ∂ǫ/∂τ is a linear combination of the individual parameter entries (∂Θ inf i /∂τ ) K i=1 , which can cancel each other without vanishing individually (here K = N (N − 1) denotes the number of model parameters). To see that not all individual derivatives can vanish simultaneously, we remind ourselves that the inferred parameters must satisfy 0 ≡ ∂PL ∂Θi (Θ inf (τ ), τ ) , i = 1, . . . , K. Additionally demanding ∂Θ inf i /∂τ = 0, i = 1, . . . , K, corresponds to solving the system of equations { ∂PL ∂Θi = 0, ∂ 2 PL ∂Θi∂τ = 0} K i=1 for the K + 1 variables (Θ i , τ ). This system of 2K nonlinear equations for K + 1 variables will in general have no solution for K > 1. Instead, we can find a good propagation time interval by plotting a single inferred parameter versus the propagation time interval τ used for inference [see Fig. 8(b)]. The regime where the inference is dominated by the error from the linearisation for large values of τ is characterised by an erratic change of the value of the inferred parameter. At small values of τ , the reconstruction is dominated by sampling fluctuations (see section IV A). These regimes are connected by a transition region, from which the propagation time interval should be chosen. We checked that this transition region stretched across the same time interval (approximately [2 × 10 5 , 2 × 10 6 ]) for all parameters and chose the logarithmic center of this transition interval as the propagation time interval τ . We found this produced a good reconstruction quality, however, a method to pinpoint the optimal value of τ is currently lacking. VI. CONCLUSIONS We study parameter inference for a non-equilibrium model from independent samples taken from the steady state. Our approach is based on a variant of the likelihood we call the propagator likelihood. In the limit of a large propagation time interval, the propagator likelihood converges to the likelihood of the model. However, for non-equilibrium system, the likelihood and the limit of large propagation time intervals is generally intractable. The propagator likelihood can be derived from a variational principle aiming to find model parameters for which the distribution of configurations sampled from the steady state is invariant under propagation in time. For systems with discrete configurations, we base our reconstruction on the single-step propagator, although increasing the propagation time interval can improve the inference when not all configurations have been sampled. This can be understood as follows: at short times, most pairs of sampled configurations have a small or even vanishing propagator, and the propagator likelihood (4) is dominated by a few pairs of close configurations. At higher values of the propagation time interval τ , more configuration pairs contribute to the propagator likelihood, which reduces sampling fluctuations. However, as the computational complexity of evaluating the propagator grows exponentially with the number of time steps, there is a competition between inference quality and computational complexity. For systems with continuous configurations, we use a short-time approximation to the propagator. Also in this case, inference improves with the propagation time interval τ until the short-time approximation becomes invalid. Inferring model parameters from the steady state requires a large number of samples: Inferring couplings of the kinetic Ising model with N = 10 spins to within a reconstruction error ǫ ≈ 0.01 requires M ≈ 10 8 samples, compared to the equilibrium case requiring approximately 10 6 samples (for couplings drawn independently from a Gaussian with mean 0 and variance 1/N ). The bottleneck in practical applications may thus well be the number of available samples. Non-equilibrium inference is also computationally expensive: evaluating the propagator likelihood takes O(M 2 ) operations for systems with continuous configurations and O(M ) operations for systems with discrete configurations (provided that only a small number of neighbouring configurations can be reached in a single step with non-zero transition probability). A challenge for the future is to find more efficient inference methods, both in terms of the number of samples required and in terms of the computational complexity. FIG. 1 . 1The propagator likelihood. The set of independent samples {x µ } M µ=1 characterize the empirical distributionp defined by equation FIG. 2. The propagator likelihood for a simple twoconfiguration system. The inset shows the single-step dynamics of the system with configurations 0 and 1, controlled by the hopping probability r ∈ (0, 1). In the main figure, the solid lines show the propagator likelihood for varying propagation time intervals τ . The dashed line shows the log-likelihood (2), which corresponds to an infinite propagation time interval. The maximum likelihood estimate of the hopping probability, r inf = 1−p(0) p(0) , is marked on the top axis and coincides with the maximum for all propagator likelihoods with an uneven number of time steps τ (see the main text for the case of even time steps). FIG. 4 . 4Parameter inference in the Ornstein-Uhlenbeck process. (a) The inset shows a schematic plot of the model describing a single particle moving in the harmonic potential U (x) = bx 2 /2. In the main figure, we show the relative reconstruction error ǫ = \|Θ inf − Θ true \|/Θ true of the parameter Θ = b/σ 2 (characterising the steady state) versus the dimensionless propagation time interval τ used in the propagator for sample sizes M = 10 3 ( ), M = 10 4 ( ), and M = 10 5 (•). The solid lines with markers show the reconstruction errors for the approximate short-time propagator, the dashed lines indicate the reconstruction errors for the exact finitetime propagator. (b) shows the estimated rate of change of the inferred parameter with respect to the propagation time interval τ . The rates of change are computed using forward difference quotients \|∂Θ inf /∂τ (τi)\| ≈ \|Θ inf (τi + ∆τ ) − Θ inf (τi)\|/∆τ and are shown on the vertical axis for the differentiation step size ∆τ = 10 −3 . The minimal rate of change corresponds to the optimal choice of the propagation time interval (see main text). The data was generated by independent sampling from the stationary distribution, i.e. a centred Gaussian with variance σ 2 /(2b) = 1/4. In order to remove fluctuations between different sample sets {xµ} M µ=1 and demonstrate the dependence of the average error on the sample size M and propagation time interval τ , the results were averaged over 50 independent sample sets. The minima of the reconstruction error and the rate of change coincide also for individual sample sets, while the position of the minima may vary across sample sets. FIG. 5 . 5The inference of couplings in the kinetic Ising model. FIG. 6 . 6Coupling inference in the sparse kinetic Ising model. FIG. 7 . 7Increasing the propagation time interval in the undersampled kinetic Ising model. (a) shows the reconstructed symmetric part of the coupling matrix J sym ij = (Jij +Jji)/2 based on the single-step propagator likelihood ( ) and on the longer propagation time interval τ = 3 (•). (b) shows the reconstructed antisymmetric part of the coupling matrix J asym ij = (Jij − Jji)/2 also based on the single-step propagator likelihood ( ) and on the longer propagation time interval τ = 3 (•).The underlying off-diagonal couplings were drawn independently from a Gaussian distribution with mean 0 and standard deviation 0.5/ √ N (we excluded self-interactions, Jii = 0), the external fields were drawn independently from a Gaussian distribution with mean 0 and standard deviation 0.5. The system size was N = 16 spins and M = 10 4 N samples were used. As a result, less than a third of the 2 16 system configurations were present in the sample. FIG. 8 . 8Reconstruction of the inter-species interactions in replicator dynamics. (a) The inset schematically shows the replicator model describing the population dynamics of different species competing for fractions of the total population size. 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[ "A characterization of some graphs with metric dimension two", "A characterization of some graphs with metric dimension two" ]	[ "Ali Behtoei \nDepartment of Mathematics\nImam Khomeini International University\n34149-16818QazvinIran\n", "Akbar Davoodi ", "Mohsen Jannesari \nUniversity of Shahreza\n86149-56841ShahrezaIran\n", "Behnaz Omoomi \nDepartment of Mathematical Sciences\nIsfahan University of Technology\n84156-83111IsfahanIran\n" ]	[ "Department of Mathematics\nImam Khomeini International University\n34149-16818QazvinIran", "University of Shahreza\n86149-56841ShahrezaIran", "Department of Mathematical Sciences\nIsfahan University of Technology\n84156-83111IsfahanIran" ]	[]	A set W ⊆ V (G) is called a resolving set, if for each pair of distinct verticesy)is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim M (G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then dim M (G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two. * [email protected] † [email protected]	10.1142/s1793830917500276	[ "https://arxiv.org/pdf/1509.02129v1.pdf" ]	41,914,155	1509.02129	984f07f7e7a889984613864a1dfbf08633fafb4e	A characterization of some graphs with metric dimension two 7 Sep 2015 Ali Behtoei Department of Mathematics Imam Khomeini International University 34149-16818QazvinIran Akbar Davoodi Mohsen Jannesari University of Shahreza 86149-56841ShahrezaIran Behnaz Omoomi Department of Mathematical Sciences Isfahan University of Technology 84156-83111IsfahanIran A characterization of some graphs with metric dimension two 7 Sep 2015 A set W ⊆ V (G) is called a resolving set, if for each pair of distinct verticesy)is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim M (G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then dim M (G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two. * [email protected] † [email protected] Introduction Throughout this paper all graphs are finite, simple and undirected. The notions δ, ∆ and N G (v) stand for minimum degree, maximum degree and the set of neighbours of vertex v in G, respectively. For an ordered set W = {w 1 , w 2 , . . . , w k } of vertices and a vertex v in a connected graph G, the k-vector r(v\|W ) := (d(v, w 1 ), d(v, w 2 ), . . . , d(v, w k )) is called the metric representation of v with respect to W , where d(x, y) is the distance between two vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W . We say a set S ⊆ V (G) resolves a set T ⊆ V (G) if for each pair of distinct vertices u and v in T there is a vertex s ∈ S such that d(u, s) = d(v, s). A minimum resolving set is called a basis and the metric dimension of G, dim M (G), is the cardinality of a basis for G. A graph with metric dimension k is called k-dimensional. The concept of the resolving set has various applications in diverse areas including coin weighing problems [10], network discovery and verification [1], robot navigation [8], mastermind game [3], problems of pattern recognition and image processing [9], and combinatorial search and optimization [10]. These concepts were introduced by Slater in [11]. He described the usefulness of these concepts when working with U.S. Sonar and Coast Guard Loran stations. Independently, Harary and Melter [6] discovered these concepts. In [8], it is proved that determining the metric dimension of a graph in general is an N P -complete problem, but the metric dimension of trees can be obtained by a polynomial time algorithm. It is obvious that for every graph G of order n, 1 ≤ dim M (G) ≤ n − 1. Chartrand et al. [5] proved that for n ≥ 2, dim M (G) = n − 1 if and only if G is the complete graph K n . They also provided a characterization of graphs of order n and metric dimension n − 2 [5]. Graphs with metric dimension n − 3 are characterized in [7]. Khuller et al. [8] and Chartrand et al. [5] proved that dim M (G) = 1 if and only if G is a path. Moreover, in [12] some properties of 2-dimensional graphs are obtained. Theorem 1.1 [12] Let G be a 2-dimensional graph. If {a, b} is a basis for G, then 1. there is a unique shortest path P between a and b, 2. the degrees of a and b are at most three, 3. the degree of each internal vertex on P is at most five. A chordal graph is a graph with no induced cycle of length greater than three. A k-tree is a chordal graph that all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. In other words, a k-tree may be formed by starting with a set of k + 1 pairwise adjacent vertices and then repeatedly adding vertices in such a way that each added vertex has exactly k neighbours that form a k-clique. By the above definition, it is clear that if G is a k-tree, then δ(G) = k. 1-trees are the same as trees; 2-trees are maximal series-parallel graphs [4] and include also the maximal outerplanar graphs. These graphs can be used to model series and parallel electric circuits. Planar 3-trees are also known as Apollonian networks [2]. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. On the other hand, regards to the recursive construction of k-trees, a k-path G can be considered as a graph with vertex set V (G) = {v 1 , v 2 , . . . , v n } and edge set E(G) = {v i v j : \|i − j\| ≤ k}. For instance, two different representations of a 2-path G with seven vertices v 1 , . . . , v 7 are shown in Figure 1. In this paper, we show that the metric dimension of each k-path (as a generalization of a path) is k. Whereas, there are some examples of 2-trees with metric dimension two that are not 2-path. This fact motivates us to study the structure of 2-dimensional 2-trees. As a main result, we characterize the class of all 2-trees with metric dimension two. v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Main Results In this section, we first prove that the metric dimension of each k-path is k. Then, we introduce a class of graphs which shows that the inverse of this fact is not true in general. Later on, we concern on the case k = 2 and toward to investigating all 2-trees with metric dimension two, we construct a family F of 2-trees with metric dimension two. Finally, as the main result, we prove that the metric dimension of a 2-tree G is two if and only if G belongs to F. Theorem 2.1 If G is a k-path, then dim M (G) = k. Proof. Let G be a k-path with vertex set V (G) = {v 1 , v 2 , . . . , v n } and edge set E(G) = {v i v j : \|i − j\| ≤ k}. Therefore, the distance between two vertices v r and v s in G is given by d(v r , v s ) = \|r−s\| k . At first, let W = {v 1 , v 2 , . . . , v k } and v i , v j be two distinct vertices of G with k < i < j. By the division algorithm, there exist integers r and s such that i = rk + s, 1 ≤ s ≤ k. Thus, we have d(v i , v s ) = \|i − s\| k = rk k = r, and d(v j , v s ) = \|j − s\| k = rk + (j − i) k = r + j − i k ≥ r + 1. This means W is a resolving set for G. Hence, dim M (G) ≤ \|W \| = k. Now, we show that dim M (G) ≥ k. Let W be a basis of the k-path G, and let X = {v 1 , v 2 , . . . , v k+1 }. Assume that \|W ∩ X\| = s and X \ W = {v i 1 , v i 2 , . . . , v i k+1−s }, where 1 ≤ i 1 < i 2 < · · · < i k+1−s ≤ k + 1. For convince, let X ′ = {x 1 , x 2 , . . . , x k+1−s }, where x r = v ir , for each r, 1 ≤ r ≤ k + 1 − s. Since each vertex v i of the k-path G is adjacent to the next k consecutive vertices {v i+1 , . . . , v i+k }, the induced subgraph on X is a (k + 1)clique. Each vertex in W ∩ X is adjacent to each vertex in X ′ . Thus, each pair of vertices in X ′ should be resolved by some element of W \ X. Assume that W ′ = {w 1 , w 2 , . . . , w t } is a minimum subset of W \ X which resolves vertices in X ′ . Thus, for each w j ∈ W ′ there exists {x r , x s } ⊆ X ′ such that d(w j , x r ) = d(w j , x s ). For each j, 1 ≤ j ≤ t, let r j = min{r : d(w j , x r ) = d(w j , x r+1 )}, and, let A j = {x 1 , x 2 , . . . , x r j }, B j = {x r j +1 , x r j +2 , . . . , x k+1−s }. Note that A j ∪ B j = X ′ , A j ∩ B j = ∅, x 1 ∈ A j and x k+1−s ∈ B j . Also, the structure of G implies that d(w j , x 1 ) = d(w j , x 2 ) = · · · = d(w j , x r j ), and d(w j , x r j +1 ) = d(w j , x r j +2 ) = · · · = d(w j , x k+1−s ). Since W ′ has the minimum size, for each 1 ≤ j < j ′ ≤ t we have A j = A j ′ (otherwise, w j and w j ′ resolve the same pair of vertices in X ′ ) and hence, \|A j \| = \|A j ′ \|. Moreover, for each r, 1 ≤ r ≤ k − s, there exists w j ∈ W ′ such that d(w j , x r ) = d(w j , x r+1 ) which implies \|A j \| = r. Therefore, t = \|{\|A 1 \|, \|A 2 \|, . . . , \|A t \|}\| = \|{1, 2, . . . , k − s}\| = k − s. Hence, \|W \| = \|W \ X\| + \|W ∩ X\| ≥ \|W ′ \| + s = (k − s) + s = k, which completes the proof. Definition 2.2 Let G and H be two 2-trees. We say that H is a branch in G on {u, v}, for convenience say a (u, v)-branch, if V (H) ∩ V (G) = {u, v}, where uv is an edge of G belonging to only one of the triangles in H. The length of a branch in a 2-tree is the number of it's triangles, which is equal to the number of vertices of branch minus 2. A cane is a 2-path with a branch of length one on a specific edge as shown in Figure 2. · · · Figure 2: A cane. In the following proposition, we provide some 2-trees with metric dimension two other than 2-paths. Proposition 2.3 If G is a 2-tree of metric dimension two with a basis whose elements are adjacent, then G is a 2-path or a cane. a (1,2) b (1,1) . . . (2,1) (t,t + 1) (t,t) (a) a . . . (b) b . . . a b (c) a (1,2) b (1,1) . . . Regards to the metric representation of vertices in G, x could be adjacent to the vertices by metric representation (t, t + 1) and (t, t) (in the case of not existence of dashed edges (t − 1, t) and (t, t)) and in the case (d) to the vertices by metric representation (1, 0) and (1, 1) as well. This concludes that G is also a path or a cane. (t,t + 1) (t,t) (d) The above proposition shows that the inverse of Theorem 2.1 is not true. Later on, we focus on the case k = 2 and construct the family F of all 2-trees with metric dimension two. Let F be the family of 2-trees, where each member G of F consists of a 2-tree G 0 and some branches on it that, in the case of existence, satisfying the following conditions. 1. G 0 is a 2-path or a 2-tree that is obtained by identifying two specific edges of two disjoint 2-paths as shown in Figure 4. 2. On every edge there is at most one branch. 3. G avoids any (a i , a i+1 )-branch. 4. Each branch is either a 2-path or a cane. 5. In each (a i , b i )-branch the degree of a i is two. 6. If G 0 is as the graph depicted in Figure 4(b), then G avoids any (a m , x)-branch. 7. G contains at most one branch on the edges of the triangle containing b i b i+1 in G 0 . 8. The degree of each b i in G is at most 7. 9. G has at most one branch of length greater than one on the edges of the triangle containing a i a i+1 in G 0 . 10. If G 0 is of the form of Figure 4(b), then (b m−1 , b m )-branch and (b m , b m+1 )-branch are 2-path and at most one of them is of length more than one. For every i, 2 ≤ i ≤ k − 1, at most one of the (b i−1 , b i )-branches and (b i , b i+1 )- branches is a cane. 12. All (a i , b i )-branches, (a i , b i+1 )-branches and (a i , b i−1 )-branches are 2-path. a 1 a 2 a 3 a k−1 a k b 1 b 2 b 3 b k−1 b k · · · (a) a 1 a 2 a 3 a m−2 a m−1 b 1 b 2 b 3 b m−2 b m−1 · · · a m a m+1 a m+2 a k−1 a k b m b m+1 b m+2 b k−1 b k · · · (b) Figure 4: Two different forms of G 0 . Theorem 2.4 If G ∈ F, then dim M (G) = 2. Proof. Let G ∈ F. Through the proof all of notations are the same as those which are used to introduce the family F and G 0 in Figure 4. Since G is not a path, dim M (G) ≥ 2. Let W = {a 1 , a k }. We show in both possible cases for G 0 that W is a resolving set for G and hence, dim M (G) = 2. Case 1. G 0 is a 2-path as shown in Figure 4(a). The metric representation of the vertices {a 1 , a 2 , . . . , a k , b 1 , b 2 , . . . , b k } are as follows. r(a i \|W ) = (i − 1, k − i), 1 ≤ i ≤ k, r(b 1 \|W ) = (1, k), r(b j \|W ) = (j − 1, k − j + 1), 2 ≤ j ≤ k. Thus, different vertices of G 0 have different metric representations. Moreover, note that {d 1 −d 2 : (d 1 , d 2 ) = r(a i \|W ), 1 ≤ i ≤ k} = {1−k, 3−k, 5−k, . . . , 2i−k−1, . . . , k−3, k−1}, and {d 1 −d 2 : (d 1 , d 2 ) = r(b i \|W ), 1 ≤ i ≤ k} = {1−k, 2−k, 4−k, . . . , 2i−k−2, . . . , k−4, k−2}. If G = G 0 , then we are done. Suppose that G = G 0 and let H be a branch of G on an edge e of G 0 . Regards to the structures of graphs in F, we consider the following different possibilities. • H is a branch on the vertical edge e = a i b i , 2 ≤ i ≤ k − 1. Note that by the definition of F, H is a 2-path and deg H ( a i ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = a i , x 2 = b i , and E(H) = {x r x s : \|r − s\| ≤ 2}. If j is odd, then d(x j , a 1 ) = d(x j , a i ) + d(a i , a 1 ) and d(x j , a k ) = d(x j , a i ) + d(a i , a k ). If j is even, then d(x j , a 1 ) = d(x j , b i ) + d(b i , a 1 ) and d(x j , a k ) = d(x j , b i ) + d(b i , a k ). Hence, we have r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i − 1 + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋) j is even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2i − k − 1, 2i − k − 2}. • H is a branch on the oblique edge e = a i b i+1 , 2 ≤ i ≤ k − 1. By the definition of F, H is a 2-path and deg H (a i ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = a i , x 2 = b i+1 , and E(H) = {x r x s : \|r − s\| ≤ 2}. If j is odd, then d(x j , a 1 ) = d(x j , a i ) + d(a i , a 1 ) and d(x j , a k ) = d(x j , a i ) + d(a i , a k ). If j is even, then d(x j , a 1 ) = d(x j , b i+1 ) + d(b i+1 , a 1 ) and d(x j , a k ) = d(x j , b i+1 ) + d(b i+1 , a k ). Hence, we have r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋ − 1) j is even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2i − k − 1, 2i − k}. • H is a branch on the horizontal edge e = b i b i+1 , 1 ≤ i ≤ k − 1. Using the definition of F, H is either a 2-path or a cane. Generally, assume that {x 1 , x 2 , . . . , x t } ⊆ V (H) ⊆ {x 1 , x 2 , . . . , x t } ∪ {x}, where the induced subgraph of H on {x 1 , x 2 , . . . , x t } is a 2-path with the edge set {x r x s : \|r − s\| ≤ 2}. We consider two different possibilities. a) x 1 = b i , x 2 = b i+1 . Hence, if H is a cane, then we have N H (x) = {b i , x 3 }. Similar to the previous cases, we have r(x 1 \|W ) = (i − 1, k − i + 1), r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j ≥ 3 is odd (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋ − 1) j is even. Also, if H is a cane, then r(x\|W ) = (i − 1 + 1, k − i + 2). b) x 1 = b i+1 , x 2 = b i . Hence, if H is a cane, then we have N H (x) = {b i+1 , x 3 }. Similarly, we have r(x 1 \|W ) = (i − 1 + 1, k − i), r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i − 1 + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋) j is even. Also, if H is a cane, then r(x\|W ) = (i − 1 + 2, k − i + 1). Note that in both states (and regardless of being a 2-path or a cane), we have {d 1 − d 2 : (d 1 , d 2 ) = r(v\|W ), v ∈ V (H)} = {2i − k − 2, 2i − k − 1, 2i − k}. Therefore, in all the above cases, distinct vertices of H have different metric representations. Also, the metric representation of the vertices in V (H) are different from the metric representations of the vertices in V (G 0 ) \ {x, y}, where H is a (x, y)-branch. Moreover, using the subtraction value of two coordinates in the metric representation of each vertex, it is easy to check that vertices of different (possible) branches on G 0 (satisfying the conditions mentioned in the definition of F) have different metric representations. Thus, in this case W is a resolving set for G. Case 2. G 0 is a 2-tree of the form Figure 4 r(a i \|W ) = (i − 1, k − i), 1 ≤ i ≤ k, r(b j \|W ) =    (j, k − j) 1 ≤ j ≤ m − 1 (m, k − m + 1) j = m (j − 1, k − j + 1) m + 1 ≤ j ≤ k. Therefore, different vertices of G 0 have different metric representations. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(a i \|W ), 1 ≤ i ≤ k} = {1 − k, 3 − k, 5 − k, . . . , 2m − k − 3, 2m − k − 1, 2m − k + 1, . . . , k − 3, k − 1}, and {d 1 − d 2 : (d 1 , d 2 ) = r(b j \|W ), 1 ≤ j ≤ k} = {2 − k, 4 − k, 6 − k, . . . , 2m − k − 2, 2m − k − 1, 2m − k, . . . , k − 4, k − 2}. If G = G 0 , then we are done. Hence, suppose that G = G 0 and let H be a branch of G on an edge e of G 0 . Again, using the possible structures of H according to the definition of F, we consider the following different cases. • H is a branch on the vertical edge e = a i b i , 2 ≤ i ≤ m − 1. Note that by the definition of F, H is a 2-path and deg H ( a i ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = a i , x 2 = b i , and E(H) = {x r x s : \|r − s\| ≤ 2}. It is straightforward to check that r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋ − 1) j is even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2i − k − 1, 2i − k}. • H is a branch on the vertical edge e = a i b i , m + 1 ≤ i ≤ k − 1. By the definition of F, H is a 2-path and deg H ( a i ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = a i , x 2 = b i , and E(H) = {x r x s : \|r − s\| ≤ 2}. We have r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i + ⌊ j 2 ⌋ − 2, k − i + ⌊ j 2 ⌋) j is even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2i − k − 1, 2i − k − 2}. • H is a branch on the oblique edge e = a i b i−1 , 2 ≤ i ≤ m − 1. Since G ∈ F, H is a 2-path and deg H ( a i ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = a i , x 2 = b i−1 , and E(H) = {x r x s : \|r − s\| ≤ 2}. We have r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i + ⌊ j 2 ⌋ − 2, k − i + ⌊ j 2 ⌋) j is even. Moreover, {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2i − k − 1, 2i − k − 2}. • H is a branch on the oblique edge e = a i b i+1 , m + 1 ≤ i ≤ k − 1. We know that H is a 2-path and deg H ( a i ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = a i , x 2 = b i+1 , and E(H) = {x r x s : \|r − s\| ≤ 2}. Similarly, it can be easily checked that r(x j \|W ) = (i − 1 + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋) j is odd (i + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋ − 1) j is even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2i − k − 1, 2i − k}. • H is a branch on the horizontal edge e = b i b i+1 , 1 ≤ i ≤ m − 2. Using the definition of F, H is either a 2-path or a cane. Generally, assume that {x 1 , x 2 , . . . , x t } ⊆ V (H) ⊆ {x 1 , x 2 , . . . , x t } ∪ {x}, where the induced subgraph of H on {x 1 , x 2 , . . . , x t } is a 2-path with the edge set {x r x s : \|r − s\| ≤ 2}. We consider two different possibilities. a) x 1 = b i , x 2 = b i+1 . Hence, if H is a cane, then we have N H (x) = {b i , x 3 }. Similar to the previous cases, we have r(x 1 \|W ) = (i, k − i), r(x j \|W ) = (i + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋ − 1) j ≥ 3 is odd (i + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋ − 2) j is even. Also, if H is a cane, then r(x\|W ) = (i + 1, k − i + 1). b) x 1 = b i+1 , x 2 = b i . Hence, if H is a cane, then we have N H (x) = {b i+1 , x 3 }. Similarly, we have r(x 1 \|W ) = (i + 1, k − i − 1), r(x j \|W ) = (i + ⌊ j 2 ⌋, k − i + ⌊ j 2 ⌋ − 1) j ≥ 3 is odd (i + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋ − 1) is even. Also, if H is a cane, then r(x\|W ) = (i + 2, k − i). Note that in the both states (and regardless of being a 2-path or a cane) we have {d 1 − d 2 : (d 1 , d 2 ) = r(v\|W ), v ∈ V (H)} = {2i − k, 2i − k + 1, 2i − k + 2}.r(x j \|W ) = (m + ⌊ j 2 ⌋ − 1, k − m + ⌊ j 2 ⌋ + 1) j is odd (m + ⌊ j 2 ⌋ − 1, k − m + ⌊ j 2 ⌋) j is even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2m − k − 2, 2m − k − 1}. • H is a branch on the horizontal edge e = b m b m+1 . By the definition of F, H is a 2-path and deg H (b m+1 ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = b m+1 , x 2 = b m , and E(H) = {x r x s : \|r − s\| ≤ 2}. We have r(x j \|W ) = (m + ⌊ j 2 ⌋, k − m + ⌊ j 2 ⌋) j is odd (m + ⌊ j 2 ⌋ − 1, k − m + ⌊ j 2 ⌋) j even. Moreover, note that {d 1 − d 2 : (d 1 , d 2 ) = r(x j \|W ), 1 ≤ j ≤ t} = {2m − k − 1, 2m − k}. • H is a branch on the horizontal edge e = b i b i+1 , m + 1 ≤ i ≤ k − 1. Using the definition of F, H is either a 2-path or a cane. Generally, assume that {x 1 , x 2 , . . . , x t } ⊆ V (H) ⊆ {x 1 , x 2 , . . . , x t } ∪ {x}, where the induced subgraph of H on {x 1 , x 2 , . . . , x t } is a 2-path with the edge set {x r x s : \|r − s\| ≤ 2}. Again, we consider two different possibilities. a) x 1 = b i , x 2 = b i+1 . Hence, if H is a cane and N H (x) = {b i , x 3 }, then We have r(x 1 \|W ) = (i − 1, k − i + 1), r(x j \|W ) = (i + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋) j ≥ 3 is odd (i + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋ − 1) j is even. Also, if H is a cane, then r(x\|W ) = (i, k − i + 2). b) x 1 = b i+1 , x 2 = b i . Hence, if H is a cane, then we have N H (x) = {b i+1 , x 3 }. Similarly, we have r(x 1 \|W ) = (i, k − i), r(x j \|W ) = (i + ⌊ j 2 ⌋ − 1, k − i + ⌊ j 2 ⌋) j ≥ 3 is odd (i + ⌊ j 2 ⌋ − 2, k − i + ⌊ j 2 ⌋) j is even. Also, if H is a cane, then r(x\|W ) = (i + 1, k − i + 1). Note that in the both states (and regardless of being a 2-path or a cane) we have {d 1 − d 2 : (d 1 , d 2 ) = r(v\|W ), v ∈ V (H)} = {2i − k − 2, 2i − k − 1, 2i − k}. Therefore, in all of above cases, distinct vertices of H have different metric representations. Also, the metric representation of the vertices in V (H) are different from the metric representations of the vertices in V (G 0 ) \ {x, y}, where H is a (x, y)-branch. Moreover, using the subtraction value of two coordinates in the metric representation of each vertex, it is easy to check that vertices of different (possible) branches on G 0 (satisfying the conditions mentioned in the definition of F) have different metric representations. Thus, in this case W is a resolving set for G. To prove the converse of Theorem 2.4, we need the following lemma. This contradicts that {a, b} is a resolving set for G ∪ H. Now, we prove that every 2-dimensional 2-tree belongs to the family F. If ∆(H) ≤ 4, then H is a 2-path as shown in Figure 4(a). Otherwise ∆(H) = 5. If there exists a vertex b j of degree 5, then it can be easily checked that b j and a j have the same representation with respect to {a 1 , a k }. Also, existence of two vertices a i and a i ′ both of degree 5, i ≤ i ′ , implies that there exists some vertex b j , i ≤ j ≤ i ′ , of degree 5, which is impossible. Thus, there exists a unique a i of degree 5. Therefore, H is the graph shown in Figure 4(b). Thus, H is a 2-path or a 2-tree obtained by identifying the specific edge, say a m b m , of two 2-paths (see Figure 4(b)), where B = {a 1 , a k }. Thus, G satisfies property (1). Clearly, on every edge there is at most one branch; thus, property (2) follows. Also, G avoids any (a i , a i+1 )-branch, because each vertex adjacent to both a i and a i+1 has the same metric representation as b i or b i+1 . Thus, G contains only (a i , b i )-branches, (a i , b i+1 )branches, (a i+1 , b i )-branches or (b i , b i+1 )-branches; which implies property (3). Moreover, by Proposition 2.3 and Lemma 2.5, each of these branches is a 2-path or a cane. Therefore, property (4) holds. Also, by Theorem 1.1, for every i, 1 ≤ i ≤ k, there is at most one (a i , x)-branch in G. Moreover, in each (a i , b i )-branch the degree of a i is two, which shows trueness of property (5). To see property (6), first note that by property (3) there is no (a m−1 , a m )-branch or (a m , a m+1 )-branch. Moreover, in each (a m , x)-branch, for x ∈ {b m−1 , b m , b m+1 }, the unique neighbour of a m on the branch has the same metric representation as b m . To show that G has property (7), suppose that a triangle a i b i b i+1 has more than one branch. By Theorem 1.1, at most one of (a i , b i )-branch and (a i , b i+1 )-branch exists. Therefore, b i b i+1 has a branch H 1 and one of the edges a i b i or a i b i+1 has another branch H 2 . Let x and y be the vertices of distance one from G 0 on branches H 1 and H 2 , respectively. Hence, d(a 1 , x) = d(a 1 , y) = i and d(a k , x) = d(a k , y) = k − i + 1. That is, {a 1 , a k } is not a basis of G, which is a contradiction. A similar reason works for triangle a i b i−1 b i . Hence, G has property (7). Let (d 1 , d 2 ) be metric representation of b i . Then metric representations of each neighbour of b i which is out of G 0 could be one of (d 1 + 1, d 2 + 1), (d 1 + 1, d 2 ) or (d 1 , d 2 + 1). Thus, b i has at most three neighbours out of G 0 . Hence, the degree of b i in G is at most 7 that is property (8). If there are two branches of length at least 2 on a triangle containing a i a i+1 , then the metric representation of the second vertices on these branches are the same, a contradiction. Thus, G satisfies property (9). If H is a (b m−1 , b m )-branch of cane type, then one can find two vertices in N G (b m ) ∪ N G (b m−1 ) with the same metric representation. A similar argument holds whenever H is a (b m , b m+1 )-branch of cane type. If there is a (b m−1 , b m )-branch, say H 1 , and a (b m , b m+1 )branch, say H 2 , both of length at least two, then b m has a neighbour in H 1 with the same metric representation as a neighbour of b m in H 2 . Hence, property (10) holds. Suppose that two branches on (b i−1 , b i ) and (b i , b i+1 ) are canes. In this case, it can be checked that in the set of neighbours of b i in these branches there are two vertices with the same metric representation. Thus, G satisfies property (11). Figure 1 : 1Two different representations of a 2-path. Figure 3 : 3The possible cases for basis {a, b} in 2-tree G Proof. We prove the statement by induction on n, the order of G. If n = 3, then G = K 3 and the statement holds. Let G be a 2-tree of order n > 3 with a basis B = {a, b}, such that d(a, b) = 1. Since each 2-tree of order greater than three has two non-adjacent vertices of degree two, there exists a vertex x ∈ V (G) \ B of degree two. Moreover, B is a basis for G \ {x}. Now, by the induction hypothesis, G \ {x} is a path or a cane and by Theorem 1.1 (2), the degrees of a and b are at most three. Therefore, B = {a, b} is one of the possible cases shown inFigure 3. Note that dashed edges could be absent. It can be checked that in cases (b) and (c) the bold vertices get the same metric representation with respect to B. Thus, B is one of the cases (a) or (d), where the metric representations of vertices are denoted inFigure 3. (b). The metric representation of the vertices {a 1 , a 2 , . . . , a m , . . . , a k } ∪ {b 1 , b 2 , . . . , b m , . . . , b k } are as follows. • H is a branch on the horizontal edge e = b m−1 b m . By the definition of F, H is a 2-path and deg H (b m−1 ) = 2. Let V (H) = {x 1 , x 2 , . . . , x t } where x 1 = b m−1 , x 2 = b m , and E(H) = {x r x s : \|r − s\| ≤ 2}. We have Lemma 2. 5 5Let H be a {u, v}-branch of G and let {a, b} be a basis for G ∪ H. If {a, b} ∩ V (H) ⊆ {u, v}, then {u, v} is a metric basis for H.Proof. Suppose on the contrary, there are two different vertices x and y in H such thatd(x, u) = d(y, u) = r, d(x, v) = d(y, v) = s.Since H is a branch on {u, v}, each path connecting a vertex in H with a vertex inV (G) \ V (H) passes through u or v. Assume that d(u, a) = r 1 , d(v, a) = s 1 , d(u, b) = r 2 , d(v, b) = s 2 .Hence, d(x, a) = min{r + r 1 , s + s 1 } = d(y, a), d(x, b) = min{r + r 2 , s + s 2 } = d(y, b). Theorem 2. 6 6If G is a 2-tree of metric dimension two, then G ∈ F.Proof. Let G be a 2-tree and {a, b} be a basis of G. If d(a, b) = 1, then by Proposition 2.3, G is a 2-path or a cane which belongs to F. Thus, assume that d(a, b) > 1 and let H be a minimal induced 2-connected subgraph of G as shown inFigure 5, containing a and b. Since the clique number of G is three, in each square exactly one of the dashed edges are allowed. Moreover, by the minimality of H we have deg H (a) = deg H (b) = 2, where a ∈ {a 1 , b 1 } and b ∈ {a k , b k }. Hence, one of two vertices a 1 , b 1 or one of two vertices a k , b k may not exist. One can check that {a, b} = {a 1 , b k } and {a, b} = {b 1 , a k }, otherwise, two neighbours of a or b get the same metric representation. Thus, by the symmetry, we may assume {a, b} = {a 1 , a k }. Figure 5 : 5A minimal induced 2-connected subgraph of G. Note that deg(a i ) ∈ {4, 5}. Now suppose that H is a branch on the edge {a i , b i }, {a i , b i+1 } or {a i , b i−1 }. If H is a cane, then deg G (a i ) ≥ 6 or two neighbours of b i−1 , b i or b i+1 in H get the same metric representation, which both are contradictions. Thus, each branch on the edge {a i , b i−1 }, {a i , b i } or {a i. in G, 1 < i < n, is at most five. b i+1 } is a 2-path and G satisfies property (12in G, 1 < i < n, is at most five. Note that deg(a i ) ∈ {4, 5}. Now suppose that H is a branch on the edge {a i , b i }, {a i , b i+1 } or {a i , b i−1 }. If H is a cane, then deg G (a i ) ≥ 6 or two neighbours of b i−1 , b i or b i+1 in H get the same metric representation, which both are contradictions. Thus, each branch on the edge {a i , b i−1 }, {a i , b i } or {a i , b i+1 } is a 2-path and G satisfies property (12). 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[ "The µ-τ reflection symmetry of Dirac neutrinos and its breaking effect via quantum corrections", "The µ-τ reflection symmetry of Dirac neutrinos and its breaking effect via quantum corrections" ]	[ "Zhi-Zhong Xing \nInstitute of High Energy Physics\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nCenter for High Energy Physics\nPeking University\n100080BeijingChina\n", "Di Zhang [email protected]†email:[email protected] \nInstitute of High Energy Physics\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "Jing-Yu Zhu \nInstitute of High Energy Physics\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n" ]	[ "Institute of High Energy Physics\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Center for High Energy Physics\nPeking University\n100080BeijingChina", "Institute of High Energy Physics\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Institute of High Energy Physics\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina" ]	[]	Given the Dirac neutrino mass term, we explore the constraint conditions which allow the corresponding mass matrix to be invariant under the µ-τ reflection transformation, leading us to the phenomenologically favored predictions θ 23 = π/4 and δ = 3π/2 in the standard parametrization of the 3 × 3 lepton flavor mixing matrix. If such a flavor symmetry is realized at a superhigh energy scale Λ µτ , we investigate how it is spontaneously broken via the one-loop renormalization-group equations (RGEs) running from Λ µτ down to the Fermi scale Λ F . Such quantum corrections to the neutrino masses and flavor mixing parameters are derived, and an analytical link is established between the Jarlskog invariants of CP violation at Λ µτ and Λ F . Some numerical examples are also presented in both the minimal supersymmetric standard model and the type-II two-Higgs-doublet model, to illustrate how the octant of θ 23 , the quadrant of δ and the neutrino mass ordering are correlated with one another as a result of the RGE-induced µ-τ reflection symmetry breaking effects.	10.1007/jhep11(2017)135	[ "https://arxiv.org/pdf/1708.09144v2.pdf" ]	119,345,446	1708.09144	1eb97d6c6c352dfd5027f3e207caa0613a0c05c5	The µ-τ reflection symmetry of Dirac neutrinos and its breaking effect via quantum corrections 15 Nov 2017 Zhi-Zhong Xing Institute of High Energy Physics School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina Center for High Energy Physics Peking University 100080BeijingChina Di Zhang [email protected]†email:[email protected] Institute of High Energy Physics School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina Jing-Yu Zhu Institute of High Energy Physics School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina The µ-τ reflection symmetry of Dirac neutrinos and its breaking effect via quantum corrections 15 Nov 20171number(s): 1460Pq1130Hv1110Hi * Given the Dirac neutrino mass term, we explore the constraint conditions which allow the corresponding mass matrix to be invariant under the µ-τ reflection transformation, leading us to the phenomenologically favored predictions θ 23 = π/4 and δ = 3π/2 in the standard parametrization of the 3 × 3 lepton flavor mixing matrix. If such a flavor symmetry is realized at a superhigh energy scale Λ µτ , we investigate how it is spontaneously broken via the one-loop renormalization-group equations (RGEs) running from Λ µτ down to the Fermi scale Λ F . Such quantum corrections to the neutrino masses and flavor mixing parameters are derived, and an analytical link is established between the Jarlskog invariants of CP violation at Λ µτ and Λ F . Some numerical examples are also presented in both the minimal supersymmetric standard model and the type-II two-Higgs-doublet model, to illustrate how the octant of θ 23 , the quadrant of δ and the neutrino mass ordering are correlated with one another as a result of the RGE-induced µ-τ reflection symmetry breaking effects. Introduction The discoveries of solar, atmospheric, reactor and accelerator neutrino oscillations [1] have demonstrated that the standard model (SM) of electroweak interactions is incomplete and must be extended in a proper way so as to accommodate tiny neutrino masses and significant lepton flavor mixing. The simplest way to do so is to introduce three right-handed (or SU(2)singlet) neutrino fields N αR (for α = e, µ, τ ) into the SM and write out a gauge-invariant, Lorentz-invariant and lepton-number-conserving mass term of the form −L Dirac = ℓ L Y νH N R + h.c. ,(1) whereH = iσ 2 H * with H being the SM Higgs doublet, ℓ L denotes the left-handed lepton doublet column vector, and N R represents the right-handed neutrino column vector with the N αR components. After spontaneous gauge symmetry breaking, the above Dirac neutrino mass term turns out to be 1 −L ′ Dirac = ν L M ν N R + h.c. ,(2) where M ν = Y ν H with H = v/ √ 2 and v ≃ 246 GeV. The three neutrino masses m i (for i = 1, 2, 3) can therefore be achieved from diagonalizing M ν if its texture is specified in a given model, but the smallness of m i is not really explained in this manner. While many theorists believe that the neutrinos should be Majorana fermions [4], by which their small masses can be naturally understood via a seesaw mechanism [5,6], the simplicity of the Dirac neutrino mass generation mechanism do attract quite a lot of attention [7,8]. Before the Majorana nature of massive neutrinos is ultimately determined with the help of a measurement of the neutrinoless double-beta decay or other lepton-number-violating processes [9], it makes sense to study the phenomenology of Dirac neutrinos as well. Assuming the massive neutrinos to be the Dirac fermions, we shall begin with Eq. (2) to explore the µ-τ reflection symmetry of L ′ Dirac and the resulting texture of M ν in the basis where the flavor eigenstates of three charged leptons are identified with their mass eigenstates. The motivation for this study is simply because such a flavor symmetry may naturally lead us to the phenomenologically favored predictions θ 23 = π/4 and δ = 3π/2 in the standard parametrization of the 3×3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix U [10] that is used to diagonalize M ν M † ν . Provided the µ-τ reflection symmetry is realized at a superhigh energy scale Λ µτ , we shall investigate how it is spontaneously broken due to the running of M ν from Λ µτ down to the Fermi scale Λ F ∼ v ∼ 10 2 GeV through the one-loop renormalization-group equations (RGEs) in the framework of either the MSSM or the type-II 2HDM. Such quantum corrections to the three neutrino masses and four flavor mixing parameters will be derived, and an analytical link will be established between the Jarlskog invariants of leptonic CP violation at Λ µτ and Λ F . We shall also present some numerical 1 If the minimal supersymmetric standard model (MSSM) is concerned, the charged-lepton and neutrino sectors are associated with the Higgs doublets H 1 (with the hypercharge +1/2 and the vacuum expectation value v cos β/ √ 2) and H 2 (with the hypercharge −1/2 and the vacuum expectation value v sin β/ √ 2), respectively [2]. But for the type-II two-Higgs-doublet model (2HDM), the Higgs doublet H 1 is coupled to both the charged-lepton and neutrino sectors [3]. These two interesting scenarios will be used to illustrate quantum corrections to the µ-τ reflection symmetry in section 4. examples in both the MSSM and the type-II 2HDM to illustrate how the octant of θ 23 , the quadrant of δ and the neutrino mass ordering are correlated with one another as a result of the RGE-triggered µ-τ reflection symmetry breaking effects. The content of this work is new in several aspects. First, applying the µ-τ reflection symmetry to the Dirac neutrino mass term, in which M ν is in general neither symmetric nor Hermitian, has not been tried before. Second, the integral form of the RGE corrections to M ν is derived for the first time, so is the integral form of the RGE effects on the neutrino masses and flavor mixing parameters. Third, a concise analytical relationship between the Jarlskog invariants of CP violation at Λ µτ and Λ F is derived for the first time. Fourth, a comparison is made between the MSSM and the type-II 2HDM, which leads to the opposite deviations of θ 12 , θ 13 , θ 23 and δ from their corresponding values in the µ-τ reflection symmetry limit. The remaining parts of this paper are organized as follows. In section 2 we shall find out the constraint conditions which allow the Dirac neutrino mass matrix M ν to be invariant under the µ-τ reflection transformation. Section 3 is devoted to the derivation of the integral form of the RGE corrections to M ν M † ν when it runs from Λ µτ down to Λ F , and to the derivation of an analytical relationship between the Jarlskog invariants at Λ µτ and Λ F . In section 4 we calculate the RGE-induced corrections to the neutrino masses and flavor mixing parameters in a perturbation way, and illustrate their salient features by taking a few numerical examples in both the MSSM and the type-II 2HDM. Finally, we summarize our main results and make a conclusion in section 5. µ-τ reflection symmetry Given the Dirac neutrino mass term in Eq. (2), let us consider the following transformations of the six neutrino fields 2 : ν eL ↔ ν c eL , N eR ↔ N c eR , ν µL ↔ ν c τ L , N µR ↔ N c τ R , ν τ L ↔ ν c µL , N τ R ↔ N c µR ,(3) where ν c αL ≡ Cν αL T and N c αL ≡ CN αL T (for α = e, µ, τ ) with T denoting the transpose and C being the charge-conjugation operator and satisfying C −1 = C † = C T = −C [12]. Under such transformations, Eq. (2) turns out to be −L ′ Dirac = ν c L SM ν SN c R + N c R SM † ν Sν c L = −ν T L SM ν SN R T − N T R SM † ν Sν L T = ν L SM * ν SN R + N R SM T ν Sν L ,(4) 2 In this work we focus on a possible µ-τ reflection symmetry of the Dirac neutrino mass matrix after spontaneous gauge symmetry breaking. Otherwise, the neutrino field transformations made in Eq. (3) would affect some other parts of the Lagrangian of the electroweak interactions. To build a consistent lepton mass model with the µ-τ flavor symmetry in the neutrino sector instead of the charged-lepton sector, one should introduce some extra scalar fields coupling to the two sectors in a different way [11]. But here we simply assume that the µ-τ reflection symmetry does not apply to the charged-lepton sector. In this sense the invariance of L ′ Dirac under the transformations in Eq. (3) can just serve as a phenomenological guiding principle to obtain the special texture of M ν in Eq. (9). in which the property of L ′ as a Lorentz scalar has been used, and S =    1 0 0 0 0 1 0 1 0    .(5) If L ′ is required to be invariant under the above µ-τ reflection transformations [13], then the Dirac neutrino mass matrix M ν ≡    m ee m eµ m eτ m µe m µµ m µτ m τ e m τ µ m τ τ    .(6) must satisfy the relationship M ν = SM * ν S .(7) In other words, the elements of M ν must satisfy m ee = m * ee , m eµ = m * eτ , m µe = m * τ e , m µτ = m * τ µ , m µµ = m * τ τ .(8) Then the texture of M ν can be simply parametrized as M ν =    a b b * e c d e * d * c *    ,(9) where a is real, and the other four parameters are in general complex. To diagonalize M ν in Eq. (9), one may do a bi-unitary transformation of the form U † M ν Q =M ν ,(10) where U and Q are the unitary matrices, andM ν ≡ Diag{m 1 , m 2 , m 3 } with m i (for i = 1, 2, 3) being the neutrino masses. In the basis where the flavor eigenstates of three charged leptons are identified with their mass eigenstates, the unitary matrix U is just the PMNS flavor mixing matrix which manifests itself in the leptonic weak charged-current interactions. It proves more convenient to consider the Hermitian matrix H ν ≡ M ν M † ν = UM 2 ν U † =    A B B * B * C D B D * C    ,(11) where A = a 2 + 2\|b\| 2 , B = ae * + bc * + b * d * , C = \|e\| 2 + \|c\| 2 + \|d\| 2 , D = e 2 + 2cd .(12) Moreover, let us parametrize U as U ≡ P V , where P = Diag{e iφ e , e iφ µ , e iφ τ } is an unphysical phase matrix associated with the charged-lepton fields 3 , and with c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 13, 23). At a given energy scale, one may rotate away P and then express the four flavor mixing parameters of V in terms of the elements of H ν ≡ P † H ν P = VM 2 ν V † =    A B B * B * C D B D * C    ,(14) where B = Be i(φ µ −φ e ) and D = De i(φ τ −φ µ ) . In this way the unphysical phases hidden in B and D will be cancelled by φ µ − φ e and φ τ − φ µ , respectively. Then we do a similar diagonalization of H ν as that done in Ref. [14] and obtain θ 12 = 1 2 arctan   2 ReB 2 ReB 2 + ImD 2 ImB ImD − 2ReB ReD   , θ 13 = arctan 1 √ 2 ImD ReB ;(15) together with the typical predictions θ 23 = π 4 , δ = π 2 or 3π 2 .(16) These two numerical predictions, which have been well known for the Majorana neutrino mass matrix with the µ-τ reflection symmetry [15], are now achieved in the Dirac case with the same flavor symmetry. It is easy to see that Eq. (16) leads us to the equalities V µ1 = \|V τ 1 \| , V µ2 = \|V τ 2 \| , V µ3 = \|V τ 3 \| ,(17) which are sometimes referred to as the µ-τ reflection symmetry at the PMNS matrix level. One may therefore define the asymmetries A i ≡ \|V µi \| 2 − \|V τ i \| 2 (for i = 1, 2, 3) to measure the effects of µ-τ symmetry breaking in a rephasing-invariant way [16]. Of course, it is more fundamental to understand how the µ-τ reflection symmetry of M ν or H ν can be spontaneously or explicitly broken, both for the model-building purpose and for explaining currently available neutrino oscillation data [17]. Following the discussions about the µ-τ symmetry breaking of the Majorana neutrino mass matrix [15,18], one can similarly introduce the most general perturbation to the Dirac neutrino mass matrix with the µ-τ reflection symmetry. But we find that it is more convenient to focus on the perturbation to H ν in Eq. (11) instead of M ν in Eq. (9), simply because the former is always Hermitian. In this case the perturbation matrix ∆H ν can also be arranged to be Hermitian, and it can be decomposed into two parts: one part conserves the original µ-τ reflection symmetry and the other part violates this symmetry. Namely, ∆H ν =    δ ee δ eµ δ eτ δ * eµ δ µµ δ µτ δ * eτ δ * µτ δ τ τ    = 1 2    2δ ee δ eµ + δ * eτ δ * eµ + δ eτ δ * eµ + δ eτ δ µµ + δ τ τ 2δ µτ δ eµ + δ * eτ 2δ * µτ δ µµ + δ τ τ    + 1 2    0 δ eµ − δ * eτ δ eτ − δ * eµ δ * eµ − δ eτ δ µµ − δ τ τ 0 δ * eτ − δ eµ 0 δ τ τ − δ µµ    ,(18) where δ ee , δ µµ and δ τ τ are real, and all the parameters are expected to be reasonably small in magnitude. Because the symmetry-conserving part can be absorbed into H ν via a redefinition of its initial matrix elements, we are then left with H ′ ν = H ν + ∆H ν =    A ′ B ′ (1 + ǫ 1 ) B ′ * (1 − ǫ * 1 ) B ′ * (1 + ǫ * 1 ) C ′ (1 + ǫ 2 ) D ′ B ′ (1 − ǫ 1 ) D ′ * C ′ (1 − ǫ 2 )    ,(19) where A ′ = A + δ ee , B ′ = B + δ eµ + δ * eτ 2 , C ′ = C + δ µµ + δ τ τ 2 , D ′ = D + δ µτ(20) and ǫ 1 = δ eµ − δ * eτ 2B ′ , ǫ 2 = δ µµ − δ τ τ 2C ′ .(21) It is obvious that ǫ 1 and ǫ 2 are complex and real, respectively. These two dimensionless parameters will vanish, if ∆H ν respects the µ-τ reflection symmetry. Although the above formulism can provide us with a generic picture of the µ-τ symmetry breaking, it has to be specified so as to see the explicit symmetry-breaking effects. In the following we shall assume that the µ-τ reflection symmetry is realized at a superhigh energy scale Λ µτ , and examine its breaking at the Fermi scale Λ F via the one-loop RGEs. RGE corrections to H ν From the point of view of model building, a specific flavor symmetry is usually realized at a superhigh energy scale where some fundamental new physics beyond the SM can naturally manifest itself. In this case the phenomenological consequences of such a flavor symmetry should be confronted with the low-energy experimental data by running the relevant physical quantities down to the Fermi scale Λ F via the RGEs. In Ref. [16] the one-loop RGEs of the µ-τ asymmetries A i of the PMNS matrix U have been derived. Here we are going to derive the integral form of the RGE corrections to M ν and H ν . The differential form of the one-loop RGE for the Dirac neutrino mass matrix M ν in the framework of the MSSM or the 2HDM is known as [19,20] 16π 2 dM ν dt = G + C ν Y ν Y † ν + C l Y l Y † l M ν ,(22) where t ≡ ln Λ/Λ µτ with Λ being a renormalization scale, Y ν and Y l are the Yukawa coupling matrices of the neutrinos and charged leptons, respectively. Given the MSSM, one has C ν = 3, C l = 1, and G ≃ −0.6g 2 1 − 3g 2 2 + 3y 2 t with g 1,2 being the gauge couplings and y t being the top-quark Yukawa coupling in the y 2 u ≪ y 2 c ≪ y 2 t approximation. If the type-II 2HDM is taken into account, one has C ν = 3/2, C l = −3/2, and G ≃ −0.45g 2 1 − 2.25g 2 2 + y 2 τ + 3y 2 b with y τ and y b being the tau-lepton and bottom-quark Yukawa couplings in the y 2 e ≪ y 2 µ ≪ y 2 τ and y 2 d ≪ y 2 s ≪ y 2 b approximations. Since the neutrino masses m i are extremely small as compared with their charged partners, it is very safe to neglect the Y ν Y † ν term in Eq. (22). In the basis that we have chosen (i.e., the mass eigenstates of three charged leptons are identified with their flavor eigenstates), Y l Y † l = D 2 l ≡ Diag{y 2 e , y 2 µ , y 2 τ } holds, where y 2 α = 2 (1 + tan 2 β) m 2 α /v 2 (for α = e, µ, τ ) with tan β being the ratio of the vacuum expectation value of H 2 to that of H 1 in the MSSM or the type-II 2HDM. Then Eq. (22) leads us to the RGE of H ν as follows: 16π 2 dH ν dt = 2 GH ν + D 2 l H ν + H ν D 2 l .(23) Integrating Eq. (23) from Λ µτ to Λ F , we immediately arrive at H ′ ν = I 2 G T l H ν T l ,(24) where H ν and H ′ ν are associated respectively with the scales Λ µτ and Λ F , T l ≡ Diag{I e , I µ , I τ }, and the evolution functions are I G = exp 1 16π 2 t ′ 0 G dt , I α = exp C l 16π 2 t ′ 0 y 2 α dt ,(25) where t ′ ≡ ln(Λ F /Λ µτ ), and α runs over e, µ and τ . If one is more interested in the relationship between M ′ ν at Λ F and M ν at Λ µτ , then it is straightforward to obtain M ′ ν = I G T l M ν ,(26) either from integrating Eq. (22) or from decomposing Eq. (24). Note that y 2 e ≪ y 2 µ ≪ y 2 τ 0.25 holds at the Fermi scale Λ F for tan β 50, and their values decrease as the energy scale grows up [21]. It is therefore an excellent approximation to take T l ≃ 1 − Diag{0, 0, ∆ τ } with 1 being the 3 × 3 unitary matrix and ∆ τ = C l 16π 2 0 t ′ y 2 τ dt ,(27) which is a small quantity of O(0.1) or much smaller. To illustrate, Figure 1 shows the numerical changes of I G and ∆ τ with the energy scale Λ in the MSSM and the type-II 2HDM by fixing Λ µτ = 10 14 GeV as the initial point and taking tan β = 10 and 30 as two typical inputs. One can see that the signs of ∆ τ are opposite in these two scenarios, and thus they are distinguishable at low energies. Now let us assume that the µ-τ reflection symmetry of M ν in Eq. (9) or H ν in Eq. (11) is realized at Λ µτ . Then at the electroweak scale Λ F we have ∆ τ I G ∆ τ I G Λ (GeV) Λ (GeV)H ′ ν ≃ I 2 G    H ν − ∆ τ    0 0 B * 0 0 D B D * 2C       ,(28) or equivalently, M ′ ν ≃ I G    M ν − ∆ τ    0 0 0 0 0 0 e * d * c *       ,(29) in which the smallness of ∆ τ has been taken into account. It is clear that the term proportional to ∆ τ measures the strength of µ-τ symmetry breaking. Even if M ν is taken to be Hermitian, the RGE-induced quantum correction will violate that Hermiticity at Λ < Λ µτ . In comparison, the Hermiticity of H ν is preserved in the whole RGE evolution from Λ µτ down to Λ F . At this point it is worth comparing the generic expression of H ′ ν in Eq. (19) with the explicit one in Eq. (28). Of course, it is straightforward to decompose the latter into a part respecting the µ-τ reflection symmetry and a part violating this flavor symmetry, from which one can easily obtain the dimensionless perturbation parameters ǫ 1 ≃ 1 2 ∆ τ , ǫ 2 ≃ ∆ τ ,(30) implying that the only source of µ-τ reflection symmetry breaking in our example is the RGEinduced ∆ τ term. In practice, it should be more convenient to directly use Eq. (28) to do a perturbation calculation of the neutrino masses and flavor mixing parameters. Before we start from Eq. (28) to derive the analytical expressions of three neutrino masses and four flavor mixing parameters at Λ F in the next section, let us first derive two interesting relations with no need of doing any perturbation calculation. Eq. (11) m ′ 1 m ′ 2 m ′ 3 = I 3 G I e I µ I τ m 1 m 2 m 3 ,(31) with m i and m ′ i (for i = 1, 2, 3) stand for the neutrino masses at Λ µτ and Λ F , respectively. Considering the traces of H ν and H ′ ν in Eq. (24), we obtain i m ′2 i = I 2 G α I 2 α i m 2 i \|V αi \| 2(32) with α and i running over (e, µ, τ ) and (1,2,3), respectively. But it is more interesting to establish an instructive relationship between the Jarlskog invariant of CP violation J at Λ µτ , defined through [22] Im V αi V βj V * αj V * βi = J γ ǫ αβγ k ǫ ijk(33) with the subscripts (α, β, γ) and (i, j, k) running respectively over (e, µ, τ ) and (1,2,3), and its counterpart J ′ at Λ F . To do so, we first write out the elements of H ′ ν in Eq. (24) in terms of the neutrino masses and the PMNS matrix elements: i m ′2 i U ′ αi U ′ * βi = I 2 G I α I β i m 2 i U αi U * βi ,(34) in which both α and β run over e, µ and τ . Note that U = P V (or U ′ = P ′ V ′ ) contains three unphysical phases. To eliminate them, let us focus on the following rephasing invariant [23]: Im i m 2 i U ei U * µi · j m 2 j U µj U * τ j · k m 2 k U τ k U * ek = i j k m 2 i m 2 j m 2 k Im V ei V µj V τ k V * ek V * µi V * τ j = J i j m 2 i m 4 j k ǫ ijk = J ∆m 2 21 ∆m 2 31 ∆m 2 32 ,(35) where the three neutrino mass-squared differences are defined as ∆m 2 ij ≡ m 2 i − m 2 j (for i, j = 1, 2, 3 which concisely connects the strength of leptonic CP violation at Λ µτ to that at Λ F . Given the parametrization of V in Eq. (13), the Jarlskog invariant J reads as J = 1 8 sin 2θ 12 sin 2θ 13 cos θ 13 sin 2θ 23 sin δ . If θ 23 = π/4 and δ = π/2 or 3π/2 are taken into account in the µ-τ reflection symmetry limit, then we arrive at \|J \| = sin 2θ 12 sin 2θ 13 cos θ 13 /8. Taking a similar parametrization for V ′ , one may express J ′ in terms of the corresponding flavor mixing parameters as J ′ = 1 8 sin 2θ ′ 12 sin 2θ ′ 13 cos θ ′ 13 sin 2θ ′ 23 sin δ ′ .(38) In the next section we shall establish the analytical relations between (θ 12 , θ 13 , θ 23 , δ) at Λ µτ and (θ ′ 12 , θ ′ 13 , θ ′ 23 , δ ′ ) at Λ F in a perturbation approach. RGE corrections to U Let us start from Eq. (28) to do a perturbation calculation in order to derive the analytical expressions of three neutrino masses and four flavor mixing parameters at Λ F . Similar to H ν in Eq. (11), H ′ ν can also be reconstructed in the same way: H ′ ν ≡ M ′ ν M ′ † ν = U ′M ′2 ν U ′ † ,(39) in which U ′ = P ′ V ′ with P ′ being a diagonal phase matrix, andM ′ ν ≡ Diag{m ′ 1 , m ′ 2 , m ′ 3 } with m ′ i being the neutrino masses at Λ F . Then the approximate relationship between H ′ ν and H ν in Eq. (28) can be rewritten aŝ M ′2 ν ≃ I 2 G U ′ †       UM 2 ν U † − ∆ τ       0 0 i m 2 i U ei U * τ i 0 0 i m 2 i U µi U * τ i i m 2 i U * ei U τ i i m 2 i U * µi U τ i 2 i m 2 i \|U τ i \| 2             U ′ .(40) Treating ∆ τ as a small perturbation parameter, let us define the RGE-induced deviations of the relevant flavor mixing angles and phase parameters at Λ F from their original counterparts at Λ µτ as follows: ∆θ 12 = θ ′ 12 − θ 12 , ∆δ = δ ′ − δ , ∆θ 13 = θ ′ 13 − θ 13 , ∆φ eµ = (φ ′ e − φ ′ µ ) − (φ e − φ µ ) , ∆θ 23 = θ ′ 23 − θ 23 , ∆φ eτ = (φ ′ e − φ ′ τ ) − (φ e − φ τ ) ,(41) which are expected to be small enough in magnitude as compared with their respective starting values at Λ µτ . Note that θ 23 = π/4 and δ = π/2 or 3π/2 at the µ-τ reflection symmetry scale Λ µτ will be implied in the subsequent perturbation calculations. Note also that only two combinations of the three unphysical phases in P or P ′ , as indicated in Eq. (41), are associated with our derivation of the RGEs for the physical parameters. They ought not to be ignored in the course of the calculations, but of course they do not show up in the final results of ∆θ 12 , ∆θ 13 , ∆θ 23 and ∆δ. Next we expand the elements ofM ′2 ν in terms of the above perturbation parameters and only keep their first-order contributions. First of all, it is straightforward to obtain the analytical results of three neutrino masses from the diagonal elements ofM ′2 ν . Namely, m ′ 1 ≃ I G m 1 1 − 1 2 ∆ τ s 2 12 c 2 13 + s 2 13 , m ′ 2 ≃ I G m 2 1 − 1 2 ∆ τ c 2 12 c 2 13 + s 2 13 , m ′ 3 ≃ I G m 3 1 − 1 2 ∆ τ c 2 13 .(42) Obviously but interestingly, m ′ i /m i ≃ I G holds in the leading-order approximation, implying that the three neutrino masses almost run in step. Given I e ≃ I µ ≃ 1 and I τ ≃ 1 − ∆ τ and the µ-τ reflection symmetry at Λ µτ , it is easy to check that the product of m ′ 1 , m ′ 2 and m ′ 3 in Eq. (42) can successfully reproduce the elegant relationship achieved in Eq. (31). Moreover, Eq. (42) leads us to the sum rule i m ′2 i ≃ I 2 G i m 2 i 1 − 2∆ τ \|V τ i \| 2 ,(43) which is consistent with the more generic one derived in Eq. (32) if the same approximations are made and the µ-τ reflection symmetry is taken into account. Second, the off-diagonal elements ofM ′2 ν in Eq. (40) must vanish, yielding the following six constraint equations in our analytical approximations: 2∆m 2 21 ∆θ 12 + ηs 13 ∆m 2 21 ∆φ eµ − ∆φ eτ − c 12 s 12 c 2 13 m 12 ∆ τ ≃ 0 , 2 c 2 12 − s 2 12 s 13 ∆m 2 21 ∆θ 23 − ηc 12 s 12 ∆m 2 21 2s 2 13 ∆δ + c 2 13 ∆φ eµ + ∆φ eτ + s 13 m 12 ∆ τ ≃ 0 , 2s 12 c 13 ∆m 2 31 ∆θ 23 + ηc 12 c 13 s 13 ∆m 2 31 2∆δ − ∆φ eµ − ∆φ eτ − s 12 c 13 m 13 ∆ τ ≃ 0 , 2c 12 ∆m 2 31 ∆θ 13 − ηs 12 c 13 ∆m 2 31 ∆φ eµ − ∆φ eτ − c 12 c 13 s 13 m 13 ∆ τ ≃ 0 , 2c 12 c 13 ∆m 2 32 ∆θ 23 − ηs 12 c 13 s 13 ∆m 2 32 2∆δ − ∆φ eµ − ∆φ eτ − c 12 c 13 m 23 ∆ τ ≃ 0 , 2s 12 ∆m 2 32 ∆θ 13 + ηc 12 c 13 ∆m 2 32 ∆φ eµ − ∆φ eτ − s 12 c 13 s 13 m 23 ∆ τ ≃ 0 , where η ≡ sin δ = ±1 in the µ-τ reflection symmetry limit, and m ij ≡ m 2 i + m 2 j (for i, j = 1, 2, 3). Solving the above equations, we obtain ∆θ 12 ≃ ∆ τ s 12 c 12 m 2 1 + m 2 22∆ms 12 c 12 ,(46) where t 12 ≡ tan θ 12 . One can see that the RGE-induced corrections to all the four flavor mixing parameters are proportional to ∆ τ , a fact which is under rational expectation. Among the three angles, θ 12 is more sensitive to the quantum corrections than θ 13 and θ 23 in most cases, mainly because of \|∆m 2 31 \| ≃ \|∆m 2 32 \| ∼ 30∆m 2 21 [24]. On the other hand, the smallness of s 13 [25] implies that the magnitude of ∆θ 13 must be smaller than that of ∆θ 23 . But the expression of ∆δ contains three terms proportional to s 13 and one term proportional to 1/s 13 , and hence the overall running effect of δ is generally expected to be more significant than those of three flavor mixing angles, or at least than those of θ 13 and θ 23 . Note that φ µ + φ τ = 2φ e holds at Λ µτ due to the µ-τ reflection symmetry of H ν , and hence 2φ ′ e − φ ′ µ − φ ′ τ = ∆φ eµ + ∆φ eτ ∝ ∆ τ is not vanishing at Λ F , providing us with another (unphysical) measure of the RGE-induced µ-τ reflection symmetry breaking of H ′ ν . There are two ways to calculate the Jarlskog invariant J ′ at Λ F : one is to apply Eq. (42) to the elegant relationship between J and J ′ in Eq. (36) with I e ≃ I µ ≃ 1 and I τ ≃ 1 − ∆ τ , and the other is to do a direct perturbation calculation of J ′ by using Eqs. (38), (45) and (46). After doing such a calculation, we obtain the ratio of J ′ at Λ F to J at Λ µτ as follows: J ′ J ≃ 1 + ∆ τ Different from δ, J evolves in a way insensitive to the smallness of θ 13 . We proceed to numerically illustrate the RGE-induced corrections to the neutrino masses and flavor mixing parameters in the MSSM and the type-II 2HDM by using the program advocated in Ref. [26] and taking Λ µτ = 10 14 GeV as a typical choice, where θ 23 = π/4 and δ = 3π/2 are input. For the sake of simplicity, we adjust the initial values of m 1 (or m 3 ), ∆m 2 21 , ∆m 2 31 , θ 12 and θ 13 to make sure that all the neutrino oscillation parameters can be compatible with current experimental data at Λ F [24]. The main numerical results are summarized in Tables 1 and 2 as well as Figures 2, 3 and 4, in which two possibilities of the neutrino mass spectrum have been taken into account -the normal hierarchy (NH) with m 1 < m 2 < m 3 or ∆m 2 31 > 0 and the inverted hierarchy (IH) with m 3 < m 1 < m 2 or ∆m 2 31 < 0. Some comments and discussions are in order. (1) In the MSSM, Table 1 and Figure 2 show that the values of three flavor mixing angles increase in the NH case as the energy scale Λ decreases, but θ 13 and θ 23 decrease in the IH case as Λ decreases. In either case a larger value of tan β will enhance the running effects. Such a direction of evolution of ∆θ ij (for ij = 12, 13 or 23) can easily be understood from our analytical approximations made in Eq. (45). In comparison, the CP-violating phase δ decreases in both NH and IH cases when Λ becomes lower. The reason for this behavior can be seen in Eq. (46) -namely, δ = 3π/2 (or η = −1) has been input at Λ µτ , and ∆δ Table 1: An illustration of the neutrino oscillation parameters at Λ µτ and Λ F in the MSSM with tan β = 10 or 30, where both NH and IH cases are considered. MSSM NH, tan β = 10 NH, tan β = 30 IH, tan β = 10 IH, tan β = 30 is essentially insensitive to the sign of ∆m 2 31 which is always the same as the sign of ∆m 2 32 . Moreover, both Table 1 and Figure 3 tell us that the magnitude of the Jarlskog invariant (i.e., \|J \|) increases as Λ decreases, no matter whether the neutrino mass hierarchy is normal or inverted. Eq. (47) shows that the ratio J ′ /J must be slightly larger than one if the term proportional to m 2 2 /∆m 2 21 is dominant. Although the above observations are more or less subject to the limited parameter space that we have taken into account, our analytical results in Eqs. (45), (46) and (47) are certainly more general and more useful. Parameter Λ µτ Λ F Λ µτ Λ F Λ µτ Λ F Λ µτ Λ F m (2) In the type-II 2HDM, the running behaviors of θ 12 , θ 13 , θ 23 and δ take the opposite directions as compared with those in the MSSM. The reason is simply that the signs of ∆ τ are opposite in these two scenarios. Because of C l = 1 in the MSSM and C l = −3/2 in the type-II 2HDM, the magnitude ∆ τ in the latter case is about 1.5 times larger than that in the former case. That is why we have taken the type-II 2HDM scenario for our numerical illustration, in contrast with the MSSM scenario. Note, however, that the evolution of ∆J with Λ is a bit subtle in the type-II 2HDM case when tan β is sufficiently large. For example, the minimum of ∆J shown in the right-bottom panel of Figure 3 is expected to arise from a significant cancellation among the terms on the right-hand side of Eq. (47). (3) It is worth highlighting that the RGE-induced effect of µ-τ reflection symmetry breaking provides a model-independent way to connect three burning issues in today's neutrino physics: the neutrino mass ordering, the octant of θ 23 and leptonic CP violation. Some interesting works have been done in this regard in the case that the massive neutrinos are the Majorana particles [15,16,18,27]. Here we have discussed how the µ-τ reflection symmetry of Dirac neutrinos can be spontaneously broken by the RGE evolution from Λ µτ down to Λ F in the MSSM and the type-II 2HDM, and how this symmetry breaking affects the octant of θ 23 and the quadrant of δ in both NH and IH cases. As shown in Figure 2, the type-II 2HDM scenario seems to be somewhat favored if we stick to the best-fit value of θ 23 at low energy scales [24], which lies in the first octant in the NH case but in the second octant in the IH case 4 . For the time being, however, the "best-fit" values of θ 23 from a global analysis of current neutrino oscillation data should not be taken too seriously, because their statistical significance remains rather poor [24]. It is more appropriate to consider the 2σ or 3σ intervals of those neutrino oscillation parameters, in which case the octant of θ 23 is not yet fixed 5 . (4) As a by-product, Figure 4 illustrates the evolution behaviors of three neutrino masses in both NH and IH cases. Since we have intended to take m lightest = 0.05 eV at Λ F in our numerical calculations so as to reasonably magnify the RGE running effects, the neutrino mass spectrum is not far away from the nearly degenerate case with a fine split between m 1 and m 2 even if it is normal. Our numerical results are consistent with the analytical ones obtained in Eq. (42) -namely, the evolution of m i is mainly governed by that of I G and thus insensitive to the value of tan β. For the same reason, the results of m i in the MSSM are not very different from those in the type-II 2HDM. Summary While the nature of massive neutrinos (i.e., whether Dirac or Majorana) remains an intriguing puzzle in particle physics, it is largely believed that there should exist an approximate µ-τ reflection symmetry behind the observed pattern of lepton flavor mixing. In this work we have studied such a simple but interesting flavor symmetry for the Dirac neutrino mass matrix, which can naturally predict θ 23 = π/4 and δ = π/2 or 3π/2 in the standard parametrization of the PMNS matrix U. Assuming the µ-τ reflection symmetry is realized at a superhigh energy scale Λ µτ , we have investigated how it is spontaneously broken via the one-loop RGEs running from Λ µτ down to the Fermi scale Λ F in two interesting scenarios: the MSSM and the type-II 2HDM. Such quantum corrections to the neutrino masses and flavor mixing parameters have been derived in a perturbation approach, and an analytical link has also been established between the Jarlskog invariants of leptonic CP violation at Λ µτ and Λ F . In addition, we have illustrated the running behaviors of relevant physical quantities by taking a few typical numerical examples in the MSSM and the type-II 2HDM. A particularly striking point of view associated with this kind of study is that the octant of θ 23 , the quadrant of δ and the neutrino mass ordering might be correlated with one another thanks to the RGE-triggered breaking of µ-τ reflection symmetry. We have illustrated this observation both analytically and numerically by considering the massive Dirac neutrinos in the MSSM and the type-II 2HDM, and found that these two scenarios lead us to the opposite deviations of θ 12 , θ 13 , θ 23 and δ from their corresponding values in the µ-τ reflection 4 If one works on the RGEs in the SM framework, then ∆θ ij and ∆δ will evolve with the energy scales in a similar way as in the type-II 2HDM scenario. In this case, however, the running effects of relevant parameters are expected to be much milder because of the lack of the tan β enhancement. More seriously, the SM-like RGEs may suffer from the vacuum-stability problem as the energy scale is above 10 10 GeV [21]. 5 At this point it is worth mentioning that the latest T2K neutrino oscillation result provides a very preliminary hint that θ 23 might lie in the second octant in the NH case [28], a possibility compatible with our results in the MSSM scenario shown in Table 1 and Figure 2. and δ = 3π/2 at Λ µτ = 10 14 GeV are fixed by the µ-τ reflection symmetry. symmetry limit. Therefore, the future experimental data on the neutrino mass ordering and flavor mixing angles will allow us to make a choice between the MSSM and the type-II 2HDM, at least in this connection. Our results are also expected to be useful for building explicit Dirac neutrino mass models and explaining upcoming neutrino oscillation data. 13 e −iδ −s 12 c 23 − c 12 s 13 s 23 e iδ c 12 c 23 − s 12 s 13 s 23 e iδ c 13 s 23 −s 12 s 23 + c 12 s 13 c 23 e iδ c 12 s 23 + s 12 s 13 c 23 e iδ −c Figure 1 : 1Changes of I G and ∆ τ versus the energy scale Λ in the MSSM or the type-II 2HDM. Figure 2 : 2The changes of ∆θ 12 , ∆θ 13 , ∆θ23 and ∆δ with the energy scale Λ in the MSSM and the type-II 2HDM, where m lightest = 0.05 eV at Λ F = 10 2 GeV is typically input and θ 23 = π/4 and δ = 3π/2 at Λ µτ = 10 14 GeV are fixed by the µ-τ reflection symmetry. Figure 3 : 3An illustration of the change of ∆J ≡ J ′ − J with the energy scale Λ in the MSSM and the type-II 2HDM, where m lightest = 0.05 eV at Λ F = 10 2 GeV is typically input and θ 23 = π/4 and δ = 3π/2 at Λ µτ = 10 14 GeV are fixed by the µ-τ reflection symmetry. Figure 4 : 4The three neutrino masses evolving with the energy scale Λ in the MSSM and the type-II 2HDM, where m lightest = 0.05 eV at Λ F = 10 2 GeV is typically input and θ 23 = π/4 tells us that H ν and H ′ ν can be diagonalized by the unitary matrices U and U ′ , respectively. So the determinants of H ν and H ′ ν are proportional to each other, giving rise to ). Applying Eq. (34) to Eq. (35), we are then left with the elegant resultJ ′ ∆m ′2 21 ∆m ′2 31 ∆m ′2 32 = I 6 G I 2 e I 2 µ I 2 τ J ∆m 2 21 ∆m 2 31 ∆m 2 32 , Table 2 : 2An illustration of the neutrino oscillation parameters at Λ µτ and Λ F in the type-II 2HDM with tan β = 10 or 30, where both NH and IH cases are considered.2HDM NH, tan β = 10 NH, tan β = 30 IH, tan β = 10 IH, tan β = 30 Parameter Λ µτ Λ F Λ µτ Λ F Λ µτ Λ F Λ µτ Λ F m lightest [10 −2 eV] 4.29 5.0 4.53 5.01 4.29 5.0 4.5 5.0 ∆m 2 21 [10 −5 eV 2 ] 5.44 7.56 5.94 7.56 5.34 7.56 8.31 7.56 \|∆m 2 31 \| [10 −3 eV 2 ] 1.88 2.55 2.04 2.55 1.84 2.49 2.04 2.49 θ 12 [ • ] 36.43 34.50 54.36 34.55 38.45 34.52 70.6 34.52 θ 13 [ • ] 8.47 8.44 8.72 8.44 8.38 8.41 8.17 8.41 θ 23 [ • ] 45 44.81 45 43.18 45 45.18 45 46.77 δ [ • ] 270 271.29 270 282.71 270 272.52 270 295.34 J [10 −2 ] −3.44 −3.35 −3.51 −3.27 −3.47 −3.34 −2.18 −3.01 Note that these unphysical phases should not be ignored in the course of deriving the RGEs of the neutrino masses and flavor mixing parameters, as one can see in section 4. 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[ "Strong lensing cross sections for isothermal models -I. Finite source effects in the circular case", "Strong lensing cross sections for isothermal models -I. Finite source effects in the circular case" ]	[ "Vanessa P De Freitas \nCentro Brasileiro de Pesquisas Físicas\nRua Dr. Xavier Sigaud 15022290-180Rio de JaneiroRJ, CEPBrazil\n", "Martin Makler \nCentro Brasileiro de Pesquisas Físicas\nRua Dr. Xavier Sigaud 15022290-180Rio de JaneiroRJ, CEPBrazil\n", "Habib S Dúmet-Montoya \nUniversidade Federal do Rio de Janeiro -Campus Macaé\nRua Aloísio Gomes da Silva\n5027930-560MacaéRJ, CEPBrazil\n" ]	[ "Centro Brasileiro de Pesquisas Físicas\nRua Dr. Xavier Sigaud 15022290-180Rio de JaneiroRJ, CEPBrazil", "Centro Brasileiro de Pesquisas Físicas\nRua Dr. Xavier Sigaud 15022290-180Rio de JaneiroRJ, CEPBrazil", "Universidade Federal do Rio de Janeiro -Campus Macaé\nRua Aloísio Gomes da Silva\n5027930-560MacaéRJ, CEPBrazil" ]	[ "MNRAS" ]	The strong galaxy-galaxy lensing produces highly magnified and distorted images of background galaxies in the form of arcs and Einstein rings. Statistically, these effects are quantified, for example, in the number counts of highly luminous sub-millimeter galaxies and of gravitational arcs. Two key quantities to model these statistics are the magnification and the arc cross sections. These are usually computed using either the circular infinitesimal source approximation or ray-tracing simulations for sources of finite size. In this work, we use an analytic solution for gravitational arcs to obtain these cross sections as a function of image magnification and length-to-width ratio in closed form, for finite sources. These analytical solutions provide simple interpretations to the numerical results, can be employed to test the computational codes, and can be used for fast a computation of the abundance of distant sources and arcs. In this paper, the lens is modeled by a Singular Isothermal Sphere, which is an excellent approximation to radial density profile of Early-Type galaxies, and the sources are also axisymmetric. We derive expressions for the geometrical properties of the images, such as the area and several definitions of length and width. We obtain the magnification cross section in exact form and derive a simple analytic approximation covering the arc and Einstein ring regimes. The arc cross section is obtained down to the formation of an Einstein ring and given in terms of elementary functions. Perturbative expansions of these results are worked out, showing explicitly the correction terms for finite sources.	10.1093/mnras/sty2412	[ "https://arxiv.org/pdf/1809.06869v1.pdf" ]	119,473,521	1809.06869	b025b174869e9b12be8610119ab7214e5591861a	Strong lensing cross sections for isothermal models -I. Finite source effects in the circular case 2018 Vanessa P De Freitas Centro Brasileiro de Pesquisas Físicas Rua Dr. Xavier Sigaud 15022290-180Rio de JaneiroRJ, CEPBrazil Martin Makler Centro Brasileiro de Pesquisas Físicas Rua Dr. Xavier Sigaud 15022290-180Rio de JaneiroRJ, CEPBrazil Habib S Dúmet-Montoya Universidade Federal do Rio de Janeiro -Campus Macaé Rua Aloísio Gomes da Silva 5027930-560MacaéRJ, CEPBrazil Strong lensing cross sections for isothermal models -I. Finite source effects in the circular case MNRAS 0002018Preprint 20 September 2018 Compiled using MNRAS L A T E X style file v3.0gravitational lensing: strong -methods: analytical -galaxies: general The strong galaxy-galaxy lensing produces highly magnified and distorted images of background galaxies in the form of arcs and Einstein rings. Statistically, these effects are quantified, for example, in the number counts of highly luminous sub-millimeter galaxies and of gravitational arcs. Two key quantities to model these statistics are the magnification and the arc cross sections. These are usually computed using either the circular infinitesimal source approximation or ray-tracing simulations for sources of finite size. In this work, we use an analytic solution for gravitational arcs to obtain these cross sections as a function of image magnification and length-to-width ratio in closed form, for finite sources. These analytical solutions provide simple interpretations to the numerical results, can be employed to test the computational codes, and can be used for fast a computation of the abundance of distant sources and arcs. In this paper, the lens is modeled by a Singular Isothermal Sphere, which is an excellent approximation to radial density profile of Early-Type galaxies, and the sources are also axisymmetric. We derive expressions for the geometrical properties of the images, such as the area and several definitions of length and width. We obtain the magnification cross section in exact form and derive a simple analytic approximation covering the arc and Einstein ring regimes. The arc cross section is obtained down to the formation of an Einstein ring and given in terms of elementary functions. Perturbative expansions of these results are worked out, showing explicitly the correction terms for finite sources. INTRODUCTION Gravitational arcs and Einstein rings (Saslaw et al. 1985) are highly distorted and magnified images of distant galaxies (sources) due to the light deflection produced by foreground galaxies acting as lenses. These images may be used to probe the mass distribution in the lens galaxies (e.g. Treu & Koopmans 2002;Koopmans et al. 2009), including substructures and tests of the Cold Dark Matter paradigm (e.g. Vegetti et al. 2012;Xu et al. 2015;Li et al. 2016); to find and study high-redshift galaxies through the gravitational telescope effect (Caminha et al. 2016;Negrello et al. 2017;Goobar et al. 2017;Zavala et al. 2017); to constrain cosmological models (Suyu et al. 2010;Cao et al. 2015;Treu & Marshall 2016); and to test modified gravity theories (Schwab et al. 2010;Enander & Mörtsell 2013). For a review on galaxyscale strong lensing, see Treu (2010). The many applications of strong lensing, in particular by galaxies, motivated the search for arcs and Einstein rings in Hubble Space Telescope (HST) images (e.g., Hogg et al. E-mail: [email protected] coming J-PAS 4 project. These numbers will increase even further in the near future, with the operation of LSST 5 and Euclid, 6 which are both expected to detect O 10 5 systems with arcs (Collett 2015). In recent years, strongly lensed systems started to be discovered in wide-field surveys at sub-millimeter and millimeter wavelengths, such as from the South Pole Telescope (Vieira et al. 2013) and the Herschel-ATLAS survey (Negrello et al. 2017). In this case, the systems are not identified by their arc or ring shape, but by the large magnification of the sources (in this case dusty star-forming galaxies). The surveys carried out with Herschel are expected to deliver a sample of more than a hundred of sub-mm bright strongly lensed galaxies (Negrello et al. 2017), and could reach around a thousand, depending on the detection technique (Lapi et al. 2012). With such large numbers of objects from optical and (sub-)mm surveys, it becomes intractable to perform a detailed modeling of each system, in particular considering the difficulty in obtaining high-resolution imaging and redshifts for the sources. An alternative approach, known as strong lensing statistics, is to calculate probability distributions of observable properties of arcs, such as their length-to-width ratio (L/W) and magnification, and compare them to observations (Grossman & Narayan 1988;Miralda-Escude 1993;Wu & Hammer 1993;Bartelmann & Weiss 1994;Fedeli & Berciano Alba 2009;Lima et al. 2010a,b). For an excellent review on arc statistics, see Meneghetti et al. (2013). To compute the abundance of arcs as a function of their properties, one needs to know the number densities of lens and sources as a function of some of their properties, including their redshift. These can be empirically determined from observational data or be represented by families of models with parameters to be determined from the arc statistics observations. Strong lensing statistics has been applied to several problems, in particular involving galaxy scale lenses, such as to predict the number of arcs found in the SLACS survey (Dobler et al. 2008) and to explain the abundance of highly luminous sub-millimeter galaxies (Lima et al. 2010a,b;Hezaveh & Holder 2011;Lapi et al. 2012). Another key ingredient in strong lensing statistics is the efficiency to produce images with given properties -such as magnification and length-to-width ratio. This efficiency is encoded in the cross section and the main aim of the paper is to compute this quantity for a specific lens and source model. During the past decade, several studies using a diverse set of observables -specially weak and strong lensingand simulations have shown that the radial density profile of galaxy-scale lenses (i.e. Early-Type galaxies) is surprisingly close to the Singular Isothermal Sphere (SIS) profile (see e.g., Gavazzi et al. 2007;Koopmans et al. 2009;van de Ven et al. 2009;Blundell et al. 2010;Treu 2010;Bolton et al. 2012;Grillo 2012;Lapi et al. 2012;Sonnenfeld et al. 2013b;Dye 4 Javalambre Physics of the Accelerating Universe Astrophysical Survey, Benitez et al. (2014) 5 Large Synoptic Survey Telescope, LSST Science Collaboration et al. (2009), http://www.lsst.org/ 6 Refregier et al. (2010); Laureijs et al. (2011), http://www. euclid-ec.org/ et al. 2018, and references therein), which is given by (Turner et al. 1984;Binney & Tremaine 1987;Schneider et al. 1992): ρ(r) = σ 2 v 2πG 1 r 2 ,(1) where σ 2 v is the one-dimensional velocity dispersion. Remarkably, lens models based on this solution and including external shear and/or ellipticity, allow one to derive analytic solutions for several lensing related quantities (see e.g., Inoue & Chiba 2005;Dobler & Keeton 2006;Dobler et al. 2008;Chu et al. 2013;Er et al. 2013, Dúmet-Montoya et al., in prep.), including gravitational arcs. Generally, two approaches have been used for computing the cross section for arc statistics: either the source is considered infinitesimal and several calculations can be carried out analytically (see e.g., Oguri et al. 2001;Meneghetti et al. 2003;Caminha et al. 2013;Dúmet-Montoya et al. 2013;Er et al. 2013) or ray-tracing simulations are carried out producing images of finite sources (Wu & Hammer 1993;Miralda-Escude 1993;Bartelmann & Weiss 1994;Ho & White 2005;Hezaveh & Holder 2011;Lapi et al. 2012;Redlich et al. 2012). The latter are more realistic, but also more time consuming. Furthermore, the simulations have to be carried out again for each change of parameter and the results cannot always be interpreted in a transparent way. Here we take an alternative approach, which is to use the analytical solutions for arcs in the SIS case to derive the cross sections. In this way we are able to introduce and study the finite source effects in an analytic or semi-analytic way. Furthermore, the treatment enables us to tackle the problem down to the formation of Einstein rings, which cannot be addressed with infinitesimal sources. An advantage of analytic solutions is that they offer the possibility of a more clear physical interpretation of the results. They enable to probe the whole parameter space involved and can be used for fast calculations. They can also be used to test the accuracy of numerical codes that are developed for more generic models, in the specific situations where the analytical results hold. Therefore, there is a complementarity with fully numerical approaches and it is worth to search for such analytical solutions. In this work we consider the simple case of a circular SIS model with circular sources (elliptical sources are addressed in a separate paper). We start by investigating the geometrical properties of the images, seeking to obtain the magnification and L/W in a closed form. We test several definitions of length and width and apply the results to compute the cross sections of magnification and arc formation, which can be used to predict the abundance of distant sources as a function of flux and arcs as a function of L/W, respectively. Remarkably, this problem can be treated analytically all the way down to the computation of the cross sections and we are able to express them, under some approximations, in terms of elementary functions. We also obtain perturbative solutions that explicitly show the correction terms for finite sources. From the solutions obtained, we are able to clarify some properties empirically found in more general situations using simulations. This paper is organized as follows: in Section 2, we present a brief review of SIS lenses and the solution for finite sources. In Section 3, we derive expressions for the magni-fication, length and widths of the images, which are used in Section 4 to obtain the magnification and arc formation cross sections. In Sections 5 and 6, we summarize and discuss our results. In Appendix A, we compare the solutions for the SIS arcs to the ArcEllipse geometrical figure. Finally, in Appendix B, we discuss the semi-analytic method introduced by Fedeli et al. (2006) for finite sources in the context of the arc cross sections obtained in this paper. ARCS IN THE SIS MODEL In this section, we present a brief overview of axially symmetric singular isothermal lens models and circular sources to fix the notation and provide the basic expressions to be used in the paper. Lensing by a Singular Isothermal Sphere The lensing properties are encoded in the lens equation, which relates the position of the observed images ξ to those of the source η. By choosing a characteristic length-scale ξ 0 and defining x ≡ ξ/ξ 0 and y ≡ η/η 0 , where η 0 ≡ D OS ξ 0 /D OL , and D OS and D OL are the angular diameter distances from the observer to the source and the lens, respectively, lens equation can be written in dimensionless form (Schneider et al. 1992;Petters et al. 2001;Mollerach & Roulet 2002): y = x − α(x),(2) where α(x) is the dimensionless deflection angle. The local distortion in the lens plane is described by the Jacobian matrix of the transformation (2) J = ∂y ∂x i j = δ i j − ∂ i α j (x).(3) The eigenvalues of the Jacobian matrix give the inverse of the magnification in the tangential and radial directions and can be written as λ r,t (x) = µ −1 r,t (x) = 1 − κ(x) ± γ(x),(4) where κ(x) and γ(x) are the convergence and the shear. The positive sign gives the eigenvalue associated to the tangential eigenvector and the negative sign corresponds to radial one. For axially symmetric lens models we have κ(x) = 1 2 α(x) x + dα(x) dx , γ(x) = 1 2 α(x) x − dα(x) dx ,(5) where x = \|x\| is the radial coordinate. The sets of points for which λ r,t (x) = 0 determine the radial and tangential critical curves, respectively. Mapping these curves onto the source plane give us the caustics. For the SIS density profile (equation (1)), if we choose the length-scale ξ 0 as the Einstein radius ξ 0 = R E = σ 2 v GΣ crit ,(6) where Σ crit is the critical surface mass density Σ crit = c 2 4πG D OS D OL D LS ,(7) and D LS is the angular diameter distance from the lens to the source, then the convergence, shear and deflection angle are κ(x) = γ(x) = 1 2x , α(x) =x,(8) and the lens equation (2) is y = (x − 1)x,(9) wherex is the radial unit vector. For y = \|y\| < 1 this equation has two solutions, one with x < 1, i.e., the image is inside the Einstein ring, and one with x > 1, which we call the internal and external images, respectively. For y > 1 there is only one solution and the boundary where the multiplicity changes (the curve y = 1) is often referred to as radial pseudo-caustic (Dobler & Keeton 2006) and we will keep this terminology along the text. Substituting expressions (8) in equation (4), the eigenvalues of the Jacobian matrix, are given by λ r = 1, λ t = 1 − 1 x .(10) Therefore, the tangential critical curve is given by x = 1, i.e., the Einstein radius, and there is no radial critical curve. The change in shape of infinitesimal sources is given by a linear transformation defined by the Jacobian in Eq. (3). In particular an infinitesimal circular source of radius R 0 will be mapped into an ellipse whose semi-axes in the tangential and radial directions will be given by, respectively a = R 0 \| µ t \| and b = R 0 \| µ r \|. Therefore, the axial ratio of the image, R λ , and the magnification, µ, which is the ratio between the areas of the image and the source, will be given by R λ = µ t µ r , µ = \| µ t µ r \| .(11) For the SIS lens, the infinitesimal axial ratio and magnification are the same, as the radial eigenvalue is unity. From Eqs. (4) and (10) they are given by R λ = µ =            1 x − 1, if x ≤ 1 1 − 1 x , if x > 1.(12) Using the lens equation (9), these quantities are expressed in terms of the source plane variables as R ex,in λ = µ ex,in = ±1 + 1 s ,(13) were s = y is the position of the source and the positive sign corresponds to the external image and the minus sign to the internal one. Analytic solutions for arcs from circular sources We use the lens equation to map a set of points representing the contour of a circular source so as to obtain its images. Consider a circular source with radius R 0 centered at (s 1 , s 2 ). We may write its boundary in the source plane as R 0 = y − s, with s = s 1ŷ1 + s 2ŷ2 .(14) Substituting the lens equation (9) in the expression above and changing to polar coordinates yields R 0 = (x − 1 − s 1 cos φ − s 2 sin φ)x + (s 1 sin φ − s 2 cos φ)φ,(15) where φ is the polar angle. Using R 2 0 = \|R 0 \| 2 and solving for x, we obtain x (±) = 1 + s 1 cos φ + s 2 sin φ ± R 2 0 − (s 1 sin φ − s 2 cos φ) 2 . (16) Rewriting s 1 and s 2 as s 1 = s cos θ, s 2 = s sin θ,(17) Eq. (16) becomes x (±) = 1 + s cos(φ − θ) ± R 2 0 − s 2 sin 2 (φ − θ),(18) which gives the outer and inner parts of the images, as indicated in Fig. 1. The points where the discriminant in Eq. (18) is zero define the arc extremities. Two ranges of φ may have a positive discriminant, indicating the existence of two solutions (as in the example of Fig.1), one inside the tangential critical curve and the other outside it, which we refer to as the internal and external arcs, representing them with the upper labels "in" and "ex" along the paper. Eq. (18) is the well known analytic solution for circular sources and the SIS lens (see e.g. Inoue & Chiba 2005;Dobler & Keeton 2006;Dobler et al. 2008, and Dúmet-Montoya et al., in prep., for more generic solutions in isothermal models). An expression of similar form is obtained as an approximate solution for arcs for generic radial profiles in the perturbative method of Alard (2007), which is exact in the SIS case (Dúmet-Montoya et al. 2013). Therefore, we expect that the approach of this paper can be extended for more generic lens models, either using exact or perturbative solutions. We define the arc ridgeline 7 as the mean of the inner and outer parts of the arc, which is independent of the source radius and is given by x(φ) = x (+) + x (−) 2 = 1 + s cos(φ − θ).(19) This curve is also shown in Fig. 1 (dotted line), which contains the tangential critical curve as well (dashed circle). The curve given by expression (19) is known as the Pascal limaçon (Lawrence 1972). If s ≤ 1/2, the limaçon is convex; if 1/2 < s < 1, the limaçon is dimpled; if s = 1, the limaçon degenerates to a cardioid and if s > 1, the limaçon has an inner loop. The limaçon is not a circumference, therefore the arc ridgeline is not an arc segment. On the other hand, the portion passing across the external arc is quite similar to an arc of a circle, but with curvature center shifted from the lens' center (see Sec. 3 and the Appendix A). Since the lens is axially symmetric we can choose the source position along the (positive) x-axis, such that we set θ = 0 without loss of generality. In this case the center of the external and internal arcs will be at φ c = 0 and π, respectively. From the discriminant in Eq. (18) we obtain the 7 Rigorously speaking the arcs we are considering do not have a ridgeline, as they represent only a boundary (or the image of a uniform brightness source). However, for a source with radial brightness distribution (i.e. with concentric circular isophotes) the brightness peak along any radial direction will be given by the curve defined in Eq. (19). Therefore we employ this nomenclature even in the current case. angular position for the arcs extremities, which, for the external arc, are given by φ i = −φ 0 and φ f = φ 0 , where φ 0 = arcsin R 0 s .(20) For the internal arc we simply add π to both angles, since they are complementary. In this paper we are interested in the case of images that can have large magnifications and length-to-width ratios, which implies that the sources must be smaller than the Einstein radius, R 0 < 1. There are three possible image configurations for the finite sources in this case: Einstein ring, two images or one image. These configurations depend on the position s of the center of the source relative to its radius R 0 : • for s ≤ R 0 the source includes the tangential caustic, i.e. the lens center, and we have an Einstein ring. If s = 0 the ring is centered at the origin. The equality indicates the limit between the formation of two images and an Einstein ring, where the two images touch at their extremities. • for R 0 < s < 1+R 0 the source is inside the pseudo-caustic and in this case we have two images. • for s ≥ 1 + R 0 the source is completely outside the pseudo-caustic and in this case we have one image. The external image is always arc-shaped, until the formation of the Einstein ring. The situation with the internal image is a bit more tricky, as its shape will depend on the values of s and R 0 . The arc rigdeline (i.e. the limaçon) can provide a guideline to classify this image. For s < 1/2, i.e. when the source center position is smaller than half the pseudo-caustic radius, the limaçon is smooth and has positive curvature in all positions. In this case, the image will have an arc shape, as in Fig. 1 (left panel). At s = 1/2 the limaçon has zero curvature at the image position and the arc ridgeline will be straight for small sources. In the intermediate region, 1/2 < s ≤ 1, where the limaçon has a cusp towards the center, this image will lose its arc shape and starts looking like a "droplet" (round on one side and more pointy towards the center of the lens), becoming larger in the radial than in the tangential direction. For s > 1 only the outer solution of the internal arc will be real, as this image will contain the lens center. This second image would not exist in the case of an infinitesimal source, but is present for a finite source, provided that the previously discussed condition, R 0 < s < 1 + R 0 , holds. Understanding the shape of the internal image is important for interpreting the results of Section 3. In particular, some length and width definitions will start to have an odd behavior for s 1/2. The magnification of the internal image is close to unity at s = 1/2 and this image becomes highly demagnified as s approaches 1. In the present study we shall focus only on high magnifications and distortions, which occur for s < 1/2. However, we will still show some results for larger values of s for completeness. Both the external arc and the internal arc in the s < 1/2 regime have smooth extremities, as in the left panel of Fig. 1. However, on the verge or merging and forming an Einstein ring, the extremities become sharp (see e.g., Fig. 2 of Liebes 1964, which considered finite circular sources and a point lens). One may define the center of curvature of the arc(s) as the center of the circumference that passes through the arc extremities, P i and P f , and its center, P c (see Fig. 1). The position of this center is given by 8 x 0 = s 2 1 + 1 1 ± s (1 + cos φ 0 ) ,(21) such that the center of curvature of the arc(s) is offset with respect to the lens center. For s 1 this offset is simply given by s, i.e., the center of curvature is at the source center. The arc radius of curvature will be given by r 0 = x(0) ∓ x 0 = 1 ± (s − x 0 ) .(22) GEOMETRICAL PROPERTIES OF THE IMAGES In order to compute the magnification and arc cross sections, we need to determine the area and the axial ratio of the images. The next sections are devoted to the computation of these geometrical quantities and the search for accurate expressions in closed form. Area and finite source magnification The area of the image(s) can be written as: A = ∫ φ f φ i ∫ x (+) x (−) x dx dφ = ∫ φ f φ i W(φ)x(φ) dφ,(23) where x(φ) is given in Eq. (19) and W(φ) = x (+) − x (−) = 2R 0 1 − (s/R 0 ) 2 sin 2 φ,(24) i.e., W(φ) is the arc width measured in the radial direction at the angular position φ. This quantity is the same for the internal and external arcs, as the SIS has no radial magnification. For s > R 0 there are two ranges of φ (spanning 2φ 0 each), corresponding to the two images, and the area for the external and internal arcs is given by: A = 4R 0 E φ 0 , (s/R 0 ) 2 ± πR 2 0 ,(25) where φ 0 is given in Eq. (20) and E(α, m) is the incomplete elliptic integral of the second kind (Byrd & Friedman 1971), which is given by E(α, m) = ∫ α 0 1 − m sin 2 θdθ = ∫ sin α 0 1 − mt 2 1 − t 2 dt.(26) The magnification of each image is simply the ratio of the image and source areas µ ex,in A = A πR 2 0 = ±1 + 4 πR 0 E φ 0 , s R 0 2 .(27) This result is equivalent to the one found by Inoue & Chiba 8 From here on we will use the convention that the first sign (i.e. the + in ± and the − in ∓) will refer to the external arc, while the second one will correspond to the internal image. The superscripts "ex" and "in" will only be kept whenever needed for clarity. (2005, their Eq. (5)), which was expressed in terms of complete elliptical integrals of the first and second kinds. The finite source magnification (27) can be expanded for low values of R 0 /s, i.e., far from the Einstein ring formation, as µ ex,in A ≈ ±1 + 1 s + R 2 0 8s 3 + O R 0 s 4 ,(28) which is valid even for very large magnifications (s 1). As expected, at zeroth order in the source size, the finite source magnification is exactly the Jacobian of the transformation, µ (Eq. 13). The first correction for finite size is quadratic in R 0 . We notice that, for small sources and before the formation of an Einstein ring, the effect of finite source is always to increase the magnification with respect to the infinitesimal case, for both the internal and external arcs. This same qualitative result was found in Bontz (1979) for a point mass lens. The total magnification of the source is the ratio of the area of all images to the source area, which is simply the sum of the magnifications for each arc µ A = A in + A ex A s = 8 πR 0 E φ 0 , s R 0 2 .(29) In this work we are interested in highly magnified (and distorted) sources. For the SIS this happens only in the regime where there are two images or an Einstein ring and for sources smaller than the Einstein radius (R 0 < 1). 9 Therefore, we will use expression (29) throughout the paper. In the case of Einstein rings (s ≤ R 0 ) the integral in (23) runs from 0 to 2π, so that φ 0 = π/2 in the expression above, and the area is A E = 8R 0 E s R 0 2 ,(30) where E(m) = E(π/2, m) is the complete elliptic integral of the second kind. Therefore the Einstein ring magnification is µ E A = 8 πR 0 E s R 0 2 ,(31) which is again equivalent to the result in Inoue & Chiba (2005). Close to a centered Einstein ring (s/R 0 1) the magnification can be expanded as µ E A ≈ 4 R 0 − s 2 R 3 0 + O s R 0 4 .(32) The first term corresponds to the magnification of a perfectly aligned observer-lens-source and gives the maximum magnification for a finite circular source, µ max = 4/R 0 (Peacock 1982). In this case the image is an annulus of circumference 2π and width 2R 0 . Combining Eqs. (29) and (31), for s/R 0 > 1 and ≤ 1, respectively, gives the total magnification for the whole range . The points P i , P f and P c are the extremities and center of the arc, whose radial position is the ridgeline evaluated at the angular positions for these points (x (φ i ), x φ f and x (φ c )). On the left panel we also show the circumference of radius r 0 passing through these 3 points, which defines the curvature center, and the arc aperture 2θ 0 with respect to this center (black solid line). of source positions s. The maximum value of this function occurs for s = 0 and is given by µ max . In the boundary between the two arc and the Einstein ring solutions the magnification is µ trans = 8/(πR 0 ). Length Contrarily to the area, there is no unique definition of length for generic shapes. In the case of gravitational arcs a few choices have been used in the literature for both simulated and real images. They all use the arc extremities, points P i and P f in Fig. 1, and involve the determination of an arc center in a way or another. This center is usually chosen as the image of the center of the source, in the case of simulated images, corresponding to P c in this figure. Below we test several length definitions, seeking at the same time expressions that are accurate to describe the arc shape and that are written in a simple form in terms of elementary functions. 1) Geometrically, the simplest length definition is to consider the sum of the segments connecting the arc extremities to its center (see Fig. 1 right panel): L 1 = P i P c + P c P f .(33) This definition has been applied to both real and simulated arcs (see e.g., Oguri 2002;Ho & White 2005;Xu et al. 2016). The angular positions of P i , P f and P c are simply φ i = −φ 0 , φ f = φ 0 and φ c = 0, and their radial position is simply the ridgeline evaluated at these angles, x (φ i ), x φ f and x (φ c ). Therefore, the lengths of the external and internal arcs will be given by L 1 = 2 2 + R 2 0 − 2 cos φ 0 ± 2R 0 sin φ 0 .(34) 2) A simple way to define a length that follows the shape of the image is to integrate the tangential part of the ridgeline (Eq. 19) along the arc: L 2 = ∫ φ f φ i x(φ)dφ = 2 (φ 0 ± R 0 ) = 2 arcsin R 0 s ± R 0 ,(35) which has also a very simple expression. 3) More rigorously, the length of the arc ridgeline is given by L 3 = ∫ dl = ∫ φ f φ i x 2 + d x dφ 2 dφ.(36) This is the most natural definition of an "exact" arc length in this context and will be taken as a reference when we compare the different expressions for the length that will be tested. As far as we know, the only application of this definition to arcs is given by the so-called Mediatrix method (Bom et al. 2012, 2017, andBom et al., in prep.). The expression above is the length of the limaçon between the image extremities and is given by L ex 3 = 4 (1 + s) E φ 0 2 ,4s (1 + s) 2 and L in 3 = 4 (1 + s) E π + φ 0 2 , 4s (1 + s) 2 − E 4s (1 + s) 2 .(38) Although several numerical methods exist for the fast computation and inversion of these functions (see e.g., Fukushima 2013Fukushima , 2015, the expressions above do not allow us to obtain the arc cross section in a simple form. Therefore, we will seek other definitions that provide results close to the one above, but can be expressed in terms of simple functions. 4) Currently, the most commonly used length definition (see e.g., Miralda-Escude 1993; Bartelmann & Weiss 1994;Meneghetti et al. 2008) is given by the arc of circumference passing through the image extremities and its center (i.e. the circle containing the points P i , P c and P f ): L 4 = 2θ 0 r 0 ,(39) where r 0 is the curvature radius given by Eqs. (21) and (22) and θ 0 is half of the arc aperture with respect to the curvature center, as indicated in Fig. 1, and is given by θ 0 = arcsin 1 ± s 2 − R 2 0 r 0 R 0 s .(40) As we shall see, L 4 provides an excellent approximation to L 3 and is written explicitly in terms of simple functions. Nevertheless, it does not allow one to obtain the cross section in closed form. Therefore, we test two alternative length definitions using the arc of a circle, but now centered at the origin (lens center) instead of the curvature center, such that the arc spans the angle 2 φ 0 . The points P i /P f and P c are located at different radii with respect to that center and we test with these two radii, defining L 5 and L 6 . 5) Arc of a circle with aperture 2 φ 0 and radius at P c : L 5 = 2φ 0 x (0) = 2φ 0 (1 ± s) = 2 arcsin R 0 s (1 ± s) ,(41) which has indeed a simpler expression than L 4 . 6) Same as above, but using as radius the distance between lens center and the arc extremities: L 6 = 2φ 0 x (φ i ) = 2φ 0 (1 ± s cos φ 0 ) (42) = 2 arcsin R 0 s 1 ± s 2 − R 2 0 ,(43) which is also more tractable than L 4 . In Fig. 2 we show the fractional difference of all length definitions above as compared to L 3 , which we take as the "exact" length. Let us focus first on the external arc (upper panel). We see that all proposed definitions are in excellent agreement in the whole range of s and R 0 . The highest deviations occur close to the Einstein ring limit and are at most of the order of 10% for R 0 = 0.2. In the case of L 1 , as it approximates the arc by two chords, even for infinitesimal sources we have L 1 → 2 √ 2 close to the formation of the Einstein ring, such that the fractional difference with respect to L 3 is ∼ 10% at this point. The approximations become better for smaller arcs (higher s, lower R 0 ), as will be discussed below. We see that L 1 and L 6 always underestimate the arc length, while L 5 overestimates it, which is the expected behavior from their definitions. The expressions L 2 and L 4 are in striking agreement with L 3 all the way down to very close to the Einstein ring formation. For the internal image the situation is a bit more complicated, since its shape can deviate substantially from an arc, which happens somewhere in the interval 1/2 < s < 1, depending on the source size, as discussed in Section 2.2. The comparison of the length definitions in this case is shown in Fig. 2 (lower panel). The changes in shape are clear from the behaviors as a function of s in the bottom left panel. We recall that all length definitions are based on points along the limaçon between the extremities of the arc, which loses its concavity and becomes dimpled for s > 1/2. For s < 1/2 all length definitions are well behaved and similar, with less than 5% deviation, except close to the Einstein ring formation. In the intermediate region the deviations become larger. For s > 1 the length starts to decrease and some definitions cease to be valid. To compute the arc cross section, we are interested in highly elongated and magnified images. The infinitesimal source L/W (Eq. 13), which gives an order of magnitude of the finite source value, is unity at s = 1/2 and decreases for higher values of s. Therefore, we are only interested in the regime where the lengths are well defined and well behaved. In this regime, all approximations agree to within ∼ 10%. Again, the best approximations are L 2 and L 4 , specially for lower values of s, which is the relevant regime for the cross section. In the lower right panel of Fig. 2 we show the lengths as a function of R 0 for s = 0.2. The behavior is qualitatively similar to that of the external arc, but now L 5 underestimates and L 6 overestimates the length, also as expected from their definitions. It is clear that, by far, L 2 and L 4 are the best approximations to L 3 in all the relevant range of s for both the external and internal arcs. In particular, L 4 is almost indistinguishable from our reference length definition. Nevertheless, we will use L 2 to compute the arc cross section in Sec. 4.3 owing to its simplicity, which will allow us to obtain an expression in closed form. The expressions (34-43) that we have obtained above for L 1 -L 6 are valid for any source size R 0 and position s > R 0 . However, it is useful to obtain perturbative solutions for small sources. Expanding these expressions up to third order in R 0 we obtain: 10 L 1 ≈ 2 ±1 + 1 s R 0 + R 3 0 4s 3 (1 ± s) ,(44)L 2 ≈ 2 ±1 + 1 s R 0 + R 3 0 3s 3 ,(45)L 3 = L 4 ≈ 2 ±1 + 1 s R 0 + (1 ± s + s 2 )R 3 0 3s 3 (1 ± s) ,(46)L 5 ≈ 2 ±1 + 1 s R 0 + (1 ± s)R 3 0 3s 3 ,(47)L 6 ≈ 2 ±1 + 1 s R 0 + (1 ∓ 2s)R 3 0 3s 3 .(48) These approximations are valid for arbitrary magnifications, as long as we are far from the Einstein ring formation (R 0 s). The first order term in R 0 yields exactly the infinitesimal source approximation L = 2R 0 \| µ t \| x=1±s as expected (see Eqs. 10 and 13). The expansions above provide the lowest order corrections for finite sources to the various length definitions. For small values of s, i. e. high magnifications, all proposed measures of L (except L 1 ) also agree up to third order in R 0 . Interestingly the expansions for L 3 and L 4 agree exactly up to this order, for any value of s, which is in agreement with what we see in Fig. 2. Neglecting the quadratic term in s in expression (46), we see that L 2 = L 3 (L 4 ), which explains why L 2 is so close to L 3 (L 4 ) in the plots of Fig. 2. It is also clear from the expressions (46), (47), and (48) why L 5 is larger and L 6 is smaller than L 3 for the external arc, and the other way around for the internal one. In brief, all the qualitative behaviors pointed out in Fig. 2 are clearly seen in the perturbative expansions above. Width If the arc length has not a unique definition, the determination of the width is even more ambiguous. Several methods have been proposed and tested in the literature (see e.g., Redlich et al. 2012;Meneghetti et al. 2013, for reviews). As in the previous section, our aim here is to test several definitions of W seeking expressions that are at the same time representative of the arc shape and that can be expressed in simple analytical form. A natural definition in the context of the smooth SIS arcs with a well defined boundary is to choose the width along the direction perpendicular to the ridgeline at the arc center: W c = W(0) = 2R 0 ,(49) where W(φ) is given in Eq. (24). The result above, which is the same for internal and external arcs, is easy to interpret as the lensing by a SIS does not change the radial positions and thus the width of the image defined as above is the same as the source diameter. For more realistic arcs, from ray-tracing simulations or real data, the shapes can be less symmetrical and the object boundary is subject to irregularities. It is therefore suitable to use information from the whole object, instead of a measurement across a single direction, as above. One approach that has been often used in the literature is to derive a width from the object area A and length L, W ∝ A/L. The proportionality constant depends on the shape of the object. In Bartelmann & Weiss (1994) and subsequent works, the images are fitted by simple geometric figures, such as rectangles, ellipses or rings. The figure that best fits the objects defines the constant, which is, for example, 1 for rectangles and 4/π for ellipses. It turns out that the arcs that we consider in this paper are very well fit by a figure known as ArcEllipse (Furlanetto et al. 2013, see Appendix A). The A-L-W relation for the ArcEllipse is identical to that of an ellipse and therefore we define the width as W i = 4A πL i ,(50) where L i represents the definitions of lengths used previously. As in the previous section, we take the length along the ridgeline L 3 and define W 3 as our reference value to compare the different width definitions. Another width definition that has been used more recently (Meneghetti et al. 2008;Redlich et al. 2012) is to consider the mean (or the median) of the width of the object along the radial direction with respect to its center (or in the direction orthogonal to the object ridgeline). This is akin to computing W = 1 φ f − φ i ∫ φ f φ i W(φ)dφ = 2R 0 φ 0 E φ 0 , s 2 R 2 0 .(51) In Fig. 3 we show the relative difference of the various definitions tested with respect to the reference value, W 3 , as a function of s and R 0 . All expressions, except W, agree reasonably well in the whole interval of s and R 0 (we recall that only the range s < 1/2 is relevant for the internal arc). The behavior of W i traces back to the behavior of L i seen in Section 3.2, as A is the same for all width definitions. We see that the ratio W/W 3 is almost constant in the whole range of parameters (considering s < 1/2). The proportionality factor is discussed below and in Appendix A. The expression W c is closer to W 3 for the external than for the internal one. The difference is at most ∼ 10% (∼ 20%), close to the Einstein ring limit, for the external (internal) arc, and less then ∼ 2% (∼ 5%) in most of the parameter range. Owing to the simplicity of W c , we will use this expression to obtain the arc cross section in an analytical form. On one hand, the differences pointed out above with respect to W 3 are much smaller than the finite source effect on the cross section. On the other hand, in a practical application, as long the arcs are measured in the same way as used to compute the cross section, any definition of W is valid. As we did for the length, it is illustrative to derive the perturbative expansions of the width for small source sizes, which are given by W 1 ≈ 2R 0 ± R 3 0 4s (1 ± s) 2 ,(52)W 2 ≈ 2R 0 − R 3 0 12s 2 (1 ± s) ,(53)W 3 = W 4 ≈ 2R 0 − (1 ± s + 4s 2 )R 3 0 12s 2 (1 ± s) 2 ,(54)W 5 ≈ 2R 0 − (1 ± 4s)R 3 0 12s 2 (1 ± s) ,(55)W 6 ≈ 2R 0 − (1 ∓ 8s)R 3 0 12s 2 (1 ± s) ,(56)W c = 2R 0 ,(57)W ≈ πR 0 2 1 − (4 − 3s)R 2 0 24s 2 .(58) Here again the first term is derived from the eigenvalue of the Jacobian matrix of the transformation, W = 2R 0 µ r = 2R 0 , and the first correction is quadratic in R 0 with respect to this term (except for W c ). The differences among the W i are due to the differences in L i and all W 2 -W 6 agree for high magnifications (s 1) up to third order in R 0 . It is clear that W, when corrected by a factor 4/π, gives the same result as the other definitions (to first order in R 0 ). The same happens for an ArcEllipse (Appendix A), for which the relation W c /W = 4/π holds exactly. The same correction factor was found by Redlich et al. (2012) to relate the mean width to the one based on the area of the ellipse. To derive a perturbative expression for the arc cross section for small sources, it is useful to write the length-towidth ratio from the expansions that were obtained before. In particular, as we will compute the cross section based on L 2 and W c , we obtain the ratio L 2 W c ≈ ±1 + 1 s + R 2 0 6s 3 ,(59) which yields the infinitesimal axial ratio R λ (Eq. 11) at zeroth order in R 0 and the lowest order correction for finite sources. As in the case of the magnification, the axial ratio is always increased with respect to the infinitesimal source, as long as we are far from the Einstein ring formation (s R 0 ). CROSS SECTIONS The cross section is defined as the area in the source plane that generates images with some specified properties, e.g. axial ratio or magnification above a certain threshold (Schneider et al. 1992;Bartelmann et al. 1995): 11 σ = ∫ Ω d 2 y,(60) where the domain Ω is the region in the source plane satisfying the condition, for example, magnification above a given threshold (µ > µ th ) or images with length-to-width ratio above a given value (L/W > R). The cross section can be expressed in terms of the lens plane variable x using the Jacobian (3) σ = ∫ Ω x \| det J(x)\|d 2 x,(61) where now the domain of integration is defined for the quantities (e.g. length-to-width ratio or magnification) expressed in terms of the lens plane coordinates. As the local magnification and axial ratio (Eqs. 4 and 11) are naturally obtained in the lens plane, the form of the cross section above is the most often used when considering infinitesimal sources. Working in the source plane is computationally more expensive, as it requires solving the lens equation (2). Care must be taken when working in the lens plane as multiple regions in this plane (corresponding 11 We recall that all distances in this paper are given in units of the Einstein radius. Therefore, to convert the cross section to physical units, one must multiply this expression by ξ 2 0 (from Eq. 6). In terms of the solid angle in steradians: to different images) can be mapped to the same region in the source plane. This multiplicity has to be accounted for in the cross section computation. σ sr = σ (ξ 0 /D OL ) 2 . When finite sources are considered, the lens equation has to be solved (either numerically or analytically) to obtain the images, and the cross section is computed as in Eq. (60). 12 In the case of this paper, not only the lens equation has a simple analytical solution, but also we have derived expressions for µ and L/W as a function of the position of the center of the source s in closed form. This will enable us to compute the cross section (Eq. 60) in a simple form. In the SIS case with circular sources, due to the axial symmetry and since µ is a monotonically decreasing function of s, the cross section (60) is simply given by σ µ = πs 2 th ,(62) where µ (s th ) = µ th . For the arc cross section there is another condition, as for s < R 0 an Einstein ring is formed. Therefore, the domain in s where arcs with L/W > R are formed is given by s th ≥ s > R 0 , where L/W (s th ) = R. The cross section is thus the area of the annulus defined by this condition σ L/W = πs 2 th − πR 2 0 .(63) 12 See, however, Fedeli et al. (2006) and the discussion in Appendix B. Infinitesimal Cross Section In the infinitesimal circular source approximation the axial ratio is given by Eq. (12), so that the condition L/W = R λ > R yields two solutions for x: x λ =            x max = R R − 1 , x min = R R + 1 .(64) Therefore, the domain of integration in Eq. (61) is an annulus with radii determined by the values above. Using the Jacobian given in Eq. (12), the cross section (61) is σ λ = ∫ 2π 0 ∫ 1 x min 1 x − 1 xdxdφ + ∫ 2π 0 ∫ x max 1 1 − 1 x xdxdφ = π (R + 1) 2 + π (R − 1) 2 = 2π R 2 + 1 R 2 − 1 2 ,(65) as obtained in Bartelmann et al. (1995); Dúmet-Montoya et al. (2013); Er et al. (2013). Notice that the region with x < 1 corresponds to the internal image and the region with x > 1 corresponds to the external image. Therefore, we can split the cross section into two, one for the internal image having R λ > R and the other for the external image satisfying this condition: σ in,ex λ = π (R ∓ 1) 2 .(66) This result is the same one would have obtained working in the source plane Eq. (60) and considering the axial ratio for each image in the source plane (Eq. 13) to define the integration domain. The total magnification is given, in the source plane (Eq. 13), simply by 13 µ = µ in + µ ex = 2 s .(67) The condition µ > µ th sets the cross section (Eq. 62) as (Schneider et al. 1992): σ µ = 4π µ 2 th .(68) Finite Source Magnification Cross Section The exact magnification cross section for finite circular sources is found from (62) by solving µ A (s th ) = µ th for s th , where µ A is given by Eq. (29) for s ≥ R 0 and by Eq. (31) for s ≤ R 0 . The result from the numerical inversion of the elliptic integrals is shown in Fig. 4, along with the results for infinitesimal sources (Eq. 68) and an approximate solution discussed below. For R 0 s we may obtain an analytical solution by using the approximation (28), such that the total magnification is µ P tot = 2 s + R 2 0 4s 3 .(69) To determine s th we solve the third order equation µ P tot = µ th and expand the solution to the lowest non-trivial order in R 0 , to obtain the cross section σ P µ = 4π µ 2 th + π 4 R 2 0 .(70) As expected, in this regime the cross section for finite sources is enhanced with respect to the infinitesimal one, as the magnification is also higher in this case (Eq. 69). We may also obtain a perturbative solution close to the perfectly aligned Einstein ring from Eq. (32), which can be easily solved for s th to obtain σ E µ = π 4R 2 0 − µ th R 3 0 = π (µ max − µ th ) R 3 0 .(71) The cross section vanishes for µ th ≥ µ max = 4/R 0 , as no image can have a magnification above this value. This is in contrast to the infinitesimal source case, for which the magnification is unbounded and the cross section (Eq. 68) never vanishes. Thus, for high magnifications, within the Einstein ring regime, the finite source cross section is smaller than the infinitesimal one. While the approximation (70) is still good close to the onset of the Einstein ring formation (i.e. at µ = µ trans ), the approximation (71) breaks down at this point. However, it is easy to improve the Einstein ring cross section, considering that the expression above is linear in (µ max − µ th ). We add a correction term that is quadratic in this quantity and fix the cross section at µ trans to its exact value (πR 2 0 ). In other words, we build an extreme perfect quadratic approximant to the cross section between the onset of the Einstein ring formation and the perfect Einstein ring solution, such that the cross section is σ E µ = π 4R 2 0 − µ th R 3 0 − 1 R 2 0 3π − 8 (4 − 8/π) 2 4R 2 0 − µ th R 3 0 2 .(72) In fact, this expression provides an excellent approximation for the magnification cross section in the full range from µ trans to µ max , as can be seen in fig. 4. By joining this solution with the expression (70) we may construct a single continuous approximation to σ µ . These two curves match at µ J = 2.15/R 0 < µ trans . Therefore, we build a single approximate magnification cross section for finite sources in the full range of the magnification threshold by using σ P µ (Eq. 70) for µ th < µ J and σ E µ (Eq. 72) for µ th ≥ µ J . This is shown as the dotted line curves in fig. 4. As we can see from Eqs. (29) and (31), the magnification can be written as µ A = f (s/R 0 ) /R 0 , where f is expressed in terms of the incomplete and complete elliptic integrals for s ≥ R 0 and s ≤ R 0 , respectively. Therefore, the cross section will be given by σ µ = π f −1 (µ th R 0 ) R 2 0 . This form is explicit in Eqs. (70) and (72) and this is why µ J , µ trans , µ max , etc. are all ∝ R −1 0 . Given this form, the accuracy of approximations (70) and (72), more specifically their fractional deviation with respect to the exact result, will be a function of the combination µ th R 0 . The highest discrepancy between the exact and approximate solutions occurs at µ J and is 6.2%. Outside the range 1.6 µ th R 0 2.3 the approximations deviate less than 2%. In particular, this precision holds in the whole interval of magnifications within the Einstein ring formation. The perturbative solution is practically exact for µ th R −1 0 . Of course the approximate solutions can be improved arbitrarily by considering higher order expansions. However, the error achieved with expressions (70) and (72) is already much smaller than other uncertainties involved in the modeling of the statistics of highly magnified sources (e.g., Hezaveh & Holder 2011;Lapi et al. 2012). Cross Section for Arc Formation In sections 3.2 and 3.3 we have tested several definitions and approximations for L and W, seeking expressions that are at the same time accurate and written in a simple analytic form. In particular, we have found that L 2 (Eq. 35) is extremely accurate in the whole parameter space of the problem for the external image. For the internal image L 2 is also very accurate in the region where it has an arc shape. This is the relevant region for the arc cross section as it is the only configuration where the image can have a large L/W in the tangential direction. We found that W c (Eq. 49) is a good approximation to the width of the image, except perhaps close to the Einstein ring. Using these two choices the axial ratio takes a very simple form L W = L 2 W c = 1 R 0 arcsin R 0 s ± 1 .(73) This expression shows explicitly the existence of a maximum value for the length-to-width ratio given by (L/W) max = π/(2R 0 ) ± 1, which corresponds to the formation of an Einstein ring (s = R 0 ). This value is easy to understand, as the two images are touching at their extrema on the verge to form the ring, such that the maximum value for L can be approximated by π, and (L/W) max π/(2R 0 ). From the expression (73) above it is easy to find the threshold value s th such that L/W ≥ R. For s ≤ R 0 an Einstein ring is formed and this region does not contribute to the arc cross section. We can compute the cross section for the formation of each arc (internal and external) individually. The total cross section will simply be the sum of the two cross sections (as in Eq. 65). If the two arcs have length-to-width ratios above the threshold, that source position will count twice for the total cross section, if only one arc satisfies this condition, it will be counted once. Below we show the results for the individual arc cross sections, which are determined by R 0 ≤ s ≤ s th and are given by σ L/W = πR 2 0 csc 2 (R 0 (R ∓ 1)) − 1 .(74) This cross section is shown in fig. 5 (dashed line), along with the cross section for infinitesimal sources (Eq. 66, solid line). We see that the cross section for finite sources goes to zero for R ≥ (L/W) max as no arcs can be formed with length-towidth ratio above this value, as pointed out in Rozo et al. (2008), in contrast to the infinitesimal source case. For small values of R 0 (and far from the Einstein ring formation) we may use the expression (59) for L/W. Solving the third order equation for s th for a given R, and taking the lowest order in R 0 leads to the cross section σ P L/W = π (R ∓ 1) 2 − 2 3 πR 2 0 ,(75) where again, we have subtracted the region where Einstein rings are formed (πR 2 0 ). This expression shows the first order correction for finite sources to cross section for infinitesimal circular sources (Eq. 66). Notice that, although the finite source L/W (Eq. 59) is higher than the infinitesimal one (Eq. 12), there is a lower limit in s such that Einstein rings are formed. We are excluding this region from the cross section, whereas this effect is not present for infinitesimal sources. In Fig. 5 we show the perturbative cross section as a function of R and R 0 , along with the complete cross section (Eq. 74) and the infinitesimal one (Eq. 12). We see that the first order correction for finite source size is a very good approximation for lower values of the length-to-width threshold and captures the behavior of the full cross section until close to the formation of the Einstein ring. SUMMARY In this paper we investigated the geometrical properties of the images of finite circular sources lensed by a SIS, aiming to compute the magnification and arc cross sections. First we obtained their area A, length L, and width W, testing several expressions for the latter two. The area is written in terms of elliptic integrals covering all possible source positions (Eqs. 27 and 30). We found that the length L 4 (Eq. 39), which is currently the most commonly adopted to measure arcs, is virtually indistinguishable from the exact integration along the arc ridgeline. The alternative definition L 2 (Eq. 35) also provides an excellent approximation to better than 1% precision for the external arc and has a simple analytical form. The width W i defined from the area and length is in good agreement with the width at the arc center and with the mean width W, as long as the right correction factor is applied in the later case. For the SIS the correction factor proposed in Redlich et al. (2012) is manifest. We obtain perturbative expressions to the lowest nontrivial order in R 0 for the area and all length and width definitions, which show clearly the finite source corrections to the solutions for infinitesimal sources. From these quantities, we derive the total magnification µ A and the length-to-width ratio L/W of the arcs. We obtain an approximate solution for µ A well inside the Einstein ring regime. Far from this regime, both µ A and L/W are enhanced for finite sources with respect to the infinitesimal case. The length-to-width ratio is expressed in a very sim- ple form for L 2 /W c (Eq. 73), which is a good approximation until the onset of Einstein ring formation. Finally we apply these results to derive the cross sections. For the magnification, σ µ , we obtain the exact solution from the numerical inversion of the elliptical function, which is valid for sources of arbitrary size (as long as their radius is smaller than the Einstein one), including both arcs and rings. We also obtain an approximation in simple form, valid for all magnification thresholds (Eqs. 70 and 72). For the arc cross section, σ L/W , we obtain a solution in terms of elementary functions (Eq. 74), for a specific choice of the length and width definitions, valid for all length-to-width thresholds until the formation of an Einstein ring. We also derive a perturbative solution (Eq. 75) showing explicitly the finite source correction. We show that the cross sections vanish for thresholds above a given value (R max π/(2R 0 ), µ max = 4/R 0 ), which is a behavior also noted in simulations (see e.g., Bartelmann & Weiss 1994;Rozo et al. 2008;Hezaveh & Holder 2011;Lapi et al. 2012). This is easy to understand as µ A and L/W are bounded in the case of finite sources. In Appendix A we compare the geometrical properties of the SIS arcs with those from the ArcEllipse. The results justify the use of the A-L-W and W-W relations valid for ellipses to the case of gravitational arcs (at least those from SIS and circular sources), as has been done in previous works using simulations (e.g., Bartelmann & Weiss 1994;Oguri 2002;Redlich et al. 2012). In Appendix B we discuss a formalism by Fedeli et al. (2006) to include finite source effects in the cross section computation, showing that it yields a good approximation to the results of this paper for small sources. CONCLUDING REMARKS We have presented a first study of the magnification and arc cross sections as computed from the exact solution for the images of finite sources. The choice of a simple lens and source model allowed us to work all expressions up to the cross section in analytical form. Despite the simplifying assumptions of SIS lens and circular sources, this example is not of purely pedagogical interest. Indeed, this combination of models has successfully been used to reproduce the observed abundance of sub-millimeter sources (Lapi et al. 2012). Furthermore our approach clarifies the results obtained empirically using ray-tracing simulations, such as the scaling of the maximum magnification µ max with the source size (Lapi et al. 2012). It becomes clear why the magnification cross section for finite sources is enhanced for moderate magnifications and has a cutoff for µ > µ max . This behavior is also seen in the ray-tracing results (e.g. compare Fig. 3 of Hezaveh & Holder 2011, with our Fig. 4). Analytical solutions for the magnification of finite sources have been obtained in the literature (e.g. Refsdal 1964;Inoue & Chiba 2005;Dobler & Keeton 2006). However, to the best of our knowledge, these results have not been used previously to obtain the magnification cross section and its applications. Also, we are not aware of analytic studies on the length-to-width ratio of arcs, except the approximative method of Fedeli et al. (2006). Finite source effects in the magnification and arc cross sections have been studied through ray-tracing techniques (e.g., Oguri 2002;Hezaveh & Holder 2011;Lapi et al. 2012). When applicable, our results are in agreement with these studies and provide a clear interpretation of some finite source effects addressed by them. The approach laid out in this paper paves the way for similar studies using more generic lens and source models. For example, some results can be readily extended for elliptical sources and SIS lenses (de Freitas et al, in prep.). This approach can be applied to other known analytical solutions for arcs from singular isothermal models, including external shear (Dobler & Keeton 2006), elliptical mass distributions (Dobler et al. 2008), and the combination of the two including source ellipticity (Dúmet-Montoya, et al., in prep.). Furthermore, the method can be applied to lenses with more generic mass distributions using the analytic solutions for arcs from the perturbative approach of Alard (2007, see also Peirani et al. (2008)). Even if these cases do not lead to analytic expressions all the way down to the cross sections, the approach employed in this paper can speed up numerical computations by orders of magnitude as compared to raytracing methods (Dobler et al. 2008) and provide hindsight on the solutions, and we expect it to be employed in realistic applications of arc statistics. rection is obtained as in Eq. (51) and is given by W AE = π 2 b = π 4 W AE .(A6) We may define another form factor of a figure by the ratio f W = W/W c . Again, arcs with thiner/sharper extremities will have lower values of f W . In Fig. A1 (lower panels) we show the relative difference of the width geometrical factor for the ArcEllipse and the SIS arcs, ∆ f W / f W = f SI S W − f AE W / f AE W . In this case, the behavior is the same for the internal and external arcs. We see that the differences are very small, except close to the Einstein ring limit. However, even when the two images are touching, the differences are 15%, as for f A . A simple recipe to obtain an ArcEllipse that matches the image of circular source lensed by a SIS is: i) center the ArcEllipse at the curvature center of the image, from Eqs. (21) and (22), such that r c = r 0 , ii) choose the ArcEllipse length such that a = r c θ 0 , with θ 0 given by Eq. (40), iii) choose the ArcEllipse width such that b = R 0 . The resulting figure is almost identical to the SIS arc, except close to the formation of an Einstein ring. In the case of the internal image its shape is very similar to an ArcEllipse in the regime in which the image is arc shaped. The deviations from the ArcEllipse shape are well described by the differences in the form factors shown in Fig. A1. The smaller the ratio R 0 /s, the SIS arc solution is closer to the ArcEllipse. In brief, for most configurations, the ArcEllipse is an excellent representation to SIS arcs. APPENDIX B: APPROXIMATE COMPUTATION OF FINITE SOURCE EFFECTS IN THE LENS PLANE As mentioned in Sec. 4, to obtain the cross sections for finite sources one has to obtain their images, which usually implies solving the lens equation numerically and is computationally expensive. The conditions L/W > R or µ > µ th then define the area in the source plane for which the images satisfy these conditions (Eq. 60). On the other hand, the computation for infinitesimal sources can be carried out on the lens plane, from Eq. (61), without the need of inverting the lens equation, as the local magnification and axial ratios are naturally defined in this plane. Fedeli et al. (2006) have proposed an approximate method to compute the cross section for finite sources in the lens plane. According to their proposal, extended sources are taken into account by convolving the eigenvalue ratio with a suitable window function quantifying the source size. In their method the axial ratio L/W is approximated by h = ∫ R 2 R λ (y)g(y)d 2 y = ∫ R 2 R λ (x)g(x) d 2 x \| µ(x)\| ,(B1) where R λ is the axial ratio for infinitesimal circular sources, as defined as in Eq. (11), and g(y) is a window function representing the surface brightness distribution of the source. For a uniform circular source g(x) = g(y(x)) is zero outside the image of the source and takes the value 1/ πR 2 0 inside. In Fedeli et al. (2006) the expression for L/W is further approximated so as to avoid explicitly carrying out the integral (Eq. B1). By assuming that the eigenvalues of the mapping do not change significantly across a single source, they find an approximate solution for h in terms of deriva-tives of R λ (Eq. 11) and the mean values of the eigenvalues λ r,t (Eq. 4) (see their expression A.16). These authors apply their formalism to lenses with Navarro-Frenk-White (Navarro et al. 1996) profiles and elliptical potentials to represent merging clusters (Fedeli et al. 2006). They find a good agreement between these approximations and raytracing simulations, but with a computation time reduced by a factor of ∼ 30 with respect to the latter. In the case of the SIS, it is simple to obtain an analytical expression for h without the need of any further approximation. In this case we have R λ (x) = µ(x), such that the integral (B1) is simply A ex,in / πR 2 0 , which is exactly the magnification of each image (Eq. 27). Therefore, in this approximation, the arc cross section is obtained by replacing L/W with µ ex,in A . This cross section, which will be referred to as σ h , is computed as in section 4.3 applying the condition µ ex,in A ≥ R (and excluding the arc formation region s < R 0 ). In this case the cross section does not have an analytic solution and has to be obtained numerically from the inversion of the elliptic integral. In Fig. B1 we show the resulting σ h as a function of R for a few values of R 0 , together with the cross section σ L/W (Eq. 74). We see that replacing the axial ratio L/W by h does capture the dependence of the finite source cross section with R 0 and R and provides a good approximation to the arc cross section for R 10. The approximate cross section is systematically lower than σ L/W , which is qualitatively consistent with the results of Fedeli et al. (2006), where the proposed approximation appears to underestimate the cross section as compared to the ray-tracing simulations (see their Fig. 1). We may obtain an explicit expression for σ h for R 0 1 by using the perturbative expansion of the magnification for each arc (Eq. 28) and following the same procedure as in section 4.3, which gives σ P h = π (µ th ∓ 1) 2 − 3 4 πR 2 0 .(B2) This result is shown in Fig. B2, for R = 10, along with σ h , σ L/W , its perturbative expansion in R 0 (Eq. 75) and the infinitesimal cross section. As expected all four expressions for finite sources are similar for R 0 1. By comparing Eqs. (75) and (B2) we see that the first correction terms to the cross section for finite sources differ by about 10% from using the exact L/W and h. Therefore, the difference between the two cross sections is less than 10% at the perturbative level. We conclude that the approximation based on equation (B1) is accurate, at least in the case of a SIS lens and circular sources. Figure B1. Comparison between the arc cross sections using L/W for finite sources (dotted), the convolution h of the local eigenvalue ratio on the image (dashed), and the infinitesimal source approximation (black solid line) for fixed values of the source radius. Left: external arc. Right: internal arc. Figure B2. Arc cross sections using L/W , σ L/W (dash-dotted curve) and σ P L/W (densely dotted), using the approximation h, σ h (dashed curve) and σ P h (dotted line) and the infinitesimal source approximation (black solid line) for R = 10. Left: external arc. Right: internal arc. Figure 1 . 1Sketch showing the inner and outer parts of the external and internal arcs (magenta and blue curves), arc rigdeline (dotted line on the right panel) and critical curve (dashed line) Figure 2 . 2Relative difference (∆L/L 3 = (L i − L 3 )/L 3 ) between the various length measurements considered. The parameters used were: R 0 = 0.2 (left) and s = 0.2 (right). Figure 3 . 3Relative difference (∆W /W 3 = (W i − W 3 )/W 3 ) between the various width measurements considered. The parameters used were: R 0 = 0.2 (left) and s = 0.2 (right). Figure 4 . 4Magnification cross section as a function of the magnification threshold (left) and source radius (right). The dashed lines represent the exact result from the numerical inversion of the elliptic integral. The small circles on the left panel correspond to the value of µ trans , at which the solution switches from two arcs to an Einstein ring. The dotted lines show the results of the perturbative expansions(70)and(72). In the right panel, the larger dot on the dotted line corresponds to the value µ J at which the approximate cross section transitions from the two perturbative expressions. The solid line is the infinitesimal source approximation (Eq. 68). Figure 5 . 5Arc formation cross sections (dashed curves), perturbed arc formation cross sections (dotted curves) and infinitesimal approximation (black solid line) for fixed value of the source radius (left) and for a fixed value of the threshold (right). The first and second rows correspond to the external and internal arc, respectively. Figure A1 . A1Relative difference of the area form factor f A = A/LW (top panel) and the width form factor f W = W /W c (bottom panel) between the ArcEllipse and the images of circular sources lensed by a SIS. The parameters are R 0 = 0.2 (left) and s = 0.2 (right). MNRAS 000, 1-17(2018) The magnification of large sources (R 0 > 1) is discussed inInoue & Chiba (2005).MNRAS 000, 1-17(2018) Considering s < 1/2 for the internal arc, for the reasons discussed previously on the behavior of the internal image around this region.MNRAS 000, 1-17(2018) This expression is valid for s ≤ 1, so that there are two images, and therefore for µ ≥ 2. APPENDIX A: GRAVITATIONAL ARCS AND THE ARCELLIPSE SHAPEAppendix AThe ArcEllipse(Furlanetto et al. 2013) is a simple geometrical figure to represent arc shapes. It is constructed by distorting an ellipse, such that its major axis is bent into an arc of a circle. Therefore, instead of keeping constant the weighted squared sum of the distances to the Cartesian axes of coordinates, as in a standard ellipse, the ArcEllipse considers the distances perpendicular and tangential to a circle. The ArcEllipse is thus the set of points whose distances from a point on the circumference along the tangential direction (r c ∆θ) and along the radial direction (∆r) satisfywhere r c is the radius of curvature of the circle (and of the constructed arc), a is the length along the circle and b is the width at the center in the radial direction (akin to the semi-axes of an ellipse).Choosing the curvature center to coincide with the center of the polar coordinates we have ∆r = x−r c and ∆θ = φ−θ, whereθ is the orientation of the ArcEllipse center. Solving the quadratic expression above, we havewhere x (+) and x (−) delimit the inner and outer boundaries of the ArcEllipse, respectively. The extremities of the arc occur when x (+) = x (−) , similarly to the SIS case, and are given byEq. (A2) is akin to expression(18), except that now the ridgeline x (+) + x (−) /2 is, by construction, a segment of a circle and the curvature center is the center of the coordinate system. Therefore, all length definitions from L 2 to L 6 (Eqs. 35-43) coincide for the ArcEllipse and are given byThe width at the center of the arc is given as in Eq.(49):Therefore, the ratio L/W is given by L AE /W AE = a/b, exactly as in the ellipse case. The area is computed in a similar fashion as in Eq. (23) and is given by(Furlanetto et al. 2013)which is identical to the area of an ellipse with semi-axes a and b. We may define the ratio f A = A/(LW) as a form factor of a geometrical figure. In the case of the ArcEllipse this factor is π/4. Of course, this factor will be smaller the sharper the extremities of the figure, i.e., when the width decreases significantly away from the center. For an annulus segment, for example, f A = 1. InFig. A1(upper panels) we show the relative difference of the area form factors for the ArcEllipse and for the arcs produced by a SIS with circular sourcesWe consider two definitions of L for making the comparison with the SIS case: the exact length along the arc ridgeline L 3 and the approximation L 2 that we used to compute the arc cross section. We use the same width definition as for the ArcEllipse, i.e., the width at the center of the arc W c = 2R 0 . For the external arc, the difference in the form factor is remarkably small until close to the formation of the Einstein ring. At that point the arc extremities become sharper and the shape deviates more substantially from the ArcEllipse. As expected, the difference between the two length definitions is negligible. For the internal image, in the regime in which it is arc shaped (s ≤ 1/2) the form factors are also very similar to the Ar-cEllipse, except close to the Einstein ring formation. 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[ "The balance between diffusion and absorption in semilinear parabolic equations ", "The balance between diffusion and absorption in semilinear parabolic equations " ]	[ "Andrey Shishkov \nInstitute of Applied Mathematics and Mechanics\nLaboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083\nNAS of Ukraine\nR. Luxemburg str. 7483114DonetskUkraine\n", "Laurent Véron \nUniversité François-Rabelais\n37200ToursFrance\n" ]	[ "Institute of Applied Mathematics and Mechanics\nLaboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083\nNAS of Ukraine\nR. Luxemburg str. 7483114DonetskUkraine", "Université François-Rabelais\n37200ToursFrance" ]	[]	Let h : [0, ∞) → [0, ∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m, q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = u k of ∂tu − ∆u m + h(t)u q = 0 in Ω × (0, T ) satisfying u k (x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the associated ordinary differential equation u ′ + h(t)u q = 0 which blows-up at t = 0. 1991 Mathematics Subject Classification. 35K60.	10.4171/rlm/481	[ "https://arxiv.org/pdf/0805.3789v1.pdf" ]	15,908,193	0805.3789	453f1b652da881288a4bfd0ac9c91131e13154f5	The balance between diffusion and absorption in semilinear parabolic equations * 24 May 2008 Andrey Shishkov Institute of Applied Mathematics and Mechanics Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083 NAS of Ukraine R. Luxemburg str. 7483114DonetskUkraine Laurent Véron Université François-Rabelais 37200ToursFrance The balance between diffusion and absorption in semilinear parabolic equations * 24 May 2008Parabolic equationsSaint-Venant principlevery singular solutionsasymptotic expansions Let h : [0, ∞) → [0, ∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m, q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = u k of ∂tu − ∆u m + h(t)u q = 0 in Ω × (0, T ) satisfying u k (x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the associated ordinary differential equation u ′ + h(t)u q = 0 which blows-up at t = 0. 1991 Mathematics Subject Classification. 35K60. Introduction Let m and q positive parameters and h : [0, ∞) → [0, ∞) a nondecreasing continuous. If one consider a reaction-diffusion equation such as ∂ t u − ∆u m + h(t)u q = 0 (1.1) (u > 0 for simplicity) in a cylindrical domain Q T = R N × (0, T ) (N ≥ 1), the behaviour of u is subject to two competing features: the diffusion associated to the partial differential operator, here −∆, and the absorption which is represented by the term h(t)u q . When q > 1 and h(t) > 0 for t > 0, the absorption term is strong enough in order positive solution to satisfy an universal bound 0 ≤ u(x, t) ≤ U h (t) = (q − 1) t 0 h(s) ds −1/(q−1) (1.2) for every (x, t) ∈ Q T . In addition, the function U h which appears above is a particular solution of (1.1 ). The associated diffusion equation ∂ t v − ∆v m = 0 (1.3) admits fundamental solutions v = v k (k > 0) which satisfy v k (x, 0) = kδ 0 if m > (N − 2) + /N . If T 0 BR h(t)v q k dx dt < ∞, B R := {\|x\| < R},(1.4) for any R ∈ (0, ∞], it is shown that (1.1 ) admits fundamental solutions u = u k in Q T which satisfy initial condition u k (x, 0) = kδ 0 . The maximum principle holds and therefore the mapping k → u k is increasing. If h > 0 on (0, ∞) then due to universal bound(1.2) there exists u ∞ = lim k→∞ u k , and u ∞ is a solution of (1.1 ) in Q T . A natural question is whether u ∞ admits a singularity only at the origin (0, 0) or at other points too. Actually, in the last case it will imply u ∞ ≡ U since the following alternative occurs: (i) either u ∞ = U . (complete initial blow-up); (ii) or u ∞ is a solution singular at (0, 0) and such that lim t→0 u(x, t) = 0 for all x = 0. (single-point initial blow-up). This phenomenon is observed for the first time by Marcus and Véron. They considered the semilinear equation ∂ t u − ∆u + h(t)u q = 0 (1.5) and proved [8,Prop. 5 .2] Theorem 1.1 If h(t) = e −κ/t (κ > 0), then the complete initial blow-up occurs. However they raised the question whether this type of degeneracy of the absorption is sharp or not. The method of [8] relies on the construction of subsolutions associated to very singular solutions of equations ∂ t u − ∆u + c ǫ t α u q = 0 (1. 6) for suitable α > 0 and c ǫ > 0, and on the study of asymptotics of these solutions. One the main result of present paper states that if the degeneracy of the absorption terms is lightly smaller respectivelly to Th. 1.1, then localization occurs. then u ∞ has single-point initial blow-up at (0, 0). The method of the proof is totally different from the one of Marcus and Véron and based upon local energy estimates in the spirit of the famous Saint-Venant 's principle (see [5,12,13]). Using appropriate test functions we prove by induction that the energy of the fundamental solutions u k remains uniformly locally bounded in Q T \ {(0, 0)}. In the case of equation ∂ t u − ∆u + h(t)(e u − 1) = 0 (1.8) the same type of phenomenon occurs, but at a different scale of degeneracy. We prove the following Theorem 1.3 1) If h(t) = e −e κ/t for some κ > 0, then the complete initial blow-up occurs. 2) If h(t) = e −e ω(t)/t for some ω ∈ C(0, ∞) positive, nondecreasing and satisfying (1.7 ), then u ∞ has single-point initial blow-up at (0, 0). In this paper we also extend the study of equation (1.1 ) to the case m = 1. The situation differs completely corresponding to m > 1, the porous media equation with slow diffusion, and to (N − 2) + /N < m < 1, the fast diffusion equation. Concerning the porous media equation, we prove Theorem 1.4 If q > m > 1 and h is nondecreasing and satisfies h(t) = O(t (q−m)/(m−1) ) as t → 0, then u ∞ ≡ U h . We give two proofs. The first one, valid only in the subscritical case 1 < m < q < m + 2/N , is based upon the construction of suitable subsolutions, as in the semilinear case. The second one, based upon scaling transformations, is valid in all the cases q + 1 > 2m > 2 where the u k exists. It reduces to proving that the equation Actually, the method is applicable to a much more general class of equations. In the fast diffusion case there is always localization. Theorem 1.6 Assume (N − 2) + /N < m < 1 and q > 1, in Equation (1.1 ). Then u ∞ (x, t) ≤ min    U h (t), C * t \|x\| 2 1/1−m)    (1.10) where C * = (1 − m) 3 2m(mN + 2 − N 1/(1−m) . This type of problem has an elliptic counterpart which is initiated in [10] where the following question is considered: suppose Ω is a C 2 bounded domain in R N , q > 1 and h ∈ C(0, ∞) is positive. What is the limit, when k → ∞ of the solutions (when they exist) u = u k of the following problem −∆u + h(ρ(x))u q = 0 in Ω u = kδ 0 in ∂Ω, (1.11) where ρ(x) = dist (x, ∂Ω). It is proved in [10] that, if h(t) = e −1/t , then u ∞ (:= lim k→∞ u k ) is the maximal solution of the equation in Ω, that is the function which satisfies −∆u + h(ρ(x))u q = 0 in Ω lim ρ(x)→0 u(x) = ∞. (1.12) On the contrary, if h(t) = t α , for α > 0 and 1 < q < (N + 1 + α)/(N − 1), it is proved in [11] that u ∞ has an isolated singularity at 0, and vanishes everywhere outside 0. In a forthcoming article we shall study this localization of singularity phenomenon for the complete nonlinear elliptic problem, replacing the powers by more general functions, and the ordinary Laplacian by the p-Laplacian operator. Our paper is organized as follows: §1 Introduction. In §2 we study sufficient conditions of complete initial blow-up for semilinear heat equation. In §3 we prove sharp sufficient condition of existence of single point initial blow-up for heat equation with power nonlinear absorption. In §4 local energy method from §3 is adapted to the heat equation with nonpower absorption nonlinearity. §5 deals with porous media equation with power nonlinear absorption, §6 -the fast diffusion equation with nonlinear absorption. 2 Complete initial blow-up for semilinear heat equation We recall the standard result concerning the existence of a fundamental solution u = u k (k > 0) to the following problem ∂ t u − ∆u + g(x, t, u) = 0 in Q T = R N × (0, T ) u(x, 0) = kδ 0 . (2.1) If v is defined in Q T , we denote byg(v) the function (x, t) → g(x, t, v(x, t)). By a solution we mean a function u ∈ L 1 loc (Q T ) such thatg(u) ∈ L 1 loc (Q T ), which verifies Q T (−u∂ t φ − u∆φ +g(u)φ) dxdt = kφ(0, 0), (2.2) for any φ ∈ C 2,1 0 (R N × [0, T ) × R). We denote by E(x, t) = (4πt) −N/2 e −\|x\| 2 /4t the fundamental solution of the heat equation in Q ∞ , by B R (a) an open ball of center a and radius R, and B R (0) = B R . The following result is classical Theorem 2.1 Let g ∈ C(R N × [0, T ] × R) such that g(x, t, r) ≥ 0 on R N × [0, T ] × R + , and assume that g = g 1 + g 2 where g 1 and g 2 are respectively nondecreasing and locally Lipschitz continuous with respect to the r-variable functions. Let k > 0 be such that T 0 BR g(x, t, kE(x, t))dxdt < ∞. (2.3) for any R > 0. Then there exists a solution u = u k to problem (2.1). Furthermore, if g 2 = 0, then u k is unique. Function g(x, t, r) = e −κ/t \|r\| q−1 r, with κ > 0 and q > 1, satisfies (2.3). Thus the problem ∂ t u − ∆u + e −κ/t \|u\| q−1 u = 0 in Q ∞ u(x, 0) = kδ 0 . (2.4) admits a unique solution. The next result is proved in [8], but we recall the proof both for the sake of completeness and to present the key-lines of the method in a simple case. Theorem 2.2 For k > 0, let u k denote the solution of (2.4 ) in Q ∞ . Then u k ↑ U S as k → ∞, where U S (t) = (q − 1) t 0 e −κ/s ds 1/(1−q) , ∀t > 0. (2.5) Proof. Case 1. 1 < q < 1 + 2/N . For any ǫ > 0, u k = u satisfies ∂ t u − ∆u + e −κ/ǫ u q ≥ 0 (2.6) on Q ǫ . Therefore if v = v k is the solution of ∂ t v − ∆v + e −κ/ǫ v q = 0 in Q ∞ v(x, 0) = kδ 0 ,(2.7) there holds u k ≥ v k . Passage to the limit k → ∞, yields lim k→∞ u k := u ∞ ≥ v ∞ = lim k→∞ v k in Q ǫ . (2.8) If we write v ∞ (x, t) = e κ/ǫ(q−1) t −1/(q−1) f (x/ √ t), then f is radial and satisfies    f ′′ + N − 1 r + r 2 f ′ + 1 q − 1 f − f q = 0 on (0, ∞), f ′ (0) = 0 , lim r→∞ r 2/q−1) f (r) = 0. Furthermore the asymptotics of f is given in [2], f (r) = Cr 2/(q−1)−N e −r 2 /4 (1 + •(1))) , as r → ∞, for some C = C(N, q) > 0. Therefore f (r) ≥C(r + 1) 2/(q−1)−N e −r 2 /4 ∀r ≥ 0, (2.9) for someC =C(N, q) > 0. If we take t = ǫ, we derive from (2.8 ) u ∞ (x, t) ≥ e κ/t(q−1) t −1/(q−1) f (x/ √ t) in R N . (2.10) Let 0 < ℓ < 2 κ/(q − 1). Inequalities (2.9 ) and (2.10 ) imply u ∞ (x, t) ≥Ct −1/(q−1) e (κ/(q−1)−ℓ 2 /4)t −1 , ∀x ∈B ℓ . (2.11) Therefore lim t→0 u ∞ (x, t) = ∞ , ∀x ∈B ℓ . We pick some point x 0 in B ℓ . Since for any k > 0, the solution u kδx 0 of (2.4 ) with initial value kδ x0 can be approximated by solutions with bounded initial data and support in B σ (x 0 ) (0 < σ < ℓ − \|x 0 \|), the previous inequality implies u ∞ (x, t) ≥ u ∞ (x − x 0 , t). Reversing the role of 0 and x 0 yields to u ∞ (x, t) = u ∞ (x − x 0 , t). If we iterate this process we derive u ∞ (x, t) = u ∞ (x − y, t) , ∀y ∈ R N . (2.12) Since u kδy is radial with respect to y, (2.12 ) implies that u ∞ (x, t) is independent of x and therefore it is solution of z ′ + e −κ/t z q = 0 on (0, ∞) lim t→0 z(t) = ∞. (2.13) Thus u ∞ = U S where U S is defined by (2.5 ). Case 2. q ≥ 1 + 2/N . Let α > 0 such that q < q c,α = 1 + 2(1 + α)/N . We write e −κ/t = t αh (t) withh(t) = t −α e −κ/t . The functionh is increasing on (0, κ/α] and we extend it byh(0) = 0. Let 0 < ǫ ≤ κ/α, then the solution u = u k of (2.4 ) verifies ∂ t u − ∆u +h(ǫ)t α u q ≥ 0, in R N × (0, ǫ]. As in Case 1, u is bounded from below on R N × (0, ǫ] by h (ǫ) −1/(q−1) v ∞ where v ∞ = v is is the very singular solution of ∂ t v − ∆v + t α v q = 0. (2.14) Then v ∞ (x, t) = t −(1+α)/(q−1) f α (\|x\| / √ t), and f α = f satisfies    f ′′ + N − 1 r + r 2 f ′ + 1 + α q − 1 f − f q = 0 on (0, ∞), f ′ (0) = 0 , lim r→∞ r 2(1+α)/q−1) f (r) = 0. The asymptotics of f α is given in [9] f α (r) = Cr 2(1+α)/(q−1)−N e −r 2 /4 (1 + •(1)) as r → ∞, thus f α (r) ≥C(1 + r) 2(1+α)/(q−1)−N e −r 2 /4 ∀r ∈ R + . Consequently u(x, t) ≥Ce (κ/(q−1)−ℓ 2 /4)t −1 , ∀x ∈B ℓ . (2.15) Taking again 0 < ℓ < 2 κ/(q − 1), we derive lim t→0 u(x, t) = ∞ , ∀x ∈B ℓ . As in the Case 1, it yields to u ∞ (x, t) = u ∞ (x − y, t) for any y ∈ R N , and finally u ∞ (x, t) = U S (t). Next we consider Cauchy problem for diffusion equation with an exponential type absorption term ∂ t u − ∆u + h(t)e u = 0 in Q ∞ u(x, 0) = kδ 0 (2.16) where h ∈ C(R + ) is nonnegative. Theorem 2.1 yields the following existence result: Proposition 2.3 Assume h satisfies lim t→0 t N/2 ln h(t) = −∞. (2.17) Then for any k > 0 problem (2.16 ) admits a unique solution u = u k . Furthermore u k (x, t) ≤ V S (t) := − ln t 0 h(s)ds ∀(x, t) ∈ Q ∞ .(2.∂ t v − ∆v + c n t αn v n = 0. (2.19) The necessary and sufficient condition for the existence of a V n is n < 1 + N (α n + 1)/2. This function is obtained in the form V n (x, t) = t −(1+αn)/(n−1) F (x/ √ t), where F solves ∆F + 1 2 ξ.DF + 1 + α n n − 1 F − c n F n = 0. We fix 1 + α n n − 1 = 1 + N 2 ⇐⇒ α n = (2 + N )(n − 1)/2 − 1, (2.20) and set f n = c 1/(n−1) n F. Then f n solves ∆f n + 1 2 ξ.Df n + N + 2 2 f n − f n n = 0. We prove that f n has an asymptotic expansion essentially independent of n, in the following form f n (ξ) ≥ δ(\|ξ\| 2 + 1)e −\|ξ\| 2 /4 =⇒ V n (x, t) ≥ δc −1/(n−1) n t −2−N/2 (\|x\| 2 + t)e −\|x\| 2 /4t (2.21) It order to see that, we putf n = 2 N + 2 1/(n−1) f n then ∆f n + 1 2 ξ.Df n + N + 2 2f n − N + 2 2f n n = 0. By the maximum principle 0 ≤f n ≤ 1 so that 0 ≤f n ′ n ≤f n n for n ′ > n. Thus ∆f n + 1 2 ξ.Df n + N + 2 2f n − N + 2 2f n ′ n ≥ 0, which implies thatf n is a subsolution of the equation forf n ′ and therefore, n ′ > n =⇒f n ≤f n ′ ⇐⇒ f n ≤ N + 2 2 (n ′ −n)/(n−1)(n ′ −1) f n ′ . (2.22) In the particular case n = n * = (N + 4)/(N + 2), the equation falls into the scoop of Brezis-Peletier-Terman study since it can also be written in the form ∆f n * + 1 2 ξ.Df n * + 1 n * − 1 f n * − f n * n * = 0. and their asymptotic expansion applies (with 2/(n * − 1) − N = 2) as \|ξ\| → ∞: f n * (ξ) = C \|ξ\| 2 e −\|ξ\| 2 /4 (1 + •(1)) =⇒ f n * (ξ) ≥ δ * (\|ξ\| 2 + 1)e −\|ξ\| 2 /4 ∀ξ. (2.23) Combining (2.22 ) with n = n * and n ′ replaced by n, and (2.23 ), we get f n (ξ) ≥ δ * 2 N + 2 (n−n * )/(n−1)(n * −1) (\|ξ\| 2 + 1)e −\|ξ\| 2 /4 ∀ξ. (2.24) Since n → (2/(N + 2) (n−n * )/(n−1)(n * −1) is bounded from below independently of n > n * , we get (2.21 ). Step 2. Some estimates from below for a related problem. In order to have v n ≤ u in the range of value of u, which is u(t) ≤ V S (t) = − ln t 0 h(s)ds ∀t > 0, (2.25) we need v = v n to be a subsolution near t = 0 of the equation that u verifies. Furthermore this can be done up to some bounded function. It is sufficient to have c n t αn (x n + 1) ≥ h(t)e x , ∀t ∈ (0, τ n ], x ∈ [0, V S (t)] (2.26) where τ n has to be defined. In particular, at the end points of the interval,      (i) c n t α k ≥ h(t) (ii) c n t αn ln n 1 t 0 a(s)ds + 1 ≥ h(t) t 0 h(s) ds . (2.27) We write (2.26 ) in the form e x 1 + x n ≤ c n t αn h(t) , (2.28) and set φ(x) = e x 1 + x n . Then φ ′ (x) = e x 1 + x n − nx n−1 (1 + x n ) 2 . The sign of φ ′ is the same as the one of ψ(x) = 1 + x k − nx n−1 , a function which decreasing then increasing, is positive near 0, vanishes somewhere between 0 and 1 and again between n − 1 and n. The first maximum of φ is less than e/2. This is not important in (2.28 ) since we can always assume that the minimum of c k t α k /h(t) is larger than e/2. Therefore, it is sufficient to have e VS (t) 1 + V n S (t) ≤ c n t αn h(t) ,(2.h(t) = −ω ′ (t)e −ω(t) , then (2.27 )-ii is equivalent to c n t αn (ω n (t) + 1) ≥ −ω ′ (t). (2.30) Since ω n (t) + 1 ≥ 2 1−n (ω(t) + 1) n , we associate the following O. D. E. on R + c n t αn = 2 1−n −η ′ (η + 1) n , the maximal solution of which is η(t) = 1 2 1 c n (n − 1) 1/(n−1) t −(αn+1)/(n−1) = 1 2 1 c n (n − 1) 1/(n−1) t −1−N/2 . If we write ω in the form ω(t) = e α(t) , with α(0) = ∞, α ′ < 0, then (2.27 )-ii becomes c n t αn e nα(t) + 1 ≥ −α ′ (t)e α(t) , and this inequality is ensured provided c n t αn e (n−1)α(t) ≥ −α ′ (t) ⇐⇒ c n ≥ −α ′ (t)e (1−n)α(t)−αn ln t = −tα ′ (t)e (1−n)(α(t)+2 −1 (N +2) ln t) , (2.31) by replacing α n by its value. Next we fix α(t) = α σ (t) = σ t ∀t > 0 (2.32) where σ > 0 is a parameter, thus −tα ′ (t)e (1−n)(α(t)+2 −1 (N +2) ln t) = e (1−n)σ/t−(2 −1 (n−1)(N +2)+1) ln t = e ρ(t) . In order to have (2.31 ) it is sufficient to have the monotonicity of the function ρ and ρ ′ (t) = σ(n − 1) t 2 − n(N + 2) − N 2t Then there exist γ > 0, independent of k and σ such that ρ ′ (t) > 0 on (0, σγ]. Consequently, inequality (2.31 ) is ensured on (0, ǫ] ⊂ (0, σγ] as soon as c n ≥ e ρ(ǫ) = e (1−n)σ/ǫ−2 −1 (n(N +2)−N ) ln ǫ . (2.33) Step 3. Complete initial blow-up for a related problem. Assume now h(t) =σt −2 eσ t −1 −eσ /t (2.34) for someσ > 0. For n > 2, we fix ǫ <σγ and take c n = e ρ(ǫ) . On (0, ǫ] we have c n t αn (e nα(t) + 1) ≥ −α ′ (t)e α(t) . Therefore, if u = u k is the solution of (2.16 ) with h(t) given by (2.34 ), it satisfies u(t) ≤ V S (t), where V S is given by (2.25 ), and ∂ t u − ∆u + c n t αn (u n + 1) ≥ 0 in Q ǫ . Therefore u is larger that the solution v =ṽ k of ∂ t v − ∆v + c n t αn (v n + 1) = 0 in Q ǫ , withṽ k (0) = kδ 0 . Furthermoreṽ k ≥ v k − c n t αn+1 /(α n + 1), where v = v k solves ∂ t v − ∆v + c n t αn v n = 0 in Q ǫ , with v k (0) = kδ 0 . If we let k → ∞, we derive from (2.21 ) and by replacing c n = e ρ(ǫ) by its precise value e (1−n)σ/ǫ−2 −1 (n(N +2)−N ) ln ǫ , that u ∞ (x, t) ≥ V n (x, t) − c n t αn+1 α n + 1 ≥ δt −2−N/2 (\|x\| 2 + t)e σ ǫ + (n(N +2)−N ln ǫ n−1 − \|x\| 2 4t on (0, ǫ]. In particular u ∞ (x, ǫ) ≥ δǫ −2−N/2 (\|x\| 2 + ǫ)e σ ǫ + (n(N +2)−N ln ǫ n−1 − \|x\| 2 4ǫ . (2.35) Taking \|x\| 2 < σ/4 yields to lim ǫ→0 ǫ −2−N/2 (\|x\| 2 + ǫ)e σ ǫ + (n(N +2)−N ln ǫ n−1 − \|x\| 2 4ǫ = ∞. Thus lim ǫ→0 u ∞ (x, ǫ) = ∞, ∀x ∈ B √ σ/2 . As in the proof of Theorem 2.2, it implies u ∞ = V S . Step 4. End of the proof. Since for any σ >σ > 0 there exists an interval (0, θ] on which σt −2 e σ ′ t −1 −e σ ′ /t ≥ e −e σ/t , any solution of (2.16 ) with h(t) given by (2.34 ) is a subsolution in Q θ of the same equation with h(t) = e −e −σ/t . This implies the claim. Single point initial blow-up for semilinear heat equation We consider the following Cauchy problem ∂ t u − ∆u + h(t) \|u\| q−1 u = 0 in Q ∞ u(x, 0) = kδ 0 . (3.1) The first result dealing with the localization of the blow-up that we prove is the following. Theorem 3.1 Assume h(t) = e −ω(t)/t where ω ∈ C([0, ∞)) is positive, nondecreasing function which satisfies ω(s) ≥ s α0 for some α 0 ∈ [0, 1) and any s > 0, and the following Dini like condition holds: 1 0 ω(s) s ds < ∞. (3.2) Then u k always exists and u ∞ := lim k→∞ u k has a point-wise singularity at (0, 0). Proof. The proof is based on the study of asymptotic properties as k → ∞ of solutions u = u k of the regularized Cauchy problem u t − ∆u + h(t)\|u\| q−1 u = 0 in Q T , u(x, 0) = u 0,k (x) = M 1/2 k k −N/2 δ k (x) ∀x ∈ R N , (3.3) where δ k ∈ C(R N ), supp δ k ⊂ \|x\| ≤ k −1 , δ k ⇀ δ(x) weakly in the sense of measures as k → ∞ and {M k } is some sequence tending to ∞ as k → ∞ fast enough so that M 1/2 k k −N/2 → ∞ as k → ∞. (3.4) Without loss of generality we will suppose that δ k (x) 2 L2(R N ) ≤ c 0 k N ∀ k ∈ N, c 0 = const. (3.5) Our method of analysis is some variant of the local energy estimates method (also called Saint-Venant principle), developed, particulary, in [12,13,[15][16][17] (see also review in [5]). Let introduce the families of subdomains Ω(τ ) = R N ∩ {\|x\| > τ } ∀ τ > 0, Q r (τ ) = Ω(τ ) × (0, r) ∀ r ∈ (0, T ), Q r (τ ) = Ω(τ ) × (r, T ) ∀ r ∈ (0, T ). Step 1. The local energy framework. We fix arbitrary k ∈ N and consider solution u = u k of (3.3), but for convenience we will denote it by u. Firstly we deduce some integral vanishing properties of solution u in the family of subdomains Q r := R N × (r, T ). Multiplying (3.3 ) by u(x, t) exp − t − r 1 + T − r and integrating in Q r , we get 2 exp T − r 1 + T − r −1 R N \|u(x, T )\| 2 dx + Qr \|D x u\| 2 + h(t)\|u\| q+1 exp − t − r 1 + T − r dxdt + 1 1 + T − r Qr \|u\| 2 exp − t − r 1 + T − r dxdt = 2 −1 Ω(τ ) \|u(x, r)\| 2 dx + 2 −1 R N \Ω(τ ) \|u(x, r)\| 2 dx, (3.6) where τ > 0 is arbitrary parameter. Using Hölder's inequality, it is easy to check that R N \Ω(τ ) \|u(x, r)\| 2 dx ≤ cτ N (q−1) q+1 h(r) − 2 q+1 R N \Ω(τ ) \|u(x, r)\| q+1 h(r) dx 2 q+1 . (3.7) Here and further we will denote by c, c i different positive constants which do not depend on parameters k, τ, r, but the precise value of which may change from one ocurrence to another. Let us consider now the energy functions I 1 (r) = Qr \|D x u\| 2 dx dt, I 2 (r) = Qr h(t)\|u(x, t)\| q+1 dxdt, I 3 (r) = Qr \|u\| 2 dxdt. (3.8) It is easy to check that − dI 2 (r) dr = R N h(r)\|u(x, r)\| q+1 dx ≥ R N \Ω(τ ) h(r)\|u(x, r)\| q+1 dx ∀ τ > 0. Therefore it follows from (3.6) and (3.7) R N \|u(x, T )\| 2 dx + I 1 (r) + I 2 (r) + I 3 (r) ≤ cτ N (q−1) q+1 h(r) − 2 q+1 (−I ′ 2 (r)) 2 q+1 + c Ω(τ ) \|u(x, r)\| 2 dx ∀ τ > 0, ∀ r : 0 < r < T. (3.9) Next we introduce additional energy functions f (r, τ ) = Ω(τ ) \|u(x, r)\| 2 dx, E 1 (r, τ ) = Q r (τ ) \|D x u\| 2 dxdt, E 2 (r, τ ) = Q r (τ ) \|u\| 2 dxdt. (3.10) Now we deduce some vanishing estimates of these energy functions. Let µ be some nondecreasing smooth function defined on (0, ∞), µ(τ ) > 0 for τ > 0 (a more precise definition will be fixed later on). Then multiplying the equation (3.3) by u(x, t) exp(−µ 2 (τ )t) and integrating in domain Q r (τ ) with τ > k −1 (remember that supp u 0,k ⊂ \|x\| < k −1 ) we deduce easily 2 −1 f µ,r (τ ) + J µ,r (τ ) := 2 −1 Ω(τ ) \|u(x, r)\| 2 exp(−µ 2 (τ )r) dx+ Q r (τ ) \|∇ x u\| 2 + µ 2 (τ )\|u\| 2 exp(−µ 2 (τ )t) dxdt ≤ µ(τ ) −1 ∂Ω(τ )×(0,r) \|∇ x u\| 2 + µ 2 (τ )\|u\| 2 exp(−µ 2 (τ )t) dsdt ∀ τ > k −1 . (3.11) Clearly there holds dJ µ,r (τ ) dτ = − ∂Ω(τ )×(0,r) \|∇ x u\| 2 + µ 2 (τ )\|u\| 2 exp(−µ 2 (τ )t) dsdt + Q r (τ ) 2µµ ′ (τ )\|u\| 2 exp(−µ 2 (τ )t) dxdt − 2 Q r (τ ) µµ ′ (τ )t \|∇ x u\| 2 + µ 2 (τ )\|u\| 2 exp(−µ 2 (τ )t) dxdt. Since µ ′ (τ ) > 0, it follows from (3.11), 2 −1 f µ,r (τ ) + J µ,r (τ ) ≤ µ(τ ) −1 − d dτ J µ,r (τ ) + 2 Q r (τ ) µ(τ )µ ′ (τ )\|u\| 2 exp(−µ 2 (τ )t) dxdt . (3.12) If we suppose 1 − 2µ ′ (τ ) µ 2 (τ ) ≥ 2 −1 ,(3.13) we derive from (3.12) f µ,r (τ ) + J µ,r (τ ) ≤ −2µ(τ ) −1 dJ µ,r (τ ) dτ . It is easy to check that this last inequality is equivalent to µ(τ ) 2 exp τ τ1 µ(s) 2 ds f µ,r (τ ) ≤ − d dτ J µ,r (τ ) exp τ τ1 µ(s) 2 ds ∀ τ > τ 1 > k −1 . By integrating this inequality and using monotonicity of the function f µ,r (τ ) we get f µ,r (τ 2 ) τ2 τ1 µ(τ ) 2 exp τ τ1 µ(s) 2 ds dτ +J µ,r (τ 2 ) exp τ2 τ1 µ(s) 2 ds ≤ J µ,r (τ 1 ) ∀ τ 2 > τ 1 > k −1 . Since µ(τ ) 2 exp τ2 τ1 µ(s) 2 ds = d dτ exp τ τ1 µ(s) 2 ds , it follows from last the relation f µ,r (τ 2 ) exp τ2 τ1 µ(s) 2 ds − 1 + J µ,r (τ 2 ) exp τ2 τ1 µ(s) 2 ds ≤ J µ,r (τ 1 ) ∀ τ 2 > τ 1 > k −1 . (3.14) Now we have to define µ(τ ). Let ε > 0 and µ(τ ) = εr −1 (τ − k −1 ) ∀ τ > k −1 . (3.15) One can easily verify that condition (3.13) is equivalent to τ ≥ k −1 + 2ε −1/2 r 1/2 . (3.16) Now from (3.14) follow two inequalities A(τ 2 ) := Q r (τ2) \|∇ x u\| 2 + ε 2 (τ 2 − k −1 ) 2 r 2 \|u\| 2 dxdt ≤ A(τ 1 ) × exp − ε (τ 2 − k −1 ) 2 − (τ 1 − k −1 ) 2 4r + ε 2 (τ 2 − k −1 ) r ∀ τ 2 > τ 1 > k −1 + 2ε −1/2 r 1/2 , (3.17) and f (r, τ 2 ) ≤ A(τ 1 ) exp ε (τ 2 − k −1 ) 2 − (τ 1 − k −1 ) 2 4r − 1 −1 exp ε 2 (τ 2 − k −1 ) 2 r ∀ τ 2 > τ 1 > k −1 + 2ε −1/2 r 1/2 . (3.18) In particular, for ε = 8 −1 we obtain from (3.17) and (3.18), Q r (τ ) \|∇ x u\| 2 + (τ − k −1 ) 2 64r 2 \|u\| 2 dxdt ≤ e exp − (τ − k −1 ) 2 64r Q r (τ (k) 0 ) \|∇ x u\| 2 + \|u\| 2 2r dxdt ∀ τ ≥ τ (k) 0 (r) := k −1 + 4 √ 2 √ r, (3.19) and f (r, τ ) ≤ e 2 e − 1 exp − (τ − k −1 ) 2 64r Q r (τ (k) 0 ) \|∇ x u\| 2 + u 2 2r dxdt ∀ τ ≥ τ (k) 0 (r) := k −1 + 8 √ r. (3.20) In order to have an estimate from above of the last factor in the right-hand side of (3.19), (3.20), we return to the equation satisfied by u, multiply it by the test function u k (x, t) exp (−t) and integrate over the domain Q r = R N × (0, r). As result of standard computations we obtain, using (3.5), R N \|u k (x, r)\| 2 dx + Q r \|∇ x u k \| 2 + \|u k \| 2 + h(t)\|u k \| q+1 dxdt ≤ c u 0,k 2 L2(R N ) ≤ cM k → ∞ as k → ∞, ∀ r ≤ T. (3.21) Due to (3.20), (3.21) it follows from (3.9) Then we choose τ k , r k such that the following relation is true, R N \|u(x, T )\| 2 dx + I 1 (r) + I 2 (r) + I 3 (r) ≤ c 1 τ N (q−1) q+1 h(r) − 2 q+1 (−I ′ 2 (r)) 2 q+1 + c 2 M k r −1 exp − (τ − k −1 ) 2 64r ∀ τ ≥ τ (k) 0 (rf (r, τ ) + E 1 (r, τ ) + (τ − k −1 ) 2 64r 2 E 2 (r, τ ) ≤ c 2 M k r −1 exp − (τ − k −1 ) 2 64r ∀ τ > τc 2 r −1 k exp − τ 2 k 64r k M k = M ε0 k , 0 < ε 0 < e −1 (3.25) where c 2 is from (3.22), (3.23). As consequence of (3.25) and (3.24) we get τ k = 8r 1/2 k (1 − ε 0 )e k + ln r −1 k + ln c 2 1/2 . (3.26) In inequality (3.22) we fix τ = τ k + k −1 , then due to definition (3.25) it follows from (3.22), R N \|u(x, T )\| 2 dx + I 1 (r) + I 2 (r) + I 3 (r) ≤ c 1 (k −1 + τ k ) N (q−1) q+1 h(r) − 2 q+1 (−I ′ 2 (r)) 2 q+1 + M ε0 k ∀ r : 0 < r ≤ r k . (3.27) I 1 (r), I 2 (r), I 3 (r) are nonincreasing functions which satisfy, due to global a' priori estimate (3.21), I 1 (0) + I 2 (0) + I 3 (0) ≤ cM k . (3.28) Let us define the number r k by r k = sup {r : I 1 (r) + I 2 (r) + I 3 (r) ≥ 2M ε0 k } . (3.29) Then it follows from (3.27) the following differential inequality I 1 (r) + I 2 (r) + I 3 (r) + R N \|u(x, T )\| 2 dx ≤ 2c 1 (τ k + k −1 ) N (q−1) q+1 h(r) − 2 q+1 (−I ′ 2 (r)) 2 q+1 ∀ r ≤ r k . (3.30) Solving it, we get I 1 (r) + I 2 (r) + I 3 (r) ≤ c 3 (τ k + k −1 ) N H(r) − 2 q−1 ∀ r ≤ r k ,(3.31) where H(r) = r 0 h(s) ds and c 3 = 2 q − 1 2/(q−1) (2c 1 ) (q+1)/(q−1) Next we will use more specific functions h(t) = exp − ω(t) t , where ω(t) is nondecreasing and satisfies the following technical assumption t α0 ≤ ω(t) ≤ ω 0 = const ∀ t : 0 < t < t 0 , 0 ≤ α 0 < 1. (3.32) It is easy to show by integration by parts the following relation r 0 exp − aω(t) t dt ≥ 1 − δ(r) (1 − α 0 )a · r 2 ω(r) exp − aω(r) r ∀ r > 0, where δ(r) → 0 if r → 0. Therefore H(r) ≥ c r 2 ω(r) h(r), c = const > 0. (3.33) As a consequence we derive from (3.31), using (3.26), I 1 (r) + I 2 (r) + I 3 (r) ≤ c 4 8r 1 2 k (1 − ε 0 )e k + ln r −1 k + ln c 2 1 2 + k −1 N × ω(r) 2 q−1 r 4 q−1 exp 2ω(r) (q − 1)r ∀ r ≤ r k . (3.34) Comparing (3.29) and estimate (3.34) we deduce that r k satisfies r k ≤ b k ,(3.35) where b k is solution of equation c 4 8b 1 2 k (1 − ε 0 )e k + ln b −1 k + ln c 2 1 2 + k −1 N ω(b k ) 2 q−1 b − 4 q−1 k exp 2ω(b k ) (q − 1)b k = 2M ε0 k = 2 exp(ε 0 e k ). This equation may be rewritten in the form ln c 4 + 2 q − 1 ln ω(b k ) b k + 2 q − 1 · ω(b k ) b k + N ln 8b N (q−1)−4 2(q−1)N k (1 − ε 0 ) exp k + ln b −1 k + ln c 2 1 2 + k −1 b − 2 (q−1)N k = ln 2 + ε 0 e k ∀ k ∈ N. (3.36) Since s −1 ln s → 0 as s → ∞, it follows from equality (3.36) that (1 + cγ(k))ε 0 e k ≥ A k + 2 q − 1 ω(b k ) b k := N ln 8b N (q−1)−4 2(q−1)N k (1 − ε 0 )e k + ln b −1 k + ln c 2 1 2 + k −1 b − 2 N (q−1) k + 2 q − 1 ω(b k ) b k ≥ (1 − γ(k))ε 0 e k ∀ k ∈ N, (3.37) where 0 < γ(k) < 1, γ(k) → 0 as k → ∞. Keeping in mind condition (3.32), we obtain easily ω(b k ) b k ≥ b −(1−α0) k , \|A k \| ≤ c (\| ln b k \| + k) ∀ k ∈ N. (3.38) Due to properties (3.38), it follows from (3.37) ce k > ω(b k ) b k ≥ d 1 e k ∀ k ∈ N, d 1 > 0. (3.39) As a consequence of (3.39), (3.38) we obtain also ln b −1 k ≤ ck ∀ k ∈ N.(3.τ k ≤ cb 1/2 k exp k 2 ≤ c exp k 2 ω(b k ) d 1 exp k 1/2 = c d 1/2 1 ω(b k ) 1/2 . Using again estimate (3.39) and the monotonicity of the function ω(s), we deduce from the above relation τ k ≤ c ω ω 0 d 1 e k 1/2 , ω 0 is from (3.32 ). (3.41) Therefore, from inequalities (3.23) and (3.34), definitions (3.25), (3.29) and property (3.35), we derive the following estimates I 1 (r k ) + I 2 (r k ) + I 3 (r k ) ≤ 2M ε0 k where r k is from (3.35 ), (3.29 ), (3.42) f (r k , τ k + k −1 ) + E 1 (r k , τ k + k −1 ) + τ 2 k 64r 2 k E 2 (r k , τ k + k −1 ) ≤ M ε0 k ,(3.43) where τ k is from (3.26 ), (3.41 ). Because ε 0 < e −1 , it follows from definition (3.24) of sequence M k that 3M ε0 k < cM k−1 ∀ k ≥ k 0 (c),(3.44) where c > 0 is arbitrary constant. Therefore, adding estimates (3.42) and (3.43), we obtain thanks to (3.44) and the fact that τ k ≫ r k (which follows from (3.25)), the inequality f (r k , τ k + k −1 ) + 3 i=1 I i (r k ) + 2 i=1 E i (r k , τ k + k −1 ) < cM k−1 ∀ k ≥ k 0 (c). (3.45) Step 3. The second round of computations. Next we introduce the terms r k−1 , τ k−1 . Firstly we come back to inequality (3.14). Fixing here the function µ(t) = εr −1 (τ − k −1 − τ k ) ∀ τ > k −1 + τ k (3.46) instead of (3.15) and using estimates (3.16)-(3.20), we obtain Q r (τ ) \|∇ x u\| 2 + (τ − k −1 − τ k ) 2 \|u\| 2 64r 2 dxdt ≤ e exp − (τ − k −1 − τ k ) 2 64r Q r (τ (k−1) 0 (r)) \|∇ x u\| 2 + \|u\| 2 2r dxdt ∀ τ > τ (k−1) 0 (r) := k −1 + τ k + 4 √ 2 √ r,(3.47) and f (r, τ ) ≤ e 2 e − 1 exp − (τ − k −1 − τ k ) 2 64r Q r (τ (k−1) 0 (r)) \|∇ x u\| 2 + \|u\| 2 2r dxdt ∀ τ ≥ τ (k−1) 0 := k −1 + τ k + 8 √ r. (3.48) The integral term in the right-hand side of (3.47), (3.48) is estimated now by using estimate (3.45) obtained in the first round of computation. So, we have Q r (τ (k−1) 0 (r)) \|∇ x u\| 2 + u 2 2r dxdt ≤ (2r) −1 3 i=1 I i (r k ) + 2 i=1 E i (r k , τ k + k −1 ) ≤ c(2r) −1 M k−1 ∀ k > k 0 (c), ∀ r ≥ r k . (3.49) Using this estimate we deduce from (3.47) and (3.48) f (r, τ ) + E 1 (r, τ ) + (τ − τ k − k −1 ) 2 64r 2 E 2 (r, τ ) ≤ c 2 r −1 M k−1 exp − (τ − τ k − k −1 ) 2 64r ∀ τ ≥ τ (k−1) 0 (r). (3.50) This estimate is similar to estimate (3.23) from first round. Now we have to deduce the analogue of estimate (3.31). For this we return to the starting relation (3.9), where we now estimate last term in right-hand side by estimate (3.48), using additionally (3.49). As a result we have 3 i=1 I i (r) ≤ c 1 τ N (q−1) q+1 h(r) − 2 q+1 (−I ′ 2 (r)) 2 q+1 + c 2 M k−1 r −1 exp − (τ − τ k − k −1 ) 2 64r ∀ r ≥ r k , ∀ τ ≥ τ (k−1) 0 (r), (3.51) which is analogous of estimate (3.22) from first round. Next we define the numbers τ k−1 and r k−1 by inequalities analogous to (3.26) and (3.29), c 2 r −1 k−1 M k−1 exp − τ 2 k−1 64r k−1 = M ε0 k−1 , 0 < ε 0 < e −1 (3.52) r k−1 = sup{r : I 1 (r) + I 2 (r) + I 3 (r) ≥ 2M ε0 k−1 }. (3.53) Now combining inequalities (3.30) and (3.44), and using definitions (3.52), (3.53), we obtain the following differential inequality 3 i=1 I i (r) ≤ 2c 1 (τ k−1 + τ k + k −1 ) N (q−1) q+1 h(r) − 2 q+1 (−I ′ 2 (r)) 2 q+1 ∀ r ≤ r k−1 . (3.54) Solving this differential inequality, we obtain an estimate similar to (3.31). Using property (3.33) we arrive to 3 i=1 I i (r) ≤ c 4 (τ k−1 + τ k + k −1 ) N ω(r) 2 q−1 r 4 q−1 exp 2ω(r) (q − 1)r ∀ r ≤ r k−1 . (3.55) As in first round we express from (3.52) τ k−1 as function τ k−1 (r k−1 ) (the analogue of (3.26)) τ k−1 = 8r 1/2 k−1 [(1 − ε 0 ) exp(k − 1) + ln r −1 k−1 + ln c 2 ] 1/2 . (3.56) Inserting this expression of τ k−1 into (3.55) and then comparing the obtained inequality with definition (3.53), we deduce an estimate similar to (3.35), r k−1 ≤ b k−1 , (3.57) where b k−1 is solution of equation c 4 8b 1/2 k−1 (1 − ε 0 ) exp(k − 1) + ln b −1 k + ln c 2 1/2 + τ k + k −1 N × ω(b k−1 ) 2 q−1 b 4 q−1 k−1 exp 2ω(b k−1 ) (q − 1)b k−1 = 2M ε0 k−1 = 2 exp(ε 0 exp(k − 1)). (3.58) From (3.50), and due to definition (3.52), it follows f (r k−1 , τ k−1 + τ k + k −1 ) + τ 2 k−1 64r k−1 E 2 (r k−1 , τ k−1 + τ k + k −1 ) + E 1 (r k−1 , τ k−1 + τ k + k −1 ) ≤ M ε0 k−1 .I 1 (r k−1 ) + I 2 (r k−1 ) + I 3 (r k−1 ) ≤ 2M ε0 k−1 . (3.60) Summing (3.59), (3.60) and using property (3.44), we deduce new global a priori estimate (the analogous of (3.45)) which is the main starting information for the next round of computation f (r k−1 , τ k−1 + τ k + k −1 ) + 3 i=1 I i (r k−1 ) + 2 i=1 E i (r k−1 , τ k−1 + τ k + k −1 ) ≤ cM k−2 . (3.61) We are ready now for the next round of computations, introducing the function µ(t) = εr −1 (τ − k −1 − τ k − τ k−1 ) ∀ τ > k −1 + τ k + τ k−1 instead of (3.46) and estimate (3.61) instead of (3.45). We realize j rounds of such computations. As result we obtain f r k−j , j l=0 τ k−l + k −1 + 3 i=1 I i (r k−j ) + 2 i=1 E i r k−j , j l=0 τ k−l + k −1 ≤ cM k−j−1 , (3.62) which was our main aim. Step 4. The control of r k−j , j l=0 τ k−l as j → k with arbitrary k ∈ N. It is clear that r k−j , τ k−j are defined by the conditions (see (3.52), (3.53)) c 2 r −1 k−j M k−j exp − τ 2 k−j 64r k−j = M ε0 k−j , 0 < ε 0 < e −1 . (3.63) r k−j = sup r : I 1 (r) + I 2 (r) + I 3 (r) ≥ 2M ε0 k−j . (3.64) Similarly to (3.56)-(3.58) we deduce that τ k−j = 8r 1/2 k−j (1 − ε 0 )e k−j + ln r −1 k−j + ln c 2 1/2 , (3.65) r k−j ≤ b k−j , (3.66) where b k−j satisfies c 4 8b 1/2 k−j (1 − ε 0 )e k−j + ln b −1 k−j + ln c 2 1/2 + j−1 i=0 τ k−i + k −1 N × ω(b k−j ) 2 q−1 b 4 q−1 k−j exp 2ω(b k−j ) (q − 1)b k−j = 2M ε0 k−j = 2 exp(ε 0 e k−j ). (3.67) In the first round of computations we have obtained the upper estimate (3.41) for τ k . Let us suppose by induction that the following estimate is true τ k−i ≤ c ω ω 0 d 1 exp(k − i) 1/2 ∀ i ≤ j − 1. (3.68) We have to prove that estimate (3.68) holds also for i = j. Obviously condition (3.67) is equivalent to (see (3.36)) ln c 4 + 2 q − 1 ln ω(b k−j ) b k−j + 2 q − 1 · ω(b k−j ) b k−j + A (j) k = ln 2 + ε 0 e k−j , (3.69) where A (j) k = N ln      b N (q−1)−4 2(q−1)N k−j (1 − ε 0 )e k−j + ln(b −1 k−j ) + ln c 2 1/2 + k −1 + j−1 i=0 τ k−i b 2 (q−1)N k−j      . Because of the induction assumption (3.68) j−1 i=0 τ k−i ≤ c j−1 i=0 ω ω 0 d 1 exp(k − i) 1/2 ≤ c 1 0 ω(s) 1/2 s ds := cL, therefore \|A (j) k \| ≤ c (\| ln b k−j \| + (k − j) + ln L) .(ce k−j ≥ ω(b k−j ) b k−j ≥ d 1 e k−j ∀ j : k − j ≥ k 0 = k 0 (L), (3.71) where k 0 < ∞ do not depend on k. From (3.71) it follows in particular ln b −1 k−j ≤ c(k − j) ∀ j : k − j ≥ k 0 . (3.72) Thanks to (3.66) and properties (3.71), (3.72), we derive from (3.65), τ k−j ≤ 8b 1/2 k−j (1 − ε 0 )e k−j + ln b −1 k−j + ln c 2 1/2 ≤ cb 1/2 k−j exp k − j 2 ≤ c d 1/2 1 [ω(b k−j )] 1/2 ∀ j : k − j ≥ k 0 (L). (3.73) Using again estimate (3.71) and monotonicity of ω(s) we deduce from (3.73) Step 5. Completion of the proof. We fix now n > k 0 (L) and take j = k − n in (3.62). This leads to f r n , τ k−j ≤ c ω ω 0 d 1 e k−j 1/2 ∀ j : k − j ≥ k 0 (L).k−n l=0 τ k−l + k −1 + 3 i=1 I i (r n )+ 2 i=1 E i r n , k−n l=0 τ k−l + k −1 ≤ cM n−1 ∀ n > k 0 (L). (3.75) Next we have k−n l=0 τ k−l ≤ ∞ i=n τ i ≤ c ∞ i=n ω ω 0 d 1 exp i 1/2 ≤ c ω 0 d 1 exp(n−1) 0 ω(s) 1/2 s ds → 0 as n → ∞. (3.76) Therefore, for arbitrary small δ > 0, we can find and fix n = n(δ) < ∞ such that from (3.75) follows uniform with respect to k ∈ N a priori estimate, sup t>0 \|x\|>δ \|u k (x, t)\| 2 dx + T 0 \|x\|>δ \|∇ x u k \| 2 + \|u k \| 2 dxdt ≤ C = C(δ) < ∞ ∀ k ∈ N. (3.77) Since u k (x, 0) = 0 ∀ \|x\| > k −1 ∀ k ∈ N, it follows from (3.77) that u ∞ (x, 0) = 0 ∀ x = 0, which ends the proof. Proof. We will consider the family u k (x, t) of solutions of regularized problems: u t − ∆u + h(t)(e u − 1) = 0 in Q T , u(x, 0) = u 0,k (x) = M 1/2 k k −N/2 δ k (x) ∀x ∈ R N , (4.2) where δ k is nonnegative, continuous with compact support in B k −1 , satisfies estimate (3.5) and converges weakly to δ 0 as k → ∞, {M k } satisfies condition (3.2). Let us introduce the energy functions (we omit index k in u k ): I 1,0 (r) = Qr \|∇ x u\| 2 dxdt, I q (r) = (q!) −1 Qr h(t)\|u\| q+1 dxdt, I 3,0 (r) = Qr \|u\| 2 dxdt. (4.3) Multiplying (4.2) by u(x, t) exp − t − r 1 + T − r , integrating in Q r and using equality s(e s − 1) = ∞ q=1 s q+1 q! , we obtain easily I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ c(q!) 2/(q+1 τ N (q−1)/(q+1) h(r) −2/(q+1) (−I ′ q (r)) 2/(q+1) + c Ω(τ ) \|u(x, r)\| 2 dx ∀ τ > 0, ∀ r : 0 < r < T, ∀ q ∈ N. (4.4) We introduce the additional energy functions f (r, τ ) from (3.10 ), E 1,0 (r, τ ) = Q r (τ ) \|D x u\| 2 dxdt, E 2,0 (r, τ ) = Q r (τ ) \|u\| 2 dxdt. (4.5) Instead of (3.21) we derive the following global a priori estimate: R N \|u k (x, r)\| 2 dx + Q r \|∇ x u\| 2 + \|u k \| 2 + h(t) ∞ l=1 \|u k \| l+1 l! dxdt ≤ c u 0,k 2 L2(R N ) ≤ cM k ∀ r < T. (4.6) Using estimate (4.6) instead of (3.21) in a similar way as in the proof of Theorem 3.1, we obtain the following inequality, analogous to (3.23), f (r, τ ) + E 1,0 (r, τ ) + (τ − k −1 ) 2 64r 2 E 2,0 (r, τ )+ ≤ c 2 M k r −1 exp − (τ − k −1 ) 2 64r ∀ τ ≥ τ (k) 0 (r) = k −1 + 8 √ r. (4.7) Using this estimate we deduce from (4.4) I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ c(q!) 2 q+1 τ N (q−1) q+1 h(r) − 2 q+1 (−I ′ q (r)) 2 q+1 + c 2 M k r −1 exp − (τ − k −1 ) 2 64r ∀ τ ≥ τ (k) 0 (r), ∀ q ∈ N. (4.8) Next, we define the numbers τ k , r k . Firstly, set r k := sup r : I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 ≥ 2M ε0 k , 0 < ε 0 < e −1 . (4.9) Then we fix the sequence {M k } by (3.24) again and τ k by inequalities (3.25), (3.26). Thanks to these definitions we derive the following series of inequalities from relations (4.8) I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ 2c 1 (q!) 2 q+1 (τ k + k −1 ) N (q−1) q+1 h(r) − 2 q+1 (−I ′ q (r)) 2 q+1 ∀ q ∈ N, ∀ r ≤ r k . (4.10) Solving these differential inequalities we obtain the estimates I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ c 3 (τ k + k −1 ) N (q!) 2 q−1 H(r) − 2 q−1 ∀ r ≤ r k , ∀ q ∈ N,(4.11) where H(r) is from (3.31). We have now to optimize estimate (4.11) with respect to parameter q. By integration by parts, it is easy to check the following inequality H(r) ≥ c r 2 ω(r) exp − ω(r) r h(r) ∀ r > 0, c > 0. (4.12) Using Stirling formula q! ∼ q e q and estimate (4.12), we deduce from (4.11) I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ c 4 (τ + k −1 ) N F q (r) ∀ r ≤ r k ,(4.13) where F q (r) = q 2 ω(r) 2 q−1 r − 4 q−1 exp 2 q − 1 · ω(r) r exp 2 q − 1 exp ω(r) r . Fixing here the optimal value of the parameter q: q = q := 2 exp ω(r) r , where [a] denotes the enteger part of a, we obtain easily F e q ≤ c exp 2ω(r) r . Therefore it follows from (4.13), I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ c 5 (τ k + k −1 ) N exp 2ω(r) r ∀ r ≤ r k . (4.14) Comparing now definition (4.9) of r k and estimate (4.14), and using additionally the expression (3.26) of τ k , we obtain r k ≤ b k ,(4.15) where b k is defined by the equation c 5 8b 1/2 k ((1 − ε 0 )e k + ln b −1 k + ln c 2 ) 1/2 + k −1 N exp 2ω(b k ) b k = 2M ε0 k = 2 exp(ε 0 exp k), 0 < ε 0 < e −1 . (4.16) By an analysis similar to Step 2 in the proof of Theorem 3.1, we obtain estimates (3.37)-(3.40) for b k . Then we prove the validity of estimate (3.41) for τ k . As a consequence of estimates (4.7), (4.14), thanks to to definitions (3.26), (4.9) of τ k , r k and the previous estimates of τ k , r k , we get I 1,0 (r) + ∞ l=1 I l (r) + I 3,0 (r) ≤ 2M ε0 k , f (r k , τ k + k −1 ) + E 1,0 (r k , τ k + k −1 ) + τ 2 k 64r 2 k E 2,0 (r k , τ k + k −1 ) ≤ M ε0 k . Summing these inequalities, and using definition of {M k } and property τ k ≫ r k , we obtain an analogue of estimate (3.45), namely, f (r k , τ k + k −1 ) + I 1,0 (r k ) + ∞ l=1 I l (r k ) + I 3,0 (r k ) + E 1,0 (r k , τ k + k −1 ) + E 2,0 (r k , τ k + k −1 ) ≤ cM k−1 . (4.17) Using (4.17) as global a priori estimate instead of (4.6) and providing a second round of computations similar to (3.46)-(3.57) we derive a second global a priori estimate analogous to (3.61), f (r k−1 , τ k−1 + τ k + k −1 ) + I 1,0 (r k−1 ) + ∞ l=1 I l (r k−1 ) + I 3,0 (r k−1 ) + E 1,0 (r k−1 , τ k−1 + τ k + k −1 ) + E 2,0 (r k−1 , τ k−1 + τ k + k −1 ) ≤ cM k−2 . Repeating such rounds j-times we derive a corresponding analogue of relation (3.62). It is easy to see that estimate (3.76) for constructed shifts τ k−i remains valid. This fact, similar to what was used in the proof of Theorem 3.1, yields to the conclusion. The porous media equation with absorption In this section we consider the following problem dealing with fundamental solutions of the porous media equation with time dependent absorption, ∂ t u − ∆(\|u\| m−1 u) + h(t)\|u\| q−1 u = 0 in Q T u(x, 0) = kδ 0 . (5.1) It is standard to assume that h ≥ 0 is a continuous function and m, q are positive real numbers. By a solution we mean a function u ∈ L 1 loc (Q T ) such that u m ∈ L 1 loc (Q T ), hu q ∈ L 1 loc (Q T ) and Q T −u∂ t φ − \|u\| m−1 u∆φ + h(t)\|u\| q−1 uφ dxdt = kφ(0, 0) (5.2) for any φ ∈ C 2,1 0 (R N × [0, T )). If h ≡ 0 and m > (N − 2) + /N this problem admits a solution for any k > 0. When m > 1 this solution has the following form B k (x, t) = t −ℓ C k − (m − 1)ℓ 2mN \|x\| 2 t 2ℓ/N 1/(m−1) + , (5.3) where ℓ = N N (m − 1) + 2 and C k = a(m, N )k 2(m−1)ℓ/N . (5.4) Since B k is a supersolution for problem (5.1 ), a sufficient condition for existence (and uniqueness) of u k is Q T B q k (x, t)h(t)dxdt < ∞. (5.5) By the change of variable y = t ℓ/N x this condition is independent of k > 0 and we have Proof. We first notice that q + 1 > 2m > 2 =⇒ q > m > 1 and q − m m − 1 > N (q − m) − 2 N (m − 1) + 2 . Step 1. Case q < m + 2/N . In this range of value we know [14] that there exists a nonnegative very singular solution v = v ∞ to ∂ t v − ∆v m + v q = 0 in Q T ,(5.9) and v ∞ = lim k→ v k , where the v k are solutions of the same equation with initial data kδ 0 . Furthermore, v ∞ is unique [6], radial with respect to x and has the following form v ∞ (x, t) = t −1/(q−1) F (\|x\| /t (q−m)/2(q−1) ), where F solves    (F m ) ′′ + N − 1 η (F m ) ′ + q − m 2(q − 1) ηF ′ + 1 q − 1 F − F q = 0 in (0, ) F ′ (0) = 0 and lim η→∞ η 2/(q−m) F (η) = 0. (5.10) Actually F has compact support in [0, ξ 0 ] for some ξ 0 > 0. Let γ = (q − m)/(m − 1), then for any ǫ > 0, u = u ∞ satisfies, for some c > 0, ∂ t u − ∆u m + cǫ γ u q ≥ 0 in Q ǫ . If we set w ǫ (x, t) = a θ v ∞ (x, at) with θ = 1/(m − 1)− and a = ǫ −1 c −(q−1)/(q−m) , then ∂ t w ǫ − ∆w m ǫ + cǫ γ w q ǫ = 0 in Q T . By comparison u ∞ ≥ w ǫ in Q ǫ . If we take in particular t = ǫ, it implies u ∞ (x, t) ≥ c −1/(q−m) t −1/(m−1) v ∞ (x, c −(m−1)/(q−m) ) = c −1 t −1/(m−1) F (c (m−1)/2(q−1) \|x\|) (5.11) If \|x\| < ξ c = c −(m−1)/2(q−1) ξ 0 , we derive that lim t→0 u ∞ (x, t) = ∞, locally uniformly in B ξc . This implies u ∞ = U h . Step 2. Case q ≥ m + 2/N . We give an alternative proof valid for all q. We first observe that it is sufficient to prove the result when h(t) is replaced by t γ . If we look for a family of transformations u → T ℓ (u) under the form T ℓ (u)(x, t) = ℓ α u(ℓ β x, ℓt) ∀(x, t) ∈ Q ∞ , ∀ℓ > 0 which leaves the equation ∂ t u − ∆\|u\| m−1 u + t γ \|u\| q−1 u = 0 (5.12) invariant, we find α = (1 + γ)/(q − 1) and β = (q − m − γ(m − 1))/2(q − 1). Due to the value of γ, we have β = 0. Because of uniqueness and the value of the initial mass T ℓ (u k ) = u ℓ α k ∀ℓ > 0, ∀k > 0 =⇒ T ℓ (u ∞ ) = u ∞ ∀ℓ > 0. (5.13) Therefore ℓ α u ∞ (x, ℓt) = u ∞ (x, t) ∀(x, t) ∈ Q ∞ , ∀ℓ > 0. In particular, if we take ℓ = t −1 , u ∞ (x, t) = t −α u ∞ (x, 1) = t −α φ(x). Plugging this decomposition into (5.12 ) yields to −αt −α−1 φ − t −αm ∆φ m + t γ−αq φ q = 0, where all the exponents of t coincide since αm = m m − 1 , αq − γ = m m − 1 and α + 1 = m m − 1 . Therefore φ is a positive and radial (as the u k are) solution of −αφ − ∆φ m + φ q = 0 in R N . Setting ψ = φ m yields to −∆ψ − 1 m − 1 ψ 1/m + ψ q/m = 0 in R N . (5.14) Clearly ψ = ψ 0 = (m − 1) −m/(q−1) is a solution. By a standard variation of the Keller-Osserman estimate, any solution is bounded from above by ψ 0 . Puttingψ(x) = Aψ(a), it is easy to find A > 0 and a > 0 such that Ifψ q/m is not constant with value 1, the right-hand side of the above inequality is decreasing with respect to r, and the only possible nonnegative limit is 0, by La Salle principle. Thus −∆ψ −ψ 1/m +ψ q/m = 0 in R N ,(5.ψ ′′ + N − 1 rψ ′ + 1 2ψ 1/m ≤ 0 for r ≥ r 0 , large enough. If N = 2, we set τ = ln r, Ψ(τ ) =ψ(r) and get Ψ ′′ + 1 2 e 2τ Ψ 1/m ≤ 0 for τ ≥ ln r 0 . The concavity of Ψ yields a contradiction. If N ≥ 3, we set τ = r N −2 /(N − 2) and Ψ(τ ) = r N −2ψ (r). Then Ψ satisfies Ψ ′′ + c N τ (4−N )/(N −2)−1/m Ψ 1/m ≤ 0. Again the concavity yields a contradiction. In any case we obtain that Ψ = 1, or, equivalently ψ = ψ 0 and finally, u ∞ = t −1/(m−1) ψ 1/m 0 . Theorem 5.3 Assume q > m > 1 and h ∈ C((0, ∞)) is nondecreasing, positive. If h(t) = t (q−m)/(m−1) ω −1 (t) with ω(t) → 0 as t → 0, and 1 0 ω θ (s) ds s < ∞, (5.16) where θ = m 2 − 1 [N (m − 1) + 2(m + 1)](q − 1) , then u ∞ := lim k→∞ u k has a point-wise singularity at (0, 0) Proof. The structure of the proof is similar to the one of Theorem 3.1. We study the asymptotic behaviour as k → ∞ of solutions u = u k (x, t) of the regularized Cauchy problem    u t − ∆(\|u\| m−1 u) + h(t)\|u\| q−1 u = 0 in Q T u(x, 0) = u 0,k (x) = M 1 m+1 k k − mN m+1 δ k (x) x ∈ R N ,(5.17) where δ k is as in Theorem 3.1. Let us rewrite problem (5.17) in the form        (\|v\| p−1 v) t − ∆v + h(t)\|v\| g−1 v = 0, in Q T v = v k = \|u\| m−1 u, p = 1/m, g = q/m \|v(x, 0)\| p−1 v(x, 0) = \|v 0,k \| p−1 v 0,k := u 0,k (x) = M p p+1 k k − N p+1 δ k (x). (5.18) Without loss of generality we may suppose δ k (x) p+1 p L p+1 p (R N ) = R N \|δ k (x)\| p+1 p dx ≤ c 0 k N p ∀ k ∈ N. (5.19) Now sequence {M k } is such that M p p+1 k k − N p+1 → ∞ as k → ∞. (5.20) Step 1. The local energy framework. Consider the following energy functions I 1 (τ ) = Qr \|∇ x v\| 2 dxdt, I 2 (τ ) = Qr h(t)\|v\| g+1 dxdt, I 3 (τ ) = Qr \|v\| p+1 dxdt. (5.21) Analogously to (3.9) we deduce the inequality R N \|v(x, T )\| p+1 dx+I 1 (r)+I 2 (r)+I 3 (r) ≤ cτ N (g−p) g+1 h(r) − p+1 g+1 (−I ′ 2 (r)) p+1 g+1 +c Ω(τ ) \|v(x, r)\| p+1 dx ∀ τ > 0, ∀ r : 0 < r < T. (5.22) This inequality will control the spreading of energy with respect to the r-variable (the time direction). As to vanishing property of energy in variable τ , we will use the finite speed propagation of support property for porous media equation with slow diffusion. In the domain Q (r) (τ ) we will use the energy function E 1 (r, τ ) = Q (r) (τ ) \|∇ x v\| 2 dxdt from (3.12). Since supp v(·, 0) = supp v k (·, 0) = supp v 0,k = {x : \|x\| < k −1 }, multiplying equation (5.18) on v(x, t) and integrating in the domain Q (r) (τ ), τ ≥ k −1 , we obtain after simple computations (see, for example [1,4]) the following differential inequality Ω(τ ) \|v(x, r)\| p+1 dx + E 1 (r, τ ) ≤ cr (p+1)(1−θ 1 ) p+1−(1−θ 1 )(1−p) − d dτ E 1 (r, τ ) p+1 p+1−(1−θ 1 )(1−p) , (5.23) ∀ τ ≥ k −1 , ∀ r > 0 where θ 1 = N (1 − p) + (p + 1) N (1 − p) + 2(p + 1) , 1 − θ 1 = p + 1 N (1 − p) + 2(p + 1) . Solving this inequality and keeping in mind that E 1 (r, τ ) ≥ 0 ∀ r > 0, ∀ τ > 0, we deduce easily v(x, r) ≡ 0 ∀ x : \|x\| > k −1 + c 0 r 1−θ1 E 1 (r, k −1 ) (1−θ 1 )(1−p) 1+p := k −1 + c 0 χ(r), ∀ r > 0. (5.24) Here the constant c 0 > 0 depends on the parameters of the problem under consideration, but do not on r and k. Analogously to (3.25) we deduce the following global a priori estimate Q (r) (\|∇ x v\| 2 + r −1 \|v\| p+1 + h(t)\|v\| g+1 ) dxdt ≤ c v 0,k p+1 Lp+1(R N ) .(5.I 1 (r) + I 2 (r) + I 3 (r) ≤ c(k −1 + χ(r)) N (g−p) g+1 h(r) − p+1 g+1 (−I ′ 2 (r)) p+1 g+1 ∀ r > 0. (5.27) Remark that due to (5.26) we have χ(r) ≤ c 1 r 1−θ1 M (1−θ 1 )(1−p) 1+p k . (5.28) Step 2. The first round of computations. Now we have to define τ k , r k . First we impose the relation Moreover, we will find the pair τ k , r k such that the following property holds k −1 + τ k ≤ 1. (5.34) Then the next inequality is a sufficient condition for validity of (5.33): and r k → 0 as k → ∞. Therefore, since ω(s) → 0 as s → 0, it follows from (5.39) that τ k → 0 as k → ∞. Consequently we can suppose k so large that condition (5.34) is satisfied. Thus, we have pair (τ k , r k ) for large k ∈ N. c 2 ω(r k ) Step 3. The second round of computations. As a starting global a priori estimate of solution we will use now, instead of (5.25), (5.26), the following estimate I 1 (r k ) = {t≥r k , x∈R N } \|∇ x v\| 2 dxdt ≤ I(r k ) ≤ cM k−1 ,(5.41) which follows from (5.32), due to definition (5.33), (5.36) of r k . Using property (5.24), estimate (5.28) and property (5.29), it ensues from (5.41) E 1 (r, k −1 + τ k ) ≤ I 1 (r) ≤ I 1 (r k ) < cM k−1 ∀ r ≥ r k . (5.42) Since v(x, r k ) = 0 ∀ x : \|x\| ≥ k −1 + τ k we deduce similarly to (5.23) Ω(τ ) \|v(x, r k + r)\| p+1 dx + E 1 (r k + r, k −1 + τ k + τ ) ≤ cr (p+1)(1−θ 1 ) (p+1)−(1−θ 1 )(1−p) × − d dτ E 1 (r k + r, k −1 + τ k + τ ) p+1 p+1−(1−θ 1 )(1−p) ∀ r > 0, ∀ τ > 0. (5.43) Solving this differential inequality, we obtain v(x, r k + r) ≡ 0 ∀ x : \|x\| ≥ k −1 + τ k + c 0 χ 1 (r), (5.44) where χ 1 (r) := r 1−θ1 E 1 (r k + r, k −1 + τ k ) . This solution has a persisting singularity and is called a razor blade [18]. It has also the property that lim t→0 W (x, t) = 0 ∀x = 0. This phenomenon is at the origin of the work of Chasseigne and Vàzquez on extended solutions of the fast diffusion equation [3]. Concerning problem (5.1 ), Proposition 5.1 is still valid provided m > (1 + 2/N ) + . We shall denote by u = u k the solutions of (5.1 ). Furthermore estimate (5.8 ) holds. Combining this with the fact that the B k are super solutions for the u k , we derive the following Theorem 6.1 Assume (1 − 2/N ) + < m < 1 and h ∈ C(0, ∞) is positive. Assume also that (5.6) holds. Then u ∞ := lim k→∞ u k has a point-wise singularity at (0, 0) and the following estimate is verified u ∞ (x, t) ≤ min    C * t −ℓ \|x\| 2 t 2ℓ/N −1/(1−m) , (q − 1) t 0 h(s) ds −1/(q−1)    (6.4) Remark. The profile of u ∞ near (x, t) = (0, 0) is completely unknown. In particular a very chalenging question could be to give precise estimates on the quantity min {W (x, t), U h (t)} − u ∞ (x, t). Theorem 1. 2 2If h(t) = exp(−ω(t)/t),where ω is continuous, nondecreasing and satisfies − ∆Ψ − Ψ 1/m + Ψ q/m = 0 in R N admits only one positive solution, the constant 1. The localization counter part is as follows,Theorem 1.5 Assume q > m > 1, in Equation (1.1 ). If h(t) = t (q−m)/(m−1) ω −1 (t) with ω(t) → 0 as t → 0,then u ∞ has single-point initial blow-up at 0, 0). y ′ + h(t)e y = 0 in (0, ∞), with infinite initial value. Our main result concerning nonexistence of localized singularities for equation (2.16) is Theorem 2.4 Let h(t) = e −e σ/t for some σ > 0 and any t > 0. Then u k ↑ V S as k → ∞. Proof. Step 1. Construction of an approximate very singular solution. For n > 1 and c n > 0 to be defined later on, let v = V n be the very singular solution of Step 2 . 2The first round of computations. Next we construct some sequences {τ j }, {r j }, j = k, k − 1, . . . , 1. First we explicit the choice of M k from condition (3.3), let namely M k = e e k . (3.24) 3.55), due to (3.56), (3.57), (3.58), it follows we have proved by induction estimate (3.68), for arbitrary k−j ≥ k 0 (L) with r i , τ i satisfying (3.66), (3.67) and (3.74). 4 Regional initial blow-up for equation with exponential absorption.The local energy method we have used in the proof of Theorem 3.1 is based on the sharp interpolation theorems for functional Sobolev spaces, which are natural tool for the study of solutions of equations with power nonlinearities. Here we propose the adaptation of mentroned method to the equations with nonpower nonlinearities.Thus, we consider the Cauchy problem∂ t u − ∆u + h(t)(e u − 1) = 0 in Q ∞ u(x, 0) = kδ 0 ,(4.1) Theorem 4.1 Assume h(t) = e −e ω(t)/t where ω ∈ C([0, ∞)) satisfies the same asumptions as in Theorem 3.1. Then solution u k always exists and u ∞ := lim k→∞ u k has a point-wise singularity at (0, 0). . 2 2t)t ℓ−ℓq dt < ∞, (5.6) then problem (5.1 ) admits a unique positive solution u = u k . In the particular case where h(t) = O(t α ) (α ≥ 0), the condition is α > N (q − m) − 2 N (m − 1) + 2 . (5.7)We recall that if q > 1 and m > (N − 2) + /N , any solution of the porous media equation with absorption is bounded from above by the maximal solution U h expressed by U h (t) = (q − 1) Assume q + 1 > 2m > 2 and h ∈ C((0, ∞)) is nondecreasing, positive and satisfies h(t) = O(t (q−m)/(m−1) ) as t → 0. Then for any k > 0 u k exists and lim k→∞ u k := u ∞ = U h . q/m −ψ 1/m )σ n−1 ds ∀r > 0. r) := I 1 (r) + I 2 (r) + I 3 (r) ≤ c(k −1 + τ k ) the function h(s) has the form h(s) = s (g−1)/(1−p) ω(s) −1, therefore estimate (as second relation, which defines our pair τ k , r k , we suppose the condition c 2 ω(r k ) have to choose the sequence {M k }. Namely, we setM k := e k ∀ k ∈ N,(5.37)and we define τ k , in accordance with assumption (5.29), by τ k = c 1 r (5.30)-(5.32) we deduceI(r) ≤ c 2 ω(r) p+1 g−p (k −1 + τ k + τ k−1 ) N r p+1 1−p ∀ r : 0 < r ≤ r k + r k−1 . 18 ) 18Notice that estimate (2.18 ) is a consequence of the fact that V S satisfies the associated O.D.E. 40 ) 40Now using estimate (3.39) we are able to obtain suitable upper estimate of τ k .Thanks to (3.35), (3.39) and (3.40) we deduce from (3.26) Next we come back to the inequality (5.22). Due to (5.24) it ensues from (5.22) the inequality25) Thus, due to (5.18)-(5.20), it follows from (5.25) E 1 (r, 0) ≤ cM k ∀ r > 0. (5.26) The second relation for defining the pair τ k−1 , r k−1 is analogous to(5.33)c 2 ω(r k + r k−1 ) p+1 g−p (k −1 + τ k + τ k−1 ) N (r k + r k−1 ) p+1 1−p ≤ cM k−2 , c is from (5.26 ).(5.48)Supposing that k −1 + τ k + τ k−1 ≤ 1,(5.49)we can define r k−1 by the following analogue of (5.36)And in accordance with (5.46) let us define τ k−1 byDue to (5.50) we havewhere S is from (5.39 ). Notice that, due to (5.47), (5.48), we have alsoand, analogously to (5.42),Step 4. Completion of the proof. Estimates (5.52), (5.53) we can use instead of (5.41), (5.42) for third round of computations. After j such rounds we deduce that Estimates (5.54) will remain true as long as the following analogue of relation (5.49) is validNow we will check this condition. Due to (3.32), it follows from (5.57)Therefore, from (5.56), it followsThus we have, using in particular the monotonicity of function ω(s),Due to condition (5.16) and estimate (5.58) we can find k 0 ∈ N, which depends on parameters of problem under consideration, but does not depend on k ∈ N, such thatAt end, our estimates (5.54)-(5.57) are true for all j ≤ k − k 0 . Therefore the proof of Theorem 5.3 follows from estimates (5.54)-(5.57), in the same way as Theorem 3.1 from estimates (3.75)-(3.77).The fast diffusion equation with absorptionWhen (1 − 2/N ) + < m < 1, it is known that the mere fast diffusion equationadmits a particular fundamental positive solution with initial data kδ 0 (k > 0) called the Barenblatt -Zeld'dovich-Kompaneets solution, expressed bywhere ℓ and C k are given in(5.4 ). The main feature of this expression is that lim k→∞ C k = 0, therefore On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. S N Antontsev, Dokl. Akad. 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E., Dead cores and instanteous compactification of the support of energy solutions of quasilinear parabolic equations of arbitrary order, Sbornik: Mathematics 190:12 (1999), 1843-1869. Different kinds of singular solutions of nonlinear parabolic equations. J L Vàzquez, L Véron, Nonlinear Problems in Applied Mathematics. SIAM, Philadelphia, PAVàzquez J. L. and Véron L., Different kinds of singular solutions of nonlinear parabolic equations, in: Nonlinear Problems in Applied Mathematics, SIAM, Philadelphia, PA, 1996, pp. 240-249	[]
[ "DRIFT OF PANCAKE ICE FLOES IN THE WINTER ANTARCTIC MARGINAL ICE ZONE DURING POLAR CYCLONES A PREPRINT", "DRIFT OF PANCAKE ICE FLOES IN THE WINTER ANTARCTIC MARGINAL ICE ZONE DURING POLAR CYCLONES A PREPRINT" ]	[ "Alberto Alberello ", "Luke Bennetts ", "Petra Heil ", "Marcello Vichi ", "Keith Machutchon ", "Miguel Onorato ", "Alessandro Toffoli ", "\nUniversity of Adelaide\n5005AdelaideAustralia\n", "\nAustralian Antarctic Division & ACE-CRC 7001\nUniversity of Adelaide\n5005Adelaide, HobartAustralia, Australia\n", "\nUniversity of Cape Town Rondenbosch\n7701South Africa\n", "\nUniversity of Cape Town Rondenbosch\n7701South Africa\n", "\nUniversità di Torino & IFNF Torino\n10125Italy\n", "\nThe University of Melbourne\n3010ParkvilleAustralia\n" ]	[ "University of Adelaide\n5005AdelaideAustralia", "Australian Antarctic Division & ACE-CRC 7001\nUniversity of Adelaide\n5005Adelaide, HobartAustralia, Australia", "University of Cape Town Rondenbosch\n7701South Africa", "University of Cape Town Rondenbosch\n7701South Africa", "Università di Torino & IFNF Torino\n10125Italy", "The University of Melbourne\n3010ParkvilleAustralia" ]	[]	High temporal resolution in-situ measurements of pancake ice drift are presented, from a pair of buoys deployed on floes in the Antarctic marginal ice zone during the winter sea ice expansion, over nine days in which the region was impacted by four polar cyclones. Concomitant measurements of wave-in-ice activity from the buoys is used to infer that pancake ice conditions were maintained over at least the first seven days. Analysis of the data shows: (i) unprecedentedly fast drift speeds in the Southern Ocean; (ii) high correlation of drift velocities with the surface wind velocities, indicating absence of internal ice stresses >100 km in from the edge in 100% remotely sensed ice concentration; and (iii) presence of a strong inertial signature with a 13 h period. A Langrangian free drift model is developed, including a term for geostrophic currents that reproduces the 13 h period signature in the ice motion. The calibrated model is shown to provide accurate predictions of the ice drift for up to 2 days, and the calibrated parameters provide estimates of wind and ocean drag for pancake floes under storm conditions.	10.1029/2019jc015418	[ "https://arxiv.org/pdf/1906.10839v1.pdf" ]	195,658,050	1906.10839	1ef07c7e172131e297466ec1cf5a50594ea382cc	DRIFT OF PANCAKE ICE FLOES IN THE WINTER ANTARCTIC MARGINAL ICE ZONE DURING POLAR CYCLONES A PREPRINT June 27, 2019 Alberto Alberello Luke Bennetts Petra Heil Marcello Vichi Keith Machutchon Miguel Onorato Alessandro Toffoli University of Adelaide 5005AdelaideAustralia Australian Antarctic Division & ACE-CRC 7001 University of Adelaide 5005Adelaide, HobartAustralia, Australia University of Cape Town Rondenbosch 7701South Africa University of Cape Town Rondenbosch 7701South Africa Università di Torino & IFNF Torino 10125Italy The University of Melbourne 3010ParkvilleAustralia DRIFT OF PANCAKE ICE FLOES IN THE WINTER ANTARCTIC MARGINAL ICE ZONE DURING POLAR CYCLONES A PREPRINT June 27, 2019Clare Eayrs New York University Abu Dhabi Abu Dhabi, United Arab Emirates High temporal resolution in-situ measurements of pancake ice drift are presented, from a pair of buoys deployed on floes in the Antarctic marginal ice zone during the winter sea ice expansion, over nine days in which the region was impacted by four polar cyclones. Concomitant measurements of wave-in-ice activity from the buoys is used to infer that pancake ice conditions were maintained over at least the first seven days. Analysis of the data shows: (i) unprecedentedly fast drift speeds in the Southern Ocean; (ii) high correlation of drift velocities with the surface wind velocities, indicating absence of internal ice stresses >100 km in from the edge in 100% remotely sensed ice concentration; and (iii) presence of a strong inertial signature with a 13 h period. A Langrangian free drift model is developed, including a term for geostrophic currents that reproduces the 13 h period signature in the ice motion. The calibrated model is shown to provide accurate predictions of the ice drift for up to 2 days, and the calibrated parameters provide estimates of wind and ocean drag for pancake floes under storm conditions. Introduction Sea ice extent modulates energy, mass and momentum exchanges between the ocean and atmosphere, thereby playing a pivotal role in the global climate system (McPhee et al. 1987, Notz 2012, Vihma et al. 2014. During the winter sea ice advance around Antarctica, pancake ice floes-small, roughly circular floes that form in wavy conditions-represent most of the sea ice mass budget . Dynamics and thermodynamics of pancake floes dominate the evolution of the Antarctic marginal ice zone (Doble et al. 2003, Doble & Wadhams 2006, Roach, Horvat, Dean & Bitz 2018, and also the emerging Arctic marginal ice zone (Pedersen & Coon 2004, Roach, Smith & Dean 2018, i.e. the 5-100 km wide outer ice belt, where atmosphere-ocean-sea ice interactions are most intense (Wadhams 1986, Strong et al. 2017. Contemporary numerical models struggle to predict the spatial variability of advance/retreat of sea ice around Antarctica (Hobbs et al. 2015, Kwok et al. 2017, Roach, Dean & Renwick 2018, resulting in strong biases in ocean-atmosphere heat fluxes and salt input to the ocean (Doble 2009). Except for few sectors around Antarctica, trends in sea ice duration and extent are dominated by storms rather than large atmospheric modes (Matear et al. 2015, Kwok et al. 2017, Schroeter et al. 2017. Vichi et al. (2019) have shown that intense winter polar cyclones continuously reshape the edge of the Antarctic marginal ice zone by advecting warm air on the sea ice and forcing ice drift. This generates strong coupling between thermodynamics and dynamics (Stevens & Heil 2011). Strong coupling also exists in the Arctic, where storms have been shown to reverse the winter sea ice advance by melting ) and drifting , Lund et al. 2018) newly formed pancakes. Moreover, intense storm events cause rapid ice drift that enhances mixing and deepens the surface mixed layer, thus promoting heat exchanges with the water sublayers (Ackley et al. 2015, Zippel & Thomson 2016, Castellani et al. 2018. Knowledge of the dynamical response of pancake ice floes to the frequent and intense storm events that impact the winter Antarctic marginal ice zone is required to model the evolution of the marginal ice zone and improve climate predictions (Schroeter et al. 2017, Barthélemy et al. 2018, particularly now that prognostic floe size information is being included in models , Roach, Horvat, Dean & Bitz 2018. For over a century, atmospheric drag has been identified as the main driver of ice drift (Nansen 1902, Shackleton 1920); a rule-of-thumb indicates that the wind factor (the ice to wind speed ratio) is 2% (Thorndike & Colony 1982, Leppäranta 2011). In the Arctic marginal ice zone, Wilkinson & Wadhams (2003) calculated an average wind factor of 2.7% for pancake ice, and noted a correlation with the ice concentration (i c ), with a larger wind factor of 3.9% towards the ice edge where i c < 25%, and a smaller value of 2.2% towards the interior of the marginal ice zone where i c > 75%. For the Antarctic, Doble & Wadhams (2006) calculated a wind factor 3-3.5% in pancake ice conditions. Doble & Wadhams (2006) used a far shorter sampling rate of 0.33 h than the 24 h rate used by Wilkinson & Wadhams (2003), possibly resulting in the larger wind factor (see §5). More recently, in the Arctic and for a low ice concentration (i c = 33%), Lund et al. (2018) reported wind factors up to 5% for pancake ice, but defined for wind at 17 m height, as opposed to the standard 10 m height. They showed low correlation with the wind forcing, and suggested that currents (not measured) contribute significantly to sea ice drift. The wind factors calculated by Wilkinson & Wadhams (2003), Doble & Wadhams (2006), Lund et al. (2018) and others, do not separate out the effect of currents and Coriolis-they assume is wind is the only forcing. As a result, wind stresses are likely to be underestimated (Leppäranta 2011). The Nansen number, i.e. the ratio between ice drift and wind speed for an ocean at rest, explicitly indicates the role of wind stresses only (Leppäranta 2011), but, to the best of our knowledge, the Nansen number has not previously been reported for pancake ice. In sophisticated contemporary models, sea ice drift is governed by a general horizontal momentum equation in which wind stresses act together with other external stresses (ocean drag, Coriolis forcing, waves and ocean tilt as external forcing), and with a rheology term used to model internal stresses (Heil & Hibler 2002, Leppäranta 2011. Granular rheologies have been developed for the marginal ice zone (Shen et al. 1987, Feltham 2005, in which internal stresses are generated by floe-floe collisions, and the magnitude of the internal stresses depends on concentration of the floes and their granular temperature (a measure of the turbulent kinetic energy of the floes). However, in low ice concentration internal stresses are small, and the rheology term typically neglected (Hunke et al. 2010, Herman 2012. Modelled windinduced stresses are defined by a standard drag formulation (Martinson & Wamser 1990), i.e. proportional to air-sea ice drag coefficient-usually larger close to the ice edge because of an increase in surface roughness (Johannessen et al. 1983)-and the relative velocity between wind and ice. Similarly, ocean-induced stresses are defined by water-sea ice drag coefficient and ocean currents, seldom available in ice covered regions (Nakayama et al. 2012). Scarcity of in-situ observation (conducted for different seasons, regions and ice types) and heterogeneity of ice conditions, particularly in the highly dynamical marginal ice zone (Doble 2009), has led to a wide range of sea ice drag coefficients (Leppäranta 2011), undermining predictive capabilities. We report and analyse a new set of pancake ice drift measurements during the winter expansion of the Antarctic marginal ice zone, and during intense storm conditions that reshaped the edge of the marginal ice zone at synoptic scales . In 100% ice concentration, ≈ 60% pancake floes and ≈ 40% interstitial frazil ice , we report the fasted ice drift recorded in the Southern Ocean. We develop a Lagrangian free-drift model, based on the general sea ice horizontal momentum equation, and quantify the reciprocal effect of winds and currents on pancake ice drift by providing the Nansen number and the derived current. Field experiment and prevailing conditions The instruments were deployed during a winter voyage to the Antarctic marginal ice zone by the icebreaker SA Agulhas II (Fig. 1a). The voyage departed from Cape Town, South Africa, on the 1st of July along the WOCE I06 transect and reached the marginal ice zone on the 4th of July at 62.5 • S and 30 • E, at which time a polar cyclone was crossing the ice edge. At midday on the 4th of July, a pair of waves-in-ice observation systems (Kohout et al. 2015), hereafter simply referred to as buoys, were deployed on separate pancake ice floes (Fig. 1b) at 62.8 • S and 29.8 • E; they were ≈100 km from the ice edge and ≈1 km apart. The buoys are expendable devices that record position and wave spectral characteristics. One of the buoys, B1, recorded data continuously at a sampling rate of 15 mins for almost 9 days-8 days and 18 h, from 12:00 on the 4th of July until 06:00 on the 13th of July-until signal was lost (most likely due to the battery running out). The other buoy, B2, recorded at 15 mins for the first 6 days from deployment, after which, to save battery life, the sampling rate was reduced to 2 h, which allowed it to record data for 3 weeks. In this study, we only consider the period over the 9 days in which both buoys were operational to allow analysis of the buoys' relative motion and ice internal stresses. Sustained winds over the open ocean, up to 33 m s −1 according to the on-board met-station, generated large waves in the open ocean, with significant wave height up to 14 m and peak period ≈ 12 s according to the ERA5 reanalysis data (Copernicus Climate Change Service (C3S) 2017), and propagating towards the ice edge. The buoys indicated the wave field maintained ≈ 50% of its energy after 100 km of propagation into the marginal ice zone. Sea ice concentration was i c = 100%, as sourced from AMSR2 (Beitsch et al. 2014) and confirmed by ASPeCt observations (de Jong et al. 2018). Deck observations (see Fig. 1) and automatic camera measurements revealed the marginal ice zone was an unconsolidated mixture of pancake ice floes covering ≈ 60% of the surface and of characteristic diameter 3.2 m , and interstitial frazil ice. In-situ observations in the marginal ice zone lasted ≈ 24 h, after which the ship headed back to Cape Town. Environmental conditions were retrieved from satellite data and reanalysis products over the 9 days both buoys returned data. AMSR2 (Spreen et al. 2008) provided ice concentration at 3.125 km spatial resolution as daily mosaics, averaged over two swaths within 24 h. ERA5 reanalysis (Copernicus Climate Change Service (C3S) 2017) was used to retrieve surface wind velocities (at 10 m height) at 0.25 • spatial resolution and hourly frequency. ERA5 also provides wave properties, but these are only available where the ice concentration is below 30% (Doble & Bidlot 2013). (Fig. 2b). The other two had cyclogenesis over the marginal ice zone. The third cyclone (Fig. 2c) was short lived and its cyclolysis was south of the buoys in the marginal ice zone. The last cyclone transited to the north-west of the buoys before progressing over open water (Fig. 2d). All observed polar cyclones strongly affected the evolution of the edge of the marginal ice zone at synoptic scale : the asymmetric cyclonic structure transports moist warm air over the sea ice while the opposite side drags ice toward the open ocean. Concurrently, strong winds associated with polar cyclones generated large waves (larger during the first two polar cyclones that developed over open water) that impacted the edge of the marginal ice zone. Fig. 3 shows wave-in-ice intensity measured by buoy B1, the peak period was 15-20 s. Peaks in wave activity are associated with the transit of cyclones and they show high correlation with the open water wave height . Intense wave-in-ice activity after deployment suggests that the marginal ice zone was comprised of pancake floes, at least until the 11th of July, when waves ceased. Fig. 3 also shows the distance between buoy B1 and the ice edge, which is defined as the daily mean position of the AMSR2 15% ice concentration in the sector 29 • -33 • E and the buoy, and denoted d 15% . The buoys are 100-200 km from the ice edge, noting that sector averaging smears ice-edge features and so the distance must be interpreted with care. 3 Drift measurements and analysis 3.1 Buoy drift Fig. 4 shows the track of buoy B1 from deployment superimposed on the AMSR2 ice concentration, where each subplot is two days apart. (At the scale shown, the track of buoy B2 would overlap the buoy B1 track.) Over the 9 days we identified three distinct phases of ice movement: i. over the first 2 days, and driven by the first cyclone during which the wind speed reached ≈15 m s −1 (to the east), the drift was predominantly eastward, initially with a slight southward drift, followed by a slight northward one; ii. over the next 2 days, affected by the second cyclone that generated sustained wind of maximum speed ≈15 m s −1 over a period of 7 h at the buoys location, the drift was mainly westward with a slight northward component; iii. over the last 4-5 days, affected by the third and fourth cyclone that generated winds of speed ≈10 m s −1 , the drift was eastward, first slight southward and then slight northward, similarly to the first phase. The phases are divided by sharp turning and looping, hence undergoing significant meandering (Gimbert et al. 2012). In total, buoy B1 drifted 262 km, mainly zonally (≈ 70 km for each of three phases), and exhibits a net northward translation (≈ 80 km). Over the 8 days and 18 h the average speed was 0.35 m s −1 which is over 50% greater than previously reported daily averages for this sector of the Southern Ocean (Heil & Allison 1999). The maximum instantaneous speed was 0.75 m s −1 , which is the fastest recorded for Antarctic pancake ice drift. Fig. 4 shows the development of an ice-edge feature over time, in the form of a localised protrusion that complicates the interpretation of the distance from the edge shown in Fig. 3. We note, however, that the buoys are always in 100% ice concentration according to remotely sensed AMSR2 ice concentration. On 11th of July (panel d), there are large areas covered by intermediate ice concentration around the protrusion (0% < i c < 100%), likely due to thermodynamic ice formation, which resulted in the sharp increase in distance between buoy B1 and ice edge on the 11th of July shown in Fig. 3. Ice deformations Fig. 5a shows the time series of the buoy separation distance, d. Over the first 4 days from deployment (4th-8th July, phases i-ii), the buoys slowly drifted apart, reaching a maximum separation of d =1.5-2 km. Over 8th-9th July, at the beginning of phase (iii), the buoys rapidly drifted apart, from 1.5 km to 2.5 km in less than a day, after which they moved slightly closer. Overall, the distance between the buoys grew by 2 km over the nine-day measurement period. Deformations of the sea ice cover are commonly reported in terms of the strain rates (Lindsay 2002) ε d = ∂u i ∂x + ∂v i ∂y ,ε s = ∂u i ∂x − ∂v i ∂y 2 + ∂u i ∂y + ∂v i ∂x 2 1/2 andε t = (ε 2 s +ε 2 d ) 1/2 ,(1) which are the divergence rate, shear rate and total deformation rate, respectively. In the equation u i and v i denote the ice velocity in the positive east (x) and north (y) directions, respectively, and the spatial derivatives are evaluated using buoy B1 and B2 velocities (u B1,B2 and v B1,B2 ) and position (x B1,B2 and y B1,B2 ), e.g. ∂u i /∂x = (u B1 − u B2 )/(x B2 − x B1 ). Figs. 5b-d show time series of the strain rates. Divergence and shear intensify at the same time; shear contributes the most to the total strain rate, suggesting that, at the buoy distance length scale, rotational motion dominates over the compression/expansion. The strain rates are highly intermittent, which provides further evidence that the ice cover remained unconsolidated, at least until the 9th of July. The dashed black vertical lines denote the time at which the B2 sampling rate was lowered to 2 h, thus reducing the accuracy of the calculations. Beyond this time, the calculated deformations are significantly lower and intermittent properties disappear due to the coarse temporal resolution. The root mean square (RMS) of the strain rates for the time during which both buoys were recording at a sampling rate of 15 mins areε d = 7 × 10 −4 s −1 ,ε s = 1 × 10 −3 s −1 andε t = 1.2 × 10 −3 s −1 .(2) These are 2-3 orders of magnitude greater than those reported by Doble & Wadhams (2006) for pancake ice in the Weddell Sea, at similar sampling rate of 20 mins, but from an array of six buoys with characteristic separation distance of 50 km. The large difference is likely due to the disparate characteristic distance between buoys, as rates of deformation are inversely proportional to the distance between buoys (Doble & Wadhams 2006), and the strain rates would be comparable if the characteristic distances were equivalent. Also, computation of the strain rates over the array of six buoys is more accurate than two. Correlation between buoy drift and wind Fig . 6 shows the velocity of buoy B1 in the zonal (east) and meridional (north) directions, compared to the ERA5 wind co-located at buoy B1 time and position using a tri-linear interpolation (2D in space, 1D in time). The ice drift velocity qualitatively follows the wind velocity, but the ice drift is characterised by oscillations of period ≈13 h. In Fig. 6a, the wind (U 10 ) and ice velocity (u i ) are positive (to the east) during phase (i), and become negative (to the west) during phase (ii). 7 shows the spectra of the wind velocity components (orange) and ice drift velocity (blue). The wind velocity spectrum forms a continuous energy cascade, but the ice velocity spectrum exhibits an energy peak (highlighted by the arrow) at a frequency just below two cycles per day (cpd; the exact value is 13.1±0.85 h). The period of these oscillations is close to the inertial range at 62-63 • S (13.5±0.05 h defined by the Earth rotation) that are clearly seen in the east and north ice velocity components (see Fig. 6). Fig. 8a shows buoy B1 speed compared to the ERA5 wind speed. The instantaneous drift speed peak of 0.75 m s −1 occurs during phase (ii), at midnight between 7th-8th of July. The wind speed at that instant is 14 m s −1 , noting that the peak wind speed of 15 m s −1 occurs at 16:00 on the 7th of July. During the transition between phase (i) and phase (ii), denoted by first vertical line in Fig. 8, the wind stops, i.e. both the north and east component of the wind velocity approach 0 m s −1 (see Fig. 6), and the ice drift almost stops. The correlation between the wind and buoy B1 speed is R 2 = 0.56; this increases to R 2 = 0.66 when inertial-like oscillations are filtered out (black curves in Fig. 6 and Fig. 8). Figure 7: (a) Spectra corresponding to the zonal buoy B1 velocity (blue) and zonal wind velocity (orange). An arbitrary vertical shift is applied to the wind spectra to aid comparison. Black arrows denote the peak associated to inertial-like oscillations (≈13 h). (b) As in (a), but for meridional velocity component. The wind factor, i.e. the ratio between ice speed and wind speed, estimated with a standard least square regression is 3.3%. For comparison, Doble & Wadhams (2006) report wind factor 3-3.5% with R 2 = 0.5 in pancake ice. Little to no correlation is found between the ice drift and the wave-in-ice activity (R 2 < 0.1), even during the periods of large significant wave heights (H S > 1.25 m), suggesting that wave-induced drift of pancake floes is negligible in comparison to wind-induced drift. The Arctic Ice Dynamics Joint Experiment (AIDJEX) model for sea-ice drift (Coon et al. 1974, Feltham 2008, Leppäranta 2011 is m i du i dt = A i S a + A i S w + m i S c + m i S g + ∇ · σ,(3) where m i , A i and u i are, respectively, the mass, area and velocity of the ice, S a , S w and S c and S g are external stresses due to wind, ocean currents, Coriolis and ocean tilt, respectively, and ∇ · σ is the rheology that defines internal stresses. The ice mass is m i = ρ i h i A i , where ρ i is the ice density and h i its thickness. As stated in §2, in-situ observations of the pancake floes concentration during deployment was ≈60%, and the remaining 40% was interstitial frazil ice (Alberello, Onorato, Bennetts, Vichi, Eayrs, MacHutchon & Toffoli 2019). The wave-in-ice activity measured by the buoys during the subsequent days indicates that similar unconsolidated conditions were maintained. On this basis, the free drift regime is assumed (∇ · σ = 0), as is standard for low ice concentration (Hunke et al. 2010, Herman 2012. A quadratic wind stress is used, of the form S a = C a ρ a \|u a \|u a exp(iθ a ), where C a is the wind drag coefficient over ice, u a the velocity difference between the wind and the ice (u a = u a − u i ), ρ a the air density, θ a is the angle between the wind direction and the wind-induced stress, and i is the imaginary unit. Linear formulations and calibrated exponents have been used yielding to similar accuracy (Martinson & Wamser 1990), but only the more common quadratic formulation is discussed. For consistency, a quadratic ocean drag is adopted (Leppäranta 2011), with S w = C a ρ w \|u w \|u w exp(iθ w ). where u w = u w + u g − u i(6) assuming ocean and geostrophic currents of velocity u w and u g , respectively. We note that in absence of currents, u w = u g = 0, the ocean drag would be proportional to the ice speed and act in the opposite direction to the ice drift, i.e. u w = −u i , so that it produces damping. The term S c denotes the Coriolis stress; in absence of other external forces, it produces rotation, which is leftward in the southern hemisphere (with respect to the direction of the ice drift). The Coriolis stress is expressed as (Cushman-Roisin & Beckers 2011) S c = −if u i ,(7) where f = 2ω sin(ψ) is the Coriolis parameter, in which ω = 7.2921 × 10 −5 rad s −1 denotes the Earth's rotation rate and ψ is latitude. The stress due to the ocean slope, S g , is written (Leppäranta 2011) S g = −∇ζ,(8) where ζ denotes the sea surface height. In deep water this term can be expressed as a function of the surface geostrophic current (Cushman-Roisin & Beckers 2011), with S g = if u g(9) which is similar in form to S c , but does not depend on the ice velocity. The AIDJEX model, Eqn. 3, becomes du i dt = αu a \|u a \| exp(iθ a ) + βu w \|u w \| exp(iθ w ) − if u i + if u g ,(10)where α = ρ a C a ρ i h i and β = ρ w C w ρ i h i .(11) The Nansen number (the ratio between wind and ocean stresses) can be expressed in terms of the coefficients α and β, as N a = ρ a C a ρ w C w = α β ,(12) which indicates the wind stresses, explicitly accounting for the air and water drag ratio. Eqn. 10 is equivalent to the free drift model given by Leppäranta (2011), Eqn. 6.3, with the advective acceleration conserved to maintain the generality of our formulation. Model setup Eqn. 10 is numerically solved in a Lagrangian frame of reference to simulate the buoy drift, using a finite difference, time stepping method; at each step (time steps of 60 s were found to give sufficient convergence) the velocity and displacement are computed, and the buoy advanced in space. Wind forcing is retrieved from ERA5 and, at each time step, interpolated in space and time onto the simulated buoy position. No data are available on ocean currents-in general, currents are rarely available in ice-covered regions (Nakayama et al. 2012)-thus, u w = 0 is set. The strong signature of the measured drift at periods close to the inertial range is likely related to the geostrophic current or eddies rather than the Coriolis term, as the contribution of m i S c is negligible for the thin sea ice in the Antarctic marginal ice zone (Martinson & Wamser 1990), and only becomes relevant for multi-year ice (h i > 1 m). Based on the buoy drift measurements, we adopt the geostrophic term to be a rotational term of the type u g = U g exp(if t),(13) where the amplitude of the near-inertial oscillations U g is estimated from the measurements to be U g = 0.125 m s −1 , which is the mean amplitude of the oscillations in 12-14 h identified utilising a band-pass filter. We set θ a = 0 and θ w = −25 • in agreement with the AIDJEX formulation when surface winds are used (Leppäranta 2011), noting that θ w ≈ θ 0 (Leppäranta 2011). The remaining free parameters, α and β, are calibrated by matching model velocity outputs with the measurements, by minimising the difference \|u O i −u M i \|+\|v O i −v M i \|, where superscripts O and M denote the observations (measurements) and the model, respectively. We test values of α in the range 0.012-0.015×10 −3 m −1 , which corresponds to C a in 3.0-3.7×10 −3 , as identified by Overland (1985) for the marginal ice zone, and ρ i = 910 kg m −3 , ρ a = 1.3 kg m −3 and h i = 0.35 m, from visual observations during in-situ operations. Similarly, we test values of β in the range 5-16×10 −3 m −1 , which corresponds to C w in 1.6-5.0×10 −3 and ρ w = 1028 kg m −3 . The lower and upper limits for C w are taken from values reported by Martinson & Wamser (1990) in the Weddel Sea, and McPhee (1982) in the Beaufort Sea, respectively. The calibrated parameters are α = 0.0128 × 10 −3 m −1 and β = 8.9 × 10 −3 m −1 . Fig. 9 shows model results against measurements for the zonal (a) and meridional (b) velocity components. Model outputs are shown for both U g = 0.125 m s −1 (full model) and U g = 0, to highlight the effect of the geostrophic term. Suppression of the geostrophic term (U g = 0) eliminates near-inertial, 13 h-period oscillations, and the time-series resembles the band-pass filtered measurements (black line in Fig. 6). Model predictions when the geostrophic term is included reproduce the measurements, although some of the high frequency oscillations observed in the measurements are not captured, likely due to relatively low temporal and spatial resolution of the input ERA5 wind data, which results in a smooth wind field, without small scale variability. The root mean square error over the entire duration of the measurements is 0.095 m s −1 for the full model and grows to 0.125 m s −1 by suppressing the geostrophic term. Fig. 10 shows the measured and simulated buoy tracks. The full model accurately reproduces buoy B1 drift during phase (i), in which the drift is eastward. After a loop at the end of phase (i), i.e. at the location denoted by a green cross in Fig. 10, the model under-predicts the maximum westward movement remaining to the east of the measured buoy position during phases (ii) and (iii). Meanders, cycloids (the half-moon shaped part of the track connected by cusps) and loops (during which the rotational component of the motion, driven by the geostrophic forcing, dominates over momentarily weak wind drag) of buoy B1 track are qualitatively reproduced only when the geostrophic current is included. Fig. 11a shows model results that start at t S = 0, 2.5 days and 5 days from the start of measurements, noting that t S = 2.5 days is ≈ 6 hrs into phase (ii) and t S = 5 days is ≈ 18 hrs into phase (iii). For the three different start times, parameters α and β are calibrated over: t S = 0 till the end of phase (i); t S = 2.5 days till the end of phase (ii); and Table 1: Calibrated parameters for the start times t s = 0, 2.5 days and 5 days (roughly phases i-iii, respectively). The values in parenthesis for phases (ii-iiii) show the variation compared to phase (i). Model results α × 10 −3 [m −1 ] β × 10 −3 [m −1 ] N a × 10 −2 [-] Phase (i) 0.0128 (-) 8.9 (-) 3.81 (-) Phase (ii) 0.0225 (+75.8%) 11.6 (+30.3%) 4.40 (+15.5%) Phase (iii) 0.0061 (−52.3%) 7.4 (−16.9%) 2.87 (−24.7%) t S = 5 days till the end of phase (iii). Parameters α and β are given in Table 1, noting that the parameters for t S = 0 (phase i) are almost identical to the ones calibrated over the entire track. Compared to phase (i), the coefficients α and β increase during phase (ii) and decrease during phase (iii), noting that α, which is related to the ice surface roughness (Johannessen et al. 1983), is the parameter with the highest variability (> ±50%). Fig. 11b shows the time series of model errors corresponding to Fig. 11a, i.e. distances between the model and measured positions, for the start times t S = 0, 2.5 days and 5 days. It also includes errors for the start times t S = 2.5 days and 5 days, without re-calibration of α and β. The error for t S = 0 days (α = 0.0128×10 −3 m −1 and β = 8.9×10 −3 m −1 ) is < 5 km during phase (i) and only exceed this threshold at 2.75 days. The error then steadily grows (during phases iiiii), up to ≈40 km at 8 days, due to error propagation in the time integration, and changes in the optimal values of α and β. The error for t S = 2.5 days (α = 0.0225 × 10 −3 m −1 and β = 11.6 × 10 −3 m −1 ) never exceeds 7.5 km during phase (ii) and at the end of phase (ii) the model error is < 3 km. For comparison, utilising parameters calibrated over the entire track the error at the end of phase (ii) is ≈11 km, i.e. 4 times larger than the model error with dedicated parameters. The error for t S = 5 days (α = 0.0061 × 10 −3 m −1 and β = 7.4 × 10 −3 m −1 ) is only 3.4 km after 2 days, and remains < 11 km till the end of phase (iii). This is significantly better than model prediction for t S = 5 days and parameters calibrated over the entire track, which, for example, result in an error of 27 km after 3 days. Discussion Pancake ice constitutes most the winter ice mass budget around Antarctica and it is becoming more common in the emerging Arctic marginal ice zone . Thermodynamics and dynamics of pancake ice govern the atmosphere-ocean-sea ice momentum and mass exchanges over vast ice covered areas, thus playing a role in the global climate system (Doble et al. 2003). This study is the first to measure and analyse both the drift of pancake ice floes and concomitant wave activity, during a series of intense winter polar cyclones. The analysis is based on the assumption that pancake ice conditions persisted over the nine days following deployment. Cyclonic activity and associated intense wave-in-ice activity prevents consolidation of pancake ice floes , Doble & Wadhams 2006. Their absence has been used to infer consolidation of the pancakes into a compact ice cover (Doble & Wadhams 2006). Our measurements of energetic waves (H S > 1.25 m) and intermittent internal sea ice deformations 100-200 km from the ice edge suggest that pancake ice conditions similar to the ones at deployment persisted for at least the first 7 days following deployment, beyond which ice conditions may have transformed as the ice edge advanced. Despite the significant wave-in-ice activity, no evidence was found of wave-induced ice drift, as might be caused by Stokes drift (Yiew et al. 2017), slope-sliding (Grotmaack & Meylan 2006) or wave radiation stresses (Masson 1991). Williams et al. (2017) and Boutin et al. (2019) recently integrated wave radiation stresses into large-scale numerical models that include wave attenuation and wave-induced ice breakup, based on the wave-ice interaction model of Williams et al. (2013a,b). They found that large wave radiation stresses, proportional to the wave attenuation rate, remain concentrated at the edge (Williams et al. 2017); wind and ocean stresses dominate ice drift over longer distances. Moreover, Williams et al. (2017) found that wave-radiation stresses are appreciable only for wave periods < 10 s; the measurements reported here have dominant periods > 15 s, and also for smaller floes than tested by Williams et al. (2017), for which radiation stresses are even weaker. Although wave-induced drift is negligible, it is expected that the significant waves measured will have induced turbulence in the water sublayers (Zippel & Thomson 2016, Alberello, Onorato, Frascoli & Toffoli 2019, Smith & Thomson 2019, enhancing mixing and heat fluxes under sea ice (Ackley et al. 2015). Shen et al. (1987) proposed a granular rheology for the marginal ice zone based on momentum transfer through floe-floe collisions. Feltham (2005) used the collisional rheology in a compositive marginal ice zone/pack ice rheology, and Bateson et al. (2019) included wave stresses in the rheology. In comparison, Sutherland & Dumont (2018) used a rheology based on Mohr-Coulomb granular theory, and their model outputs and field measurements showed strong wave attenuation and ice deformation that resulted in rafting of the floes. Notably, the ice drift was constrained by the coast, allowing for the internal stresses to build up (Dai et al. 2004). The model-data agreement shown in §4.3, without a rheology term, indicates internal stresses are negligible for pancake ice during intense cyclones conditions, and no collisions or rafting were observed during deployment. This is consistent with laboratory wave basin experiments reported by Bennetts & Williams (2015), which showed negligible attenuation, and although regular floe-floe collisions occurred, they were weak and did not result in rafting. Discrete element models of pancake ice floes in waves (Hopkins & Shen 2001, Sun & Shen 2012) also show that no rafting occurs in open boundary configuration, but it does when waves push the floes against a fixed boundary (Dai et al. 2004). Drift measurements conducted at high temporal resolution generated accurate estimates of the drift speed (Thorndike 1986) under cyclonic activity. The speed reached ≈0.75 m s −1 , which is the highest ice speed ever recorded in the Southern Ocean. Evaluation of the drift speed is sensitive to the sampling rate (Thorndike 1986) and daily or sub-daily measurements, available using remote sensing products (e.g. OSI-SAF; Lavergne et al. (2010)), can underestimate the maximum ice drift speed by over 20%, making them unsuitable to study drift at small temporal scales. A detailed analysis of our data indicates that the maximum speed is reduced by ≈ 5% when the sampling is lowered to 6 h, and by ≈20% for 12 h sampling. Velocity components in the north and east directions show larger reductions. Previously reported measurements at a 6 h sampling rate or greater (Martinson & Wamser 1990, Vihma et al. 1996, Heil & Allison 1999 might have underestimated the instantaneous drift speed and, consequently, provided lower estimates of the drag coefficients and wind factors over sea ice. Low temporal resolution drift measurements would not have captured the oscillations with period close to the inertial range; at least a 3 h resolution is needed to capture these oscillations. The rotational motion significantly contributes to instantaneous ice speed, and induces instantaneous ice drift in opposition to the wind direction, when the wind intensity drops. The model outputs indicate Coriolis forcing is not responsible for the observed oscillations, i.e. model results including Coriolis forcing and omitting geostrophic forcing do not reproduce the observed velocity oscillations, even for thicker ice, up to 1 m. Moreover, tidal currents have previously been found to affect ice drift only in limited water depth conditions (Meyer et al. 2017, Peterson et al. 2017, Padman et al. 2018, especially in shelf seas and coastal areas, and, therefore, are unlikely to be the source of the periodic oscillations since the study area is located in deep waters (Arndt et al. 2013). Instead, combined measurements and model outputs support the existence of a geostrophic-like forcing at period close to 13 h, similarly to the indirect observations of Lund et al. (2018) in the Arctic. The rotational motion period and amplitude (≈2 km in diameter) are consistent with sub-mesoscale eddies that have been found to form at the edge of the marginal ice zone in the Arctic (Lund et al. 2018) and in numerical experiments (Manucharyan & Thompson 2017, Dai et al. 2019). Our analysis indicates that the ratio C a /C w , using a quadratic drag formulation, is order unity for pancake ice in the Southern Ocean winter marginal ice zone-this value is obtained using ρ i = 910 kg m −3 , ρ a = 1.3 kg m −3 , ρ w = 1028 kg m −3 and h i = 0.35 m estimated at deployment, which gives C a = 0.0032, C w = 0.0027. The values for the drag coefficients are close to those found in the marginal ice zone by Overland (1985), Martinson & Wamser (1990), McPhee (1982) and Leppäranta (2011), noting that none of previously reported drag coefficients explicitly refers to pancake ice. Moreover, sea ice drag coefficients do not account for roughness due to ocean waves propagating in the marginal ice zone, which remains an open problem (Zippel & Thomson 2016). Leppäranta (2011) argues that the ratio C a /C w does not vary for all ice types because the ice roughness on the air and water sides are correlated, and hence the Nansen number N a = α/β ∝ C a /C w does not vary (assuming air and water densities are constant). Model calibrations of the parameters α and β indicate variation over the nine days following deployment: N a = 0.0381 in phase (i); N a = 0.0440 in phase (ii); and N a = 0.0287 in phase (iii). Evolution of the Nansen number indicates a corresponding change of the ratio C a /C w , suggesting the ice conditions modified two days after deployment, i.e. at the transition from phase (i) to phase (ii), and again after four days, i.e. at the transition from phase (ii) to phase (iii), noting that phase (iii) is characterised by less intense wave-in-ice activity and significantly slower drift. However, model calibrations crucially depend on the input wind and currents, and we recall that no data on currents are available for our experiments, and a thorough analysis of ERA5 wind bias in ice covered region is needed. Conclusions High temporal resolution measurements of drift of a pair of buoys deployed on pancake floes, initially 100 km into the marginal ice zone, during the Antarctic winter expansion were analysed over a 9-day period, over which four polar cyclones impacted the ice cover. The measurements, and comparisons with a calibrated Lagrangian free drift model, revealed that: • Pancake ice floes in the marginal ice zone are extremely mobile, even in 100% ice concentration (60% pancake ice and 40% interstitial frazil ice). The maximum instantaneous ice drift speed was 0.75 m s −1 , measured during intense storm conditions (winds up to ≈15 m s −1 ), and exceeding previously reported values for the ice-covered Southern Ocean. • Pancake ice drift velocity correlates very well with wind velocity, indicating that wind is the dominant forcing, except for a strong inertial-like signature at ≈13 h in the drift, which was attributed to geostrophic currents (or sub-mesoscale eddies). Despite the strong wave-in-ice activity, no correlation was found with the measured ice drift. • A free drift model accurately predicts pancake ice drift velocities, indicating that internal stresses are negligible. This finding was backed by the relative motion between the buoys, which was two orders of magnitude smaller than the total drift. • The Nansen number varied considerably over the nine days period at the scale of synoptic events (2-3 days) suggesting that ice conditions and, consequently, ocean and wind drag have changed, although it may also be due to inaccuracies in the model forcings used. Present results highlight the need for better understanding and models of α and β (equivalently, the drag coefficients and ice thickness) and their temporal and spatial variation, together with reliable wind and current data. This will empower accurate predictions of pancake ice drift in the marginal ice zone at the 2-3 day temporal scale of synoptic events, particularly during polar cyclones continuously reshape the marginal ice zone and have the largest effect on the advance and retreat of pancake ice around Antarctica. Figure 1 : 1(a) Ice conditions on the 4th of July 2017, and (b) deployment of the buoys on pancake floes using the ship crane. ERA5 reanalysis shows that another three cyclones, albeit less intense, impacted the marginal ice zone surrounding of the buoys over the 9 days.Figs. 2a-d show the tracks of the four polar cyclones overlaid on the significant wave height (in open water) and the ice concentration. The cyclogenesis of the first and most intense polar cyclone took place over open water and its cyclolysis over the marginal ice zone south-east of the buoys (see track in Fig. 2a). The second polar cyclone skirted the ice edge, travelling over open water to the north of the buoys Figure 2 : 2Environmental conditions when the polar cyclones were close to the buoys: (a) at 15:00 on 4th of July; (b) at 18:00 on 7th of July; (c) at 21:00 on 10th of July; (d) at 15:00 on 12th of July. The cyclone tracks are shown in red, and the buoy B1 with the magenta circle. The shadings show the AMSR2 ice concentration, and the ERA5 significant wave height. The contour lines (in black) denote the ERA5 isobars in hPa. Figure 3 : 3Significant wave height from buoy B1 (left axis; blue), distance between buoy B1 and the ice edge defined by 15% concentration (right axis; orange). Figure 4 :Figure 5 : 45Buoy B1 track over 9 days following deployment on the 4 July 2017, superimposed on ice concentration. Intervals intense wave-in-ice activity (H S > 1.25 m) are highlighted (red). The green cross and the green dot indicate the time of transition between phases (a) Distance between buoys, (b) divergence rate, (c) shear rate, and (d) total deformation rate. The vertical black dashed line denotes the time when the acquisition rate of the second buoy B2 was dropped from 15 mins to 2 h to preserve battery life. The green vertical lines indicate the time of transition between phases. Figure 6 : 6(a) Zonal buoy B1 velocity (on the left axis) and zonal wind velocity (on the right axis), in black (on the left axis) buoy B1 zonal velocity when ≈13 h oscillations are excluded. The green vertical lines indicate the time of transition between phases. (b) As in (a), but for meridional velocity component. Fig. Fig. 7 shows the spectra of the wind velocity components (orange) and ice drift velocity (blue). The wind velocity spectrum forms a continuous energy cascade, but the ice velocity spectrum exhibits an energy peak (highlighted by the arrow) at a frequency just below two cycles per day (cpd; the exact value is 13.1±0.85 h). The period of these oscillations is close to the inertial range at 62-63 • S (13.5±0.05 h defined by the Earth rotation) that are clearly seen in the east and north ice velocity components (see Fig. 6). Fig. 8 b 8shows the angle θ 0 between the ice drift and the wind direction. Previous observations indicate the angle, on average, to be in the range 0 • -30 • , positive in the northern hemisphere and negative in the southern hemisphere(Leppäranta 2011). The present dataset gives a mean angle ≈ −25 • , as shown by the dashed line, and with generally large variations from -60 • to 10 • . The large variations are consistent with, for example,Lund et al. (2018), who reported turning angles from -23 • to +83 • during a storm event in the Arctic Basin, and in other instances reported ice drift against the wind (turning angle > 90 • ). During our measurements, the largest angles between wind and ice direction (\|θ 0 \| > 90 • ) occur sporadically and are always observed for low wind speed (\|u 10 \| < 6 m s −1 ), when the wind stresses become small. Figure 8 : 8(a) Buoy B1 speed (left axis) in total (blue) and with ≈13 h oscillations are excluded (black), and total wind speed (right axis; orange). (b) Difference between ice and wind direction θ 0 (blue), difference when ≈13 h oscillations are excluded (black), and the mean difference −25 • (yellow dashed). The green vertical lines indicate the time of transition between phases. Figure 9 : 9Time series of buoy B1 measurements (blue) and model simulations, for full model (orange) and U g = 0 (black): (a) zonal velocity and (b) meridional velocity. The green vertical lines indicate the time of transition between phases. Figure 10 : 10Buoy B1 tracks from measurements (blue) compared to model simulations (full model, orange; U g = 0, black). The green cross and the green dot indicate the time of transition between phases. Figure 11 : 11(a) As in Fig 10, but for three different start times (t S , indicated by filled squares), and corresponding tuned parameters α and β. The green cross and the green dot indicate the time of transition between phases. (b) Time series of errors for model calibrated for each new start time (solid curves), compared to errors when overall calibrated parameters are used (dashed curves). The green vertical lines indicate the time of transition between phases. AcknowledgmentsThe expedition was funded by the South African National Antarctic Programme through the National Research Foundation. This work was motivated by the Antarctic Circumnavigation Expedition (ACE) and partially funded by the ACE Foundation and Ferring Pharmaceuticals. AA, LB, AT and PH were supported by the Australian Antarctic Science Program (all by project 4434, and PH by projects 4301 and 4390). MO was supported by the Departments of Excellence 2018-2022 Grant awarded by the Italian Ministry of Education, University and Research (MIUR) (L.232/2016). CE was supported under NYUAD Center for global Sea Level Change project G1204. 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[ "Magnetic field amplification by the small-scale dynamo in the early Universe", "Magnetic field amplification by the small-scale dynamo in the early Universe" ]	[ "Jacques M Wagstaff \nHamburger Sternwarte\nGojenbergsweg 11221029HamburgGermany\n", "Robi Banerjee \nHamburger Sternwarte\nGojenbergsweg 11221029HamburgGermany\n", "Dominik Schleicher \nInstitut für Astrophysik\nGeorg-August-Universität Göttingen\nFriedrich-Hund-Platz 137077GöttingenGermany\n", "Günter Sigl \nII Institut für Theoretische Physik\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n" ]	[ "Hamburger Sternwarte\nGojenbergsweg 11221029HamburgGermany", "Hamburger Sternwarte\nGojenbergsweg 11221029HamburgGermany", "Institut für Astrophysik\nGeorg-August-Universität Göttingen\nFriedrich-Hund-Platz 137077GöttingenGermany", "II Institut für Theoretische Physik\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany" ]	[]	In this paper we show that the Universe is already strongly magnetized at very early epochs during cosmic evolution. Our calculations are based on the efficient amplification of weak magnetic seed fields, which are unavoidably present in the early Universe, by the turbulent small-scale dynamo. We identify two mechanisms for the generation of turbulence in the radiation dominated epoch where velocity fluctuations are produced by the primordial density perturbation and by possible first-order phase transitions at the electroweak or QCD scales. We show that all the necessities for the small-scale dynamo to work are fulfilled. Hence, this mechanism, operating due to primordial density perturbations, guarantees fields with comoving field strength B0 ∼ 10 −6 ε 1/2 nG on scales up to λc ∼ 0.1 pc, where ε is the saturation efficiency. The amplification of magnetic seed fields could be even larger if there are first-order phase transitions in the early Universe. Where, on scales up to λc ∼ 100 pc, the comoving field strength due to this mechanism will be B0 ∼ 10 −3 ε 1/2 nG at the present time. Such fields, albeit on small scales, can play an important role in structure formation and could provide an explanation to the apparently observed magnetic fields in the voids of the large-scale structure.	10.1103/physrevd.89.103001	[ "https://arxiv.org/pdf/1304.4723v2.pdf" ]	118,323,945	1304.4723	ea09d166f743f514943a6abb59a1d12dda58652e	Magnetic field amplification by the small-scale dynamo in the early Universe (Dated: May 22, 2014) Jacques M Wagstaff Hamburger Sternwarte Gojenbergsweg 11221029HamburgGermany Robi Banerjee Hamburger Sternwarte Gojenbergsweg 11221029HamburgGermany Dominik Schleicher Institut für Astrophysik Georg-August-Universität Göttingen Friedrich-Hund-Platz 137077GöttingenGermany Günter Sigl II Institut für Theoretische Physik Universität Hamburg Luruper Chaussee 14922761HamburgGermany Magnetic field amplification by the small-scale dynamo in the early Universe (Dated: May 22, 2014) In this paper we show that the Universe is already strongly magnetized at very early epochs during cosmic evolution. Our calculations are based on the efficient amplification of weak magnetic seed fields, which are unavoidably present in the early Universe, by the turbulent small-scale dynamo. We identify two mechanisms for the generation of turbulence in the radiation dominated epoch where velocity fluctuations are produced by the primordial density perturbation and by possible first-order phase transitions at the electroweak or QCD scales. We show that all the necessities for the small-scale dynamo to work are fulfilled. Hence, this mechanism, operating due to primordial density perturbations, guarantees fields with comoving field strength B0 ∼ 10 −6 ε 1/2 nG on scales up to λc ∼ 0.1 pc, where ε is the saturation efficiency. The amplification of magnetic seed fields could be even larger if there are first-order phase transitions in the early Universe. Where, on scales up to λc ∼ 100 pc, the comoving field strength due to this mechanism will be B0 ∼ 10 −3 ε 1/2 nG at the present time. Such fields, albeit on small scales, can play an important role in structure formation and could provide an explanation to the apparently observed magnetic fields in the voids of the large-scale structure. I. INTRODUCTION Magnetic fields of strengths of order a few µG have been observed in galaxies at high and low redshifts, in galaxy clusters and in superclusters [1][2][3][4][5]. There is also evidence for strong extragalactic magnetic fields coming from γ-ray observations. These studies place a lower bound on intergalactic magnetic fields at 3 × 10 −7 nG [6], although plasma effects may complicate the propagation of electromagnetic cascades [7,8]. Theoretically, magnetic fields are very likely to have been generated at some level in the early Universe through a variety of mechanisms. On the largest scales, magnetic fields can be generated during inflation giving today B 0 ∼ (10 −25 − 10 −1 ) nG on a scale of 1 Mpc [9]. However, such mechanisms require some modification to the Maxwell theory in order to break its conformal invariance. Without such modifications, fields of present strengths B 0 ∼ (10 −20 − 10 −11 ) nG could have been generated at the electroweak (EW) and QCD phase transitions respectively [10]. In this case, the magnetic field coherence length is limited by the particle horizon size at the time of generation, typically much smaller than 1 Mpc. At later times, magnetic fields could have been generated through the generation of vorticity in the primordial plasma [11][12][13][14][15] (originally proposed by Harrison [16]). This mechanism is very natural, since vorticity in the plasma is unavoidably generated in the late radiation era through the nonlinear couplings of first-order density perturbations. The seed fields generated here are of order B 0 ∼ 10 −20 nG. * [email protected] In many cases, the observed magnetic fields are much stronger than fields predicted by theories. Therefore, in order to explain observations, some amplification of the generated seed fields must have occurred at some point in the history of the Universe. A popular mechanism for such amplification is known as the dynamo mechanism. The dynamo mechanism comes in two broad classes (see Ref. [17] for a review). The large-scale dynamo converts kinetic energy on large scales into magnetic energy. This mechanism can act if the conducting fluid flow is highly helical, inhomogeneous or anisotropic, the typical example being the differential rotation of galaxies. This galactic dynamo, which operates only for spiral galaxies, requires seed fields of order B 0 ∼ 10 −21 nG to obtain µG strengths today [18]. However, this type of dynamo cannot explain strong fields in much younger galaxies, in galaxy clusters and superclusters or indeed in the voids of the large-scale structure. The second class of dynamo works with stationary, homogeneous and isotropic turbulence. This mechanism, known as the small-scale dynamo (SSD), also converts kinetic energy from turbulent motions into magnetic energy and typically operates on much smaller scales. Magnetic field lines, which are frozen into the conducting plasma, are stretched, twisted and folded by the random motions of fluid elements leading to exponential field amplification. The SSD mechanism has been applied to the formation of the first stars and galaxies in the matter dominated Universe [19,20] (see Ref. [21] for an early discussion on this subject), where the turbulent motions arise from gravitational collapse, accretion and supernovae explosions (e.g. Refs. [20,[22][23][24][25]). If turbulence is predominantly injected by supernova explosions, this mechanism may further explain the observed corre-lation between the star formation rate and the magnetic field strength in spiral galaxies [26]. This mechanism can be highly effective at magnetizing structures in the early Universe. However, a problem evades explanation; the large field strengths apparently observed in the voids of the large-scale structure [6]. In this paper we investigate the SSD amplification of magnetic seed fields in the radiation dominated Universe. If significant turbulence is generated in this early epoch, then small magnetic seed fields could be amplified very efficiently by the mechanism. Unlike velocity perturbations, magnetic fields survive through the viscous damping and free-streaming regimes. Therefore, the SSD could be an effective mechanism to strongly magnetize the early Universe prior to structure formation, leading to strong intergalactic magnetic fields. We demonstrate that the conditions are right for efficient dynamo amplification leading to large magnetic fields, albeit on very small scales, which could explain observations and have an impact on early structure formation. The structure of our paper is as follows. In Sec. II we give a brief review of the small-scale dynamo mechanism and the conditions necessary for its action. In Sec. III we look at two mechanisms for the generation of turbulence in the radiation dominated era. In Sec. IV we investigate the amplification of magnetic fields due to SSD action and look at their subsequent evolution to the present time. We summarize in Sec. V and conclude in Sec. VI. II. THE SMALL-SCALE DYNAMO MECHANISM The small-scale dynamo (SSD) mechanism is a very efficient mechanism at converting kinetic energy from turbulent motions to magnetic energy [27,28]. To describe the mechanism, we first briefly review the conditions necessary for turbulence to arise. The kinetic Reynolds number R e characterizes the relative importance of the fluid advective and dissipative terms in the Euler equation. For random motions correlated on some physical scale l with root-mean-square (rms) velocity v rms l , the local kinetic Reynolds number is given by [29,30] R e (l) =      v rms l l η s if l l mfp v rms l α d l if l l mfp(1) for dissipation due to diffusing particles and freestreaming particles respectively, where l mfp is the particle mean-free-path (mfp). Here, η s is the shear viscosity and α d is a drag coefficient due to the occasional scattering of fluid particles [30,31]. On a given scale l, a viscous regime corresponds to R e (l) 1. Whereas for a turbulent regime R e (l) 1, in this case the dissipative time scale is much greater than the eddy-turnover time scale τ l , where τ l ≡ l/v rms l = al c /v rms l ,(2) and l c is a comoving length. With the injection of kinetic energy, a turbulent flow develops almost inevitably when the kinetic Reynolds numbers are large enough [32,33]. Indeed, there is a critical value for which turbulence is expected, i.e. R e (l) > ∼ R cr e ∼ 10 3 . The fundamental reasons for the transition to a turbulent flow are not completely understood. However, flow instabilities always arise when the Reynolds numbers are larger than the critical value [32]. One possible mechanism for the triggering of the flow instability is due to thermal fluctuations [34], which are important in the radiation epoch, but other mechanisms may exist too. Indeed, in this paper we will look at turbulence driven by bubble collisions in first-order phase transitions. Turbulence is characterized by a direct cascade of energy from large scales to small scales. The eddy-turnover time gives the time over which eddy flows break down to smaller scales in this direct cascade. With a continuous injection of kinetic energy (or forcing/stirring of the fluid) at the forcing/stirring scale L, turbulence becomes fully developed (or stationary) on a time scale of order the eddy-turnover time scale at the forcing scale τ L . In an expanding Universe, this time scale must be at most given by the Hubble time i.e. τ L = 1/H. Thus, the largest possible forcing scale is L = v rms L /H, where v rms L is the typical velocity fluctuation on the forcing scale L. The velocity spectrum for fully developed turbulence is then given by v rms l = v rms L l L ϑ .(3) The scaling index ϑ varies between the two extremes 1/3 ≤ ϑ ≤ 1/2, where for incompressible Kolmogorov type turbulence ϑ = 1/3 and for highly compressible Burgers type turbulence ϑ = 1/2. This spectrum is valid only on the inertial range: l diss < l < L, which is determined by the turbulent kinetic energy cascade. The dissipative scale l diss is the scale at which turbulent velocities are diffused due to viscosity at the same rate as they are replenished from larger scales. At this scale the direct cascade ends. The dissipative scale can be defined through R e (l diss ) ∼ 1, hence l diss ∼ LR e (L) −1/(ϑ+1) assuming l diss l mfp in Eq. (1). The SSD mechanism converts this turbulent kinetic energy to magnetic energy [27,28]. The effectiveness of the mechanism depends strongly on three important environmental factors (i) the Reynolds number; stronger turbulence is more effective (ii) the turbulent velocity modes; rotational modes are much more efficient than longitudinal modes [35,36] and (iii) the Prandtl number P m ≡ R m /R e = 4πση s , where the Prandtl number is a measure of the relative importance of the magnetic and kinetic diffusion. Here, the R m is the magnetic Reynolds number given by R m (l) = 4πσal c v rms l ,(4) and σ is the plasma conductivity. There are two competing effects in the turbulent dynamo mechanism; magnetic field line stretching and resistive reconnection. The critical magnetic Reynolds number R cr m defines the balance between the stretching and reconnection. For R m (L) < R cr m reconnection wins and there is no dynamo and for R m (L) > R cr m the stretching wins and the dynamo takes effect amplifying the magnetic field. Independent of the Prandtl number, the critical magnetic Reynolds number is R cr m ≈ 60 and R cr m ≈ 2700 for Kolmogorov and Burgers type turbulence respectively [17,36,37]. When the SSD takes effect, the fluctuating component of the magnetic field grows exponentially B rms ∝ exp(Γt) due to turbulence in a weakly magnetized plasma. Depending on the Prandtl number, the growth rate Γ scales with either the kinetic Reynolds number R e or the magnetic Reynolds number R m , in particular Γ ∝ R (1−ϑ)/(1+ϑ) e and Γ ∝ R (1−ϑ)/(1+ϑ) m for P m 1 and P m 1 respectively [36,37]. There is a large number of numerical studies in the literature that have demonstrated the SSD action unambiguously for a number of settings [35,[38][39][40][41]. Analytically, the Kazantsev model (following the formalism by Refs. [17,42]) was developed in order to study the evolution of magnetic fields in a conducting plasma containing turbulent motions [28]. The Kazantsev model can be used to calculate the magnetic field growth rate and the critical magnetic Reynolds number required for SSD action. The model considers random turbulent motions correlated on a scale l with velocity v rms l and spectrum given in Eq. (3) valid on the inertial range l diss < l < L. The spectrum of velocity fluctuations is assumed to be Gaussian, homogeneous and isotropic in space and instantaneously correlated in time. The kinetic Reynolds number must also be larger than some critical value for which turbulence is expected, i.e. R e (L) > ∼ R cr e ∼ 10 3 (this is a conservative estimate for R cr e , indeed the SSD action has been observed in cases where R e (L) < ∼ 100 [38,40]). By modeling the turbulent velocity spectrum so that it behaves as Kolmogorov or Burgers turbulence for scaling index ϑ = 1/3, 1/2 respectively, it can be shown that the magnetic field growth rate is given by [36,43] Γ = (163 − 304ϑ) 60 R e (L) (1−ϑ)/(1+ϑ) τ −1 L .(5) This result is valid in the large Prandtl number limit P m 1, which is relevant to cosmological plasmas. Since the Reynolds numbers are typically very large, the magnetic fields can be amplified very rapidly. We also note that this analytical result for the growth rate has been verified by Ref. [43] via a numerical integration of the Kazantsev equation. The phase of rapid exponential amplification comes to an end when the magnetic energy becomes comparable to the kinetic energy on the dissipative scale l diss . The system then enters a stage of nonlinear growth, where the magnetic field grows as some power law in time B rms (t) ∝ t ϑ/(1−ϑ) [44]. This phase lasts until the magnetic field is saturated on the forcing scale L. Saturation is given by the approximate equipartition between magnetic and kinetic energy E M /E kin ≈ ε. This occurs when B 2 (x) ≈ 4πε(ρ + p) v 2 (x) ,(6) where the parameter ε quantifies the saturation efficiency. Numerical studies (for P m ≈ 2) indicate that the SSD mechanism is more efficient for rotational modes, where the saturation efficiency ε is close to unity [35]. Whereas the saturation level is lower for compressive modes ε ∼ 10 −3 − 10 −4 [35]. However, further numerical work is required to establish the saturation level for large Prandtl numbers. So far there are no analytical results to determine the efficiency parameter ε. Here, we stress that the SSD mechanism is a rather generic phenomenon, in the sense that the mechanism works independently of the type of turbulence [36,45,46]. In particular, it is interesting to note that even purely irrotational turbulence can still drive a small-scale dynamo. This was originally shown by Ref. [46] and later Ref. [36] reached similar conclusions. Hence, the efficient amplification of magnetic fields seems unavoidable if any kind of turbulence is generated in a magnetized plasma. III. TURBULENCE IN THE EARLY UNIVERSE In this section we use Eq. (1) to calculate the Reynolds numbers in the radiation dominated (RD) era in order to identify epochs of turbulence. With large Reynolds numbers, as we have argued in the previous section that any injection of kinetic energy into the plasma will lead to a state of fully developed turbulence for a range of scales. In the RD era, the kinetic Reynolds number in the diffusive regime is given by R e (l, T ) = 5g * (T ) g ν,γ v rms l l c l ν,γ mfp,c (T ) ,(7) where the shear viscosity η s = (g ν,γ /5g * )l ν,γ mfp is determined by the particles of longest mean free path l ν,γ mfp , which are either neutrinos or photons depending on the time. Here, g * and g ν,γ are the total and component number of effective relativistic degrees of freedom. In the very early Universe, before neutrino decoupling T > ∼ 2.6 MeV, neutrinos have the longest mfp and are thus most efficient at transporting momentum and heat. At high temperatures the shear viscosity due to neutrinos is low and the plasma could be in a turbulent regime R e 1. At this time, the comoving mfp is [31] l ν mfp,c a −1 G 2 F T 2 (n l + n q ) ,(8) which is proportional to 1/T 4 . Here n l = 6g l ζ(3)T 3 /7π 2 and n q = 6g q ζ(3)T 3 /7π 2 are the lepton and quark number densities, g l,q are the number of degrees of freedom for relativistic leptons and quarks, ζ is the Riemann zeta function and G F is the Fermi constant. However, the neutrino mfp increases as the Universe expands and cools, leading to a viscous regime R e < 1 (see e.g. Ref. [30]). Eventually the neutrinos decouple at T dec 2.6 MeV. From here on, momentum and heat is effectively transported by the photons. At early times, photons generate a small shear viscosity in the plasma and the fluid flow could become turbulent once again. The comoving photon mfp is given by [47] l γ mfp,c a −1 σ T (n 2 pair + n 2 e ) 1/2 ,(9) where σ T = 8πα 2 /3m 2 e is the Thomson cross section, α ≈ 1/137 is the fine structure constant and m e is the electron mass. During this epoch, the number densities n pair and n e of e ± pairs and free electrons respectively are given by [47] n pair ≈ 2m e T π 3/2 exp − m e T 1 + 15 8 T m e ,(10)n e = X e Ω b ρ 0 m p T T 0 3 ,(11) where m p is the proton mass, the baryon fraction and present day density product is Ω b ρ 0 1.81 × 10 −12 eV 4 [48], T 0 2.725 K is the present day photon temperature and the ionization fraction is X e = 1 in the RD era. Once the temperature decreases below the electron mass T < m e 0.511 MeV, e ± pairs begin to annihilate and the photon mfp increases rapidly. The e ± annihilation completes at around T 20 keV. After this, as the temperature drops further, photons begin to diffuse followed by photon drag and the fluid is in a viscous regime once again [30,31]. Hence, there are two epochs in the RD era, before and after neutrino decoupling, where the Reynolds numbers could be large and turbulence is potentially fully developed for a range of scales. Diffusing particles also damp away velocity fluctuations (see Sec. III C). Therefore, before diffusion sets in, there is always a possibility for the plasma to be in a turbulent state if the considered scales are large enough. However, the eddy-turnover time increases for larger scales. We must therefore look at the evolution of all relevant scales carefully in order to establish whether or not turbulence is possible. To complete the calculation for the Reynolds numbers, we must estimate the turbulent velocity fluctuations in the early Universe. In the next two subsections we present two mechanisms for the generation of turbulence in the RD era. A. Turbulence from primordial density perturbations To explain the formation of the large-scale structure in the Universe observed today, a primordial density perturbation of magnitude δρ/ρ ∼ 10 −5 is required at the time of matter-radiation equality [29]. The primordial density perturbation is thought to have been generated at a much earlier time and therefore must be present in the very early Universe during the RD era. Cosmic inflation provides the most compelling explanation for the origin of the primordial perturbation [49]. Well before horizon entry, the primordial curvature perturbation, which determines the gravitational potential Φ, remains constant and given by the initial condition Φ 0 . In the Newtonian gauge, its equation of motion is Φ + 3H(1 + w)Φ − w∇ 2 Φ = 0, where p = wρ, H = a /a and ≡ ∂/∂η with conformal time η [50]. In the RD era w = 1/3, the solution for the Fourier modes reads Φ(k, η) = 3[j 1 (y)/y]Φ 0 (k), where y ≡ kη/ √ 3, k is the comoving wave number and j 1 (y) = sin y/y 2 − cos y/y is the first spherical Bessel function. As a Fourier mode of the gravitational potential reenters the horizon during the RD era, it begins to oscillate with an amplitude decreasing as 1/y 2 ∝ 1/t. The initial conditions, which are probed by observations, are given by the two-point correlation function Φ 0 (k 1 )Φ * 0 (k 2 ) = (2π) 3 P Φ (k)δ 3 (k 1 − k 2 ),(12) where P Φ (k) = (2π 2 /k 3 )(9/25)∆ 2 R (k 0 ) (k/k 0 ) ns−1 . The results of the Planck mission give ∆ 2 R (k 0 ) = 2.215 × 10 −9 and n s 0.96 for the pivot scale k 0 = 0.05 Mpc −1 [48]. Perturbations in the fluid 3-velocity field are generated by the density perturbations. At first-order in density perturbations, the fluid velocity perturbation is purely irrotational (curl free) with Fourier modes [50] v i (k, η) = − ik i 2H 2 [Φ (k, η) + HΦ(k, η)](13)= −i 3 √ 3 2k i [sin y − 2j 1 (y)] Φ 0 (k) .(14) These modes v i oscillate with the density perturbation, but have a term which does not decay with the expansion. Let us define the spectrum of velocity perturbations in Fourier space by the two-point correlation function v i (k 1 , η)v * i (k 2 , η) = (2π) 3 2π 2 k 3 P v (k)δ 3 (k 1 − k 2 ) . (15) Hence, with Eq. (14) and the spectrum P Φ we find P v (k) = 243 100 [sin y − 2j 1 (y)] 2 ∆ 2 R (k 0 ) (k/k 0 ) ns−1 . (16) The velocity spectrum oscillates rapidly for subhorizon scales y, k/H 1, therefore we can average P v (k) over many oscillations. For a scale invariant primordial spectrum n s = 1 we find P v (k) 243 200 ∆ 2 R (k 0 ). Hence, on subhorizon scales, the spectrum of velocity perturbations generated by first-order density perturbations is isotropic, homogeneous, Gaussian and to a good approximation scale invariant, see Fig. 1. The root-mean-square (rms) velocity is then given by v rms ≡ v 2 (x) = ∞ 0 P v (k) dk k 1/2 .(17) It will also be useful to define the rms velocity on a given comoving length scale l c = 2π/k l by v rms l k diss k l P v (k) dk k 1/2 ,(18) where l diss,c = 2π/k diss is a cut-off or dissipative scale. The typical value of the velocity perturbation is therefore v rms l ∼ P v (k) ∆ 2 R (k 0 ) 5 × 10 −5 ,(19) where natural units are used such that c = 1. In the radiation dominated era, new k-modes of the primordial density perturbation, which generate velocity perturbations, are continuously reentering the horizon. This continuous production of velocity perturbations can be seen as the continuous forcing of the fluid on the largest scales. Thereby, if the Reynolds numbers are large enough, turbulent flow will occur. The velocity perturbations forcing the fluid are purely longitudinal in this case. However, rotational modes will be generated at second order in cosmological perturbations [51][52][53][54][55]. In any case, with R e 1, nonlinear interactions can play a role in generating turbulence with both longitudinal and rotational fluid motions. Therefore, a state of fully developed (or stationary) turbulence can be expected on a time scale of order the eddy-turnover time scale at the forcing scale τ L i.e. τ L = 1/H in an expanding Universe (see Sec. II). Hence, for all scales below L = v rms L /H, there are many eddy-turnover times per Hubble time τ l < ∼ 1/H. This condition ensures interactions between eddy flows leading to fully developed turbulence with a spectrum given in Eq. (3). A second condition is that the Reynolds number on the forcing scale L is larger than some critical value for which turbulence is expected, i.e. R e (L) > ∼ R cr e ∼ 10 3 . With these two conditions we find the range of scales, corresponding to the inertial range, l diss < ∼ l < ∼ L, where R e (l diss ) ∼ 1. In Sec. III D we establish when, in the RD era, and on what scales, the Reynolds numbers are large given the turbulent velocity fluctuations generated by the primordial density perturbations. B. Turbulence injected from phase transitions In this section we briefly describe another mechanism for the generation of turbulence in the RD era. The mechanism occurs during first-order phase transitions when bubbles of the new phase collide and merge [56][57][58]. In the early Universe, the electroweak and QCD phase transitions are potentially first order, although under early Universe conditions with very small chemical potentials the QCD transition is a smooth transition whereas the electroweak transition could be first order in certain Standard Model extensions. These violent phenomena can inject large kinetic energy into the plasma, thereby generating turbulence and allowing the possibility of small-scale dynamo action. The characteristic time scale for the phase transition is given by the rate of bubble nucleation β −1 . Here β is expected to be β ∼ 100H [58]. The largest bubbles reach a size β −1 v b by the end of the phase transition, where v b is the bubble wall expansion velocity. Thus, we take L β −1 v b as the largest stirring scale and τ stir = β −1 as the stirring time scale [58]. The phase boundary can propagate via two modes, deflagration and detonation, where the wall velocity v b is subsonic and supersonic respectively [56,57]. It has been argued in the literature that deflagrations are unstable to becoming detonations via bubble wall instabilities [59]. Hence, for simplicity we will only consider detonations, where the wall velocity is fully determined and given by [56] v b (α) = 1 1 + α 1 √ 3 + α 2 + 2α 3 .(20) Here α ≡ ρ vac /ρ thermal determines the strength of the phase transition. In this case, the fraction of vacuum energy converted to kinetic energy κ ≡ ρ kin /ρ vac takes the form [57] κ(α) = 1 1 + 0.72α 0.72α + 4 27 3α 2 .(21) For phase transitions that give large stirring times compared to the eddy turnover time of the largest scale τ stir τ L , a direct cascade of energy is set up and a state of fully developed turbulence is established in a time scale τ L and can be expected for a duration time τ stir [58]. Since the rate of energy dissipation is equal to the mean input power in stationary turbulence, the amplitude of the Kolmogorov spectrum can be easily determined. This calculation is done in Ref. [58], where they show that v rms L (ακv b ) 1/3 and argue that v rms L < ∼ 1/ √ 3. Hence, the condition for this simpler case τ stir τ L is translated to 3v b √ 2ακ [58,60] which is satisfied only for α > ∼ 1. Therefore, in this case of strong detonation α > ∼ 1 we have v rms L ∼ 1 [57]. However, if τ stir < ∼ τ L , a state of turbulence can still be expected [58]. The stirring corresponds to an impulsive force acting on the plasma that will cascade down to smaller scales. Eddy flows on large scales L act as a source for eddies on smaller scales for a duration time τ L . Following Ref. [60], in the time scale τ L , we neglect the decay of the turbulence and assume a state of fully developed turbulence for a duration time τ L [58]. Numerical work in Ref. [61] has established that kinetic energy in the form of acoustic waves persist well beyond the time of the phase transition. The nonlinear interaction of these acoustic waves could also be a source of turbulence on larger time scales. In the weak detonation limit α < ∼ 1, we find [57,58] v rms L √ 2ακ 3(2π) 4/3 .(22) Hence, for first-order phase transitions of strengths in the range α ∼ (10 −5 − 10 −1 ), we find turbulent velocities In this case, the turbulence is expected to be of Kolmogorov type. C. Damping of turbulence Velocity perturbations of the baryonic fluid are damped below a scale l D due to particles diffusing out of overdense regions. The damping is very efficient and given by (see for example Ref. [62]) v ∝ exp − l ν,γ D l c 2 ,(24) where l c is a comoving length scale. In the RD era, the comoving damping scale due to neutrinos or photons random walking out of perturbations is given by [29,62] (l ν,γ D ) 2 t 0 l ν,γ mfp,c (t ) a(t ) dt ,(25) where l ν,γ mfp,c is the comoving particle mean-free-path (mfp). The efficient damping of velocity perturbations is seen in the spectrum in Fig. 1. This important effect must be considered carefully when we come to investigate the scales of turbulence in the RD era (see next section). Here, we briefly note that although turbulent velocity fluctuations are efficiently damped due to diffusing particles, magnetic fields become overdamped and survive through such viscous and free-streaming regimes (this effect is discussed in detail in Sec. IV B) [30,31,63]. (19) and (23) respectively, and the damping scale due to neutrino diffusion l ν D [cf. Eq. (25)]. Here, we assume that the damping scale due to neutrino diffusion is the only relevant damping scale at this time i.e. the velocity perturbations on small scales are not damped due to physical processes at higher temperatures. Indeed, this is a safe assumption since neutrinos are the most weakly interacting particles in the Standard Model and at early times are the most efficient heat transporters. Let us first consider the turbulence generated by the primordial density perturbations (PDP). From Fig. 2 we can see that for T > ∼ 0.2 GeV the stirring scale L c,PDP (the lower blue dotted line in the figure) is larger than the damping scale l ν D . Hence, the velocity perturbations are not damped and we can use the value given in Eq. (19). With this value for the v rms L we can calculate the Reynolds numbers R e (L c ) from Eq. (7), these are shown in Fig. 3 (the lower dotted blue line). We find that R e (L c ) 1 for 0.2 < ∼ T /GeV < ∼ 100. The largest stirring scale, over which large Reynolds numbers are found, is roughly given at T 0.2 GeV i.e. L c,PDP ∼ 10 −5 pc. Hence, at these times, between the damping scale l ν D and L c,PDP , we expect a state of fully developed turbulence. However, for T < ∼ 0.2 GeV, the scale L c,PDP is below the damping scale l ν D , which means that velocity perturbations generated by the primordial density perturbation are exponentially damped [cf. Eq. (24)] and so are the Reynolds numbers. Thus, below this temperature, the plasma is in a viscous regime and there is no turbulence. For turbulence generated by first-order phase transitions (PT), the stirring scale can be much larger (the upper dotdashed blue line in Fig. 2 to be very large between the EW and QCD scales, indicating a highly turbulent state. The largest stirring scale, over which large Reynolds numbers are found, is roughly given by the horizon size at that time of the phase transition: 1/aH\| QCD ∼ 0.1 pc and 1/aH\| EW ∼ 10 −4 pc for the QCD and EW phase transitions respectively. The evolution of relevant scales from the time of neutrino decoupling T dec 2.6 MeV to a time long after e ± annihilation T 100 eV is shown in Fig. 4. In this epoch the photons generate the plasma viscosity. The scales of interest are the comoving Hubble scale l H = 1/aH, the largest stirring scale L c,PDP = v rms L /aH with the value v rms L from Eq. (19) and both damping scales l ν,γ D given by Eq. (25). The largest damping scale due to neutrino diffusion (which occurs at an earlier time) is approximately given by the particle horizon at the time of neutrino decoupling i.e. l ν D ≈ 1/aH\| dec 42 pc [31]. Here, we assume that, if there is indeed a first-order phase transition, it would have occurred at a much earlier time. Therefore, in this epoch, turbulence can only be generated by the primordial density perturbations. However, with the value of the rms velocity from Eq. (19) damped only by photon diffusion, we can clearly see from Fig. 4 that the scale L c,PDP is below the largest damping scale due to neutrino diffusion l ν D ≈ 42 pc throughout this epoch. Therefore, the velocity pertur- bations are efficiently damped by particle diffusion [cf. Eq. (24)]. Indeed, on the scale L c,PDP , the velocity perturbations generated by the primordial density perturbations are completely erased due to neutrino diffusion at this time. Hence, the kinetic Reynolds numbers become vanishingly small even when the shear viscosity is very small. Thus, at these temperatures, the plasma is in a viscous regime and there is no turbulence. IV. EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS In this section we consider the cosmological evolution of magnetic fields from the time of their generation to the present day (see for example Refs. [30,31,63] ). We first consider the amplification of magnetic seed fields due to small-scale dynamo (SSD) action in the early Universe. Then, we consider the subsequent evolution to the present time. In order to compare with observations, it is important to theoretically determine the final magnetic field strength and coherence length. 19), and l γ D is the damping scale due to photons (dotdashed, red) given by Eq. (25). Velocity fluctuations below the scale l ν D ≈ 1/aH\| dec 42 pc (shaded area) are damped due to neutrino diffusion at an earlier time, see Fig. 2. Hence, below lc ≈ 42 pc, the velocity perturbations generated by the primordial density perturbations are completely erased and no turbulence can be generated in this epoch. A. Amplification by small-scale dynamo action In Sec. III we identified two mechanisms that generate turbulence in the early Universe. In a turbulent and weakly magnetized plasma, small magnetic seed fields can be amplified exponentially through the SSD action (as described in Sec. II). Let us now assume that small magnetic seed fields exist at the time of the EW phase transition. These fields may have been generated at the phase transition [10] or at an earlier time (for example during inflation [9]). We now investigate the possibility of SSD action in the RD era. The injection of kinetic energy, together with large Reynolds numbers, leads to a state of fully developed turbulence. In the previous section we found that turbulence is expected due to primordial density perturbations or first-order phase transitions at high temperatures, between 0.2 < ∼ T /GeV < ∼ 100 (see Fig. 3). Indeed, we expect fully developed turbulence below the stirring scale L c = v rms L /aH. At these temperatures, the conductivity σ is given by [64] 0.76T < ∼ σ < ∼ 6.7T ,(26) where the larger value corresponds to the upper temperature bound. With the above we can calculate the Prandtl numbers and the magnetic Reynolds numbers. These are also shown in Fig. 3. The Prandtl numbers at these times are very large, P m ∼ (10 2 − 10 12 ), which means that we can neglect dissipative effects due to finite conductivity throughout the epoch of interest. From Eq. (4), we find the magnetic Reynolds numbers R m (L c ) ∼ (10 9 − 10 12 ) for turbulence generated by primordial density perturbations and a maximum range of R m (L c ) ∼ (10 16 − 10 18 ), using the upper value v rms L ∼ 0.1 in Eq. (23), for turbulence generated by first-order phase transitions. The large magnetic Reynolds numbers, R m R cr m ≈ 60 (for Kolmogorov turbulence) [17], indicate that we are well within the regime where the SSD mechanism is expected to operate. Figure 5 shows the magnetic field growth rate Γ, where B rms ∝ exp(Γt), which is determined from the Kazantsev model of the SSD mechanism and given in Eq. (5). The growth rate depends on the type of turbulence, where ϑ = 1/3, 1/2, applicable on the inertial range l diss < l < L [cf. Eq. (3)], for Kolmogorov and Burgers type turbulence respectively. Here, we assume that the turbulence is of Kolmogorov type, which is relevant for the subsonic velocity fluctuations determined in this paper. Now, since Γ varies in time, it will be useful to consider the number of e-foldings given by N ≡ Γ(t)dt. In Fig. 5, the growth rate is shown in units of the turnover rate of the largest eddy τ −1 L and N (T ) is shown where Γ(T ) is integrated from T = 100 GeV to temperature T . Since the magnetic field grows as B rms ∝ exp(Γt), the number of e-foldings gives the total amplification factor. Due to very large Reynolds numbers, the growth rate is initially very large and the number of e-foldings quickly becomes large. Hence, we find a very rapid increase in the field strength and a huge amplification factor leading to rapid saturation. The phase of rapid exponential amplification comes to an end when the magnetic energy becomes comparable to the kinetic energy. In the radiation dominated epoch, this saturation occurs when B 2 (x) ≈ 8 45 π 3 εg * T 4 v 2 (x) ,(27) where g * is the total number of effective relativistic degrees of freedom, and the parameter ε quantifies the saturation efficiency (see Sec. II). Without further dynamical evolution, the field strength will only be diluted by the expansion B ∝ a −2 ∝ T 2 . Let us assume that the magnetic field becomes saturated on the largest forcing scale L c . Thus, redshifted to present day values we find B rms 0 = a 2 B rms ≈ 8 45 π 3 εg * (T * ) T 2 0 v rms L ,(28) where T 0 is the present day photon temperature and T * is the radiation temperature at the time of magnetic field amplification. With this simple assumption, we find that in order to saturate magnetic seed fields of strength We can now estimate the saturated field strength from Eq. (28). For turbulence generated by the primordial density perturbation, the kinetic energy is given by the typical velocity fluctuations in Eq. (19). Hence, we find a saturated magnetic field strength whose value today is B rms 0 ≈ 1 × 10 −9 ε 1/2 G. For turbulence generated by first-order phase transitions, the kinetic energy is given by the velocity fluctuations in Eq. (23) i.e. v rms L ∼ (10 −4 − 10 −1 ). Hence, we find the saturated field strengths in the range B rms 0 ≈ (10 −3 − 1)ε 1/2 µG. Here we note that the amplification up to saturation of magnetic seed fields from kinetic energy injected at firstorder phase transitions has been considered previously in the literature, see for example Ref. [10] and references within. However, we believe it is important to point out that there is a well-defined and clearly described dynamo process that does the amplification, namely the smallscale dynamo theory provides the relevant framework for predictions regarding growth rates and saturation levels. In the case where turbulence is generated by the primordial density perturbation, the fluid forcing is continuous and turbulence is fully developed throughout this epoch. Hence, as we can see from Fig. 5, the number of e-foldings quickly become very large N > ∼ O(100) and there is enough time in this epoch for tiny magnetic seed fields to saturate. However, for turbulence generated by first-order phase transitions, the fluid forcing is not continuous. In Sec. III A, we argue that for α < ∼ 1, the time scale for the duration of turbulence is approximately τ L [58]. Therefore, the number of e-foldings N ≡ Γdt is roughly given by Γ/τ −1 L ≈ R 1/2 e 1 (see Eq. (5) assuming a Kolmogorov spectrum). Hence, the magnetic fields can easily become saturated in the time scale of the phase transition. The saturated field strengths are very strong. However, the scales over which we expect saturation are very small. We expect saturation on scales from the damping scale l D up to the largest forcing scale L c . For a Kolmogorov spectrum, where most power resides on the largest scale, we can identify L c as the comoving coherence length λ c of the magnetic field. For turbulence generated by the primordial density perturbation, the largest forcing scale in the epoch considered is L c at T 0.2GeV i.e. λ c ∼ 10 −5 pc (see Fig. 2). For turbulence generated by first-order phase transitions, the length scales on which saturation is expected depends on the exact time of the phase transition. The basic constraint on the coherence length is the horizon size at the time of the phase transition; λ c 0.1 pc and λ c 10 −4 pc for the QCD and EW phase transitions respectively. To obtain larger coherence lengths, the magnetic field would need to be amplified and saturated at a later time when the Hubble horizon is larger and turbulence develops on larger scales. Unfortunately, the velocity perturbations generated by the primordial density perturbation are efficiently damped below T 0.2 GeV and therefore do not lead to a state of fully developed turbulence. Without turbulence there is no SSD action and therefore no amplification of primordial magnetic fields. For the SSD mechanism to be effective at a later time in the RD era, a different mechanism which injects kinetic energy into the plasma is required. B. Subsequent evolution In the previous section we considered the amplification of magnetic seed fields at the time when turbulent kinetic energy is injected into the primordial plasma. The integral scale of the magnetic field, which is amplified up to equipartition with the kinetic energy, is determined by the scale of the injected turbulence. In this section we consider the cosmological evolution of the magnetic field strengths and coherence lengths from the time after the injection of turbulence to the present time. We follow the works of Refs. [30,31,63,65,66] for the growth of the coherence length and the damping of turbulence in this subsequent regime in order to determine the final magnetic field strengths and coherence lengths. The most important result from such works is that magnetic fields generated in early epochs survive through viscous and free-streaming regimes, unlike velocity perturbations which are efficiently damped (see Sec. III C). In the turbulent regimes, strong magnetic fields on small scales drive turbulence in the plasma up to equipartition. The turbulence removes power on small scales thereby increasing the correlation length and reducing the field strength. This turbulent magnetohydrodynamic (MHD) effect, free turbulent decay, depends on the type of turbulence generated and on whether or not the mag-netic field is helical [30,63]. For nonhelical fields, the growth of the coherence length is purely due to the dissipation of power on small scales. In the helical case, there is an inverse cascade effect where the power on larger scales grows [30,67]. Let us only consider nonhelical magnetic fields in turbulent regimes. The growth of λ c is a power law in time with an index that depends on the magnetic field spectrum n B . For magnetic fields generated by causal processes, e.g. phase transitions, the index is n B = 2 [68]. In this case λ c ∼ t 2/7 , t 2/5 and the magnetic field strength at the scale λ c evolves as a 2 B rms ∼ t −5/7 , t −3/5 for incompressible (Kolmogorov) and compressible (Burgers) type turbulence respectively [30,63,65,66]. Hence, the evolution of the field strength and coherence length up to the time of recombination is determined by the relation a 2 B rms ∼ λ −n c ,(29) where n = 5 2 , 3 2 for Kolmogorov and Burgers turbulence respectively [30,63,65,66]. Besides the evolution of magnetic fields in the turbulent regimes, in the RD era, there are also epochs of viscous damping and free-streaming. In a magnetized plasma, there are different modes in which magnetic energy can be stored; fast, slow and Alfvén modes, as opposed to only the acoustic mode in the case of velocity fluctuations. The fast magnetosonic mode decays in the same manner as the acoustic mode due to particle diffusion, see Eq. (25). However, the slow and Alfvén modes evolve differently and can become overdamped [31]. Hence, the magnetic energy stored in these modes becomes frozen-in. The overdamping depends on the scales and magnetic field strength. Therefore, magnetic fields survive through viscous and free-streaming regimes, which is in contrast to turbulent velocity fluctuations that become efficiently damped. In the viscous damping and free-streaming regimes, the evolution of a 2 B rms and λ c is halted until free turbulent decay begins again [30,63]. The evolution due to free turbulent decay terminates when the correlation length and field strength end on the line, in the {a 2 B rms , λ c } plane, given by [30,63] a 2 B rms 10 −8 λ c Mpc G .(30) This line corresponds to the largest eddies being processed at recombination 1/(a rec H rec ) λ c /v A with v A the Alfvén speed [63]. In the matter dominated Universe, there is no further evolution of the magnetic field correlation length λ c , although strong fields on small scales can drive turbulence at much later times in the intergalactic medium and restore the turbulent decay [63]. In this sense, Eq. (30) becomes an upper bound on the present day magnetic field strength. The difference in evolution of the magnetic modes, in contrast to acoustic modes, means that fields generated and amplified in the radiation dominated era can survive to the present day. As seen in the previous subsection, turbulence generated by the primordial density perturbation can amplify tiny magnetic seed fields through the SSD mechanism to values of order a 2 B rms ∼ 1ε 1/2 nG on scales at most λ c ∼ 10 −5 pc. From eqs. (29) and (30), these fields would evolve to a 2 B rms ∼ 10 −6 ε 1/2 nG on scales λ c ∼ 10 −1 pc. Since the primordial density perturbation is necessarily present for structure formation, such fields are guaranteed by the SSD mechanism and can play an important role in structure formation. Unfortunately, these fields are too weak on too short scales to explain the Fermi observations of TeV Blazars [6]. However, turbulence generated by first-order phase transitions can amplify magnetic seed fields to values of order a 2 B rms ∼ (10 −3 − 1)ε 1/2 µG on scales λ c ∼ (10 −4 − 10 −1 ) pc. These initial field configurations will evolve to a 2 B rms ∼ (10 −6 − 10 −3 )ε 1/2 nG on scales λ c ∼ (10 −1 − 10 2 ) pc. Such fields are strong enough to explain the Fermi observations of TeV Blazars [6]. V. SUMMARY In this paper we have identified two mechanisms that generate turbulence in the radiation dominated Universe. The two mechanisms inject kinetic energy into the primordial plasma at times when the kinetic Reynolds numbers are very large R e 1. With the injection of kinetic energy, which is determined by v rms L , and large Reynolds numbers, a state a fully developed turbulence is expected on the inertial range: l diss,c < l c < L c , where L c = v rms L /aH is the largest forcing scale and l diss,c ∼ L c R e (L) −3/4 is the dissipative scale (for Kolmogorov turbulence). The two mechanisms for generating velocity fluctuations are: by the primordial density perturbation and bubble collisions during first-order phase transitions. In Sec. IV A, we have seen that turbulence is inevitably generated by the primordial density perturbation prior to the QCD scale T > ∼ 200 MeV. In Sec. III B, we investigated the possibility of turbulence generated at first-order phase transitions between the electroweak (EW) and the QCD scales 200 MeV < ∼ T < ∼ 100 GeV. For the generation of turbulence at first-order phase transitions, we have simply followed the same assumptions made in the literature regarding the dynamics of the phase transitions [57,58,60]. The turbulence generated can amplify tiny magnetic seed fields through the small-scale dynamo (SSD) mechanism. If we assume that the plasma is already weakly magnetized by the time of the EW scale, then SSD amplification will inevitable occur. The rapid amplification ends at saturation when the magnetic and kinetic energies are in approximate equipartition E M /E kin ≈ ε, where the efficiency of the mechanism is characterized by ε. The saturation efficiency parameter ε, which is determined numerically, varies depending on the type of forcing. For rotational modes ε ≈ 1, whereas the saturation level is lower for compressive modes ε ∼ 10 −3 − 10 −4 [35]. We note that these numerical studies were carried out for P m ≈ 2, further numerical work is required to establish the saturation level for large Prandtl numbers. In this paper, we show that, even for tiny seed fields of strengths B seed 0 (10 −30 − 10 −20 ) nG, the SSD mechanism can operate for a long enough period of time and be efficient enough to amplify such fields to saturation. The magnetic field strength saturates at a 2 B rms ∼ 1ε 1/2 nG on scales at most λ c ∼ 10 −5 pc for turbulence generated by the primordial density perturbation. Such fields, assumed to be nonhelical, evolve to B rms 0 ∼ 10 −6 ε 1/2 nG on scales λ c ∼ 10 −1 pc due to free turbulent decay. For turbulence generated by first-order phase transitions, the SSD mechanism can be even more effective, since the turbulent velocities can be quite large compared to those generated by the primordial density perturbation. We show that the mechanism can amplify magnetic fields to strengths a 2 B rms ∼ (10 −3 − 1)ε 1/2 µG on scales λ c ∼ (10 −4 − 10 −1 ) pc. These initial field configurations evolve to a 2 B rms ∼ (10 −6 − 10 −3 )ε 1/2 nG on scales λ c ∼ (10 −1 − 10 2 ) pc due to free turbulent decay. Unfortunately, the damping of velocity perturbations due to neutrino diffusion inhibits turbulence from developing below the QCD scale T QCD 200 MeV (unless there is an injection of kinetic energy from a firstorder phase transition prior to neutrino decoupling at T > ∼ 2.6 MeV, see Fig. 2). Hence, it is difficult to generate turbulence from these mechanisms on larger length scales than l c ∼ 10 −1 pc. Without turbulence there is no SSD action and therefore no amplification of primordial magnetic fields on larger scales. Although turbulent velocities are completely erased in viscous and free-streaming regimes, magnetic fields are overdamped and can survive to the present day. Such fields would fill the voids in the large scale structure and provide the seeds for magnetic fields generated by structure formation and galactic dynamo. Unfortunately, the saturated field strengths due to turbulence generated by the primordial density perturbation are too weak on too short scales in the voids of the large scale structure to explain the Fermi observations of TeV Blazars [6]. However, the field strengths obtained due to turbulence generated by first-order phase transitions are strong enough to explain such observations. VI. DISCUSSION AND CONCLUSION The discrepancy between theoretically generated and observed magnetic fields in the Universe needs explaining. The galactic dynamo can be a very effective mechanism at producing the µG fields observed in spiral galaxies [1]. However, strong fields in young galaxies, clusters and superclusters of galaxies and in the intergalactic medium require further explanation [2][3][4][5][6]. As noted in a number of numerical and analytical works, the rapid amplification of magnetic seed fields can occur due to the turbulent motions of the conducting plasma. This small-scale dynamo (SSD) mechanism is believed to play a crucial role in the formation of large magnetic fields in a number of astronomical settings, from stars to galaxies and the intergalactic medium [19-21, 39, 40]. For these settings, the turbulent motions arise from gravitational collapse, accretion and supernovae explosions. Hence, the SSD mechanism can be highly effective at magnetizing structures in the early Universe. However, the large field strengths apparently observed in the voids of the large-scale structure [6] still require an explanation. Magnetic seed fields will almost certainly be generated at some level in the early Universe through a variety of mechanisms. Such mechanisms include inflation [9], phase transitions [10] and the Harrison mechanism through the generation of vorticity [16]. The SSD mechanism for the amplification of such seed fields could play an important role for the explanation of the observed large magnetic fields throughout the Universe. In this paper we have demonstrated that the conditions necessary for such turbulent amplification arise in the radiation dominated Universe before the onset of structure formation. We have shown that significant turbulence is generated in this early epoch by at least two mechanisms; velocity perturbations generated by the primordial density perturbation and bubble collisions in firstorder phase transitions. Turbulent plasma motions arise inevitably from perturbations of the gravitational potential. The continuous production of velocity perturbations upon horizon entry of primordial density modes, act as a continuous forcing of the fluid on the largest scales. Therefore, in regimes of large Reynolds numbers, a state of stationary fully developed turbulence is expected. Turbulent flow can be triggered, for example, by thermal fluctuations on very small scales [34]. Turbulence can also be injected into the plasma by bubble collisions during first-order phase transitions. Although the kinetic energy injection occurs only for the duration of the phase transition, we argue, following Refs. [58,60], that a state of fully developed turbulence is also expected from this mechanism. Once fully developed turbulence is established, the Kazantsev model of the SSD mechanism can be used to estimate the magnetic field growth rate. We have demonstrated that the Prandtl numbers are very large in the regime considered. Thus, the results from the Kazantsev theory for P m 1 are applicable. The analytical work shows that the magnetic field growth rate depends on the kinetic Reynolds numbers [36,46], which are very large in our case. We have shown that, for both models of turbulence, the amplification is strong enough for small magnetic seed fields to reach a saturated state. The saturated state is given by the approximate equipartition between magnetic and kinetic energy E M /E kin ≈ ε, where the parameter ε characterizes the efficiency of the mechanism. We note that numerical studies at Prandtl numbers P m ≈ 2 indicate that the SSD mechanism is more efficient for rotational modes, where the saturation efficiency ε is close to unity [35]. Whereas the saturation level is lower for compressive modes ε ∼ 10 −3 − 10 −4 [35]. However, further numerical work is required to establish the saturation level for larger Prandtl numbers and smaller Mach numbers relevant to our settings. We also note that, although only longitudinal velocity modes are generated by first-order primordial density perturbations, rotational modes are generated at second order in cosmological perturbations [51][52][53][54][55]. Also, there is no reason not to expect rotational modes generated by firstorder phase transitions. In any case, since the Reynolds numbers are so large, nonlinear interactions can play a role leading to a state of fully developed turbulence with both rotational and longitudinal modes. In particular, we expect that, below the integral scale, Kolmogorov type turbulence is established. But we stress that the SSD mechanism works independently of the type of turbulence [36,45,46]. Indeed, even purely irrotational turbulence can still drive a small-scale dynamo [35,36,46] . Hence, the efficient amplification of magnetic fields seems unavoidable, leading to a strongly magnetized early Universe prior to structure formation. For the two mechanisms of turbulence investigated in this paper, we calculated the saturated field strengths and their subsequent evolution up to the present day. We note that although turbulence is completely erased in viscous and free-streaming regimes, magnetic fields are overdamped and can survive to the present day. Therefore, the most important epochs of evolution are due to free turbulent decay. This turbulent MHD effect decreases the field strength and increases the coherence length in nonhelical fields [30,63]. From the turbu-lence generated by the primordial density perturbation we found B rms 0 ∼ 10 −6 ε 1/2 nG on scales λ c ∼ 10 −1 pc. Unfortunately, even for a high efficiency factor ε ∼ 1, these fields are too weak on too short scales to explain the Fermi observations of TeV Blazars [6]. From the turbulence generated by first-order phase transitions, we found B rms 0 ∼ (10 −6 − 10 −3 )ε 1/2 nG on scales λ c ∼ (10 −1 − 10 2 ) pc. Such fields are strong enough to explain the apparent observations of intergalactic magnetic fields suggested by the Fermi results [6]. Thus, in this paper we have demonstrated that the conditions are right for the efficient amplification of magnetic fields via the small-scale dynamo. The mechanism generates large field strengths, albeit on very small scales, which could explain observations of magnetic fields in the voids of the large-scale structure and have an impact on early structure formation. FIG. 1 . 1The spectrum of velocity perturbations, given in Eq.(16), at a temperature T 0.21 GeV. The figure shows how the velocity perturbations are generated upon horizon entry (at around lc 0.31pc) and are scale invariant until damping due to neutrino diffusion takes over at around lc 1.5 × 10 −5 pc (see Sec. III C). ∼ (10 −4 − 10 −1 ) . D. The scales of turbulence The evolution of relevant scales from the time of the electroweak (EW) scale T EW ∼ 100 GeV to the time of neutrino decoupling T dec 2.6 MeV is shown in Fig. 2. In this early epoch the neutrinos generate the plasma viscosity. The QCD phase transition occurs at around T QCD 200 MeV. The scales of interest are the comoving Hubble scale l H = 1/aH, the largest stirring scale L c = v rms L /aH with the values v rms L from primordial density perturbations (PDP) and first-order phase transitions (PT) in eqs. ), since the velocity fluctuations can be much stronger [cf. Eq. (23)]. Hence, if the phase transition occurs at any time in the epoch between T ∼ (10 2 − 10 −3 ) GeV, a state of turbulence can be expected. In Fig. 3, the Reynolds numbers are shown FIG. 2 . 2This figure shows the evolution of relevant comoving scales from the EW scale TEW ∼ 100 GeV to the time of neutrino decoupling at T dec 2.6 MeV. In this early epoch the neutrinos generate the plasma viscosity. The QCD phase transition occurs at around TQCD 200 MeV. Here, lH = 1/aH is the Hubble scale (solid, black), l ν mfp,c is the neutrino mean-free-path (dashed, green) and l ν D is the damping scale due to neutrino diffusion given by Eq. (25) (dashed, brown). For turbulence generated by the primordial density perturbation (PDP) and first-order phase transitions (PT), the largest stirring scales Lc = v rms L /aH are shown with the values v rms L from eqs. (19) and (23) (dotted, blue) and (dotdashed, blue) respectively. In Eq. (23), the upper value for v rms L is used. Although the turbulent motions from PDP become completely damped below T 0.2 GeV, the magnetic field gets frozen-in with integral scale λc (dashed, red). shows the evolution of the different Reynolds numbers from the EW scale TEW ∼ 100 GeV to the QCD scale TQCD 200 MeV. The kinetic Reynolds number Re(Lc) (dashed, blue) is determined from eqs. (7) and (8) and the magnetic Reynolds number Rm(Lc) (dotted, red) from eqs.(4)and(26). The lower lines for the Reynolds numbers correspond to turbulence generated by the primordial density perturbations (PDP) i.e. using the (undamped) value for v rms L from Eq.(19). The upper lines for the Reynolds numbers correspond to turbulence generated by first-order phase transitions (PT) with the (undamped) value v rms L ∼ 10 −1 from Eq. (23). This figure clearly shows that Re 1 in all cases, suggesting that the plasma is in a state of fully developed turbulence during this time. The figure also shows the Prandtl numbers Pm = Rm/Re 1 (solid, brown). FIG. 4 . 4This figure shows the evolution of relevant comoving scales from the time of neutrino decoupling at T dec 2.6 MeV to a time long after e ± annihilation at T 100 eV. In this epoch the photons generate the plasma viscosity. The time of e ± annihilation occurs at me > ∼ T > ∼ 20 keV. Here, lH = 1/aH is the Hubble scale (solid, black), l γ mfp,c is the photon meanfree-path (dashed, green), Lc,PDP = v rms L /aH (dotted, blue) with the value v rms L from Eq. ( B seed 0 ( 010 −30 − 10 −20 ) nG up to O(1) nG level we require e-folding numbers N 46 − 70. shows the SSD growth rate Γ (solid) and number of e-foldings N = Γ(t)dt (dashed) from the time of the EW scale TEW ∼ 100 GeV to the QCD scale TQCD 200 MeV. Recall that the magnetic seed field is amplified as Brms ∝ exp(Γt). The growth rate, shown here in units of τ −1 L , is determined from Eq.(5)where Kolmogorov turbulence ϑ = 1/3 is assumed. The growth rate and the number of e-foldings for turbulence generated by the primordial density perturbation and first-order phase transitions are shown in (black) and (blue) respectively. ACKNOWLEDGMENTSWe wish to thank the anonymous referees for their constructive comments. This work was supported by the Deutsche Forschungsgemeinschaft through the collaborative research centre SFB 676, by the Helmholtz Alliance for Astroparticle Phyics (HAP) funded by the Initiative and Networking Fund of the Helmholtz Association. 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[ "Probing Neutrino Oscillations in Supersymmetric Models at the Large Hadron Collider", "Probing Neutrino Oscillations in Supersymmetric Models at the Large Hadron Collider" ]	[ "F De Campos \nDepartamento de Física e Química\nUniversidade Estadual Paulista\nGuaratinguetáSPBrazil\n", "O J P Éboli \nInstituto de Física\nUniversidade de São Paulo\nSão PauloSPBrazil\n", "M Hirsch \nAHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna\nApartado Postal\n22085, E-46071ValenciaSpain\n", "M B Magro \nInstituto de Física\nUniversidade de São Paulo\nSão Paulo -SPBrazil\n\nCentro Universitário Fundação Santo André\nSanto AndréSPBrazil\n", "W Porod \nAHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna\nApartado Postal\n22085, E-46071ValenciaSpain\n\nInstitut für Theoretische Physik und Astronomie\nUniversität Würzburg\nGermany\n", "¶ D Restrepo \nInstituto de Física\nUniversidad de Antioquia\nColombia\n", "J W F Valle \nAHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna\nApartado Postal\n22085, E-46071ValenciaSpain\n" ]	[ "Departamento de Física e Química\nUniversidade Estadual Paulista\nGuaratinguetáSPBrazil", "Instituto de Física\nUniversidade de São Paulo\nSão PauloSPBrazil", "AHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna\nApartado Postal\n22085, E-46071ValenciaSpain", "Instituto de Física\nUniversidade de São Paulo\nSão Paulo -SPBrazil", "Centro Universitário Fundação Santo André\nSanto AndréSPBrazil", "AHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna\nApartado Postal\n22085, E-46071ValenciaSpain", "Institut für Theoretische Physik und Astronomie\nUniversität Würzburg\nGermany", "Instituto de Física\nUniversidad de Antioquia\nColombia", "AHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna\nApartado Postal\n22085, E-46071ValenciaSpain" ]	[]	The lightest supersymmetric particle may decay with branching ratios that correlate with neutrino oscillation parameters. In this case the CERN Large Hadron Collider (LHC) has the potential to probe the atmospheric neutrino mixing angle with sensitivity competitive to its low-energy determination by underground experiments. Under realistic detection assumptions, we identify the necessary conditions for the experiments at CERN's LHC to probe the simplest scenario for neutrino masses induced by minimal supergravity with bilinear R parity violation.	10.1103/physrevd.82.075002	[ "https://arxiv.org/pdf/1006.5075v2.pdf" ]	54,693,423	1006.5075	f5da213d7ff839550474b58335ab66b7d4d399a8	Probing Neutrino Oscillations in Supersymmetric Models at the Large Hadron Collider 22 Oct 2010 F De Campos Departamento de Física e Química Universidade Estadual Paulista GuaratinguetáSPBrazil O J P Éboli Instituto de Física Universidade de São Paulo São PauloSPBrazil M Hirsch AHEP Group Instituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna Apartado Postal 22085, E-46071ValenciaSpain M B Magro Instituto de Física Universidade de São Paulo São Paulo -SPBrazil Centro Universitário Fundação Santo André Santo AndréSPBrazil W Porod AHEP Group Instituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna Apartado Postal 22085, E-46071ValenciaSpain Institut für Theoretische Physik und Astronomie Universität Würzburg Germany ¶ D Restrepo Instituto de Física Universidad de Antioquia Colombia J W F Valle AHEP Group Instituto de Física Corpuscular -C.S.I.C./Universitat de València Edificio Institutos de Paterna Apartado Postal 22085, E-46071ValenciaSpain Probing Neutrino Oscillations in Supersymmetric Models at the Large Hadron Collider 22 Oct 2010numbers: 1130Pb1260Jv1460Pq9530Cq The lightest supersymmetric particle may decay with branching ratios that correlate with neutrino oscillation parameters. In this case the CERN Large Hadron Collider (LHC) has the potential to probe the atmospheric neutrino mixing angle with sensitivity competitive to its low-energy determination by underground experiments. Under realistic detection assumptions, we identify the necessary conditions for the experiments at CERN's LHC to probe the simplest scenario for neutrino masses induced by minimal supergravity with bilinear R parity violation. I. INTRODUCTION The CERN Large Hadron Collider (LHC) will provide high enough center-of-mass energy to probe directly the weak scale and the origin of mass [1][2][3][4][5][6]. In addition to its designed potential, here we show how LHC searches for new physics at the TeV region may provide an unexpected opportunity to probe neutrino properties, currently determined only in neutrino oscillation experiments [7], shedding light on some of the key issues in neutrino physics. We illustrate how this works in a class of supersymmetric models where the lepton number is broken, together with the so-called R parity symmetry [8]. Even when the latter holds as a symmetry at the Lagrangian level, as in some SO (10) unification schemes, R parity breaking may be driven spontaneously by a nonzero vacuum expectation value of an SU(3) ⊗ SU(2) ⊗ U(1) singlet sneutrino [9][10][11][12]. In this case the low-energy theory is no longer described by the minimal supersymmetric standard model, but contains new R parity violating interactions [13][14][15]. The simplest realization of this scenario leads to an effective model with bilinear violation of R parity [16][17][18][19][20]. The latter constitutes the minimal way to break R parity in the minimal supersymmetric standard model and provides the simplest intrinsically supersymmetric way to induce neutrino masses [21][22][23][24]. Its main feature is that it relates lightest supersymmetric particle (LSP) decay properties and neutrino mixing angles [25][26][27]. Here we demonstrate that indeed, under realistic assumptions, the simplest scenario for neutrino masses in su-persymmetry (SUSY) with bilinear violation of R parity can be tested at the LHC in a crucial way and potentially falsified. We identify the regions of minimal supergravity (mSUGRA) parameters, event reconstruction efficiencies and luminosities where the LHC will be able to probe the atmospheric neutrino mixing angle with sensitivity competitive to its low-energy determination by underground experiments, both for 7 and 14 TeV center-of-mass energies. For the sake of definiteness, we consider the minimal supergravity model supplemented with bilinear R parity breaking [22][23][24] added at the electroweak scale; we refer to this scenario as RmSUGRA. In this effective model one typically finds that the atmospheric scale is generated at tree level by a weak-scale neutralino-exchange seesaw, while the solar scale is induced radiatively [22]. The LSP lacks a symmetry to render it stable and, given the neutrino mass scales indicated by oscillation experiments, typically decays inside the LHC detectors [22,23,25] 1 . As an illustration we depict the neutralino LSP decay length in Fig. 1. We can see from Fig. 1 that the expected decay lengths are large enough to be experimentally resolved, leading to displaced vertex events [33,34]. More strikingly, one finds that in such a RmSUGRA model one has a strict correlation between neutralino decay properties measurable at high-energy collider experiments and neutrino mixing angles determined in low-energy neutrino oscillation experiments, that is tan 2 θ atm ≃ BR(χ 0 1 → µ ± W ∓ ) BR(χ 0 1 → τ ± W ∓ ) .(1) The derivation of Eq. (1) can be found in [25]. In short, the relation between the neutralino decay branching ratio and the low-energy neutrino angle in the bilinear model can be understood in the following way. At tree-level in RmSUGRA the neutrino mass matrix is given by [22] m ef f = M 1 g 2 +M 2 g ′ 2 4 det(M χ 0 )    Λ 2 e Λ e Λ µ Λ e Λ τ Λ e Λ µ Λ 2 µ Λ µ Λ τ Λ e Λ τ Λ µ Λ τ Λ 2 τ   (2) where Λ i = µv i +v D ǫ i and ǫ i and v i are the bilinear superpotential parameters and scalar neutrino vacuum expectation value, respectively. Equation (2) is diagonalized by two angles; the relevant one for this discussion is the angle tan θ 23 = − Λµ Λτ . One can understand this tree-level mass as a seesaw-type neutrino mass with the right-handed neutrino and the Yukawa couplings of the ordinary seesaw replaced by the neutralinos of the minimal supersymmetric standard model and couplings of the form cΛ i , where c is some combination of (generation independent) parameters. These couplings, which determine (the generation structure of) the neutrino mass matrix, also determine the couplings [25]. Taking the ratio of decays to different generations the prefactors c drop out and one finds Eq. (1), when the angle tan θ 23 is identified with the atmospheric neutrino angle. One-loop corrections tend to modify this relation, but, as long as the loop corrections are smaller than the tree-level neutrino mass, Eq. (1) is a good approximation [25]. χ 0 i − l ± i − W ∓ and χ ± i − ν i − W ∓ In other words, as seen in Fig. 2, the LSP decay pattern is predicted by the low-energy measurement of the atmospheric angle [21,25], currently determined by underground low-energy neutrino experiments [7], as sin 2 θ atm = 0.50 +0.07 1 decay branching ratios, Br(χ 0 1 → µq ′q ) over Br(χ 0 1 → τ q ′q ) in terms of the atmospheric angle in bilinear R parity violation [25]. The shaded bands include the variation of the model parameters in such a way that the neutrino masses and mixing angles fit the required values within 3σ. In this paper we show how a high-energy measurement of LSP decay branching ratios at the LHC allows for a redetermination of θ atm and hence a clear test of the model. We provide quantitative estimates of how well this ratio of branchings should be measured at LHC in order to be competitive with current oscillation measurements. This issue has already been addressed but only at the parton level, using some semirealistic acceptance and reconstruction cuts, and for just one specific mSUGRA point [35]. II. FRAMEWORK OF OUR ANALYSIS Our goal is to present a more detailed analysis of the LHC potential to measure the LSP branching ratios required to test the relation shown in Eq. (1), going beyond the approximations made in the previous work of Ref. [35]. The generation of the supersymmetric spectrum and decays in the scope of the RmSUGRA model was carried out using the SPheno package [36] 2 . The event generation was done employing PYTHIA [37] with the RmSUGRA particle properties being passed into it in the SUSY Les Houches accord (SLHA) format [38,39]. Jets were defined using the subroutine PYCELL with a cone size of ∆R = 0.4. A striking property of RmSUGRA models is the existence of displaced vertices associated to the LSP decay [34]. We use the detached vertices to probe the LSP branching ratio relation Eq. (1). In order to mimic the LHC potential to study displaced vertices we use a toy detector based on the ATLAS technical proposal [3]. We begin our analysis demanding that the events pass some basic requirements to guarantee that they will be triggered by the experimental collaborations. This is done because the LHC experiments have not defined so far any specific strategy to trigger displaced vertices with such high invariant mass, therefore, we restricted our analysis to events that would be accepted by the ongoing analyses. We accept events passing at least one of the following requirements, denoted as cut C1, 1. the event has one isolated electron or a photon with p T > 20 GeV; 2. the event has one isolated muon with p T > 6 GeV; 3. the event has two isolated electrons or photons with p T > 15 GeV; 4. the event has one jet with p T > 100 GeV; 5. the event has missing transversal energy in excess of 100 GeV. Next, in cut C2, we require that at least one of the neutralinos in the event decays beyond the primary vertex point, that is, outside an ellipsoid [34] x 5δ xy 2 + y 5δ xy 2 + z 5δ z 2 = 1 ,(3) where the z axis is taken along the beam direction. We made a conservative assumption, since we are not performing a detailed detector simulation, that the ellipsoid dimensions are 5 times the ATLAS expected resolutions in the transverse plane (δ xy = 20 µm) and in the beam direction (δ z = 500 µm), in order to ensure that the neutralino displaced vertex is distant of the primary vertex. We also demand that all tracks must be initiated inside the pixel inner detector within a radius of 550 mm and z axis length of 800 mm. A detached vertex complying with these requirements we called signal vertex. In order to check relation Eq. (1) we looked for detached vertices presenting a W associated to them and we must isolate the LSP decays into W µ and W τ . Moreover we consider only hadronic final states of the W as a necessary condition for the identification of the lepton flavor. In cut C3, which is designed for the W reconstruction, we require two jets with charged tracks intersecting the neutralino resolution ellipsoid, and invariant mass between 60 and 100 GeV. In order to be sure that the W reconstruction is clean, we further impose that the axes of other jets of the event to be outside of a cone ∆R = 0.8 of the W jets' axes. Note that this cut should eliminate standard model (SM) backgrounds coming from displaced vertices associated to b's or τ 's. To guarantee a high quality in the reconstruction of the displaced vertices we impose that the W decay jets must be central, having pseudorapidities \|η\| < 2.5; this constitutes our cut C4. The events passing the above requirements most probably originate from LSP decay, having basically no sizable standard model background, except for instrumental backgrounds and beam-gas interactions. A signal vertex is classified as originating from the LSP decay into a µW pair if it presents a µ ± and a hadronically decaying W stemming from the displaced vertex with transverse momentum p T > 6 GeV and \|η\| < 2.5. In the τ ± case we demanded that the τ ± associated to a detached W possesses p T > 20 GeV and \|η\| < 2.5. These requirements are called C5. Detecting taus is somewhat more complicated than detecting muons, so one needs to be more careful in reconstructing the τ W pair displaced vertex. The following criteria, denoted C6, are used to separate the detached vertices exhibiting a τ ± through its 1-and 3-prong decay modes. We check also that the secondary displaced vertex from tau decay does not spoil the signal vertex; i.e., we verify that the tau decay products point towards the LSP decay vertex within the experimental resolution. We define the neutralino resolution ellipsoid as the ellipsoid centered at the displaced vertex position of neutralino, v 1 , with axes δ xy = 12 µm and δ z = 77 µm based on ref. [3]. Let p prong be the momentum of either 1-prong tau decay or the sum of momenta of the 3-prong decays. Let also v 2 be the position of the secondary vertex coming from τ . We verify whether the line along p prong , crossing v 2 intersects the neutralino resolution ellipsoid. For this we require that for each τ , the discriminant of quadratic equation for parameter t 2 i p i prong t + v i 2 − v i 1 δ xy 2 + p 3 prong t + v 3 2 − v 3 1 δ z 2 − 1 = 0 (4) be equal to or greater than zero. In previous [35] analysis only 3-prong tau decays modes were considered. An additional cut C7 was applied to 3-prong tau events, i.e. we also require that one of the prongs has a transverse momentum p T > 9 GeV while the other two have p T > 2 GeV. In addition we check if all prongs lie within a cone radius of ∆R < 0.2 around the tau direction obtained from the prongs' tracks. Finally we require that the signal lepton (µ or τ ) be isolated; cut C8. µ isolation demands that there are no other tracks whose total transverse energy satisfies E T > 5 GeV within a cone ∆R > 0.3. The τ was required to be isolated using the same criteria as for the muon, but for an annulus of outer radius ∆R = 0.4 and inner radius ∆R = 0.1. Isolation of the leptons is a needed requirement to eliminate events presenting leptons generated inside jets and constitutes an important cut to reduce potential backgrounds. III. RESULTS AND DISCUSSION In order to access the effects of the above defined cuts C1-C8 we present detailed information on their effects for the mSUGRA SPS1a benchmark point [40] characterized by m 1/2 = 250 GeV, m 0 = 100 GeV, A 0 = −100 GeV, tan β = 10, and sgn(µ) = +1. This allows us to compare our results with the one previously obtained in [35]. For the default solution of SPheno to the neutrino masses and mixings, the relevant neutralino branching ratios are BR(χ 0 1 → W ± µ ∓ ) = 5.4% BR(χ 0 1 → W ± τ ∓ ) = 6.2% BR(χ 0 1 → Zν) = 1.2% BR(χ 0 1 → e ± τ ∓ ν) = 11.5% BR(χ 0 1 → µ ± τ ∓ ν) = 24.3% BR(χ 0 1 → τ ± τ ∓ ν) = 36.4% BR(χ 0 1 → bbν) = 14.7%;(5) with the R parity parameters being ǫ 1 = 0.0405 GeV, ǫ 2 = −0.0590 GeV, ǫ 3 = 0.0506 GeV, v 1 = −0.0027 GeV, v 2 = 0.0042 GeV, v 3 = −0.0033 GeV. Furthermore, for this choice of parameters the neutralino decay length is cτ = 1.1 mm, and it travels an average of 4.4 mm in the laboratory. From Table I we see that the vast majority of the events pass the trigger requirements C1, as expected. For the SPS1a SUSY point, the LSP decay length is sufficiently long to guarantee that a sizeable fraction of its decays take place away from the primary vertex; this reflects as a high efficiency for passing the cut C2. We have focused our attention to events presenting a W ± decaying into two jets through C3. It is interesting to notice that 63% of the W hadronic decays are in the form of two jets. Additional suppression of the signal by C3 comes from the matching of the sum of momenta of the charged tracks pointing to the detached vertex and the jets reconstructed using PYTHIA. To further illustrate the W decay, we present in Fig. 3 the jet-jet invariant mass distribution. As we can see, this distribution is clearly peaked around the W mass and a good fraction of the two jets reconstructed as associated to the LSP decay pass the cut C3. The observed high efficiency of cut C4 shows that the W 's produced in the LSP decay are rather central. We also learn from Table I that detached vertices presenting a W possess around 60% of the time an energetic µ ± or τ ± complying with C5. Moreover the cuts C6 and C7, which ensure the quality of the τ reconstruction, reduce significantly the number of W ± τ ∓ events. Finally the isolation cut C8 turns out to be quite important significantly reducing the signal. For the parameter point SPS1a, the expected efficiencies for the reconstruction of µW and τ W decays are 0.107 and 0.0098 respectively, where in the last we have added 1-and 3-prong hadronic decays. When the τ decays into a µ and neutrinos, the event was computed as being a µW decay if the µ passes the cuts. This was included appropriately in our calculations. Taking into account the total SUSY production cross section (41 pb) at 14 TeV, an integrated luminosity of 100 fb −1 and these efficiencies we anticipate that the number of observed µW and τ W events after cuts to be N µ =32000 N hadron τ = 3382 where N hadron τ = N 1−prong τ →hadron + N 3−prong τ . Therefore, the statistical accuracy of the ratio R = BR(χ 0 1 → µ ± W ∓ )/BR(χ 0 1 → τ ± W ∓ ) is expected to be σ(R)/R = 1/N µ + 1/N τ ≈ 0.015. In the case one takes into account only the three-prong decays of the tau, as in Ref. [35], the statistical error of this ratio increases to ≈ 0.053. Moreover, as expected, there is a degradation of the accuracy in the determination of this ratio of branching ratios in a more realistic analysis; the result obtained in [35] is ≃ 0.028. In the evaluation of the above efficiencies we have not taken into account multiple interactions at the LHC as needed for the high luminosity run. Therefore, we reevaluated the detection efficiencies for muons and taus with multiple interactions switched on in PYTHIA. We found that these efficiencies were only slightly degraded by the occurrence of pileup, that is, we obtained that the efficiencies for muon reconstruction are reduced to 0.102 and for tau are 0.008 68 in hadronic mode and 0.000 94 in the 3-prong mode. In our analyses we took into account the effect of multiple interactions. For the sake of comparison, we present a detailed analysis for a different mSUGRA point that is m 1/2 = 500 GeV, m 0 = 500 GeV, A 0 = −100 GeV, tan β = 10, and sgn(µ) = +1. Once again using SPheno, we obtain that the neutralino branching ratios larger than 1% are: BR(χ 0 1 → W ± µ ∓ ) = 22.9%, BR(χ 0 1 → W ± τ ∓ ) = 25.2%, BR(χ 0 1 → Zν) = 25.1%, BR(χ 0 1 → νh 0 ) = 16.9%, BR(χ 0 1 → τ ± τ ∓ ν) = 3.4%, BR(χ 0 1 → bbν) = 2.9%; and the corresponding R parity parameters are ǫ 1 = 0.1507 GeV, ǫ 2 = −0.1507 GeV, ǫ 3 = 0.1507 GeV, v 1 = −0.0056 GeV, v 2 = 0.0058 GeV, v 3 = −0.0054 GeV. As we can see, the neutralino LSP decays are dominated by the two-body ones, in contrast with the SPS1a point where the three-body decays mediated by light scalars are dominant. Because of its heavier spectrum, the total SUSY production for this parameter point is smaller than the SPS1a one; however, the cross section loss is partially compensated by the higher branching ratios into µW and τ W . The total cross section for this case is 832.0 fb and our analyses indicate that the reconstruction efficiency for µW decays is 0.203 while the τ W decays are reconstructed with an efficiency of 0.035, where we did not take into account pileup. The inclusion of this effect leads to a tiny reduction of the reconstruction efficiencies that become 0.199 for µW and 0.033 for τ W . On the other hand the efficiency for reconstructing a τ W event in the 3-prong mode is 0.012. Notice that these efficiencies are larger for this mSUGRA point than for the SPS1a because the neutralino is heavier and, consequently, its decay products are more energetic and pass the cuts more easily. The expected total number of reconstructed events after cuts for this SUSY point is N µ = 5171 and N hadron τ = 933 where we have included the pileup effects. Therefore, the expected statistical error on the ratio R becomes ≈ 0.036, or ≈ 0.056 when we only use 3-prong taus as in [35]. As we can see, the statistical error on the ratio R increases as m 1/2 (LSP mass) increases due to the reduction of the SUSY production cross section despite the increase in the detection efficiencies. We evaluated the reconstruction efficiencies as a function of m 0 ⊗ m 1/2 for A 0 = −100 GeV, tan β = 10 and sgn(µ) = +1 and our results are depicted in Fig. 4. As we can see from the left panel of this figure, the µW decays exhibit a high reconstruction efficiency, i.e., between 10% and 20%, in a large area of the parameter space, degrading only at large m 1/2 . On the other hand, the τ W reconstruction (see right panel of Fig. 4) is at most 3.5%, indicating that the statistical error on the ratio R is going to be dominated by these events. We present in Fig. 5 the attainable precision σ(R)/R with which the correlation R can be measured as a function of m 0 ⊗ m 1/2 for A 0 = −100 GeV, tan β = 10, and sgn(µ) = +1 for an integrated luminosity of 100 fb −1 and a center-of-mass energy of 14 TeV. We require in all plots that at least 5 events of reconstructed taus are observed. In the left panel of this figure we present the expected statistical error on the ratio R assuming no systematic errors on the determination of the reconstruction efficiencies, while in the right panel we consider a more conservative scenario, where we anticipate a systematic error of 10% in each of the reconstruction efficiencies. One can see from this panel that the precision drops as m 1/2 grows since the neutralino production rates from squark/gluino cascade decays also decrease with increasing m 1/2 values. Therefore, if the systematic errors of the efficiency determination are negligible the LHC collaborations should be able to probe with a very good precision ( 10%) the ratio R for m 1/2 650 GeV, which correspond to an LSP mass up to ≃ 270 GeV. The inclusion of systematic errors at the level assumed in the Note that in Fig. 5 we also present results for the 7 TeV run of the LHC. For this case one can see that the LHC has a much more limited capability of probing the ratio R, since the reach of this run covers only up to m 1/2 300 GeV. Still, although large, the statistical errors in this region [0.3 σ(R)/R 0.5], due mainly to the small anticipated integrated luminosity, which we have taken to be 1 fb −1 , allow a determination of the atmospheric angle comparable to that obtained at low energies. In the left panel of Fig. 6 we show the dependence of the attained precision as a function of the neutralino mass for luminosities of 2, 10, and 100 fb −1 . For small neutralino masses the SUSY production cross section is large enough to guarantee that the statistical errors are small; therefore, the uncertainty on the ratio R is dominated by the assumed systematic errors on the reconstruction efficiencies, even for an integrated luminosity of 2 fb −1 . As the accumulated luminosity increases the LHC experiments will be able to probe higher neutralino masses; however, the precision worsens due to the increase of statistical errors. We can also see clearly that increasing the luminosity allows a more precise measurement of R as expected. Moreover, one can probe LSP masses up to 250 (320 or 370) GeV for an integrated luminosity of 2 (10 or 100) fb −1 . From the right panel of Fig. 6 we estimate the luminosity needed to measure R with a given precision for several LSP masses. For instance, let us consider mχ0 1 = 250 GeV. In this case R can only be measured with a precision σ(R)/R ≃ 50% with 2 fb −1 , while this error can be brought down to 20%, i.e., close to the limit set by the systematic uncertainties, with 50 fb −1 . IV. CONCLUSIONS We have demonstrated how the LHC may have the potential of probing neutrino mixing angles with sensitivity competitive to their low-energy determination by oscillation experiments. This analysis was carried out, for the sake of concreteness, in the simplest scenario for neutrino masses induced by minimal supergravity with R parity violation as framework. In this class of models, the smoking gun for the neutrino mass generation mechanism is the ratio of branching fractions of neutralino decaying into µW and τ W , as this fraction is related to the atmospheric neutrino mixing angle in RmSUGRA models. Under realistic detection assumptions we have made a detailed analysis of the reconstruction of neutralino decays, as well as of the cuts needed to characterize the signal. After that we determined the attainable precision on the measurements of the ratio R given in Eq. (1). Comparing with a previous parton level study, we improved the reconstruction efficiencies of muons as well as taus. We showed that the 7 TeV run of the LHC will have a somewhat weak potential for probing the RmSUGRA model, since it is statistics limited. Still, precisions comparable to the low-energy determination should be reached. In contrast, a 14 TeV run with 100 fb −1 integrated luminosity will be able to probe a large fraction of the parameter space with a good precision, as seen in Fig. 5. In fact, our analyses suggest that the error on R will be dominated by the systematic ones on the reconstruction efficiencies of the decay µW and τ W , with the statistical errors being under control. In short, we find that in this case the atmospheric mixing angle may be probed relatively neatly. In fact, a determination of R within a given error translates into a prediction of the atmospheric mixing angle with an error of very similar size. Needless to say, what we have presented is only one example of a class of LSPs. There are other variant schemes based on alternative supersymmetry and/or R parity breaking, where other states emerge as LSP and similar correlations to other neutrino mixing angles appear [41][42][43]. These would, however, require separate dedicated studies. We encourage the particle detector groups ATLAS and CMS to add the test of such possibilities to their physics agenda, as this might lead to a tantalizing synergy between high-energy accelerator and low-energy nonaccelerator searches for new physics. Studies with the real LHC data may also make it possible to probe, at some level, the mass scale characterizing atmospheric neutrino oscillations, as well as the angle characterizing solar neutrino oscillations, an issue to be taken up separately. length in the plane m0, m 1/2 for A0 = −100 GeV, tan β = 10 and µ > 0. Figure 2 : 2Ratio ofχ 0 Figure 3 : 3From top to bottom:χ 0 1 → jjX without cuts, with cut on lepton isolation (µ or τ ) and with all other cuts leaving free the invariant mass range. 015 Figure 4 : 0154ffµ < 0.02 G 0.02 < ε ffµ < 0.05 I 0.05 < ε ffµ < 0.10 ◊ 0.10 < ε ffµ < 0.20 L ε ffµ > 0.025 < ε ffτ < 0.035 I 0.015 < ε ffτ < 0.025 ◊ 0.005 < ε ffτ < 0.Reconstruction efficiencies of µW (left panel) and τ W events (right panel) as a function of m0 ⊗ m 1/2 for A0 = −100 GeV, tan β = 10 and sgn(µ) = +1 including the effect of pileup. The red (dark shaded) area corresponds to the region where stau is the LSP, while the yellow (light shaded) area represents the region excluded by LEP.right panel ofFig. 5increases the uncertainty in R; however, it is still possible to perform an accurate test of the RmSUGRA scenario. Figure 5 :$Figure 6 : 56Precision in the determination of the ratio R in the plane m 1/2 × m0 for a luminosity of 100 fb −1 , center-of-mass energy of 14 TeV, A0 = −100 GeV, tan β = 10, and sgn(µ) = +1. In the right (left) panel we did (not) include a possible systematic uncertainty in the extraction of the efficiencies for the channels µW and τ W . The stars in the right panel represent the results for the 7 TeV run with an integrated luminosity of 1 fb −1 . The shaded areas represent the same as in Fig. 4. m χ ≈ 360 GeV M m χ ≈ 335 GeV # m χ ≈ 290 GeV ◊ m χ ≈ 250 GeV I m χ ≈ 205 GeV ∆ m χ ≈ 160 GeV G m χ ≈ 120 GeV The left panel displays the achievable precision in the ratio R as a function of the neutralino mass mχ0 1 for luminosities of 2, 10, and 100 fb −1 at 14 TeV whereas the right panel contains the foreseen statistical error on R as a function of the integrated luminosity for several LSP masses. Table I: Fraction of events passing the successive cuts C1-C8 used for the event reconstruction at the SPS1a mSUGRA point.cut Nµ Nτ N 1−prong τ →all N 1−prong τ →hadron N 3−prong τ C1 0.996 0.968 0.816 0.475 0.058 C2 0.923 0.898 0.757 0.440 0.055 C3 0.391 0.407 0.344 0.199 0.025 C4 0.369 0.385 0.325 0.188 0.024 C5 0.230 0.248 0.211 0.121 0.024 C6+C7 0.230 0.078 0.057 0.033 0.014 C8 0.102 0.015 0.014 0.009 0.001 We may add, parenthetically, that such schemes require a different type of dark matter particle, such as the axion[28]. 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[ "GAP THEOREMS FOR ENDS OF SMOOTH METRIC MEASURE SPACES", "GAP THEOREMS FOR ENDS OF SMOOTH METRIC MEASURE SPACES" ]	[ "Bobo Hua ", "Jia-Yong Wu " ]	[]	[]	In this paper, we establish two gap theorems for ends of smooth metric measure space (M n , g, e −f dv) with the Bakry-Émery Ricci tensor Ric f ≥ −(n − 1) in a geodesic ball Bo(R) with radius R and center o ∈ M n . When Ric f ≥ 0 and f has some degeneration outside Bo(R), we show that there exists an ǫ = ǫ(n, sup Bo(1) \|f \|) such that such a space has at most two ends if R ≤ ǫ. When Ric f ≥ 1 2 and f (x) ≤ 1 4 d 2 (x, Bo(R)) + c for some constant c > 0 outside Bo(R), we can also get the same gap conclusion.	10.1090/proc/16022	[ "https://export.arxiv.org/pdf/2108.01969v4.pdf" ]	236,912,589	2108.01969	fee1e9bdee309739975918872ba7d90219bb374d	GAP THEOREMS FOR ENDS OF SMOOTH METRIC MEASURE SPACES 15 Aug 2022 Bobo Hua Jia-Yong Wu GAP THEOREMS FOR ENDS OF SMOOTH METRIC MEASURE SPACES 15 Aug 2022 In this paper, we establish two gap theorems for ends of smooth metric measure space (M n , g, e −f dv) with the Bakry-Émery Ricci tensor Ric f ≥ −(n − 1) in a geodesic ball Bo(R) with radius R and center o ∈ M n . When Ric f ≥ 0 and f has some degeneration outside Bo(R), we show that there exists an ǫ = ǫ(n, sup Bo(1) \|f \|) such that such a space has at most two ends if R ≤ ǫ. When Ric f ≥ 1 2 and f (x) ≤ 1 4 d 2 (x, Bo(R)) + c for some constant c > 0 outside Bo(R), we can also get the same gap conclusion. Introduction and main results The Cheeger-Gromoll's splitting theorem [4] states that if a complete Riemannian manifold (M n , g) with nonnegative Ricci curvature contains a line, then M n is isometric to N × R with the product metric, where N is a Riemannian manifold with the Ricci curvature Ric(N ) ≥ 0. As a consequence, any manifold with nonnegative Ricci curvature has at most two ends. In [2], Cai studied a complete manifold M n with Ric ≥ −(n − 1)K for some constant K ≥ 0 in a geodesic ball B o (R) with radius R and center o ∈ M n and Ric ≥ 0 outside B o (R). He proved that the number of ends of such a manifold is finite and can be estimated from above explicitly; see also Li and Tam [7] for an independent proof by a different method. Later, Cai, Colding and Yang [3] gave a gap theorem for this class of manifolds, which states that there exists an ǫ(n) such that such a manifold has at most two ends if KR ≤ ǫ(n). In this paper we will extend the Cai-Colding-Yang result and get two gap theorems on smooth metric measure spaces with the Bakry-Émery Ricci tensor. Our results may be useful for understanding the topological information of smooth metric measure spaces. Recall that a complete smooth metric measure space (for short, SMMS) is a triple (M, g, e −f dv), where (M, g) is an n-dimensional complete Riemannian manifold, dv is the volume element of metric g, f is a smooth potential function on M , and e −f dv is called the weighted volume element. On (M, g, e −f dv g ), given a constant m > 0, Bakry andÉmery [1] introduced the m-Bakry-Émery Ricci tensor Ric m f := Ric + Hess f − df ⊗ df m , where Ric is the Ricci tensor of (M, g) and Hess is the Hessian with respect to the metric g. When m = ∞, we have the (∞-)Bakry-Émery Ricci tensor Ric f = Ric ∞ f . If Ric f = λg for some λ ∈ R, then (M, g, e −f dv) is called the gradient Ricci soliton, which is a generalization of the Einstein manifold. A Ricci soliton is called shrinking, steady or expanding, if λ > 0, λ = 0, or λ < 0, respectively. Gradient Ricci soliton often arises as a limit of dilations of singularities in the Ricci flow and it plays a fundamental role in the Ricci flow [6] and Perelman's resolutions of the Poincaré Conjecture [17,18,19]. The Bakry-Émery Ricci tensor Ric f is linked with the f -Laplacian ∆ f := ∆ − ∇f · ∇ via the generalized Bochner formula ∆ f \|∇u\| 2 = 2\|Hess u\| 2 + 2g(∇u, ∇∆ f u) + 2Ric f (∇u, ∇u) for u ∈ C ∞ (M ). It plays an important role in the comparison geometry of SMMSs; see [20]. The Bakry-Émery Ricci tensor is also related to the probability theory and optimal transport. We refer the reader to [10], [11], [20] for further details. Lichnerowicz [8], and Wei and Wylie [20] independently extended the classical Cheeger-Gromoll splitting theorem to a SMMS. It states that if (M, g, e −f dv) with Ric f ≥ 0 and bounded f contains a line, then M = N × R. Fang, Li and Zhang [5] showed that the above splitting result remains true for only an upper bound on f . Lim [9] observed that the splitting result holds if ∇f → 0 at infinity. In [15] Munteanu and Wang proved a splitting result when Ric f has a positive lower bound and f satisfies certain quadratic growth of distance function. Recently, G. Wu [21] obtained splitting results for the gradient Ricci soliton when some integral of the Ricci curvature along a line is nonnegative. From these results, we see that the above mentioned manifolds all have at most two ends. Besides, Wei and Wylie [20] proved that any SMMS with Ric f > 0 for some bounded f has only one end. The second author [22] studied a SMMS with Ric f ≥ 0 outside a compact set and proved that the number of ends of such a manifold is finite if f has at most sublinear growth outside the compact set. Inspired by the gap theorem of manifolds [3] and the number estimate for ends of SMMSs [22], in this paper we first give a gap theorem for ends of a smooth metric measure space when Ric f ≥ 0 and f has some degeneration outside a compact set. Here ξ(t) is a smooth even function on R such that ξ(t) = −(n − 1), t ∈ [0, R 2 ], ξ(t) ≥ −(n − 1), t ∈ ( R 2 , R) and 0 ≤ ξ(t) ≤ \|t\| −α , t ∈ [R, ∞), for R > 0 and α > 1. One easily checks that f is smooth, even and it satisfies f ′′ (t) = ξ(t). So Ric f ≥ −(n − 1) and Ric f ≥ 0 outside B o (R). Since f (t) ≤ C\|t\| 2−α + C for \|t\| ≫ 1, f (0) = 1 and A = sup x∈Bo(1) \|f (x)\| ≥ 1, they satisfy the conditions in Theorem 1.1 for K = 1 and M has two ends. Another example is that: as in [3], by applying the metric surgery techniques to manifold M = S 1 × R × S n−2 , n ≥ 4, one can get an n-dimensional complete manifold M of infinite homotopy type with exactly two ends and with Ric ≥ −δ and with Ric ≥ 0 outside a small ball. Let f (x 1 , t, x 2 ) = −t on M , and it satisfies (1.1). Then Ric f ≥ −δ and Ric f ≥ 0 outside the small ball. Remark 1. 3. If f grows sublinearly, then (1.1) automatically holds. If ∇f → 0 at infinity, (1.1) still holds due to Lim [9]. If f is constant, the theorem recovers the Cai-Colding-Yang result [3]. Theorem 1.1 is obvious suitable to gradient steady Ricci soliton because the potential f of steady gradient Ricci solitons is negative linear outside a compact set. It is an interesting question if one can weaken the assumption of f such that it is suitable to the quadratic growth of f on gradient shrinking Ricci solitons. Remark 1.4. When we say that E is an end of the manifold M we mean that it is an end with respect to some compact subset of the manifold. If R 1 ≤ R 2 , then the number of ends with respect to B o (R 1 ) is less than the number of ends with respect to B o (R 2 ). So we assume the radius R < 1 in the theorem seems to be sensible. We can apply a similar argument to get a gap theorem under the conditions of Ric m f (without any assumption on f ). It states that when Ric m f ≥ −(n − 1)K in B o (R) and Ric m f ≥ 0 outside B o (R), there exists an ǫ = ǫ(n + m) depending only on n + m such that M has at most two ends if √ KR ≤ ǫ. Furthermore, inspired by the Munteanu-Wang splitting theorem [15], we may give another gap theorem for ends when Ric f has a positive lower bound and f grows quadratically outside a compact set. More precisely, after a suitable scaling of the metric, we may in fact assume Ric f ≥ −(n − 1) in a ball and get that Theorem 1.5. Let (M, g, e −f dv) be an n-dimensional complete smooth metric measure space. Fix a point o ∈ M and 0 < R < 1. Suppose Ric f ≥ −(n − 1) in the geodesic ball B o (R); outside B o (R), suppose Ric f ≥ 1 2 and f (x) ≤ 1 4 d 2 (x, B o (R))+c for some constant c > 0. There exists a constant ǫ(n, A) depending only on n and A, where A := sup x∈Bo(1) \|f (x)\|, such that if R ≤ ǫ(n, A), then (M, g, e −f dv) has at most two ends. Remark 1.6. We give an example satisfying Theorem 1. 5. Let M = S n−1 × R (n ≥ 3) with the standard metric. For 0 < R < 1, let f (x, t) = f (t) = 1 + a + bt + t 0 ds s 0 η(τ )dτ for (x, t) ∈ S n−1 × R, where a, b are chosen satisfying f (R) = 1, f ′ (R) = 0. Here η(t) is a smooth even function on R such that η(t) = −(n − 1), t ∈ [0, R 2 ], η(t) ≥ −(n − 1), t ∈ ( R 2 , R) and η(t) = 1 2 , t ∈ [R, ∞). One easily checks that f is smooth, even and f ′′ (t) = η(t). So Ric f ≥ −(n − 1) and Ric f ≥ 1 2 outside B o (R). Since f (t) = 1 4 (\|t\| − R) 2 + 1 for \|t\| ≥ R, f (R) = 1 and A ≥ 1, they satisfy the conditions of Theorem 1.5 and M has two ends. Remark 1.7. If the assumption R ≤ ǫ(n, A) in Theorem 1.5 is removed, we can show that the number of ends for such a space is finite. Moreover, we can provide an explicit upper bound for the number; see Appendix of the paper. We would like to point out that Munteanu and Wang systematically studied the number of ends on gradient Ricci solitons. In [13] they proved that any nontrivial steady gradient Ricci soliton has only one end. In [14] they showed that the expanding gradient Ricci soliton Ric f = − 1 2 with scalar curvature S ≥ − n−1 2 has at most two ends. They also considered a similar problem for SMMS with Ric f ≥ − 1 2 . In [16] they proved that any shrinking Kähler gradient Ricci soliton has only one end. Recently, Munteanu, Schulze and Wang [12] showed that the number of ends is finite on shrinking gradient Ricci soliton when the scalar curvature satisfies certain scalar curvature integral at infinity. The proof of our theorems uses the argument of Cai-Colding-Yang [3] and it relies on a Wei-Wylie's weighted Laplacian comparison [20] and geometric inequalities for two different ends (see Lemmas 2.8 and 2.11), which are derived by locally analyzing splitting theorems. We would like to point out that Cai-Colding-Yang's proof depends on a delicate constructional function G(r), which satisfies certain Laplacian equation with the Dirichlet boundary condition. In our case, the function G(r) constructed in Proposition 3.1 does not satisfy the f -Laplacian equation, but it is sufficient to deduce our desired results. The paper is organized as follows. In Section 2, we give some basic concepts and results on SMMSs. In Section 3, we apply the weighted Laplacian comparison and geometric inequalities for ends in Section 2 to prove our theorems. In Appendix, we give an upper bound for the number of ends of a class of SMMSs. Preliminary In this section, we introduce some results about SMMSs, which will be used in the proof of our results. We first recall a weighted Laplacian comparison due to Wei and Wylie [20]. We also have a weighted volume comparison of Wei and Wylie [20]. The weighted volume is denoted by V f (B x (R)) := Bx(R) e −f dv. Lemma 2.2. Let (M, g, e −f dv) be an n-dimensional complete smooth metric measure space. If Ric f ≥ −(n − 1)K for some constant K > 0, then (2.1) V f (B x (r 2 )) V f (B x (r 1 )) ≤ r2 0 (sinh n−1+4A √ K t)dt r1 0 (sinh n−1+4A √ K t)dt for any x ∈ M and 0 < r 1 < r 2 , where A = A(x, r 2 ) = sup y∈Bx(r2) \|f (y)\|. Then we recall some definitions of geometric quantities such as line, ray, end and asymptotic ray on Riemannian manifolds. Next we recall the definition of the Busemann function and its properties on a complete SMMS (M, g, e −f dv). The Busemann function associated to each ray γ ⊂ M is defined by b γ (x) := lim t→∞ (d(x, γ(t)) − t). By the triangle inequality, we know that b γ (x) is Lipschitz continuous with Lipschitz constant 1 and hence it is differential almost everywhere. At the points where b γ is not smooth we interpret the f -Laplacian in the following sense of barriers. A continuous function h on M satisfies ∆ f h ≥ a at p in the barrier sense, if for every ǫ > 0, there exists a lower barrier function h p,ǫ at p such that ∆ f h p,ǫ ≥ a − ǫ. A continuous function h satisfies ∆ f h ≤ a in the barrier sense is similarly defined. Definition 2.6. For a fixed point p ∈ M , let α(t) be a minimal geodesic from p to ray γ(t). As t → ∞, α(t) has a convergent subsequence which converges to a ray at p. Such a ray is called an asymptotic ray to γ(t) at p. For a line γ in M , there exist rays γ + : [0, ∞) → M by γ + (t) = γ(t) and γ − : [0, ∞) → M by γ − (t) = γ(−t). Similar to the above procedure, we can let b + γ (or b − γ , respectively) be the associated Busemann function of γ + (or γ − , respectively). Next we will introduce two geometric inequalities for two different ends under two types of curvature assumptions, which are important in the proof of Theorems 1.1 and 1.5. On one hand, recall that Fang, Li and Zhang [5] proved a Cheeger-Gromoll splitting theorem when Ric f ≥ 0 and f satisfy some degeneration condition. As in [22], we can easily apply the Fang-Li-Zhang arguments locally and get that Lemma 2.7. Let N be the δ-tubular neighborhood of a line γ on (M, g, e −f dv). Suppose that from every point p in N , there are asymptotic rays to γ ± such that Ric f ≥ 0 and (1.1) on both asymptotic rays. Then through every point in N , there exists a line α such that b + γ (α + (t)) = t, b − γ (α − (t)) = t. Sketch proof of Lemma 2.7. The proof is the same as the argument of Lemma 2.8 in [22] and we include it for the completeness. For any point p ∈ N , by [5] we firstly prove that the two asymptotic rays to γ ± are uniquely determined at p and form a line γ p . Then we can prove that Busemann functions b ± γ at p with b + γ + b − γ = 0 are smooth f -harmonic functions in the barrier sense. Meanwhile, when Ric f ≥ 0 and (1.1) hold on γ p , from [5] we get that Busemann functions b ± γ satisfy \|∇b ± γ \| = 1 and Hess b ± γ = 0 on γ p . Here the restriction of b ± γ to γ p is a linear function with derivative 1. So we can reparameterize γ p and the lemma follows. As in the proof of Lemma 3.3 in [2], we are able to apply Lemma 2.7 to prove the following property about ends. We refer the reader to Lemma 3.1 and Proposition 3.2 in [22] for the detailed discussion. (2.2) d(γ 1 (t 1 ), γ 2 (t 2 )) > t 1 + t 2 − 6R. Remark 2.9. In Proposition 3.2 in [22], we only proved (2.2) when t 1 , t 2 ≥ 3R. Checking the previous proof, we easily see that (2.2) in fact holds for all t 1 , t 2 ≥ 0. On the other hand, recall that Munteanu and Wang [15] proved another Cheeger-Gromoll type splitting theorem when Ric f ≥ 1 2 and f has certain quadratic growth. As in the preceding argument, we easily get the following result by analyzing the Munteanu-Wang's proof locally. d(γ 1 (t 1 ), γ 2 (t 2 )) > t 1 + t 2 − 6R. Gap theorems In this section we will prove Theorems 1.1 and 1.5. We start with an important proposition, which will be used in our proof of theorems. For the convenient discussion, we assume that (M, g, e −f dv) has Ric f ≥ −(n−1) by scaling the metric. v(o) = u(o) − G(d(o, x)) = −G(1 − δ) < −6ǫ. Therefore there exists a point z on the boundary of the annulus such that v(z) ≤ v(o) < −6ǫ. On the other hand, on the sphere S x (1), by the assumption (ii), we get that v = u − G(1) = u ≥ −6ǫ, where we used G(1) = 0. This implies that z ∈ S x (c). Combining this with the assumption (iii), the definitions of H(r) and v, and (3.2), we finally get u(x) ≤ u(z) + 2c = v(z) + H(c) < 2 − 2δ − 12ǫ and the result follows. We now apply Proposition 3.1 to prove Theorem 1.1, by following the argument of Cai-Colding-Yang [3]. Proof of Theorem 1.1. When K = 0, the theorem easily follows by the Fang-Li-Zhang splitting theorem [5]. Now let (M, g, e −f dv) be as Theorem 1.1 with K = 1. Let ǫ = ǫ(n, A) be as Proposition 3.1. We only need to show that when R ≤ ǫ(n, A), M n has at most two ends. Suppose the conclusion is not true. That is, there exists three different ends, denoted by [γ 1 ], [γ 2 ] and [γ 3 ]. We consider the function u(x) := b γ1 (x) + b γ2 (x). We claim that u(x) satisfies four conditions of Proposition 3.1. Indeed, (i) and (iii) are obvious. By (2.2) and the triangle inequality, u(x) : = b γ1 (x) + b γ2 (x) = lim t→∞ (d(x, γ 1 (t)) − t) + lim t→∞ (d(x, γ 2 (t)) − t) ≥ lim t→∞ (d(γ 1 (t), γ 2 (t)) − 2t) ≥ −6R ≥ −6ǫ, which implies that u satisfies (ii). Moreover, by Lemma 2.1, ∆ f u(x) = ∆ f d(x, γ 1 (∞)) + ∆ f d(x, γ 2 (∞)) ≤ 2(n + 4A − 1) lim r→∞ coth r = 2(n + 4A − 1), which implies that u satisfies (iv). Therefore, by Proposition 3.1, we conclude that (3.3) u(γ 3 (1 − δ)) < 2 − 2δ − 12ǫ. On the other hand, by Lemma 2.8, for any t > 0, u(γ 3 (1 − δ) ≥ 2t − 12R. In particular, letting t = 1 − δ, it follows that u(γ 3 (1 − δ)) ≥ 2(1 − δ) − 12R ≥ 2 − 2δ − 12ǫ. This contradicts (3.3) and hence completes the proof. Finally we give an explanation how to prove Theorem 1.5. Sketch proof of Theorem 1.5. We can apply Proposition 3.1 and Lemma 2.11 to prove Theorem 1.5. In fact the argument in this case is exactly the same as the proof of Theorem 1.1. Here we omit the details. Appendix In this part we will give a number estimate for ends of a class of SMMSs. The weight f allows to be certain quadratic growth of distance function, which improves the growth of f in [22]. The proof follows by Lemmas 2.11 and 2.2 by using the arguments of [2,22]. We include it for the readers' convenience. Proof of Theorem 4.1. For any a point o ∈ M , let γ 1 , γ 2 , ..., γ k be k rays with k different ends starting from the base point o. Then we only need to give an upper bound of the number k. For a fixed R > 0, consider the sphere S o (4R) and let {p j } be a maximal set of points on S o (4R) such that the balls B pj (R/2) are disjoint each other. Clearly, the balls B pj (R) cover S o (4R). Since the set {γ i (4R), i = 1, 2..., k} is contained in S o (4R), each γ i (4R) is contained in some B pj (R). From Lemma 2.11 with t = 4R, we know that each ball B pj (R) contains at most one γ i (4R), and hence the number of balls is not less than k. Therefore, to estimate an upper bound of k, it suffices to bound the number of balls B pj (R/2). By the weighted volume comparison (2.1), using a fact that B pj (R/2) ⊂ B o ( 9 2 R) ⊂ B pj ( 17 2 R), we have V f B pj ( 17 2 R) ≤ 17 2 R 0 sinh n+4 A−1 t dt R/2 0 sinh n+4 A−1 t dt V f B pj (R/2) , where A = sup x∈Bp j 17 2 R \|f (x)\|. Therefore, the number of balls B pj (R/2) is no more than Theorem 1. 1 .f 1Let (M, g, e −f dv) be an n-dimensional complete smooth metric measure space. Fix a point o ∈ M and 0 < R < 1. Suppose Ric f ≥ −(n − 1)K for some constant K ≥ 0 in the geodesic ball B o (R); outside B o (R), suppose Ric f (σ(t))dt ≤ 0 on any ray σ, where r is the distance function starting from σ(0). There exists a constant ǫ(n, A) depending only on n and A, where A := sup x∈Bo(1) \|f (x)\|, such that if √ KR ≤ ǫ(n, A), then (M, g, e −f dv) has at most two ends. Remark 1.2. There exist many examples satisfying Theorem 1.1. Let M = S n−1 ×R (n ≥ 3) with the standard metric and f (x, t) = f (t) = 1 + t 0 ds s 0 ξ(τ )dτ for (x, t) ∈ S n−1 × R. Lemma 2. 1 . 1Let (M, g, e −f dv) be an n-dimensional complete smooth metric measure space with a base point o ∈ M . If Ric f ≥ −(n − 1)K for some constant K > 0, then ∆ f (r) ≤ (n + 4A − 1) √ K coth( √ Kr) along any minimal geodesic segment r from o, where A = A(o, r) = sup x∈Bo(r) \|f (x)\|. Definition 2. 3 . 3On a complete Riemannian manifold (M, g), we say that a geodesic γ :(−∞, +∞) → M is called line if d(γ(s), γ(t)) = \|s − t\|for all s and t. We say that a geodesic γ :[0, +∞) → M is called ray if d(γ(0), γ(t)) = tfor all t > 0. As we all known if M is complete noncompact, it must contain a ray.Definition 2.4. On a manifold M with a base point o ∈ M , two rays γ 1 and γ 2 starting at o are called cofinal if for any R > 0 and t > R, γ 1 (t) and γ 2 (t) lie in the same component of M \ B o (R). An equivalent class of cofinal rays is called an end of M . In this paper we let [γ] be the class of the ray γ. One readily checks that this definition is independence of the base point o and the complete metric on manifold M . Thus, the number of ends is a topological invariant of M . Definition 2. 5 . 5A lower barrier for a continuous function h at the point p ∈ M is a C 2 function h p , defined in a neighborhood U of p, such that h p (p) = h and h p (x) ≤ h(x), x ∈ U. Lemma 2 . 8 . 28Under the same assumptions of Theorem 1.1, M cannot admit a line γ satisfying d(γ(t), B o (R)) ≥ \|t\| + 2R for all t. Moreover, if [γ 1 ] and [γ 2 ] are two different ends of M n , then for any t 1 , t 2 ≥ 0, Lemma 2 . 10 . 210Let N be the δ-tubular neighborhood of a line γ on (M, g, e −f dv). Suppose that from every point p in N , there are asymptotic rays to γ ± such that Ric f ≥ 1 2 and f (x) ≤ 1 4 d 2 (x, B o (1)) + c for some constant c > 0 on both asymptotic rays. Then through every point in N , there exists a line α such that b + γ (α + (t)) = t, b − γ (α − (t)) = t. Using Lemma 2.10, we can get a geometric inequality about two different ends along the above similar argument. Lemma 2.11. Under the same assumptions of Theorem 1.5, M cannot admit a line γ satisfying d(γ(t), B o (R)) ≥ \|t\| + 2R for all t. Moreover, if [γ 1 ] and [γ 2 ] are two different ends of M n , then for any t 1 , t 2 ≥ 0, Proposition 3 . 1 . 31Let (M, g, e −f dv) be an n-dimensional complete smooth metric measure space with a base point o ∈ M . Assume that Ric f ≥ −(n − 1) and let A := sup x∈Bo(1) \|f (x)\|. There exist an ǫ = ǫ(n, A) and a δ = δ(n, A) such that u(x) < 2 − 2δ − 12ǫ for all x ∈ S o (1 − δ) := {x ∈ M \|d(x, o) = 1 − δ} if u : M n → R is a continuous function satisfying the following properties: ) sup x =y \|u(x) − u(y)\|/d(x, y) ≤ 2, (iv) ∆ f u ≤ 2(n + 4A − 1) in the barrier sense. Proof of Proposition 3.1. Let H(r) := 2r + G(r), where G(r) A := sup x∈Bo(1) \|f (x)\|. We remark that in gereral G(d(x, o)) does not satisfy the f -Laplacian equation ∆ f G = 2(n + 4A − 1). But we observe that G(1) G ′ (r) ≤ 0. So H(1) = 2 and H ′ (r) > 0 when r → 1. Therefore there exists a real constant c such that c ∈ (0, 1) and H(c) < 2. Now we choose δ = δ(n, A) and ǫ = ǫ(n,A) {G(1 − δ), 2 − H(c) − 2δ} .Consider function v(y) := u(y) − G(d(x, y)) on the annulus B x (1) \ B x (c). By the weighted Laplacian comparison (Lemma 2.1) and G ′ (r) ≤ 0, we compute that∆ f G = G ′′ (r)\|∇r\| 2 + G ′ (r)∆ f r ≥ G ′′ (r) + G ′ (r) [(n + 4A − 1) coth r] = 2(n + 4A − 1),where we used the definition of G(r) in the last equality. This implies that ∆ f v ≤ 0 in the barrier sense by combining the assumption (iv). By the maximum principle, v achieves its minimum on the boundary of the annulus B x (1) \ B x (c). By (3.1), we know that o is an interior point of the domain B x (1) \ B x (c). By (3.2) and the assumption (i), we see that Theorem 4 . 1 . 41Let (M, g, e −f dv) be an n-dimensional complete smooth metric measure space. Fix a point o ∈ M and R > 0. Suppose Ric f ≥ −(n − 1) in the geodesic ball B o (R) and Ric f ≥ 1 2 outside B o (R). 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Bobo Hua: School of Mathematical Sciences, LMNS, Fudan University ; Fudan UniversityChina Email address: [email protected] Hua: School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China; Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China Email address: [email protected] . Jia-Yong Wu, ShanghaiDepartment of Mathematics, Shanghai UniversityChina Email address: [email protected] Wu: Department of Mathematics, Shanghai University, Shanghai 200444, China Email address: [email protected]	[]
[ "Ground state of a partially melted Wigner molecule", "Ground state of a partially melted Wigner molecule" ]	[ "Zoltánádám Németh \nDepartement of Physics of Complex Systems\nEötvös University\nPázmány Péter sétány 1H-1117BudapestHungary\n", "Jean-Louis Pichard ", "\nService de Physique de l'Etat Condensé\nCentre d'Etudes de SaclayF-91191Gif-sur-Yvette CedexFrance (\n" ]	[ "Departement of Physics of Complex Systems\nEötvös University\nPázmány Péter sétány 1H-1117BudapestHungary", "Service de Physique de l'Etat Condensé\nCentre d'Etudes de SaclayF-91191Gif-sur-Yvette CedexFrance (" ]	[ "EUROPHYSICS LETTERS Europhys. Lett" ]	received ; accepted ) PACS. 71.10-w -Theories and models for many-electron systems. PACS. 71.27+a -Strongly correlated electron systems. PACS. 73.20.Qt -Electron solids.Abstract. -We consider three spinless fermions free to move on 2d square lattice with periodic boundary conditions and interacting via a U/r Coulomb repulsion. When the Coulomb energy to kinetic energy ratio rs is large, a rigid Wigner molecule is formed. As rs decreases, we show that melting proceeds via an intermediate regime where a floppy two particle molecule coexists with a partially delocalized particle. A simple ansatz is given to describe the ground state of this mesoscopic solid-liquid regime.Typeset using EURO-T E X	10.1209/epl/i2002-00412-2	[ "https://arxiv.org/pdf/cond-mat/0203477v1.pdf" ]	16,464,851	cond-mat/0203477	780171317ff5f253399987074f8f2f96bbe172d1	Ground state of a partially melted Wigner molecule 22 Mar 2002 Zoltánádám Németh Departement of Physics of Complex Systems Eötvös University Pázmány Péter sétány 1H-1117BudapestHungary Jean-Louis Pichard Service de Physique de l'Etat Condensé Centre d'Etudes de SaclayF-91191Gif-sur-Yvette CedexFrance ( Ground state of a partially melted Wigner molecule EUROPHYSICS LETTERS Europhys. Lett pp. (22 Mar 2002 received ; accepted ) PACS. 71.10-w -Theories and models for many-electron systems. PACS. 71.27+a -Strongly correlated electron systems. PACS. 73.20.Qt -Electron solids.Abstract. -We consider three spinless fermions free to move on 2d square lattice with periodic boundary conditions and interacting via a U/r Coulomb repulsion. When the Coulomb energy to kinetic energy ratio rs is large, a rigid Wigner molecule is formed. As rs decreases, we show that melting proceeds via an intermediate regime where a floppy two particle molecule coexists with a partially delocalized particle. A simple ansatz is given to describe the ground state of this mesoscopic solid-liquid regime.Typeset using EURO-T E X Ordered arrays of charged particles with long range Coulomb repulsion have been a continuous subject of interest in various branches of physics, including colloidal suspensions, ion rings, atomic-ion Wigner crystals, quantum computers, biophysics, plasmas, electrons deposited on liquid Helium surfaces, charges created in semiconductor or organic field effect devices. These arrays can melt, exhibiting a transition from collective towards independent-particle motion, either as a function of the temperature (classical melting) or as a function of the charge density (quantum melting) at very low temperature. In principle, the quantum melting can be observed using electrons trapped in quantum dots [3,4] or cooled ions confined in radio frequency traps [5]. Very often, a parabolic confinement is imposed. When the confinement is weak and at a sufficiently low temperature, the Coulomb repulsion dominates the kinetic energy, the charges are ordered and a Wigner molecule is formed. If the confinement becomes stronger, the kinetic energy dominates the Coulomb repulsion, the molecule melts and one gets a Fermi system of weakly interacting particles. In a parabolic 2d trap, the molecule consists of well-separated shells. Both for the classical melting [1] (achieved by increasing the temperature for a given trap) and for the quantum melting [2] (achieved at zero temperature by reducing the size of the trap), it has been observed that melting proceeds in two stages: first neighboring shells may rotate relative to each other while retaining their internal order, second the shell broadening leads to radial melting. Wigner quantum crystallization in 2d electron dots is characterized by two distinct -radial and orientational -ordering transitions. However, a parabolic trap does not yield a uniform charge density, the low density shells at the edges could order before the high density part in the bulk, and this two stage melting could be an artifact due to the interplay between surface and bulk orderings. It is therefore interesting to study if a multi stage melting persists in a system of uniform charge density, for instance when the charges are confined on a 2d torus. One has then to take into account the translational symmetry of a 2d torus instead of the rotational symmetry of a parabolic trap. Another important issue can be mentioned, assuming that insights gained through investigations limited to small systems provide the foundations for understanding larger systems. Long ago, it was suggested by Andreev and Lifshitz [6] that low temperature localized defects change into excitations that move practically freely through a crystal. As a result, the number of sites of a quantum crystal may be smaller than the total number of particles present in the system for intermediate couplings, such a crystal being neither a solid, nor a liquid. Two kinds of motion are possible in it; one possesses the properties of motion in an elastic solid, the second possesses the properties of motion in a liquid. An intermediate regime of melting was detected using N = 4 spinless fermions in a L × L lattice, and it was observed [7,8] that a combination of a few plane waves and site orbitals was able to describe it, suggesting a liquid-solid regime consistent either with a scenarioà la Andreev-Lifshitz, or with a possible quantum liquid crystal regime [9,10]. This intermediate regime was shifted to lower ratios r s when site disorder was included. Many signatures of a novel ground state (GS) were observed for intermediate couplings, considering the structure [11] of the persistent currents when the torus was pierced by an Aharonov-Bohm flux, the statistics [9] of its low energy excitations, the failure [8] of the Hartree Fock approximation to describe the persistent currents above a first ratio r F s , the suppression of the same currents above a higher ratio r W s , the GS magnetization [12] when the spin degrees of freedom where included. Moreover, it was noticed in Ref. [11] that the ratios r F s < r s < r W s where the novel mesoscopic GS was observed were consistent with those where transport measurements using dilute electron gases in 2d field effect devices [13,14] indicate the puzzling possibility of a novel 2d metal. The purpose of this work is to reveal the exact nature of the intermediate mesoscopic GS and to describe it with a simple wave function, considering N = 3 spinless fermions with U/r Coulomb repulsion in a L × L square lattice with periodic boundary conditions. Using the operators c † j (c j ) , c k (c † k ) which create (destroy) a spinless fermion either at the lattice site j = (j x , j y ) or in a plane wave state of momentum k = (k x , k y ) of this lattice, the Hamiltonian reads: H = −t j,j ′ c † j c j ′ + U 2 j,j ′ j =j ′ n j n j ′ d j,j ′ = k ǫ k c † k c k + k,k ′ ,q U (q)c † k+q c † k ′ −q c k ′ c k .(1) n j = c † j c j , d j,j ′ is the shortest distance between sites j and j ′ , ǫ k = −2t(cos k x + cos k y ) and U (q) = U/(2L 2 ) j cos(qj)/d 0,j . The Coulomb energy to kinetic energy ratio r s = U/(2t √ πn e ) for a filling factor n e = N/L 2 . The operators c k and c j are related by the usual Fourier transform: c k = 1 L j e −i(kj) c j .(2) In the eigenbasis of the non interacting system (eigenvectors c † k1 c † k2 c † k3 \|0 , \|0 being the vacuum state), the Hamiltonian matrix is block diagonal, each block being characterized by the same conserved total momentum K = 3 i=1 k i . Moreover, only the non interacting states having in common one k out of three can be coupled by the interaction inside a K sub-block. Therefore, each K sub-block is a sparse matrix which can be exactly diagonalized using the Lanczos algorithm when L is small enough. K = ( 2π 8 ,2π In the large coupling limit (r s → ∞) the eigenstates correspond to rigid Wigner molecules. For a localized center of mass, the charges would be simply located on three lattice sites a, b and c, the location of those sites minimizing the electrostatic energy. However translational invariance implies a delocalization of the center of mass in a plane wave state of momentum K, and the Wigner molecule wave functions become: \|Ψ = 1 L j e i(Kj) c † j+a c † j+b c † j+c \|0 .(3) For a given shape (a, b, c) of the three particle molecule, one has three well defined inter particle spacings d min ≤ d int ≤ d max . The electrostatic energy becomes U (d −1 min +d −1 int +d −1 max ) while the kinetic energy ∝ t ef f (center of mass effective band width) → 0 as r s → ∞. For some arbitrary values of K and for some molecular shapes of low electrostatic energy, we have decreased r s and followed the corresponding levels, ignoring possible level crossings with other levels of different K and of different molecular shape. Two examples calculated using a L = 8 lattice are shown in Fig. 1, revealing the generic scenario for the melting of a three particle Wigner molecule. If one considers the change of the relative fluctuations ∆d/ d as one increases r s , one can see both for d min and d max a crossover from a weak coupling behavior where the fluctuations are large (the d are not well defined) towards a large coupling behavior where the fluctuations become negligible (the d become well defined). The weak (strong) coupling limits can be described using U/t (t/U ) perturbative expansions. For instance, the large coupling behavior of the Wigner molecule is given at first order of a t/U expansion by: \|Ψ(1) = \|Ψ + 12 α=1 α =0 t ∆E α \|Ψ α ,(4) where the \|Ψ α label the Wigner molecules of same K obtained by moving one of the sites a, b, c of \|Ψ by one lattice spacing, ∆E α ∝ U being the corresponding changes of electrostatic energy. This gives the t/U decays of the three ∆d/ d indicated in Fig. 1. The main point to notice is the clear separation between the crossover ratios r * s (indicated by the arrows in Fig. 1) characterizing d min and d max . As one can see in the data, there are relatively large intervals of intermediate couplings where d min is well defined while d max is not, giving rise to an intermediate behavior for d int . This tells us that the generic melting of a three particle molecule proceeds also in two stages, if one considers a 2d system of uniform density, as it had been shown using 2d parabolic traps. The intermediate regime of melting consists of a floppy two particle molecule co-existing with a third delocalized particle. We now study the ground states (GSs) of the three body problem. The GS momenta and degeneracies depend on L, as the possible existence of GS level crossings. For simplicity, let us consider the case where L is even. At U = 0, one has a sixfold GS degeneracy which is partially removed by an infinitesimal U , the energy of a set of four states with momenta K = (±2π/L, ±2π/L) and K = (±2π/L, ∓2π/L) becoming different to those of the two K = 0 states. Using a U/t expansion, one finds that the GSs remain in the first set for L ≤ 6 while they go into the second set for L ≥ 8. At t = 0, the low energy Wigner molecules are L 2 triangles (d min = d int = L/2, d max = L/ √ 2) having L 2 /4 different locations of their centers of mass and 4 different orientations. This L 2 degeneracy is removed when one turns on t. The energies E 0 (K) of the L 2 first levels are given when t/U → 0 by: E 0 (K) ≈ A − 2t ef f (cos K x + cos K y ) + 2r ef f (cos(K x L/2) + cos(K y L/2)) ,(5) where A is a K-independent energy, t ef f ∝ t(t/U ) N −1 is the effective center of mass band width while r ef f ∝ t(t/U ) L/2−1 is the effective band width coming from single particle motions which couple triangles of same center of mass but of different orientations (L/2 one particle hops). For L ≥ 8, t ef f >> r ef f , one has a non degenerate K = 0 GS when t/U → ∞, and consequently a GS level crossing as r s increases inside the K = 0 subspace between the two weak coupling GSs and the single large coupling GS. For L = 6, r ef f and t ef f are both ∝ t 3 /U 2 , the GSs keep as r s varies their momenta K = (±2π/L, ±2π/L) and K = (±2π/L, ∓2π/L) (fourfold degeneracy) and no GS level crossing occurs. Hereafter we study the L = 6 GS of momentum K = (2π/6, 2π/6). This allows us to avoid the complications coming from the GS level crossing for L ≥ 8. However, as shown by the previous examples, our results will be relevant to generically describe the multi stage quantum melting of a N = 3 low energy Wigner molecule if one continuously follows a given level from large couplings towards weak couplings. For L = 6, the degeneracy of the L 2 triangular molecules is broken when one turns on t. A t/U expansion gives for the L 2 = 36 low energy levels E 0 (K) = A 2 − 2t 3 (cos 2πK x + cos 2πK y ) + 2r 3 (cos 2π3K x + cos 2π3K y ) + 0( t 4 U 3 ).. K = ( 2π 6 ,2π A 2 = 0.9023U − 208.9(t 2 /U ), t 3 = 1000(t 3 /U 2 ) and r 3 = 1660(t 3 /U 2 ). This t/U expansion makes sense when both d min and d max are well defined, with small relative fluctuations of order t/U . This means r s ≥ 200 (see Fig. 3). To describe lower r s (40 < r s < 200), where d min ≈ 3 is well defined, while d max has still large fluctuations, we propose a simple ansatz based on the concept of partially melted triangular molecules (PMTMs). A x-oriented PMTM (x-PMTM) is a rigid two particle Wigner molecule (2PWM) with d min = L/2 combined with a third particle free to move with a wave vector k x parallel to the 2PWM at a distance L/2, as sketched in Fig. 2. The x-PMTM wave function of momentum K reads: \|Ψ x (K, k x ) = 1 6 √ 2 j e i(Kx−k3x)jx+Kyjy c † j+a c † j+b c † kx,jy+cy \|0 ,(7) where a = (0, 0), b = (3, 0), c = (0, 3), and c † k3x,jy+cy = 1 √ 6 j x ′ e ikxj x ′ c † j x ′ ,jy+cy .(8) (K x − k x ) · 6/(2π) must be odd, which leads to k x = 0, ±2π/3 for K = (2π/6, 2π/6). The y-oriented PMTM wave function \|Ψ y (K, k y ) is defined in a similar way. The final PMTM ansatz of momentum K is a normalized combination of the x and y-PMTMs, which reads: \|Ψ(K, k x , k y ) = \|Ψ x (K, k x ) − \|Ψ y (K, k y ) 2 − 2 Ψ x (K, k 3x ) \| Ψ y (K, k 3y ) = 3 8 (\|Ψ x (K, k x ) − \|Ψ y (K, k y ) ) ,(9) and the constraint k x = k y makes it invariant under x − y permutation. In Fig. 2, the three values P x (K, k x ) = \| Ψ 0 (K) \| Ψ x (K, k x ) \| 2 taken by the projections of the exact GS \|Ψ 0 (K) over the x-PMTMs \|Ψ x (K, k x ) are given as a function of r s , together with the GS projection P (K, k x , k y ) = \| Ψ 0 (K) \| Ψ(K, k x , k y ) \| 2 over the PMTM of momenta (K, k x = 0, k y = 0). Following the three projections over the x-PMTMs of different wave vector k x , one can see how the third particle gets progressively localized in the x-direction as r s increases, the rigid three particle triangular molecule corresponding to P x (K, k x ) = 1/3 for the three possible k x . P (K, k x = 0, k y = 0) ≈ 93% at r s ≈ 100. However, only the PMTMs with k x = k y = 0 contribute when r s ≤ 50. For those values of r s , the third particle is fully delocalized in the direction parallel to the oriented PMWMs. However, it is likely that the PMTM ansatz overestimates the rigidity of the remaining 2PWM when r s becomes smaller. This can be partly fixed using a t/U expansion for the 2PWM (as sketched in Fig. 2 left) and keeping the third particle in its delocalized plane wave state with k x (k y ) = 0. The improvements coming from this partial t/U expansion of the PMTM ansatz are given in Fig. 3, where one can see the behaviors of the bare ansatz, of the ansatz corrected to first order and to second order of the t/U expansion of the 2PWM. In the upper figures, the GS projections and the relative errors ∆E(p)/E are shown, E denoting the exact GS energy, ∆E(p) = E A (p) − E, E A (p) being the ansatz energy at the p th order of the partial t/U expansion. Not only the GS description is improved, but lower values of r s can now be reached. In the lower figures, the three GS interparticle spacings d min d int and d max are given, and compared to the corresponding values of the ansatz, after a second order t/U expansion of the 2PWM. As underlined by the arrows, both the averages and the variances are now well described for r s ≈ 40. However, let us underline that the mesoscopic melting process is not yet achieved at r s ≈ 40. From a study of the weak coupling limit, we have obtained precursor behaviors of the formation of the Wigner molecule at smaller r s . For instance, certain GS projections over low energy non interacting states which are close in energy to the non interacting GS, but orthogonal to the large coupling Wigner molecule begin to decay when r s > 5. Therefore, the PMTM ansatz, even improved by a t/U expansion of the 2PWM fails to describe this precursor regime (5 < r s < 30) where a floppy 2PWM takes place, but is not rigid enough to be described by a simple t/U expansion. In summary, we have shown that the quantum melting of a three particle Wigner molecule confined on a 2d torus proceeds via an intermediate regime which can be described by the simple concept of a partially melted Wigner molecule, built of a delocalized particle and of a floppy 2PWM. This is in agreement with the general multi stage picture of mesoscopic quantum melting given by other works using 2d parabolic traps. At a mesoscopic scale, this gives a simple illustration of the quantum crystal with k = 0 defectons conjectured by Andreev and Lifshitz. Notably, one can see that the number of Wigner lattice sites is smaller than the total number of charges. This shows that the multi stage melting is not a mesoscopic surface effect and suggests that dilute 2d electron gases of intermediate r s could be more complicated than usually assumed. This work was supported by the EU program "Nanoscale dynamics, coherence and computation" and the Hungarian Science Foundation OTKA TO25866 and TO34832. Fig. 1 . 1-Left sides: Scheme of two low energy Wigner molecules in the limit rs → ∞ and of total momenta K for L = 8. Right sides: Corresponding relative fluctuations ∆d/ d of the three particle spacings as a function of rs. Smallest (continuous line), intermediate (dashed dot line) and largest (dashed line) spacings. The thin dotted lines give the t/U perturbative behaviors. The arrows indicate the two crossover ratios r * s . Fig. 2 . 2-Left figure: Scheme of a x-oriented partially melted triangular molecule (x-PMTM) for L = 6. The arrows give configurations allowed by a partial t/U expansion of the 2PWM. Right figure: GS projections Px(K, kx = 0) (solid line) Px(K, kx = 2π/3) (dashed-dotted line), Px(K, kx = −2π/3) (dotted line) over x-PMTMs of momentum kx. P (K, kx = 0, ky = 0) (upper thin dashed line) gives the total projection over the PMTM of momenta kx = 0 and ky = 0. Total momentum K = (2π/6, 2π/6). Fig. 3 . 3-Upper left: GS projections over PMTM ansatz with kx and ky = 0; Upper right: Relative errors ∆E/E; The bare ansatz behaviors, corrected by a first order or a second order t/U expansion of 2PWM are given by dashed lines, dashed-dotted lines and the dotted lines respectively; Lower left: Average inter-particle spacings d ; Lower right: Fluctuations ∆d of the inter-particle spacings as a function of rs. Exact behaviors (thick lines), and ansatz behaviors corrected by a second order t/U expansion of 2PWM. K = (2π/6, 2π/6). . V M Bedanov, F M Peeters, Phys. Rev. B. 492667Bedanov V. M. and Peeters F. M., Phys. Rev. B, 49 (2001) 2667. . A V Filinov, M Bonitz, E Yu, Phys. Rev. Lett. 863851Filinov. A. V., Bonitz M. and Lozovik Yu. E., Phys. Rev. Lett., 86 (2001) 3851. . R C Ashoori, Nature. 413Ashoori R. C., Nature, 379 (1996) 413. . 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[ "Cubical Convex Ear Decompositions", "Cubical Convex Ear Decompositions" ]	[ "Russ Woodroofe [email protected] \nDepartment of Mathematics\nWashington University in St. Louis St. Louis\n63130MOUSA\n" ]	[ "Department of Mathematics\nWashington University in St. Louis St. Louis\n63130MOUSA" ]	[]	We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced by Nyman and Swartz, starts with a CL-labeling and uses this to shell the 'ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "CL-ced" or "EL-ced". We find an EL-ced of the d-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented finite group. Along the way, we construct new EL-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes.We then proceed to show that if two posets P 1 and P 2 have convex ear decompositions (CL-ceds), then their products P 1 × P 2 , P 1× P 2 , and P 1× P 2 also have convex ear decompositions (CL-ceds). An interesting special case is: if P 1 and P 2 have polytopal order complexes, then so do their products.	10.37236/83	[ "https://arxiv.org/pdf/0709.2793v3.pdf" ]	14,066,089	0709.2793	d9c795a11675d2af2f00e0485367072a385404b2	Cubical Convex Ear Decompositions 18 Jun 2009 Russ Woodroofe [email protected] Department of Mathematics Washington University in St. Louis St. Louis 63130MOUSA Cubical Convex Ear Decompositions 18 Jun 2009arXiv:0709.2793v3 [math.CO] Mathematics Subject Classification: 05E25 Dedicated to Anders Björner in honor of his 60th birthday. We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced by Nyman and Swartz, starts with a CL-labeling and uses this to shell the 'ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "CL-ced" or "EL-ced". We find an EL-ced of the d-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented finite group. Along the way, we construct new EL-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes.We then proceed to show that if two posets P 1 and P 2 have convex ear decompositions (CL-ceds), then their products P 1 × P 2 , P 1× P 2 , and P 1× P 2 also have convex ear decompositions (CL-ceds). An interesting special case is: if P 1 and P 2 have polytopal order complexes, then so do their products. Introduction Convex ear decompositions, introduced by Chari in [6], break a simplicial complex into subcomplexes of convex polytopes in a manner with nice properties for enumeration. A complex with a convex ear decomposition inherits many properties of convex polytopes. For example, such a complex has a unimodal h-vector [6], with an analogue of the gtheorem holding [31], and is doubly Cohen-Macaulay [31]. Nyman and Swartz constructed a convex ear decomposition for geometric lattices in [17]. Their proof method used the EL-labeling of such lattices to understand the decomposition's topology. Similar techniques were pushed further by Schweig [23]. In Section 2, we introduce the necessary background material, and then axiomatize the conditions necessary for these techniques. We call such a convex ear decomposition a "CL-ced", or "EL-ced. " We then show by example in Sections 3 and 4 how to use these techniques on some poset families: d-divisible partition lattices, and coset lattices of a relatively complemented group. These posets have each interval [a,1] supersolvable, where a =0. Finding the convex ear decompositions will involve constructing a (dual) EL-labeling that respects the supersolvable structure up to sign, and showing that a set of (barycentricly subdivided) hypercubes related to the EL-labeling is an EL-ced, or at least a convex ear decomposition. We will prove specifically: Although both poset families were known to be EL-shellable, the EL-labelings that we construct in these sections also seem to be new. The ideas used to find them may be applicable in other settings, as briefly discussed in Section 6. Lemma 1.3. Π d n has a dual EL-labeling; C(G) has a dual EL-labeling if G is a complemented finite group. In Section 5 we change focus slightly to discuss products of bounded posets. Our first goal is: Theorem 1.4. If bounded posets P 1 and P 2 have convex ear decompositions, then so do P 1 × P 2 , P 1× P 2 , and P 1× P 2 . This is the first result of which I am aware that links poset constructions and convex ear decompositions with such generality. A result of a similar flavor (but more restrictive) is proved by Schweig [23]: that rank selected subposets of some specific families of posets have convex ear decompositions. A special case of Theorem 1.4 has a particularly pleasing form: Lemma 1.5. If P 1 and P 2 are bounded posets such that \|P 1 \| and \|P 2 \| are isomorphic to the boundary complexes of simplicial polytopes, then so are \|P 1 × P 2 \|, \|P 1× P 2 \|, and \|P 1× P 2 \|. We then recall the work of Björner and Wachs [4, Section 10] on CL-labelings of poset products, which we use to prove a result closely related to Theorem 1.4: Theorem 1.6. If bounded posets P 1 and P 2 have CL-ceds with respect to CL-labelings λ 1 and λ 2 , then P 1 × P 2 , P 1× P 2 , and P 1× P 2 have CL-ceds with respect to the labelings λ 1 × λ 2 , λ 1× λ 2 , and λ 1× λ 2 . We close by considering some additional questions and directions for further research in Section 6. Definitions and tools All simplicial complexes, posets, and groups discussed in this paper are finite. A poset P is bounded if it has a lower bound0 and an upper bound1, so that0 ≤ x ≤1 for all x ∈ P . If P is a bounded poset, then the order complex \|P \| is the simplicial complex whose faces are the chains of P \ {0,1}. (This is slightly different from the standard definition, in that we are taking only the proper part of the poset.) Where it will cause no confusion, we talk about P and \|P \| interchangeably: for example, we say P has a convex ear decomposition if \|P \| does. We denote by M(P ) the set of maximal chains of P , which is in natural bijective correspondence with the facets of \|P \| through adding or removing0 and1. Convex ear decompositions A convex ear decomposition of a pure (d − 1)-dimensional simplicial complex ∆ is an ordered collection of subcomplexes ∆ 1 , . . . , ∆ m ⊆ ∆ with the following properties: ced-polytope ∆ s is isomorphic to a subcomplex of the boundary complex of a simplicial d-polytope for each s. ced-topology ∆ 1 is a (d − 1)-sphere, and ∆ s is a (d − 1)-ball for s > 1. ced-bdry ( s−1 t=1 ∆ t ) ∩ ∆ s = ∂∆ s for each s > 1. ced-union m s=1 ∆ s = ∆. It follows immediately from the definition that any complex with a convex ear decomposition is pure. As far as I know, no one has tried generalizing the theory of convex ear decompositions to non-pure complexes. As many interesting posets are not graded (i.e., have an order complex that is not pure), finding such a generalization could be useful. Convex ear decompositions were first introduced by Chari [6]. He used the unimodality of the h-vector of a simplicial polytope to give a strong condition on the h-vector for a complex with a convex ear decomposition. Swartz [31] showed that a 'g-theorem' holds for any (d − 1)-dimensional complex with a convex ear decomposition, as stated precisely in Theorem 2.2. We refer the reader to [28] for further background on h-vectors, M-vectors, and the (original) g-theorem. is an M-vector. Shellings An essential tool for us will be the theory of lexicographic shellability, developed by Björner and Wachs in [1,2,3,4]. We recall some of the main facts. We say that an ordering of the facets F 1 , F 2 , . . . , F t of a simplicial complex ∆ (with t facets) is a shelling if F i ∩ i−1 j=1 F j is pure (dim F i − 1)-dimensional for all 1 < i ≤ t. An equivalent condition that is often easier to use is: if 1 ≤ i < j ≤ t, then ∃k < j such that (1) F i ∩ F j ⊆ F k ∩ F j = F j \ {x} for some x ∈ F j . A simplicial complex is shellable if it has a shelling. The existence of a shelling tells us a great deal about the topology of a pure ddimensional complex: the complex is Cohen-Macaulay, with homotopy type a bouquet of spheres of dimension d. A fact about shellable complexes that will be especially useful for us is that a shellable proper pure d-dimensional subcomplex of a simplicial d-sphere is a d-ball [7,Proposition 1.2]. A cover relation in a poset P , denoted x ⋖ y, is a pair x y of elements in P such that there is no z ∈ P with x z y. Equivalently, a cover relation is an edge in the Hasse diagram of P . An EL-labeling of P (where EL stands for edge lexicographic) is a map from the cover relations of P to some fixed partially ordered set, such that in any interval [x, y] there is a unique increasing maximal chain (i.e., a unique chain with increasing labels, read from the bottom), and this chain is lexicographically first among maximal chains in [x, y]. It is a well-known theorem of Björner in the pure case [1,Theorem 2.3], and more generally of Björner and Wachs [3,Theorem 5.8], that any bounded poset P with an EL-labeling is shellable. As a result, the term EL-shelling is sometimes used as a synonym of EL-labeling. The families of posets that we study in this paper will have lower intervals [0, x] that 'look like' the whole poset, but upper intervals [x,1] of a different form. For induction, then, it will usually be easier for us to label the posets upside down, and construct dual EL-labelings, that is, EL-labelings of the dual poset. Dual EL-labelings have been used in other settings, and are natural in many contexts [2,Corollary 4.4] [24,Corollary 4.10]. A generalization of an EL-labeling which is sometimes easier to construct (though harder to think about) is that of a CL-labeling. Here, instead of labeling the cover relations (edges), we label "rooted edges." More precisely, a rooted edge, or rooted cover relation, is a pair (r, x ⋖ y), where the root r is any maximal chain from0 to x. Also, if x 0 ⋖ x 1 ⋖ · · · ⋖ x n is a maximal chain on [x 0 , x n ], and r is a root for x 0 ⋖ x 1 , then r ∪ {x 1 } is a root for x 1 ⋖ x 2 , and so on, so it makes sense to talk of a rooted chain c r on a rooted interval [x 0 , x n ] r . A CL-labeling is one where every rooted interval [x, z] r has a unique increasing maximal chain, and the increasing chain is lexicographically first among all chains in [x, z] r . An in-depth discussion of CL-labelings can be found in [2,3]: the main fact is that EL-shellable =⇒ CL-shellable =⇒ shellable. We will make real use of the greater generality of CL-labelings only in Section 5, and the unfamiliar reader is encouraged to read "EL" for "CL" everywhere else. The homotopy type of bounded posets with a CL-labeling (including an EL-labeling) is especially easy to understand, as discussed in [3]. Such a poset is homotopy equivalent to a bouquet of spheres, with the spheres in one-to-one correspondence with the descending maximal chains. These descending chains moreover form a cohomology basis for \|P \|. Supersolvable lattices The upper intervals [x,1] in the posets we look at will be supersolvable, so we mention some facts about supersolvable lattices. For additional background, the reader is referred to [26] or [15]. An element x of a lattice L is left modular if for every y ≤ z in L it holds that (y ∨ x) ∧ z = y ∨ (x ∧ z). This looks a great deal like the well-known Dedekind identity from group theory, and in particular any normal subgroup is left modular in the subgroup lattice. A graded lattice is supersolvable if there is a maximal chain1 = x 0 ⋗ x 1 ⋗ · · · ⋗ x d =0, where each x i is left modular. Thus the subgroup lattice of a supersolvable group is a supersolvable lattice. In fact, supersolvable lattices were introduced to generalize the lattice properties of supersolvable groups. A supersolvable lattice has a dual EL-labeling λ ss (y ⋗ z) = min{j : x j ∧ y ≤ z} = max{j − 1 : x j ∨ z ≥ y}, which we call the supersolvable labeling of L (relative to the given chain of left modular elements). This labeling has the property: Given an interval [x, y], every chain on [x, y] has the same set of labels (in different orders). McNamara [14] has shown that having an EL-labeling that satisfies (2) characterizes the supersolvable lattices. Cohen-Macaulay complexes If F is a face in a simplicial complex ∆, then the link of F in ∆ is link ∆ F = {G ∈ ∆ : G ∩ F = ∅ and G ∪ F ∈ ∆}. A simplicial complex ∆ is Cohen-Macaulay if the link of every face has the homology of a bouquet of top dimensional spheres, that is, ifH i (link ∆ F ) = 0 for all i < dim(link ∆ F ). The Cohen-Macaulay property has a particularly nice formulation on the order complex of a poset. A poset is Cohen-Macaulay if every interval [x, y] hasH i ([x, y]) = 0 for all i < dim(\|[x, y]\|). In particular, every interval in a Cohen-Macaulay poset is Cohen-Macaulay. It is well-known that every shellable complex is Cohen-Macaulay. For a proof of this fact and additional background on Cohen-Macaulay complexes and posets, see [28]. The Cohen-Macaulay property is essentially a connectivity property. Just as we say a graph G is doubly connected (or 2-connected) if G is connected and G \ {v} is connected for each v ∈ G, we say that a simplicial complex ∆ is doubly Cohen-Macaulay ( Thus, convex ear decompositions can be thought of as occupying an analogous role to shellings in the geometry of simplicial complexes: a shelling is a combinatorial reason for a complex to be (homotopy) Cohen-Macaulay, and a convex ear decomposition is a combinatorial reason for a complex to be doubly Cohen-Macaulay. Of course, convex ear decompositions also give the strong enumerative constraints of Theorems 2.1 and 2.2. Intervals in a poset with a convex ear decomposition are not known to have convex ear decompositions. However, intervals do inherit the 2-CM property, as intervals are links in the order complex, and intervals inherit the Cohen-Macalay property. Thus, Theorem 2.3 is particularly useful in proving that a poset does not have a convex ear decomposition. EL-ceds and CL-ceds Nyman and Swartz used an EL-labeling in [17] to find a convex ear decomposition for any geometric lattice. The condition on an EL-labeling says that ascending chains are unique in every interval, and that the lexicographic order of maximal chains is a shelling. Starting with the usual EL-labeling of a geometric lattice, Nyman and Swartz showed that descending chains are unique in intervals of an ear of their decomposition, and that the reverse of the lexicographic order is a shelling. Schweig used similar techniques in [23] to find convex ear decompositions for several families of posets, including supersolvable lattices with complemented intervals. In this subsection, we axiomatize the conditions necessary for these techniques. Although we state everything in terms of CL-labelings, one could just as easily read 'EL' for the purposes of this section, and ignore the word 'rooted' whenever it occurs. Suppose that P is a bounded poset of rank k. Let {Σ s } be an ordered collection of rank k subposets of P . For each s, let ∆ s be the simplicial subcomplex generated by all maximal chains that occur in Σ s , but not in any Σ t for t < s. (Informally, ∆ s is all "new" maximal chains in Σ s .) Recall that M(Σ s ) refers to the maximal chains of Σ s , and let M(∆ s ) be the maximal chains of ∆ s . As usual, maximal chains are in bijective correspondence with facets of the order complex via removing or adding1 and0. The ordered collection {Σ s } is a chain lexicographic convex ear decomposition (or CLced for short) of P with respect to the CL-labeling λ, if it obeys the following properties: CLced-polytope For each s, Σ s is the face lattice of a convex polytope. CLced-desc For any ∆ s and rooted interval [x, y] r in P , there is at most one descending maximal chain c on [x, y] r which is a face of ∆ s . CLced-bdry If c is a chain of length < k, such that c can be extended to a maximal chain in both of ∆ s and ∆ t , where t < s; then c can be extended to a chain in M(Σ s ) \ M(∆ s ). CLced-union Every chain in P is in some Σ s . Note 2.4. We note the resemblance of (CLced-desc) with the increasing chain condition for a CL-labeling (under the reverse ordering of labels); but though ∆ s is a simplicial complex corresponding with chains in P , it is not itself a poset. Note 2.5. By analogy with CL-labelings, it would seem that we should require the descending chain in (CLced-desc) to be lexicographically last. But this would be redundant: suppose c is the lexicographically last maximal chain in [x, y] r that is also in ∆ s , but that c has an ascent at c i . Then Lemma 2.7 below gives that we can replace the ascent with a descent, obtaining a lexicographically later chain, a contradiction. Note 2.6. As previously mentioned, we will usually refer to EL-ceds in this paper, i.e., the special case where λ is an EL-labeling. Similarly, we may refer to dual EL-ceds, that is, EL-ceds of the dual poset. Proof. Let c − = c \ {c i }, and let Σ t be the first subposet in the CL-ced that contains c − . Since Σ t is the face lattice of a polytope, it is Eulerian, so c − has two extensions in Σ t . By the uniqueness of ascending chains in CL-labelings, at most one is ascending at rank i; by (CLced-desc), at most one is descending. Thus, there is exactly one of each. The extension with the ascent is c, call the other extension c ′′ . We have shown that c is in Σ t and (since Σ t is the first subposet containing c − ) that s = t, so that c ′′ is in ∆ s . Finally, c ′′ is lexicographically later than c by the definition of CL-labeling. We also recall a useful lemma from undergraduate point-set topology [16,Exercise 17.19]: Lemma 2.8. If B is a closed subset of X, then ∂B = B ∩ X \ B. Although they did not use the terms "CL-ced" or "EL-ced" in their paper, the essence of the following theorem was proved by Nyman and Swartz in [17,Section 4], where they used it to construct convex ear decompositions of geometric lattices. Proof. (Nyman and Swartz [17,Section 4]) The property (ced-union) follows directly from (CLced-union), and (ced-polytope) follows from (CLced-polytope) because the barycentric subdivision of a polytope is again a polytope. For (ced-bdry), we first note that ∂∆ s = ∂ \|Σ s \| \ ∆ s (the topological closure), hence ∂∆ s ⊆ ∆ s ∩ ( t<s ∆ t ). Conversely, if c is in ∆ s ∩ ( t<s ∆ t ), then (CLced-bdry) gives that c is in both ∆ s and \|Σ s \| \ ∆ s . Lemma 2.8 then gives the desired inclusion. It remains to check (ced-topology). Using (CLced-desc), we show that the reverse of the lexicographic order is a shelling of ∆ s . For if c = {0 ⋖ c 1 ⋖ · · · ⋖ c k−1 ⋖1} and c ′ = {0 ⋖ c ′ 1 ⋖ · · · ⋖ c ′ k−1 ⋖1} are maximal chains in ∆ s , with c lexicographically earlier than c ′ , then (CLced-desc) and Note 2.5 give that c has an ascent on some interval where c disagrees with c ′ . So c has an ascent at i, and c i = c ′ i . Apply Lemma 2.7 on the interval [0,1] to get c ′′ in ∆ s which descends at i, and otherwise is the same as c. Then c ′ ∩ c ⊆ c ′′ ∩ c = c \ {c i }, so \|c ′′ ∩ c\| = \|c\| − 1, and so c ′′ is lexicographically later than c, as Condition (1) requires for a shelling. We now check that ∆ s is a proper subcomplex of \|Σ s \| for s ≥ 2. Suppose that ∆ s = \|Σ s \|. Then by Notes 2.4 and 2.5, λ is a CL-labeling on Σ s with respect to the reverse ordering of its label set. Since \|Σ s \| is a sphere, there is an ascending chain (descending chain with respect to the reverse ordering) in Σ s . Since the ascending chain in P is unique, we have s = 1. By definition ∆ 1 = \|Σ 1 \| is a (k − 2)-sphere. Now since ∆ s is shellable and a proper subcomplex of the (k − 2)-sphere \|Σ s \| for s ≥ 2, we get that ∆ s is a (k − 2)-ball; thus (ced-topology) holds. In the following two sections, we will exhibit an EL-ced for the d-divisible partition lattice, and (using only slightly different techniques) a convex ear decomposition for the coset lattice of a relatively complemented group. (which we denote by ≺), as in the usual partition lattice Π n (= Π 1 n ). In general, Π d n is a subposet of Π n , with equality in the case d = 1; on the other hand, intervals [a,1] are isomorphic to Π n/d for any atom a ∈ Π d n . We refer frequently to [33] for information about the d-divisible partition lattice. The d-divisible partition lattice As Π n is a supersolvable geometric lattice, and hence quite well understood, we restrict ourself to the case d > 1. It will sometimes be convenient to partition a different set S = [n]. In this case we write Π S to be the set of all partitions of S, and Π d S the set of all d-divisible partitions of S, so that Π d n = Π d [n] is a special case. Wachs found a homology basis for Π d n in [33, Section 2]. We recall her construction. By S n we denote the symmetric group on n letters. We will write a permutation α ∈ S n as a word α(1)α(2) . . . α(n), and define the descent set of α to be the indices where α descends, i.e., des α = {i ∈ [n − 1] : α(i) > α(i + 1)}. Then a split of α ∈ S n at di divides α into α(1)α(2) . . . α(di) and α(di + 1) . . . α(n). A switch-and-split at position di does the same, but first transposes ('switches') α(di) and α(di + 1). These operations can be repeated, and the result of repeated applications of splits and switch-and-splits at d-divisible positions is a d-divisible partition. For example, if α = 561234, then the 2-divisible partition 56 \| 13 \| 24 results from splitting at position 2 and switch-and-splitting at position 4. Let Σ α be the subposet of Π d n that consists of all partitions that are obtained by splitting and/or switch-and-splitting the permutation α at positions divisible by d. Let 1. Σ α is isomorphic to the face lattice of the ( n d − 1)-cube for any α ∈ S n . 2. {Σ α : α ∈ A d n } is a basis for H * (Π d n ). After some work, this basis will prove to be a dual EL-ced. A dual EL-labeling for Π d n In addition to the homology basis already mentioned, Wachs constructs an EL-labeling in [33,Section 5], by taking something close to the standard EL-labeling of the geometric lattice on intervals [a,1] ∼ = Π n/d (for a an atom), and "twisting" by making selected labels negative. While her labeling is not convenient for our purposes, we use her sign idea to construct our own dual EL-labeling starting with a supersolvable EL-labeling of [a,1]. Partition lattices were one of the first examples of supersolvable lattices to be studied [26]. It is not difficult to see that the maximal chain with jth ranked element 1 \| 2 \| . . . \| j \| (j + 1) . . . n is a left modular chain in Π n . Let y · ≻ z be a cover relation in Π n . Then y is obtained by merging two blocks B 1 and B 2 of the partition z, where without loss of generality max B 1 < max B 2 . The supersolvable dual EL-labeling (relative to the above chain of left modular elements) is especially natural: λ ss (y · ≻ z) = min{j : (1 \| . . . \| j \| (j + 1) . . . n) ∧ y ≺ z} = max B 1 . We now construct the labeling that we will use for Π d n . Let y · ≻ z be a cover relation in Π d n , where z =0. As above, y is obtained by merging blocks B 1 and B 2 of z, where max B 1 < max B 2 . Label λ(y · ≻ z) = − max B 1 if max B 1 < min B 2 , max B 1 otherwise, and λ(y · ≻0) = 0. When discussing dual EL-labelings, any reference to ascending or descending chains is in the dual poset, so that the inequalities go in the opposite direction from normal. Note 3.2. Let a ∈ Π d n be an atom. Then a has n/d blocks, and every block has d elements. Order the blocks {B i } so that max B 1 < max B 2 < · · · < max B n/d , and let B = {max B 1 , . . . , max B n/d }. Then [a, 1] ∼ = Π B , and we recognize \|λ\| as the supersolvable dual EL-labeling λ ss on Π B . Note 3.3. We also can view Π d n as a subposet of Π n . A cover relation y · ≻ z in Π d n is a cover relation in Π n unless z =0. Thus, \|λ\| is the restriction of λ ss on Π n , except at the bottom edges y · ≻0. Note 3.4. The cover relation x 0 · ≻ x 1 gets a negative label if and only if B 1 \| B 2 is a noncrossing partition of B = B 1 ∪ B 2 . We will call this a non-crossing refinement of x 0 . The poset of all non-crossing partitions has been studied extensively [25,13], although this seems to have a different flavor from what we are doing. Also related is the connectivity set of a permutation [29], the set of positions at which a split yields a non-crossing partition. Recall that if P 1 and P 2 are posets, then their direct product P 1 × P 2 is the Cartesian product with the ordering ( x 1 , x 2 ) ≤ (y 1 , y 2 ) if x 1 ≤ x 2 and y 1 ≤ y 2 . The lower reduced product P 1× P 2 of two bounded posets is (P 1 \ {0}) × (P 2 \ {0}) ∪ {0}. Although the definition of the lower reduced product may appear strange at first glance, it occurs naturally in many settings, including the following easily-proved lemma: Lemma 3.5. Let y ≻ x be elements of Π d n \ {0}, with y = B 1 \| . . . \| B k . Then 1. [0, y] ∼ = Π d B 1× Π d B 2× . . .× Π d B k . 2. [y,1] ∼ = Π k . 3. [x, y] is the direct product of intervals in Π d B i . Note 3.6. We discuss (lower/upper reduced) products of posets at much more length in Section 5. Although the situation with Π d n is simple enough that we do not need to refer directly to product labelings (introduced in Section 5.3), they are the underlying reason we can look at partitions block by block in the proofs that follow. Theorem 3.7. λ is a dual EL-labeling of Π d n . Proof. We need to show that each interval has a unique (dual) increasing maximal chain which is lexicographically first. There are two forms of intervals we must check: Case 1. Intervals of the form [0, x 0 ]. Since the bottommost label on every chain in [0, x 0 ] is a 0, every other label in an increasing chain must be negative. Hence, every edge x i · ≻ x i+1 in an increasing chain must correspond to a non-crossing refinement of x i . In such a chain, any block B of x 0 is partitioned repeatedly into non-crossing sub-blocks. At the atom level, this block B is sub-partitioned as B 1 \| . . . \| B k , where max B i < min B i+1 . Thus, any increasing chain on [0, x 0 ] passes through this single atom, and we have reduced the problem to Case 2. Case 2. Intervals of the form [x m , x 0 ], where x m =0. By Lemma 3.5 and the discussion following, it suffices to examine a single block B of x 0 . (The labels on disjoint blocks are independent of each other.) In x m , let B be subpartitioned as B 1 \| . . . \| B k , with max B s = b s and b 1 < b 2 < · · · < b k . The edges we consider correspond with subpartitioning B between itself and B 1 \| . . . \| B k . First, we show that the lexicographically first chain c = x 0 · ≻ x 1 · ≻ . . . · ≻ x m is unique. If there are any negative labels down from x i , the edge x i · ≻ x i+1 will have the label −b s with greatest absolute value among negative labels. Thus, x i+1 = x i ∧ (B 1 . . . B s \| B s+1 . . . B k ) , and hence x i · ≻ x i+1 is the unique edge down from x i with this label. Otherwise, x i · ≻ x i+1 will have the least possible (positive) label, which is unique since \|λ\| is a dual supersolvable EL-labeling on [x m , x 0 ]. Next, we show that the lexicographically first chain is increasing. Suppose that c has a descent at x i−1 · ≻ x i · ≻ x i+1 , with λ(x i−1 · ≻ x i ) = α and λ(x i · ≻ x i+1 ) = β, corresponding to dividing a block C as C · ≻ C 1 \|C 2 ∪ C 3 · ≻ C 1 \| C 2 \| C 3 . Since \|λ\| is a dual EL-labeling, both labels cannot be positive. Thus, β < 0. If then \|α\| < \|β\|, we have max C 1 < max C 2 < min C 3 , and then C · ≻ C 1 ∪ C 2 \| C 3 is noncrossing, with a β label, and so lexicographically before x i−1 · ≻ x i . Otherwise, \|α\| > \|β\|. Since we have a descent at i, we see α > 0, and so the ±β < α label on the edge obtained by partitioning C · ≻ C 1 ∪ C 2 \| C 3 is again lexicographically before x i−1 · ≻ x i . In either case, we have shown that any c with a descent is not lexicographically first. Finally, we show that any increasing chain is lexicographically first. Suppose that there is an edge x 0 · ≻ y (≻ x m ) that receives a −b s label. Then y = B 1 . . . B s \| B s+1 . . . B k is a non-crossing partition of B, and in particular B s < B s+1 , . . . , B k . We see that any subpartion of x 0 separating B s from B t for t > s is non-crossing, thus every chain on [x m , x 0 ] has a −b s label. This fact, combined with (2) shows that any increasing chain on an interval must be constructed inductively by repeatedly taking the least-labeled edge down, hence be lexicographically first. The descending chains of Wachs's EL-labeling are {r σ : σ ∈ A d n }, where r σ corresponds to successively splitting σ at the greatest possible σ(id) [33,Theorem 5.2]. It is easy to see that each r σ is also descending with respect to our dual EL-labeling, and a dimension argument shows us that {r σ : σ ∈ A d n } is exactly the set of descending chains. An EL-ced for Π d n Order {Σ α } lexicographically by the reverse of the words α according to the reverse ordering on [n]. That is, order lexicographically by the words α(n)α(n − 1) · · · α(1), where n ⊳ n − 1 ⊳ · · · ⊳ 1. We refer to this ordering as rr-lex, for "reverse reverse lexicographic." For example, 132546 is the first permutation in A 2 6 with respect to rr-lex, while 231546 < rr−lex 142536 (since 4 > 3 in position 5). We will prove the following version of Theorem 1.1. Let Σ α be as in the text preceding Theorem 3.1, and λ as in Section 3.1. Theorem 3.8. {Σ α : α ∈ A d n } ordered by rr-lex is a dual EL-ced of Π d n with respect to λ. We introduce some terms. If B 1 \| . . . \| B k is a partition of [n], then we say that α ∈ S n has the form B 1 B 2 . . . B k if the first \|B 1 \| elements in the word α are in B 1 , the next \|B 2 \| are in B 2 , and so forth. When k = 2, we say that α has switched form B 1 B 2 if α ′ has the form B 1 B 2 for α ′ = α • (\|B 1 \| \|B 1 \| + 1), that is, for α ′ equal to α composed with the transposition of adjacent elements at \|B 1 \|. We can also talk of α having form B 1 B 2 . . . B k up to switching, by which we mean some α ′ has the form B 1 . . . B k , where α ′ is α up to transpositions at the borders of some (but not necessarily all) of the blocks. Finally, if B ⊆ [n], then α\| B is the word α = α(1)α(2) . . . α(n) with all α(i)'s that are not in B removed.c in Π d n is in Σ α for some α ∈ A d n . Proof. We will in fact construct the earliest such α according to the rr-lex ordering, which will in turn help us with Corollary 3.11. The proof has a similar feel to the well-known quicksort algorithm. Let c = {1 = c 0 · ≻ . . . · ≻ c n/d =0}. Consider first the edge1 · ≻ c 1 in c. The edge splits [n] into B 1 \| B 2 , and clearly such α, if it exists, must have the form B 1 B 2 or B 2 B 1 up to switching. If max B 1 < max B 2 , then all permutations in A d n of the (possibly switched) form B 1 B 2 come before permutations of the (possibly switched) form B 2 B 1 , so the rr-lex first α with c in Σ α has the form B 1 B 2 up to switching. Apply this argument inductively down the chain. At c i , we will have shown that the rr-lex first α with c in Σ α must have the form B 1 B 2 · · · B i+1 up to switching. Then if c i · ≻ c i+1 splits block B j into B j,1 and B j,2 , with max B j,1 < max B j,2 , an argument similar to that with1 · ≻ c 1 gives that α must in fact have the form B 1 B 2 . . . B j−1 B j,1 B j,2 B j+1 . . . B i+1 up to switching. At the end, we have shown the earliest α having c in Σ α must have the form B 1 . . . B n/d up to switching. Conversely, it is clear from the above that for any α of this form, c is in Σ α . Sort the elements of each B i in ascending order to get a permutation α 0 . This α 0 is in S n but not necessarily in A d n , so we perform a switch at each d-divisible position where there is an ascent (i.e., where B i < B i+1 ). This gives us an element α ∈ A d n of the given form up to switching, and finishes the proof of the statement. We continue nonetheless to finish showing that α is the first element in A d n with c in Σ α . We need to show that if β is another element of A d n with the same form up to switching of B 1 B 2 . . . B n/d (but different switches), then β > rr−lex α. If B i < B i+1 , then both α and β are switched at id (as otherwise we are not in A d n ). Otherwise, if β is a switch at id, then the switch exchanges β(id) and β(id + 1) (up to resorting the blocks). Since β(id) > β(id + 1), "unswitching" moves a larger element of [n] later in the permutation, yielding an rr-lex earlier element of the given form up to switching. Let ∆ α be the simplicial complex generated by maximal chains that are in Σ α (α ∈ A d n ), but in no Σ β for β ∈ A d n with β < rr−lex α. In the following corollary, we summarize the information from the proof of Lemma 3.10 about the form of α with c in ∆ α . 2. Let τ id be the transposition exchanging id and id + 1. If y · ≻ x corresponds to a switch-and-split at id, then the permutation α • τ id is ascending between positions (i − 1)d + 1 and (i + 1)d. 3. α\| B 1 = . . . max B 1 , i.e., max B 1 is rightmost in α\| B 1 . Proof. (1) and (2) are clear from the proof of Lemma 3.10. For (3), suppose that max B 1 is not rightmost in α\| B 1 . Then since α is ascending on d-segments, we have that max B 1 is rightmost in some d-segment of α\| B 1 . If a switchand-split occurs at max B 1 then we have a contradiction of (2), while a split contradicts (1). Every chain passing through an atom a has the same labels up to sign, and Corollary 3.11 tells us what the labels are. It is now not difficult to prove (CLced-desc) and (CLcedbdry). Proposition 3.12. Let [x m , x 0 ] be an interval with x m , x 0 ∈ ∆ α . Then there is at most one (dual) descending maximal chain c on [x m , x 0 ] which is in ∆ α . Proof. There are two cases: Case 1. x m =0. It suffices to consider a block B of x 0 . Partitions of B corresponding to edges in Σ α must either split or switch-and-split α\| B at d-divisible positions, and as every chain on [0, x 0 ] has bottommost label 0, all other edges of a descending chain must have positive labels (and so correspond to crossing partitions). Claim 3.13. All edges of such a descending chain correspond to splittings of α. Proof. (of Claim) Suppose otherwise. Without loss of generality we can assume that x 0 · ≻ x 1 in c corresponds to a switch-and-split of B into B 1 \| B 2 , with B the block of smallest size which is switch-and-split by an edge in c. We will show that c has an ascent. Corollary 3.11 part 2 tells us that the first d letters in α\| B 2 are strictly greater than the last d in α\| B 1 , and since max B 1 is rightmost in α\| B 1 , that the first d letters of α\| B 2 are strictly greater than all of B 1 . Since λ(x m−1 · ≻0) = 0, any negative label gives an ascent. If \|B 2 \| = d, then we have shown that B 1 \| B 2 is non-crossing (giving a negative label). If on the other hand \|B 2 \| > d, then any subdivision of B 2 gives a label with absolute value > max B 1 , hence an ascent. In either case, we contradict c being a descending chain. It follows immediately that a descending chain on [0, x 0 ] is unique. Case 2. x m =0. As usual, we consider what happens to a block B of x 0 . In x m , let B partition as B 1 \| . . . \| B k , where α\| B has the form B 1 B 2 . . . B k up to switching. Then every edge in ∆ σ comes from subdividing at some B i , i.e., as shown at the dotted line here Suppose that · · · ∪ B i \| B i+1 ∪ . . . is crossing, but B i \| B i+1 is non-crossing. Corollary 3.11 part 3 tells us that max(· · · ∪ B i ) = max B i , so that · · · ∪ B j \| B j+1 ∪ · · · ∪ B i . . . B i+1 ∪ · · · ∪ B l \| . . . Let b i = max B i ,min(B i+2 ∪ · · · ∪ B k ) < max B i < min B i+1 < max B i+1 . It follows that B i+1 \| B i+2 ∪ . . . is also crossing. Thus, if B i \| B i+1 is non-crossing, then (positive) b i is not the label of the first edge of a descending chain c, since b i+1 > b i would then be the label of a later edge. That is, if B i \| B i+1 is non-crossing, then a descending chain has a −b i label. The "only if" direction is immediate, thus there is a unique permutation and set of signs for the ±b 1 , . . . , ±b k−1 that could label a descending chain. Proposition 3.14. Let c be a (non-maximal) chain with extensions in both Σ α and Σ β , β < rr−lex α. Then c has maximal extensions in M(Σ α ) \ M(∆ α ). Proof. Let c = {1 = c 0 > c 1 · · · > c m > c m+1 =0}. The first β with c in Σ β obeys the following two conditions: 1. For each c i =0, each block B in c i−1 splits into sub-blocks B 1 , . . . , B k in c i , where max B 1 < · · · < max B k . The restriction β\| B is of the form B 1 B 2 . . . B k . (By repeated application of Corollary 3.11 .) 2. For each block B of c m , β\| B is the permutation Since α is not the first permutation in A d n such that c ∈ Σ α , α must violate at least one of these. If it violates (1) for some B, then α\| B has the form B 1 . . . B k with max B j > max B j+1 . Merge B j and B j+1 to add an edge down from c i−1 that is in Σ α , otherwise extend arbitrarily in Σ α . By Corollary 3.11 part 1, the resulting chain is not in ∆ α . {b 1 b 2 . . . b d+1 b d . . . b id+1 b id . . . b k }, where B = {b 1 , . . . , b k } for b 1 < · · · < b k . That is, β\| B is If α\| B violates (2) for some B, then extend c by switch-and-splitting at every ddivisible position of B, otherwise arbitrarily in Σ α . At the bottom, B is partitioned into some B 1 \| . . . \| B k . Since (2) is violated, applying transpositions to α at d-divisible partitions gives a descent. But this contradicts the conclusion of Corollary 3.11 part 2, and the resulting chain is not in ∆ α . We check the CL-ced properties: Wachs had already proved (CLced-polytope) as presented in Theorem 3.1, Lemma 3.10 gives us (CLced-union), Proposition 3.12 gives (CLced-desc), and Proposition 3.14 gives (CLced-bdry). We have completed the proof of Theorem 3.8. The coset lattice 4.1 Group theory background The coset poset of G, denoted C(G), is the set of all right cosets of all proper subgroups of G, ordered under inclusion. The coset lattice of G, denoted C(G), is C(G) ∪ {∅, G}, that is, C(G) with a top1 = G and bottom0 = ∅ added. With our definitions, it makes sense to look at the order complex of C(G) (which is the set of all chains of C(G)), and so we talk about the coset lattice, even though "coset poset" has a better sound to it. We notice that C(G) has meet operation Hx ∧ Ky = Hx ∩ Ky and join Hx ∨ Ky = H, K, xy −1 y (so it really is a lattice.) General background on the coset lattice can be found in [22,Chapter 8.4], and its topological combinatorics have been studied in [5,19,36]. The subgroup lattice of G, denoted L(G) is the set of all subgroups of G, ordered by inclusion. General background can be found in [22], and its topological combinatorics have been studied extensively, for example in [24,32]. Notice that for any x ∈ G, the interval [x, G] in C(G) is isomorphic to L(G). It is a theorem of Iwasawa [11] that L(G) is graded if and only if G is supersolvable, hence C(G) is graded under the same conditions. As we have only defined convex ear decompositions for pure complexes, we are primarily interested in supersolvable groups in this paper. Schweig proved the following: It is easy to check that any normal subgroup N ⊳ G is left modular in L(G), so a supersolvable group has a supersolvable subgroup lattice with any chief series as its left modular chain. Let G ′ denote the commutator subgroup of G. The following collected classification of groups with every interval in their subgroup lattice complemented is presented in Schmidt's book [22,Chapter 3.3], and was worked out over several years by Zacher, Menegazzo, and Emaldi. L(G) is coatomic, i.e., every subgroup H of G is an intersection of maximal subgroups of G. G has elementary abelian Sylow subgroups, and if H 1 ⊳ H 2 ⊳ H 3 ⊆ G, then H 1 ⊳ H 3 . 5. G ′ and G/G ′ are both elementary abelian, G ′ is a Hall π-subgroup of G, and every subgroup of G ′ is normal in G. Note 4.4. The classification of finite simple groups is used in the proof that (3) is equivalent to the others. We will follow Schmidt and call such a group a relatively complemented group. We notice that relatively complemented groups are complemented, that is, satisfy the condition of Proposition 4.3 Part (2) on the interval [1, G]. On the other hand, S 3 × Z 3 is an example of a complemented group which is not relatively complemented. The complemented groups are exactly the groups with (equivalently in this case) shellable, Cohen-Macaulay, and sequentially Cohen-Macaulay coset lattice [36]. Computation with GAP [10] shows that there are 92804 groups of order up to 511, but only 1366 complemented groups, and 1186 relatively complemented groups. We summarize the situation for the subgroup lattice regarding convex ear decompositions: Corollary 4.5. The following are equivalent for a group G: 1. L(G) has a convex ear decomposition. L(G) is doubly Cohen-Macaulay. G is a relatively complemented group. As a consequence, we get one direction of Theorem 1.2. Corollary 4.6. If C(G) is doubly Cohen-Macaulay (hence if it has a convex ear decomposition), then G is a relatively complemented group. Proof. Every interval of a 2-Cohen-Macaulay poset is 2-Cohen-Macaulay, and the interval [1, G] in C(G) is isomorphic to L(G). The remainder of Section 4 will be devoted to proving the other direction. A dual EL-labeling for C(G) As with the d-divisible partition lattice, the first thing we need is a dual EL-labeling of C(G). We will construct one for the more general case where G is complemented. The main idea is to start with the EL-labeling of an upper interval and "twist" by adding signs, similarly to our EL-labeling for Π d n . The resulting labeling is significantly simpler than the one I described in [36]. Let G be a complemented group, and fix a chief series G = N 1 ⊲ N 2 ⊲ · · · ⊲ N k+1 = 1 for G throughout the remainder of Section 4. Our labeling (and later our convex ear decomposition) will depend on this choice of chief series, but the consequences for the topology and h-vector of C(G) will obviously depend only on G. For each factor N i /N i+1 , choose a complement B 0 i , i.e., a subgroup such that N i B 0 i = G but N i ∩ B 0 i = N i+1 . (Such a B 0 i exists, as every quotient group of a complemented group is itself complemented [22, Lemma 3.2.1].) From Section 2.3, the usual dual EL-labeling of the subgroup lattice of a supersolvable group is λ ss (K 0 ⊃ · K 1 ) = max{i : N i K 1 ⊇ K 0 } = min{i : N i+1 ∩ K 0 ⊆ K 1 }. Remember that λ ss labels every chain on a given interval with the same set of labels (up to permutation). We now define a labeling λ of C(G) as follows. For K 0 ⊃ · K 1 labeled by λ ss with i, let λ(K 0 x ⊃ · K 1 x) = −i if K 1 x = K 0 x ∩ B 0 i , i otherwise, and λ(x ⊃ · ∅) = 0. It is immediate from this construction that \|λ\| [x,G] = λ ss (up to the "dropping x" isomorphism), much like the situation discussed in Section 3.1 for the d-divisible partition lattice. Lemma 4.7. Let G be any supersolvable group. Then: 1. If KB = G where B ⊂ · G, then K ∩ B ⊂ · K. 2. If λ ss (K 0 ⊃ · K 1 ) = i, then for any complement B i of N i /N i+1 and K ⊇ K 0 we have For part 2, by the definition of the labeling, Proof. We need to show that every interval has a unique increasing maximal chain which is lexicographically first. There are two kinds of intervals we need to check: K 0 B i = KB i = G.N i ∩K 0 ⊆ K 1 but N i+1 ∩K 0 ⊆ K 1 . We see that N i ∩ K 0 ⊆ N i+1 , and since N i+1 ⊂ · N i , that (N i ∩ K 0 )N i+1 = N i and so K 0 N i+1 ⊇ N i . Then K 0 B i = K 0 N i+1 B i ⊇ N i B i = G.Case 1. [∅, H 0 x] As the last label of any chain on this interval is 0, in an increasing chain the others must be negative (in increasing order). Since every chain has the same labels up to permutation, uniqueness of the increasing chain is clear from the definition of λ. Existence follows from applying Lemma 4.7 to the maximal subgroups B 0 i . Finally, the chain takes the edge with the least possible label down from each Hx, so it is lexicographically first. Case 2. [H n x, H 0 x] Let S be the label set of λ ss restricted to the interval [H n , H 0 ]. We notice that a −i label is possible on [H n x, H 0 x] only if H n x ⊆ B 0 i and i ∈ S. Thus, the lexicographically first chain is labeled by all possible negative labels (in increasing order), followed by the remaining (positive) labels, also in increasing order. Such a chain clearly exists and is increasing. The negative-labeled part is unique since a −i label corresponds with intersection by B 0 i , while the positive-labeled part is unique since λ ss = \|λ\| is an EL-labeling. It remains to check that there are no other increasing chains. We have already shown that there is only one increasing chain which has a −i label for each B 0 i containing H n x, so any other increasing chain would need to have a +i label for some i ∈ S where H n x ⊆ B 0 i . Without loss of generality, let this edge H 0 x ⊃ · H 1 x be directly down from H 0 x. Then i = min S, and since λ ss is a dual EL-labeling, we have that there is a unique edge down from H 0 x with label ±i. But then H 1 x = H 0 x ∩ B 0 i , so the edge gets a −i label, giving us a contradiction and completing the proof. Though we do not need it for our convex ear decomposition, let us briefly sketch the decreasing chains of λ. Following Thévenaz [32], a chain of complements to a chief series G = N 1 ⊃ · N 2 ⊃ · . . . ⊃ · N k+1 = 1 is a chain of subgroups G = H k+1 ⊃ · H k ⊃ · . . . ⊃ · H 1 = 1 where for each i, H i is a complement to N i . Thévenaz showed that the chains of complements in G correspond to homotopy spheres in \|L(G)\|. The following proposition is the EL-shelling version of Thévenaz's result for a supersolvable group, and is a special case of [37,Proposition 4.3]. Proof. If G = H k+1 ⊃ · H k ⊃ · . . . ⊃ · H 1 = 1 is a chain of complements, then N i H i = G ⊇ H i+1 , while N i+1 H i ∩ H i+1 = (N i+1 ∩ H i+1 )H i = 1 · H i = H i by left modularity (the Dedekind identity). Thus λ ss (H i+1 ⊃ · H i ) = i, and the chain is descending. Conversely, any descending chain corresponds to a sphere in \|L(G)\|, and by Thévenaz's correspondence, there can be no others. {G = H k+1 x ⊃ · . . . ⊃ · H 1 x = x ⊃ · ∅} to the chief series G = N 1 ⊲ . . . ⊲ N k+1 = 1 such that no H i x = H i+1 x ∩ B 0 i . A convex ear decomposition for C(G) Recall that subgroups H and K commute if HK = KH is a subgroup of G. Proof. Suppose j < i. Then N i+1 N i ⊆ N j+1 N j , and N j+1 ⊆ B j . Thus, B j B i ⊇ N i B i = G. Recalling G = N 1 ⊲ N 2 ⊲ · · · ⊲ N k+1 = 1 as the chief series we fixed in Section 4.2, let B = {B i : B i is a complement to N i /N i+1 , 1 ≤ i ≤ k} be a set of complements to N i , one complement for each chief factor (so that \|B\| = k). For any x ∈ G, let Bx = {B i x : B i ∈ B}. We will call B a base-set for C(G). The first step is to show that intersections of certain cosets of B give us a cube, using a stronger version of Lemma 4.11. Lemma 4.12. If B is a base-set, then (B i 1 ∩ · · · ∩ B i l )B i ℓ+1 = G. Proof. We count \|(B i 1 ∩ · · · ∩ B i ℓ )B i ℓ+1 \| = \|(B i 1 ∩ · · · ∩ B i ℓ )\|\|B i ℓ+1 \| \|B i 1 ∩ · · · ∩ B i ℓ ∩ B i ℓ+1 \| = \|B i 1 ∩ · · · ∩ B i ℓ−1 \|\|B i ℓ \|\|B i ℓ+1 \| \|(B i 1 ∩ · · · ∩ B i ℓ−1 )B i ℓ \|\|B i 1 ∩ · · · ∩ B i ℓ ∩ B i ℓ+1 \| . By induction on ℓ, this is = \|B i 1 ∩ · · · ∩ B i ℓ−1 \|\|B i ℓ \|\|B i ℓ+1 \| \|G\|\|B i 1 ∩ · · · ∩ B i ℓ ∩ B i ℓ+1 \| , and by symmetry, \|(B i 1 ∩ · · · ∩ B i ℓ )B i ℓ+1 \| = \|(B i 1 ∩ · · · ∩ B i ℓ−1 ∩ B i ℓ+1 )B i ℓ \|. Repeating this argument shows that \|(B i 1 ∩ · · · ∩ B i ℓ )B i ℓ+1 \| is independent of the ordering of the B i j 's, or of the choice of i ℓ+1 . Then take i ℓ+1 to be the largest index of any such B i j , so that N i ℓ+1 ⊆ B i 1 ∩ · · · ∩ B i ℓ . In particular, (B i 1 ∩ · · · ∩ B i ℓ )B i ℓ+1 ⊇ N i ℓ+1 B i ℓ+1 = G. Since the ordering of the i j 's doesn't affect the cardinality, \|(B i 1 ∩ · · · ∩ B i ℓ )B i ℓ+1 \| = \|G\| for any choice of i ℓ+1 , proving the lemma. We henceforth assume that G is relatively complemented. Let B be a base-set for C(G) as above, and x ∈ G be such that B i x = B 0 i (for each i). Then we define Σ Bx to be the meet sublattice of C(G) generated by Bx ∪ {B 0 i : B i x = B 0 i x}, and the larger meet sublattice Σ + Bx to be generated by Bx ∪ {B 0 i : B i x = B 0 i x} ∪ {B i y i : B i = B 0 i }, where the y i 's are some elements such that B i y i = B i x. By Lemma 4.7 and the proof of Corollary 4.13, {i : B i x=B 0 i x} B 0 i ∩ {i : B i =B 0 i } B i y i = y (for some y), so Σ + Bx is given by all intersections of Bx ∪ By. Thus (also by Corollary 4.13) \|Σ + Bx \| is a convex polytope with subcomplex \|Σ Bx \|. Lemma 4.14. Let H 0 x ⊃ · H 1 x be an edge in C(G) with λ(H 0 x ⊃ · H 1 x) = i. Then H 1 x = H 0 x ∩ B i x for some complement B i to N i /N i+1 . Proof. Since every maximal chain in C(G) has exactly one edge with λ(H 0 x ⊃ · H 1 x) = ±i for each i ∈ [k], it suffices to show that H 1 is contained in some complement B i to N i /N i+1 . Then H 0 cannot be contained in B i , as that would give two ±i edges, and so H 1 = H 0 ∩B i . Since G is relatively complemented, every interval in L(G) is complemented. In particular, any interval of height 2 has both increasing and decreasing chains, so for any H −1 ⊃ · H 0 there is an H + 1 ⊂ · H −1 with λ ss (H −1 ⊃ · H + 1 ) = i. Repeat this argument inductively on H −1 ⊃ · H + 1 until H −1 = G. The final H + 1 is the desired B i , and the definition of λ ss shows that B i is a complement to N i /N i+1 . Now that we have a set of cubes that cover C(G), the next step is to assign an order to them. For any base-set B, let ρ i (B) be 0 if B i = B 0 i , and 1 otherwise. We put the ρ i 's together in a binary vector ρ(B), which we will call the pattern of B. Order the Bx's (and hence the Σ Bx 's) in any linear extension of the lexicographic order on ρ(B). Let ∆ Bx be the simplicial complex with facets the maximal chains that are in Σ Bx , but not in any preceding Σ B ′ x ′ . The Σ Bx 's are generally proper subsets of face lattices of convex polytopes, so (CLcedpolytope) does not hold and we do not have an EL-ced. We can use the same sort of argument, however, to prove the following refinement of Theorem 1.2: Corollary 4.15 shows that the ears cover C(G), that is, that (CLced-union) holds. Our next step is to show that an analogue of (CLced-desc) holds. It will be convenient to let S([a, b]) be the label set of \|λ\| on the interval [a, b], that is, the set of nonnegative i's such that λ gives ±i labels on cover relations in [a, b]. If a = ∅, then the interval [a, b] in Σ Bx is Boolean, with a maximal chain for each permutation of S ([a, b]). If there is a −i label on a chain in ∆ Bx , then the edge can be obtained by intersecting with B 0 i . But since Σ Bx is the first such complex containing the chain, we must have ρ i (B) = 0 (otherwise, replace B i x with B 0 i ). But this tells us that every label with absolute value i on [a, b] in ∆ Bx is negative. Every such chain thus has the same set of labels, and at most one permutation of these labels is descending. Corollary 4.18. ∆ Bx is shellable. Proof. Suppose a maximal chain c = {G = c 1 ⊃ · . . . ⊃ · c k+1 ⊃ · c k+2 = ∅} in Σ Bx has an ascent at j. If j = k + 1, then it is immediate that c \ {c j } has two extensions in Σ Bx , and we argue exactly as in Lemma 2.7 and Theorem 2.9. If j = k + 1, then the ascent at j has labels −i, 0, and hence ρ i (B) = 0 and Σ Bx is the first cube containing c \ {c k+1 }. Intersecting with B 0 i x instead of B 0 i at c k gives another chain c ′ in ∆ Bx with a descent at k + 1, and we again argue as in Theorem 2.9. Finally, we show directly that (ced-bdry) holds. We start with a lemma. Lemma 4.19. Given any chain c = {G = c 1 ⊃ · · · ⊃ c m ⊃ c m+1 = ∅}, there is an extension to a maximal chain c ++ such that if c is in Σ Bx , then c ++ is in some Σ + Bx . If Bx is the first such with c in Σ Bx , then c ++ is in Σ Bx . Proof. We make the extension in two steps. First, let c + be the extension of c by augmenting each c j ⊃ c j+1 for j = m with the chain on [c j+1 , c j ] that is increasing according to \|λ\|. Intersecting c j iteratively with B i x or B 0 i (as appropriate, for each i in S([c j+1 , c j ])) in increasing order gives this chain, thus, c + is also in Σ Bx . In a similar manner, let c ++ be the extension of c + at c m ⊃ ∅ by intersecting with each B 0 i for i ∈ S(m) in increasing order. Suppose c m = Hx. Then uniqueness of the lexicographically first chain in [1, H] gives that H ∩ B 0 i = H ∩ B i , so there is some B i y i with Hx ∩ B 0 i = Hx ∩ B i y i . Repeated use of this gives us a Σ + Bx containing c ++ : the generating elements for this cube include those for Σ Bx and the B i y i 's found here. Notice that if ρ(i) = 0 for each i ∈ S([∅, c m ]), then B 0 i is already in the generating set for Σ Bx , thus c ++ is also in Σ Bx . Proposition 4.20. ∆ Bx ∩ B ′ x ′ ≺Bx ∆ B ′ x ′ = ∂∆ Bx . Proof. Suppose that c is in ∆ Bx ∩ B ′ x ′ ≺Bx ∆ B ′ x ′ , and let c ++ be as in Lemma 4.19. Then c ++ is an extension in Σ + Bx , but since c ++ is contained in Σ B ′ x for the first such complex containing c, we get that c ++ is in M(Σ + Bx ) \ M(∆ Bx ). Lemma 2.8 then gives that c is in ∂∆ Bx . Conversely, let c be in ∆ Bx , but not in a previous Σ B ′ x ′ . Since c is not in any previous Σ B ′ x ′ , no extensions of it are either, so any extension of c that is in Σ Bx is in ∆ Bx . As we have ordered the base-sets by pattern, we get that ρ i (B) = 0 for i ∈ S([∅, c m ]), thus, by the special treatment of B 0 i in the definition of Σ Bx , every extension of c in any Σ + Bx is in Σ Bx . Combining these two statements, we see that there is no extension of c in M(Σ + Bx ) \ M(∆ Bx ), and so by Lemma 2.8 that c is not in ∂∆ Bx . We have now finished the proof of Theorem 4.16. Let us review: Corollary 4.13 gave us (ced-polytope), Proposition 4.20 was (ced-bdry), and Corollary 4.15 gave us (ced-union). We notice that the base-set with the earliest pattern is B 0 = {B 0 i }, and that each Σ B 0 x is the face lattice of a cube. Thus the first ∆ Bx is a polytope, while all subsequent ones are proper subcomplexes of polytopes. Since we proved in Corollary 4.18 that each ∆ Bx is shellable, we have (ced-topology). Note 4.21. As previously mentioned, the convex ear decomposition we have constructed is not a (dual) EL-ced. Although we would rather find an EL-ced than a general convex ear decomposition, this is not in general possible with the cubes we are looking at here. For example C(Z 2 2 ) has exactly three possible Σ + Bx 's, but the homotopy type of the wedge of 6 1-spheres, so some \|Σ + Bx \| \ \|Σ + B ′ x ′ \| must be disconnected. The example of C(Z 2 2 ) is a geometric lattice, so does have an EL-ced (for a different EL-labeling), but I have not been able to extend this to an EL-ced for other relatively complemented groups. The reader may have noticed that the constructed convex ear decomposition is not far from being an EL-ced -the difference is that each Σ + Bx gives several "new" ears -and that another possibility would be to extend the definition of EL-ced to cover this case. However, as this would make the definition more complicated, and as the gain seems relatively small, I have chosen to leave the definition as presented. Poset products Throughout this section, let P 1 and P 2 be bounded posets. In Section 3.1, we defined the product P 1 × P 2 and lower reduced product P 1× P 2 of P 1 and P 2 . It should come as no surprise that the upper reduced product P 1× P 2 of P 1 and P 2 is defined as (P 1 \ {1}) × (P 2 \ {1}) ∪ {1}. There is a natural inclusion of P 1× P 2 (and of P 1× P 2 ) into P 1 × P 2 . Our goal in Section 5 is to explain the background and give proofs for Theorems 1.4 and 1.6. The flavor and techniques of this section are different from the previous two, so we pause to justify its connection with "Cubical Convex Ear Decompositions". Lower reduced products come up fundamentally both in the d-divisible partition lattice, as we discussed in Section 3.1, as well as in the coset lattice, where C(G 1 × G 2 ) ∼ = C(G 1 )× C(G 2 ) for groups G 1 and G 2 of co-prime orders. And some of the decompositions in product posets are cubical after all: a cube is the direct product of intervals, so if C d is the boundary of the d-cube, with face lattice L(C d ), then L(C d ) =ˇ Poset products and polytopes I am told that the following proposition is folklore. It is also discussed briefly in [12]. Proposition 5.1. If Σ 1 and Σ 2 are the face lattices of convex polytopes X 1 and X 2 , then 1. Σ 1 × Σ 2 is the face lattice of the "free join" X 1 ⊛ X 2 , a convex polytope. 2. Σ 1× Σ 2 is the face lattice of the Cartesian product X 1 × X 2 , a convex polytope. 3. Σ 1× Σ 2 is the face lattice of the "free sum" of X 1 and X 2 , a convex polytope. Proposition 5.1 guides us to a proof of Lemma 1.5. Our main tool will be stellar subdivision. If ∆ is a convex polytope with a proper face σ, then a stellar subdivision of ∆ at σ, denoted stellar σ ∆, is conv (∆ ∪ {v σ }), where v σ = w σ − ε(w ∆ − w σ ) for some point w σ in the relative interior of σ, some point w ∆ in the interior of ∆, and a small number ε. In plain language, we "cone off" a new vertex lying just over σ. Note that the relative interior of a vertex is the vertex itself. Stellar subdivisions are discussed in depth in [9,III.2] and [8]. The main fact [9, III.2.1, III.2.2] that we will need is that the faces of the boundary complex of stellar σ ∆ are {τ : σ ⊆ τ } ∪ {v σ * τ : τ ∈ ∆ with τ, σ ⊆ τ ′ for some τ ′ ∈ ∆, but σ ⊆ τ } . Thus the stellar subdivision replaces the faces containing σ with finer subdivisions. If X is the boundary complex of a polytope, then let X denote conv X, that is, the polytope of which X is the boundary complex. Lemma 5.3. Suppose P 1 and P 2 are bounded posets and that \|P 1 \| and \|P 2 \| are the boundary complexes of polytopes. Then \|P 1× P 2 \| can be obtained from the boundary complex of \|P 1 \| × \|P 2 \| by a sequence of stellar subdivisions. Proof. Let ∆ 0 be the boundary complex of \|P 1 \| × \|P 2 \|. The faces of ∆ 0 are exactly the products F (1) × F (2) , where each F (i) is a non-empty face in P i , and at least one is proper. In particular the vertices are products of vertices v (1) × v (2) , where v (i) is in P i \ {0,1}. We write this product of vertices as (v (1) , v (2) ), and think of it as sitting in \|P 1× P 2 \|. We start by ordering the elements {v (2) } of P 2 by a reverse linear extension, and stellarly subdividing at each σ = \|P 1 \| × v (2) in this order. Inductively assume that the faces containing σ are those of the form (\|P 1 \| × F (2) ) * C, where F (2) is a face of \|P 2 \| with top-ranked vertex v (2) , and C is a simplex corresponding to (the simplicial join of) a chain of elements of the form (1, w (2) ) (with each w (2) > v (2) ). Subdivision replaces these faces with those of the form (\|P 1 \| × F (2) 0 ) * C * {v σ }, where F (2) 0 is a face having top-ranked vertex < v (2) . We abuse notation to call the newly introduced vertex v σ as (1, v (2) ), which puts us in the situation required to continue our induction. We next do the same procedure for the faces v (1) × \|P 2 \|. That is, we order {v (1) } by a reverse linear extension of P 1 , and repeatedly perform stellar subdivision at each such face according to this order. Since a face cannot contain both \|P 1 \| and \|P 2 \|, these stellar subdivisions are independent of the ones at \|P 1 \| × v (2) . After subdividing at all \|P 1 \| × v (2) and v (1) × \|P 2 \|, we obtain a complex ∆ 1 . The vertex set of ∆ 1 is exactly P 1× P 2 \ {0,1}. The faces of ∆ 1 are {(F (1) × F (2) ) * C}, where F (i) is a face of \|P i \|, and C is a simplex corresponding to either a chain of elements (1, w (2) ) or a chain of elements (w (1) ,1). Finally, we perform stellar subdivision at the vertices v = (v (1) , v (2) ), where v (i) ∈ P i \ {0,1}, in the order of a reverse linear extension of P 1× P 2 . We make an induction argument parallel to the one above: at the step associated with vertex v, the faces containing v are {(F (1) × F (2) ) * C}. As before, F (i) is a face of \|P i \| with top-ranked vertex v (i) , and C corresponds to (the simplicial join of) elements in a chain greater than v in P 1× P 2 . Stellar subdivision at v replaces these faces with {(F (1) 0 × F (2) 0 ) * C * {v}}, where F (i) 0 has greatest vertex < v (i) , and we continue the induction. When we have subdivided at every vertex, we obtain a complex ∆ 2 . The faces of ∆ 2 are simply {C}, where C is the simplicial join of vertices in a chain of P 1× P 2 , which is the definition of the order complex \|P 1× P 2 \|. Corollary 5.4. If If P 1 and P 2 are bounded posets such that \|P 1 \| and \|P 2 \| are the boundary complexes of polytopes, then \|P 1× P 2 \| and (by duality) \|P 1× P 2 \| are also boundary complexes of polytopes. For P 1 × P 2 , a similar result holds. Recall that the free join ∆ 1 ⊛ ∆ 2 of two polytopes ∆ 1 and ∆ 2 is obtained by taking the convex hull of embeddings of ∆ 1 and ∆ 2 into skew affine subspaces of Euclidean space (of high enough dimension). The faces of ∆ 1 ⊛ ∆ 2 , as hinted in Proposition 5.1, are F (1) ⊛ F (2) , and dim F (1) ⊛ F (2) = dim F (1) + dim F (2) + 1. Lemma 5.5. Suppose P 1 and P 2 are bounded posets and that \|P 1 \| and \|P 2 \| are the boundary complexes of polytopes. Then \|P 1 × P 2 \| can be obtained from the boundary complex of \|P 1 \| ⊛ \|P 2 \| by a sequence of stellar subdivisions. Proof. Since the details of the proof are very similar to the preceding Lemma 5.3, we provide a sketch only. Let ∆ 0 = \|P 1 \| ⊛ \|P 2 \|. Notice that the vertices of ∆ 0 are {∅ ⊛ v (2) } ∪ {v (1) ⊛ ∅}, while the edges are {v (1) ⊛ v (2) }. As in Lemma 5.3, we begin by ordering the facets \|P 1 \| ⊛ v (2) and v (1) ⊛ \|P 2 \| according to reverse linear extensions of P 2 and P 1 , and inductively performing stellar subdivision. Each such subdivision creates a vertex, which we name (1, v (2) ) or (v (1) ,1). We obtain a complex ∆ 1 with faces {(F (1) ⊛ F (2) ) * C} where F (i) is a proper face of P i (possibly empty), and C corresponds to a chain in the elements {(1, v (2) )} or {(v (1) ,1)}. We then order the edges v (1) ⊛v (2) by a linear extension of P 1 ×P 2 , and inductively perform stellar subdivision to create vertices (v (1) , v (2) ). The resulting complex is isomorphic to \|P 1 × P 2 \|. Corollary 5.6. If P 1 and P 2 are bounded posets such that \|P 1 \| and \|P 2 \| are the boundary complexes of polytopes, then \|P 1 × P 2 \| is also the boundary complex of a polytope. We now show both inclusions for (ced-bdry). If d is any chain in ∂∆ s,t with p 1 (d) in ∂∆ (1) s , then p 1 (d) is in ∆ (1) u for some u < s by (ced-bdry), so d is in ∆ u,t ; similarly if p 1 (d) is maximal and p 2 (d) is in ∂∆ (2) t . Thus ∂∆ s,t ⊆ ∆ s,t ∩ (u,v)<(s,t) ∆ u,v . In the other direction: if c is in ∆ s,t and ∆ u,v (for (u, v) < (s, t)), then p 1 (c) is in both ∆ (1) s and ∆ (1) u . If s = u, then p 1 (c) is in ∂∆ (1) s , so c is in ∂∆ s,t . A similar argument applies for p 2 when s = u. Thus, ∂∆ s,t ⊇ ∆ s,t ∩ (u,v)<(s,t) ∆ u,v , and we have shown (ced-bdry), completing the proof. Product CL-labelings In this subsection, we explicitly recall the product CL-labelings introduced by Björner and Wachs in [4,Section 10], and hinted at in Section 3.1. Since there is no particular reason to work with dual labelings in Section 5, I've chosen to work with standard (not dual) CL-labelings, so that everything is "upside down" relative to Sections 3 and 4. Since the root of an edge of the form0 ⋖ x is always ∅, we suppress the root from our notation in this case. Let P be a bounded poset with a CL-labeling λ that has label set S λ . A label s ∈ S λ is atomic if it is used to label a cover relation0 ⋖ x (for any atom x), and non-atomic if it is used to label any other rooted cover relation. (In an arbitrary CL-labeling, a label can be both atomic and non-atomic.) A CL-labeling is orderly if S λ is totally ordered and partitions into S − λ < S A λ < S + λ , where every atomic label is in S A λ , and every non-atomic label is either in S − λ or S + λ . There are similar definitions of co-atomic, non-co-atomic, and co-orderly, and of course we can generalize to talk of orderly and co-orderly chain edge labelings, even if the CL-property is not met. Then P has an orderly CL-labeling λ ′ , and a co-orderly CL-labeling λ ′′ , such that any maximal chain c in P has the same set of ascents and descents under each of the three labelings λ, λ ′ , and λ ′′ . The proof involves constructing a recursive atom ordering from λ, and then constructing a CL-labeling with the desired properties from the recursive atom ordering. To find a CL-labeling of P 1 × P 2 , we label each edge in P 1 × P 2 with the edge in P 1 or P 2 to which it projects. More formally, notice that any rooted cover relation (r, x ⋖ y) projects to a cover relationship in one coordinate, and to a point in the other. Then the product labeling, denoted λ 1 × λ 2 , labels (r, x ⋖ y) with λ i (p i (r), p i (x ⋖ y)), where i is the coordinate where projection is nontrivial. It is straightforward to show that λ 1 × λ 2 is a CL-labeling if λ 1 and λ 2 are CL-labelings of P 1 and P 2 , and where we order S λ 1 ∪ S λ 2 by any shuffle of S λ 1 and S λ 2 [4, Proposition 10.15]. The idea behind finding a CL-labeling of P 1× P 2 (or similarly P 1× P 2 ) is to restrict λ 1 × λ 2 to P 1× P 2 . For a cover relation x ⋖ y where x =0, this works very well, as x ⋖ y in P 1× P 2 is also a cover relation in P 1 × P 2 , and the roots project straightforwardly. The problem comes at cover relations0 ⋖ y, which project to a cover relation in both P 1 and P 2 . Here, we need to combine the labels λ 1 0 ⋖ p 1 (y) and λ 2 0 ⋖ p 2 (y) . The orderly labelings constructed in Lemma 5.9 are a tool to perform this combination in a manner that preserves the CL-property. Let P 1 and P 2 have orderly CL-labelings λ 1 and λ 2 , with disjoint label sets S 1 and S 2 . Suppose the label sets are shuffled together as S − 1 < S − 2 < S A 1 < S A 2 < S + 1 < S + 2 . Then the lower reduced product labeling λ 1× λ 2 labels an edge0 ⋖ y with the word λ 1 p 1 (0 ⋖ y) λ 2 p 2 (0 ⋖ y) in S A 1 S A 2 (lexicographically ordered), while all other rooted edges (r, x ⋖ y) (for x =0) are labeled with the nontrivial projection λ i (p i (r), p i (x ⋖ y)) as in λ 1 × λ 2 . Björner and Wachs proved [4, Theorems 10.2 and 10.17] that λ 1× λ 2 is a CL-labeling of P 1× P 2 . Similarly, if λ 1 and λ 2 are co-orderly CL-labelings of P 1 and P 2 , with disjoint label sets shuffled together as for the orderly labelings above, we define the upper reduced product labeling λ 1× λ 2 as follows. Label an edge of the form (r, x ⋖1) with the word λ 1 p 1 (r), p 1 (x ⋖1) λ 2 p 2 (r), p 2 (x ⋖1) in S A 1 S A 2 , and all other edges (r, x ⋖ y) (for y =1) as in λ 1 × λ 2 . Then [4, Theorems 10.2 and 10.17] gives us that λ 1× λ 2 is a CL-labeling of P 1× P 2 . Example 5.11. The labeling λ div we constructed for the d-divisible partition lattice was an co-orderly EL-labeling of the dual lattice: actually, S A was just {0}. As discussed in Lemma 3.5, intervals split as products, and the restriction of λ div to an interval splits as the appropriate product labeling. We summarize in the following theorem: 1. If λ 1 and λ 2 are CL-labelings (EL-labelings), then λ 1 × λ 2 is a CL-labeling (ELlabeling) of P 1 × P 2 . 2. If λ 1 and λ 2 are orderly CL-labelings, then λ 1× λ 2 is a CL-labeling of P 1× P 2 . 3. If λ 1 and λ 2 are co-orderly CL-labelings, then λ 1× λ 2 is a CL-labeling of P 1× P 2 . CL-ceds of product posets Fix our notation as in Section 5.2, but suppose in addition that P 1 and P 2 have CL-ceds {Σ (1) s } and {Σ (2) t } with respect to the CL-labelings λ 1 and λ 2 . Denote the resulting ears of new chains as {∆ (1) s } and {∆ (2) t }, as in Section 2.5. Then take Σ s,t to be the appropriate product of Σ (1) s and Σ (2) t , and ∆ s,t to be the associated ear of new chains. We first notice that there is no inconsistency with the notation used in Section 5.2: Proof. The statement follows straightforwardly from the fact that the maximal chains of Σ s,t are those that project to Σ (1) s and Σ (2) t . As we did in Section 5.2, order the {Σ s,t } according to the lexicographic order of (s, t). Let λ be the appropriate product CL-labeling, where we assume without loss of generality via Lemma 5.9 that λ 1 and λ 2 are orderly or co-orderly. Then we will prove: Theorem 5.14. {Σ s,t } is a CL-ced for \|P \| with respect to λ. Proof. Proposition 5.1 tells us that (CLced-polytope) is satisfied, and (CLced-union) is immediate from the definitions. For (CLced-bdry), we work backwards, and notice that we have already shown in Theorem 5.7 that ∂∆ s,t = ∆ s,t ∩ u,v<s,t ∆ u,v . Lemma 2.8 meanwhile gives that ∂∆ s,t = ∆ s,t ∩ \|Σ s,t \| \ ∆ s,t , hence that a chain c with extensions in both ∆ s,t and ∆ u,v has an extension in M(Σ s,t ) \ M(∆ s,t ), as required. (A direct proof is also straightforward.) It remains to check (CLced-desc). Although the statement of this property is very similar to [4,Theorem 10.17] (which says that λ is a CL-labeling), the proof in [4] uses some machinery. So we work from scratch, as follows. If P = P 1 × P 2 and we are considering the rooted interval [x, y] r , then the labels of a maximal chain c r on [x, y] r are the same as the labels of p 1 (c) p 1 (r) union with the labels of p 2 (c) p 2 (r) , "shuffled together" in some order. Thus, if c r is descending, then the projections must also be descending. Since the label sets S 1 and S 2 are taken to be disjoint, there is a unique way of shuffling the two label sets (and so the two chains) together to get a descending chain. For P = P 1× P 2 , the proof is the same unless x =0. In this case, the first label of a maximal chain c is in S A 1 S A 2 , while the first label of the projections are in S A 1 and S A 2 , respectively. If c is descending, then all labels after the first are from S − 1 or S − 2 , since S − 1 < S − 2 < S A 1 S A 2 < S + 1 < S + 2 , and thus p 1 (c) and p 2 (c) are descending, and we argue as before. The proof for P = P 1× P 2 is entirely similar to that for P 1× P 2 . Further questions The close relationship between the techniques used in Sections 3 and 4 leads us to ask the following question. Question 1. Are there other families of posets with similar structure to Π d n and C(G)? Can the techniques used in Sections 3 and 4 be used to construct dual EL-labelings and EL-ceds? What we mean by 'similar' here is not clear. At the least, we need a poset P where every interval of the form [a,1] is supersolvable, and where the supersolvable structure is canonically determined, i.e., such that we can label all edges of P \ {0} in a way that restricts to a supersolvable labeling on each such [a,1] interval. We then need a way to sign the edges giving an EL-labeling, and the poset has to somehow be 'wide' or 'rich' enough to have an EL-ced. One possible source of such examples is the theory of exponential structures. An exponential structure is a family of posets with each upper interval isomorphic to the partition lattice, and each lower interval isomorphic to a product of smaller elements in the same family. Exponential structures were introduced in [27], where the family of d-divisible partition lattices was shown to be one example. Shellings are constructed for some other examples in [21,35]. Question 2. Can techniques like those used in Section 3 (and Section 4) be used to construct dual EL-labelings and/or EL-ceds of exponential structures besides the d-divisible partition lattice? However, it is not a priori clear how to construct a labeling that restrict to a supersolvable labeling on any [a,1] for exponential structures. In examples even finding an EL-labeling often seems to be a difficult problem. A question suggested by the results of Section 5 is: Question 3. Are there other operations on posets that preserve convex ear decompositions and/or CL-ceds? For example, Schweig shows [23, Theorem 5.1] that rank-selected supersolvable and geometric lattices have convex ear decompositions. Do all rank-selected subposets of posets with convex ear decompositions have a convex ear decomposition? Are there any other useful constructions that preserve having a convex ear decomposition and/or ELced? A place to start looking would be in Björner and Wach's papers [2,3,4], where they answer many such questions for EL/CL-labelings. Theorem 1 . 1 . 11The d-divisible partition lattice Π d n has an EL-ced, hence a convex ear decomposition.Theorem 1.2. The coset lattice C(G) has a convex ear decomposition if and only if G is a relatively complemented finite group. I believe these convex ear decompositions to be the first large class of examples where each ear is a hypercube. Theorem 2.1. (Chari [6, Section 3]) The h-vector of a pure (d − 1)-dimensional complex with a convex ear decomposition satisfies the conditionsh 0 ≤ h 1 ≤ · · · ≤ h ⌊d/2⌋ h i ≤ h d−i , for 0 ≤ i ≤ ⌊d/2⌋.Theorem 2.2. (Swartz [31, Corollary 3.10]) If {h i } is the h-vector of a pure (d − 1)dimensional complex with a convex ear decomposition, then (h 0 , h 1 − h 0 , . . . , h ⌊d/2⌋ − h ⌊d/2⌋−1 ) Lemma 2. 7 . 7(Technical Lemma) Let {Σ s } be a CL-ced of a poset, with {∆ s } as above, and let c = {x ⋖ c 1 ⋖ · · · ⋖ c j−1 ⋖ y} be a maximal chain on a rooted interval [x, y] r , with c a face in ∆ s . Suppose that c has an ascent at c i . Then ∆ s contains a c ′′ = (c \ {c i }) ∪ c ′′ i which descends at c ′′ i , and is lexicographically later than c. Theorem 2. 9 . 9If {Σ s } is an CL-ced for P , then the associated subcomplexes {∆ s } form a convex ear decomposition for \|P \|. Note 2.10. Each non-empty ear of {∆ s } contains exactly one descending chain. This is no accident: see the discussion at the end of Section 2.2. Corollary 2 . 11 . 211The following families of posets have EL-ceds, thus convex ear decompositions. 1. (Nyman and Swartz [17, Section 4]) Geometric lattices. 2. (Schweig [23, Theorem 3.2]) Supersolvable lattices with Möbius function non-zero on every interval. 3. (Schweig [23, Theorems 5.1 and 7.1]) Rank-selected subposets of supersolvable and geometric lattices. The d-divisible partition poset, denoted Πd n , is the set of all proper partitions of [n] = {1, . . . , n} where each block has cardinality divisible by d. The d-divisible partition lattice, denoted Π d n is Π d n with a 'top'1 and 'bottom'0 adjoined. Π d n is ordered by refinement {α ∈ S n : α(n) = n, des α = {d, 2d, . . . , n − d}} .Wachs proved Theorem 3.1. (Wachs [33, Theorems 2.1-2.2]) Example 3. 9 . 9If B 1 = {1, 2, 3} and B 2 = {4, 5, 6}, then 123456, 321654, and 213465 all have the form B 1 B 2 . 124356 and 135246 have switched form B 1 B 2 , while 152346 does not have the form B 1 B 2 , even up to switching. Clearly, the d-divisible partition B 1 \| . . . \| B k is in Σ α if and only if α has the form B 1 B 2 . . . B k up to switching. Lemma 3 . 10 . 310Every maximal chain Corollary 3. 11 . 11Let c be a maximal chain in ∆ α , with y · ≻ x an edge in c which merges blocks B 1 and B 2 into block B (max B 1 < max B 2 ). Then 1. α has the form . . . B 1 B 2 . . . , up to switching. so every chain on [x m , x 0 ] has labels ±b 1 , ±b 2 , . . . , ±b k−1 . the ascending permutation of the elements of B, with transpositions applied at d-divisible positions. (The proof is by starting at the end and working to the front, greedily taking the greatest possible element for each position.) Proposition 4. 1 . 1(Schweig [23]) For a supersolvable lattice L, the following are equivalent:1. L has a convex ear decomposition.2. L is doubly Cohen-Macaulay.3. Every interval of L is complemented. Note 4.2. A construction very much like Schweig's convex ear decomposition was earlier used by Thévenaz in[32] on a subposet of L(G) to understand the homotopy type and the conjugation action on homology of L(G) for a solvable group G. Proposition 4 . 3 . 43The following are equivalent for a (finite) group G:1. Every interval of L(G) is complemented.2. If H is any subgroup on the interval[H 0 , H 1 ], then there is a K such that HK = H 1 and H ∩ K = H 0 . Proof. For part 1, count: \|K ∩ B\| = \|K\|\|B\| \|G\| = \|K\| [G:B] and by supersolvability, [K : K ∩ B] = [G : B] is a prime. Theorem 4. 8 . 8If G is a complemented group, then λ is a dual EL-labeling of C(G). Proposition 4 . 9 . 49The decreasing chains in L(G) with respect to λ ss are the chains of complements to G = N 1 ⊲ . . . ⊲ N k+1 = 1. Corollary 4. 10 . 10The decreasing chains in C(G) with respect to λ are all cosets of chains of complements Lemma 4 . 411. (Warm-up Lemma) Let G be a solvable group with chief series G = N 1 ⊲ N 2 ⊲ · · · ⊲ N k+1 = 1, and B i and B j be complements of normal factors N i /N i+1 and N j /N j+1 where i = j. Then B i and B j commute. Corollary 4. 13 . 13If B is a base-set and x is such that the elements of B and Bx are distinct from one another (i.e., B i = B i x for all i), then the meet sublattice generated by B ∪ Bx is isomorphic to the face lattice of the boundary of a k-cube.Proof. Any B j commutes with any intersection of B i 's, j = i, and the result follows from Lemma 4.7 and since B i ∩ B i x = ∅ for all i. Corollary 4 . 15 . 415Every maximal chain in C(G) is in some Σ Bx . Corollary 4 . 415 would not hold if we replaced 'relatively complemented' with any weaker condition, since the result implies coatomicity, and Proposition 4.3 tells us that relatively complemented groups are exactly those with coatomic subgroup lattice. Theorem 4 . 16 . 416{∆ Bx } is a convex ear decomposition for C(G) under the pattern ordering. Lemma 4 . 17 . 417For any interval[a, b] in C(G), there is at most one (dual) descending maximal chain c on [a, b] which is in ∆ Bx .Proof. If a = ∅, then the unique descending chain on[a, b] in Σ Bx is given by intersecting with each B i x (i ∈ S([∅, b]) \ {0}) in order. Example 5.2. [8, Section 2] The barycentric subdivision of a polytopal d-complex ∆ is the repeated stellar subdivision of ∆ along a reverse linear extension of its face lattice L(∆). That is, subdivide each d-dimensional face, then each (d − 1)-dimensional face, and so forth. Lemma 5.9. (Björner and Wachs [4, Lemma 10.18]) Let P be a bounded poset with a CL-labeling λ. Note 5.10. The result of Lemma 5.9 is not known to be true if 'CL' is replaced by 'EL'. Theorem 5.12. (Björner and Wachs [4, Proposition 10.15 and Theorem 10.17]) Let P 1 and P 2 be posets, with respective labelings λ 1 and λ 2 . Lemma 5 . 513. A maximal chain c is in ∆ s,t if and only if p 1 (c) is in ∆ 2-CM) if 1. ∆ is Cohen-Macaulay, and 2. for each vertex x ∈ ∆, the induced complex ∆ \ {x} is Cohen-Macaulay of the same dimension as ∆.Doubly Cohen-Macaulay complexes are closely related to complexes with convex ear de- compositions: Theorem 2.3. (Swartz [31]) If ∆ has a convex ear decomposition, then ∆ is doubly Cohen-Macaulay. AcknowledgementsThanks to Ed Swartz for introducing me to convex ear decompositions; and to him, Jay Schweig, and my graduate school advisor Ken Brown for many helpful discussions about them. Tom Rishel listened to and commented on many intermediate versions of the results and definitions of this paper. Sam Hsiao helped me in understanding the material of Proposition 5.1, and in looking for its extension to Lemma 1.5. Vic Reiner pointed out that the labeling based on pivots of C(G) that I used in[36]was really a supersolvable labeling, which suggested the improved EL-labeling used in Section 4. Volkmar Welker suggested exponential structures as a possible area for further exploration. The anonymous referee gave many helpful comments.This completes the proof of Lemma 1.5.Convex ear decompositions of product posetsLet P 1 and P 2 be bounded posets with respective convex ear decompositions {∆(1)s } and {∆(2)t }. Let P be either P 1 × P 2 , P 1× P 2 , or P 1× P 2 ; with coordinate projection maps p 1 and p 2 . Take d = dim \|P \|, d 1 = dim \|P 1 \|, and d 2 = dim \|P 2 \|.We define ∆ s,t to be the simplicial complex generated by the maximal chains of P that project to ∆(1)s in the first coordinate, and ∆(2)t in the second. Order these complexes lexicographically by (s, t).Proof. Lemma 1.5 gives that ∆ s,t is a subcomplex of the boundary complex of a polytope, so (ced-polytope) is satisfied.The topology of various poset products is nicely discussed in Sundaram's [30, Section 2]. There are homeomorphismswhere * is the join of topological spaces. This result goes back to Quillen [18, Proposition 1.9], although his notation was much different -Sundaram makes the connection in[where susp denotes the topological suspension. Identical proofs to Quillen's and Walker's show that ∆ s,t ≈ ∆ s * ∆ t in the upper/lower reduced case, and that ∆ s,t ≈ susp(∆ s * ∆ t ) in the direct product case. In particular, ∆ s,t is a d-ball for (s, t) > (1, 1) and a d-sphere for (s, t) = (1, 1) by results in PL-topology[20,Proposition 2.23]. We have shown that (ced-topology) is satisfied.It is clear that (ced-union) holds. It remains to check (ced-bdry).Claim 5.8. ∂∆ s,t is exactly the set of all faces in ∆ s,t that project to either ∂∆Proof. The boundary of a simplicial d-ball ∆ is generated by the d − 1 faces that are contained in only a single facet of ∆. If c is a d − 1 face of ∆ s,t (i.e., a chain of length d − 1), then at least one of p 1 (c) and p 2 (c) also has codimension 1. 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[ "Output-Feedback Stabilization of the Korteweg-de Vries Equation", "Output-Feedback Stabilization of the Korteweg-de Vries Equation" ]	[ "Agus Hasan \nDepartment of Engineering Cybernetics\nNorwegian University of Science and Technology Trondheim\nNorway\n" ]	[ "Department of Engineering Cybernetics\nNorwegian University of Science and Technology Trondheim\nNorway" ]	[]	The present paper develops boundary output-feedback stabilization of the Kortewegde Vries (KdV) equation with sensors and an actuator located at different boundaries (anti collocated set-up) using backstepping method. The feedback control law and output injection gains are found using the backstepping method for linear KdV equation. The proof of stability is based on construction of a strict Lyapunov functional which includes the observer states. A numerical simulation is presented to validate the result.	10.1109/med.2016.7536057	[ "https://arxiv.org/pdf/1603.08750v1.pdf" ]	15,472,794	1603.08750	58252f79682103628b4f516c668d07b97d1ad3c5	Output-Feedback Stabilization of the Korteweg-de Vries Equation Agus Hasan Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim Norway Output-Feedback Stabilization of the Korteweg-de Vries Equation Distributed parameter systemsStabilizationKorteweg-de Vries equation The present paper develops boundary output-feedback stabilization of the Kortewegde Vries (KdV) equation with sensors and an actuator located at different boundaries (anti collocated set-up) using backstepping method. The feedback control law and output injection gains are found using the backstepping method for linear KdV equation. The proof of stability is based on construction of a strict Lyapunov functional which includes the observer states. A numerical simulation is presented to validate the result. INTRODUCTION The Kortweg-de Vries (KdV) equation is a third-order partial differential equation (PDEs), which can be used to described weakly nonlinear shallow water surface (Kortweg and de Vries, 1895). The KdV equation can be classified as hyperbolic-type PDEs, which describes a reversible dynamical process. Furthermore, it was found to have solitary wave solutions. The KdV equation is completely integrable and has infinitely many conserved quantities. Boundary control of the KdV equation can be found in many literatures, e.g, Rosier andZhang (2006, 2009); Crepeau and Prieur (2010); Balogh and Krstic (2000); Liu and Krstic (2002); . In these literatures, the boundary control laws were found using the Lyapunov methods. Furthermore, only state-feedback was considered. Recent control design for the KdV equation includes the backstepping method (Cerpa and Coron, 2013;Tang and Krstic, 2013), where the analysis mostly done for the linear KdV equation with state-feedback. In the infinitedimensional backstepping, a Volterra integral transformation is used to transform the original system into a stable target system. Different with other approaches that require the solution of operator Riccati equations e.g., optimal control method (Hasan et al., 2013;Hasan and Imsland, 2014), backstepping yields control gain formulas which can be computed using symbolic computation and, in some cases, can even be given explicitly in terms of Bessel function (Krstic and Smyshlyaev, 2008) and Marcum Qfunction (Vazquez and Krstic, 2014). The backstepping method has been successfully used for control design of many PDEs such as the Schrodinger equation (Krstic et al., 2011), the Ginzburg-Landau equation (Aamo et al., 2005), the Navier-Stokes equation (Vazquez and Krstic, 2007), the surface wave equation , and the hyperbolic equation ( , 2016). Furthermore, the backstepping method has found several applications in oil well drilling problems, including slugging control (di Meglio, 2011), the lost circulation and kick problem (Hasan, 2014a(Hasan, , 2015, and the heave problem (Aamo, 2013;Hasan, 2014b). In this paper, we concern with the problem of outputfeedback stabilization of the KdV equation using the infinite-dimensional backstepping method with sensors and an actuator located at different boundaries (anti collocated set-up). The design utilized the result for the linear KdV equation by Marx and Cerpa (2014). The control law from the linear KdV is used to stabilize the systems, where the state is generated from a nonlinear observer of the KdV equation. The novelty of of this paper lies on the introduction of a strict Lyapunov functional for the output-feedback control problem. To prove the stability, we introduce a strict Lyapunov functional which equivalent to the H 3 norm. This paper is organized as follow. In section 2, we state the problem. Notations and definitions are presented in section 3. Output-feedback control for the linear KdV equation, which was solved by Marx and Cerpa (2014), is presented in section 4. The main result for output-feedback stabilization of the KdV equation is presented in section 5. In section 6, we present a numerical example and the last section contains conclusions. PROBLEM STATEMENTS We consider output-feedback stabilization of the Kortwegde Vries (KdV) equation u t (x, t) + u x (x, t) + u xxx (x, t) + u(x, t)u x (x, t) = 0 (1) with boundary conditions u(0, t) = U (t), u(1, t) = 0, u x (1, t) = 0(2) where u : [0, 1]×[0, ∞) → R. The subscripts x and t denote partial derivatives with respect to x and t, respectively. The objective is to find a feedback law U (t) to make the origin of (1)-(2) locally exponentially stable, using only measurements of u xx (1, t). NOTATIONS AND DEFINITIONS For u(x, t) ∈ R, we define u ∞ = sup x∈[0,1] \|γ(x, t)\| (3) u L 1 = 1 0 \|γ(x, t)\| dx (4) u H i = 1 0 i j=0 ∂ j ∂x j γ(x, t) dx(5) Furthermore, to simplify our notation, we denote \|u\| = \|u(x, t)\| and u = u(·, t) . For u ∈ H 3 ([0, 1]), recall the following well-known inequalities u L 1 ≤ c 1 u L 2 ≤ c 2 u ∞ ,(6)u ∞ ≤ c 3 ( u L 2 + u x L 2 ) ≤ c 4 u H 1 ,(7)u x ∞ ≤ c 5 ( u x L 2 + u xx L 2 ) ≤ c 6 u H 2 ,(8)u xx ∞ ≤ c 7 ( u xx L 2 + u xxx L 2 ) ≤ c 8 u H 3 , (9) where c 1 , · · · , c 8 > 0. OUTPUT FEEDBACK STABILIZATION OF THE LINEAR KDV EQUATION To stabilizes (1)-(2), we use the design presented in Marx and Cerpa (2014) for the linear system. Consider the following linear KdV equation u t (x, t) + u x (x, t) + u xxx (x, t) = 0 (10) with the following boundary conditions u(0, t) = U (t), u(1, t) = 0, u x (1, t) = 0 (11) We assume only u xx (1, t) is measurable. If we select the control law U (t) as U (t) = 1 0 k(0, y)û(y, t) dy(12) whereû is computed from u t +û x +û xxx + p 1 (x)[y(t) −û xx (1, t)] = 0 (13) and where p 1 (x) = p(x, 1) with boundary conditionŝ u(0, t) = U (t),û(1, t) = 0,û x (1, t) = 0(14) it can be shown that the origin of (10)-(11) is exponentially stable, where the kernels k in (12) is solution of the following kernel equation k xxx + k yyy + k x + k y = −λk (15) with boundary conditions k(x, 1) = 0, k(x, x) = 0, k x (x, x) = λ 3 (1 − x) (16) Similarly, the kernel p in (13) is solution of the following kernel equation p xxx + p yyy + p x + p y = λp (17) with boundary conditions p(x, x) = 0, p x (x, x) = λ 3 x, p(0, y) = 0 (18) Both kernel equations evolve in a triangular domain T = {(x, y)\|0 ≤ y ≤ x ≤ 1}. The output-feedback boundary stabilization for the linear KdV equation is given as follow. Theorem 1. (Marx and Cerpa, 2014) Consider systems (10)- (11) and (13)-(14) with control law (12) and initial conditions u 0 ∈ H 3 ([0, 1]) andû 0 ∈ L 2 ([0, 1]). Then, there exists λ > 0 and c > 0 such that u(·, t) H 3 + û(·, t) L 2 ≤ ce −λt ( u 0 H 3 + û 0 L 2 )(19) Remark 1. We slightly modified the proof of this theorem so it can be used to analyze the stability of the KdV equation in section 4. The contribution (novelty) of this paper is in introduction of a strict Lyapunov functional for the output-feedback problem. To ease the reader, we present the proof of the above theorem as follow. Proof 1. Let the observer error be given byũ = u −û. We define new target variablesω andω using the following transformationŝ ω(x, t) =û(x, t) − 1 x k(x, y)û(y, t) dy (20) u(x, t) =ω(x, t) − 1 x p(x, y)ω(y, t) dy(21) It can be shown that, if the kernels verify (15)-(18), then ω andω satisfy the following equationŝ ω t +ω x +ω xxx + λω = −p(x)ω xx (1, t)(22) ω(0, t) = 0,ω(1, t) = 0,ω x (1, t) = 0 (23) ω t +ω x +ω xxx + λω = 0 (24) ω(0, t) = 0,ω(1, t) = 0,ω x (1, t) = 0 (25) wherep (x) = p 1 (x) − 1 x k(x, y)p 1 (y) dy(26) Remark that, by bounding the norms in (24), we have ω t L 2 ≤ K 1 ω xxx L 2(27) ω xxx L 2 ≤ K 2 ω t L 2 (28) where K 1 , K 2 > 0. Thus, the ω t L 2 is equivalent to ω H 3 . Let us consider the following Lyapunov functional (29) where A and B are positive constants. We can observe that the last term of the Lyapunov functional is equivalent to ω H 3 . Calculating the derivative of the Lyapunov functional along (22)-(25), we have V (t) = A 2 1 0ω 2 dx + B 2 1 0ω 2 dx + B 2 1 0ω 2 t dxV (t) ≤ A −λ + D 2 A 1 0ω 2 dx + A 2ω2 xx (1, t) −λB 1 0ω 2 dx − λB 1 0ω 2 t dx(30) where D = max x∈[0,1] p 1 (x) − 1 x k(x, y)p 1 (y) dy . From (28) and using integration by parts, we compute the bound ofω 2 xx (1, t) as follow \|ω 2 xx (1, t)\| ≤ a ω 2 L 2 + b ω t 2 L 2 (31) Thus, we havė V (t) ≤ A −λ + D 2 A 1 0ω 2 dx +B −λ + aA 2 B 1 0ω 2 dx +B −λ + bA 2 B 1 0ω 2 t dx(32) Since we can choose arbitrary large λ, there exists > 0 such thatV (20) and (21) are given byû (t) ≤ − V (t)(33)(x, t) =ω(x, t) + 1 x l(x, y)ω(y, t) dy (34) ω(x, t) =ũ(x, t) + 1 x r(x, y)ũ(y, t) dy(35) The kernel l(x, y) satisfy l xxx + l yyy + l x + l y = λl (36) l(x, 1) = 0, l(x, x) = 0, l x (x, x) = λ 3 (1 − x) (37) while the kernel r satisfy r xxx + r yyy + r x + r y = −λr (38) r(x, x) = 0, r x (x, x) = λ 3 x, r(0, y) = 0(39) The kernel k(x, y) and l(x, y) are related by the formula l(x, y) − k(x, y) = y x k(x, ξ)l(ξ, y) dξ(40) A similar relation is also found in p(x, y) and r(x, y). The existence and uniqueness of the kernel solutions can be proved using the method of successive approximations. OUTPUT FEEDBACK STABILIZATION OF THE KDV EQUATION We will show that the linear design (10)-(11) works for the nonlinear system (1). Therefore, we design the following nonlinear observer as folloŵ u t +û x +û xxx +ûû x + p 1 (x)ũ xx (1, t) = 0 (41) with boundary conditionŝ u(0, t) = U (t),û(1, t) = 0,û x (1, t) = 0 (42) The observer error system is given byũ t +ũ x +ũ xxx + uu x −ûû x − p 1 (x)ũ xx (1, t) = 0 (43) with boundary conditions u(0, t) = 0,ũ(1, t) = 0,ũ x (1, t) = 0 (44) Let us define the following functionals (50) By direct observation, these functionals satisfy K[ω] = ω(x, t) − 1 x k(x, y)ω(y, t) dy (45) L[ω] = ω(x, t) + 1 x l(x, y)ω(y, t) dy (46) P[ω] = ω(x, t) − 1 x p(x, y)ω(y, t) dy (47) R[ω] = ω(x, t) + 1 x r(x, y)ω(x, t) dy (48) L 1 [ω] = l(x, x)ω(x, t) + 1 x l x (x, y)ω(y, t) dy (49) P 1 [ω] = p(x, x)ω(x, t) − 1 x p x (x, y)ω(y, t) dy\|K[ω]\| ≤ C 1 (\|ω\| + ω L 1 ) (51) \|L[ω]\| ≤ C 2 (\|ω\| + ω L 1 ) (52) \|P[ω]\| ≤ C 3 (\|ω\| + ω L 1 ) (53) \|R[ω]\| ≤ C 4 (\|ω\| + ω L 1 ) (54) \|L 1 [ω]\| ≤ C 5 (\|ω\| + ω L 1 ) (55) \|P 1 [ω]\| ≤ C 6 (\|ω\| + ω L 1 )(56) for C 1 , · · · , C 6 > 0. Furthermore, we calculate the derivatives of (20) with respect to x (59) and the derivative of (20) with respect to t along (41) (60) Plugging the kernel (15)-(16) into the above equations, the observeω satisfŷ (61) with boundary conditionŝ ω x (x, t) =û x (x, t) + k(x, x)û(x, t) − 1 x k x (x, y)û(y, t) dy (57) ω xx (x, t) =û xx (x, t) + d dx k(x, x)û(x, t) + k(x, x)û x (x, t) +k x (x, x)û(x, t) − 1 x k xx (x, y)û(y, t) dy (58) ω xxx (x, t) =û xxx (x, t) + d 2 dx 2 k(x, x)û(x, t) +2 d dx k(x, x)û x (x, t) + k(x, x)û xx (x, t) + d dx k x (x, x)û(x, t) + k x (x, x)û x (x, t) +k xx (x, x)û(x, t) − 1 x k xxx (x, y)û(y, t) dyω t (x, t) = −û x (x, t) −û xxx (x, t) + 1 x k(x, y)û y (y, t) dy +k(x, 1)û xx (1, t) − k(x, x)û xx (x, t) −k y (x, 1)û x (1, t) + k y (x, x)û x (x, t) +k yy (x, 1)û(1, t) − k yy (x, x)û(x, t) − 1 x k yyy (x, y)û(y, t) dy −p 1 (x)ũ xx (1, t) + 1 x k(x, y)p 1 (y) dyũ xx (1, t) −ûû x (x, t) + 1 x k(x, y)û(y, t)û y (y, t) dyω t +ω x +ω xxx + λω = −p(x)ω xx (1, t) − F [ω,ω x ]ω(0, t) = 0,ω(1, t) = 0,ω x (1, t) = 0 (62) where F [ω,ω x ] = K[L[ω] (ω x + L 1 [ω])](63) This functional satisfy \|F \| ≤ c 1 ( ω L 2 + \|ω\|) ( ω x L 2 + \|ω x \|) +c 2 ω 2 L 2 + \|ω\| 2(64) Next, computing the derivatives of (35) with respect to x, we havẽ ω x (x, t) =ũ x (x, t) − r(x, x)ũ(x, t) + 1 x r x (x, y)ũ(y, t) dy (65) ω xx (x, t) =ũ xx (x, t) − d dx r(x, x)ũ(x, t) − r(x, x)ũ x (x, t) −r x (x, x)ũ(x, t) + 1 x r xx (x, y)ũ(y, t) dy (66) ω xxx (x, t) =ũ xxx (x, t) − d 2 dx 2 r(x, x)ũ(x, t) −2 d dx r(x, x)ũ x (x, t) − r(x, x)ũ xx (x, t) − d dx r x (x, x)ũ(x, t) − r x (x, x)ũ x (x, t) −r xx (x, y)ũ(y, t) + 1 x r xxx (x, y)ũ(y, t) dy (67) Furthermore, we calculateω −r(x, 1)ũ yy (1, t) + r(x, x)ũ yy (x, t) +r y (x, 1)ũ y (1, t) − r y (x, x)ũ y (x, t) −r yy (x, 1)ũ(1, t) + r yy (x, x)ũ(x, t) + 1 x (r y (x, y) + r yyy (x, y))ũ(y, t) dy + 1 x r(x, y)p 1 (y) dyũ yy (1, t) −uu x +ûû x + 1 x r(x, y) (−uu y +ûû y ) dy (68) Plugging the kernel equation (38)-(39), we havẽ ω t +ω x +ω xxx + λω = −G[ω,ω x ,ω,ω x ](69) with boundary conditions ω(0, t) = 0,ω(1, t) = 0,ω x (1, t) = 0 (70) where G[ω,ω x ,ω,ω x ] = R[P[ω] (ω x + P 1 [ω])] +R[P[ω] (ω x + L 1 [ω])] +R[L[ω] (ω x + P 1 [ω])] −R[L[ω] (ω x + L 1 [ω])](71) This functional satisfy \|G\| ≤ c 1 ( ω L 2 + \|ω\|) ( ω x L 2 + \|ω x \|) +c 2 ω 2 L 2 + \|ω\| 2 + c 3 ω x 2 L 2 + \|ω x \| 2 +c 4 ( ω L 2 + \|ω\|) ( ω x L 2 + \|ω x \|) +c 5 ω 2 L 2 + \|ω\| 2 + c 6 ω x 2 L 2 + \|ω x \| 2 (72) We can observe that (61)-(70) are PDE-PDE cascade systems. To study its stability, first we denote η = ω t . Taking a derivative of (61)-(70) with respect to t, we havê η t +η x +η xxx + λη = −p(x)η xx (1, t) − L[η]ω x −F 1 [ω,η,η x ](73)η(0, t) = 0,η(1, t) = 0,η x (1, t) = 0 (74) η t +η x +η xxx + λη = −P[η]ω x − P[η]ω x − L[η]ω x (75) +L[η]ω x − G 1 [ω,η,η x ,ω,η,η x ] η(0, t) = 0,η(1, t) = 0,η x (1, t) = 0 (76) where F 1 [ω,η,η x ] = −L[η(1, t)]ω(1, t) + L[η(x, t)]ω(x, t) + 1 x (ω x + L 1 [η])ω dy + K[L[η]L 1 [ω]] +K[L[ω] (η x + L 1 [η])](77) and G 1 [ω,η,η x ,ω,η,η x ] = −P[η(x, t)]ω(x, t) − 1 x P x [η]ω dy +R[P[η]P 1 [ω]] + R[P[ω] (η x + P 1 [η])] −P[η(x, t)]ω(x, t) − 1 x P x [η]ω dy +R[P[η]L 1 [ω]] + R[P[ω] (η x + L 1 [η])] −L[η(x, t)]ω(x, t) − 1 x L x [η]ω dy +R[L[η]P 1 [ω]] + R[L[ω] (η x + P 1 [η])] +L[η(x, t)]ω(x, t) + 1 x L x [η]ω dy −R[L[η]L 1 [ω]] − R[L[ω] (η x + L 1 [η])](78) Bounding the norms for small ω ∞ + ω ∞ in (73) and (76), we can prove that the norm ω H 3 + ω H 3 is equivalent to η L 2 + η L 2 = ω t L 2 + ω t L 2 . The main result of this paper is stated as follow. Theorem 2. Consider systems (1)- (2) and (41)-(42) with control law (12) and initial conditions u 0 ,û 0 ∈ H 3 ([0, 1]). Then, there exists δ, λ > 0, and c > 0 such that if u 0 H 3 + û 0 H 3 ≤ δ, then u(·, t) H 3 + û(·, t) H 3 ≤ ce −λt ( u 0 H 3 + û 0 H 3 )(79) Remark 2. The difference between the linear system and the nonlinear system results lie in the smallness of the initial condition. Thus, in the nonlinear system, we only achieved local exponential stability. Proof 2. We introduce the following Lyapunov functional W (t) = A 2 1 0ω 2 dx + A 2 1 0η 2 dx + B 2 1 0ω 2 dx + B 2 1 0η 2 dx(80) Computing its derivative along (61)-(70) with respect to t, yieldṡ W (t) ≤ −µW (t) − A 1 0ω F dx − A 1 0η F 1 dx (81) −B 1 0ω G dx − B 1 0η G 1 dx − 1 0η L[η]ω x dx − 1 0η (P[η]ω x + P[η]ω x + L[η]ω x − L[η]ω x ) dx We estimate − 1 0η L[η]ω x dx ≤ K 1 η ∞ W (t) (82) − 1 0η (P[η]ω x + P[η]ω x + L[η]ω x − L[η]ω x ) dx ≤ K 2 η ∞ W (t)(83) Furthermore, we estimate −A 1 0ω F dx ≤ K 3 ω x ∞ W (t) (84) −A 1 0η F 1 dx ≤ K 4 η W (t) + W (t) 3/2 (85) −B 1 0ω G dx ≤ K 5 ω x ∞ W (t) −B 1 0η G 1 dx ≤ K 6 η W (t) + W (t) 3/2(86) where K 1 , · · · , K 6 > 0. Thus, we havė W (t) ≤ −µW (t) + CW (t) 3 2(87) for some positive µ and C. Then, for any µ 0 such that 0 < µ 0 < µ, there exists δ 0 such that C W 3/2 < (µ − µ 0 ) W, ∀W < δ 0 ,(88) which implies thaṫ W < −µ 0 W, ∀W < δ 0 .(89) Since W is equivalent to ω H 3 + ω H 3 when ω ∞ + ω ∞ is sufficiently small, this concludes the proof. NUMERICAL EXAMPLE To show our linear control law works for the nonlinear system, we simulate the closed-loop system (1)- (2) and (41)-(42) with control law (12). The initial conditions are chosen such that they are compatible with the boundary conditions. The result is presented in figure 1. We can observe in the controlled case, the controller drives the closed-loop system into its equilibrium. CONCLUSION In this paper, we have presented output-feedback boundary stabilization of the KdV equation with actuation and measurement on only one boundary. The control law was obtained using backstepping method for the linear system. 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[ "Optimization Models for Machine Learning: A Survey", "Optimization Models for Machine Learning: A Survey" ]	[ "Claudio Gambella \nIBM Research Ireland\nDublin 15MulhuddartIreland\n", "Bissan Ghaddar \nIvey Business School\nUniversity of Western Ontario\nN6G 0N1LondonOntarioCanada\n", "Joe Naoum-Sawaya \nIvey Business School\nUniversity of Western Ontario\nN6G 0N1LondonOntarioCanada\n" ]	[ "IBM Research Ireland\nDublin 15MulhuddartIreland", "Ivey Business School\nUniversity of Western Ontario\nN6G 0N1LondonOntarioCanada", "Ivey Business School\nUniversity of Western Ontario\nN6G 0N1LondonOntarioCanada" ]	[]	This paper surveys the machine learning literature and presents machine learning as optimization models. Such models can benefit from the advancement of numerical optimization techniques which have already played a distinctive role in several machine learning settings. Particularly, mathematical optimization models are presented for commonly used machine learning approaches for regression, classification, clustering, and deep neural networks as well new emerging applications in machine teaching and empirical model learning. The strengths and the shortcomings of these models are discussed and potential research directions are highlighted.	10.1016/j.ejor.2020.08.045	[ "https://arxiv.org/pdf/1901.05331v2.pdf" ]	58,007,007	1901.05331	bfcb2c35698443eeb534268717d06e8a6af61632	Optimization Models for Machine Learning: A Survey Claudio Gambella IBM Research Ireland Dublin 15MulhuddartIreland Bissan Ghaddar Ivey Business School University of Western Ontario N6G 0N1LondonOntarioCanada Joe Naoum-Sawaya Ivey Business School University of Western Ontario N6G 0N1LondonOntarioCanada Optimization Models for Machine Learning: A Survey Machine LearningMathematical OptimizationRegres- sionClassificationClusteringDeep Learning This paper surveys the machine learning literature and presents machine learning as optimization models. Such models can benefit from the advancement of numerical optimization techniques which have already played a distinctive role in several machine learning settings. Particularly, mathematical optimization models are presented for commonly used machine learning approaches for regression, classification, clustering, and deep neural networks as well new emerging applications in machine teaching and empirical model learning. The strengths and the shortcomings of these models are discussed and potential research directions are highlighted. Introduction The pursuit to create intelligent machines that can match and potentially rival humans in reasoning and making intelligent decisions goes back to at least the early days of the development of digital computing in the late 1950s [176]. The goal is to enable machines to perform cognitive functions by learning from past experiences and then solving complex problems under conditions that are varying from past observations. Fueled by the exponential growth in computing power and data collection coupled with the widespread of practical applications, machine learning is nowadays a field of strategic importance. Broadly speaking, machine learning relies on learning a model that returns the correct output given a certain input. The inputs, i.e. predictor measurements, are typically numerical values that represent the parameters that define a problem while the output, i.e. response, is a numerical value that represents the solution. Machine learning models fall into two categories: supervised and unsupervised learning [91,118]. In supervised learning, a response measurement is available for each observation of predictor measurements and the aim is to fit a model that accurately predicts the response of future observations. In unsupervised learning on the otherhand, response variables are not available and the goal of learning is to understand the underlying characteristics of the observations. The fundamental theory of these learning models and consequently their success can be largely attributed to research at the interface of computer science, statistics, and operations research. The relation between machine learning and operations research can be viewed along three dimensions: (a) machine learning applied to management science problems, (b) machine learning to solve optimization problems, (c) machine learning problems formulated as optimization problems. Leveraging data in business decision making is nowadays mainstream as any business in today's economy is instrumented for data collection and analysis. While the aim of machine learning is to generate reliable predictions, management science problems deal with optimal decision making. Thus methodological developments that can leverage data predictions for optimal decision making is an area of research that is critical for future business value [30,130]. Another area of research at the interface of machine learning and operations research is using machine learning to solve hard optimization problems and particularly NP-hard integer constrained optimization [25,41,128,126,142,185]. In that domain, machine learning models are introduced to complement existing approaches that exploit combinatorial optimization through structure detection, branching, and heuristics. Lastly, the training of machine learning models can be naturally posed as an optimization problem with typical objectives that include optimizing training error, measure of fit, and cross-entropy [42,43,72,191]. In fact, the widespread adoption of machine learning is in parts attributed to the development of efficient solution approaches for these optimization problems which enabled the training of machine learning models. As we review in this paper, the development of these optimization models has largely been concentrated in areas of computer science, statistics, and operations research however diverging publication outlets, standards, and terminology persist. The aim of this paper is thus to present a unifying framework for machine learning as optimization problems. For that, in addition to publications in classical operations research journals, this paper surveys artificial intelligence conferences and journals, such as the conference on Neural Information Processing Systems and the International Conference on Machine Learning. Furthermore, since machine learning research has rapidly accelerated with many important papers still in the review process, this paper also surveys a considerable number of relevant papers that are available on the arXiv repository. This paper also complements the recent surveys of [43,72,191] which have largely focused on the methodological developments for solving several classes of machine learning optimization problems. Particularly this paper presents optimization models for regression, classification, clustering, and deep learning (including adversarial attacks), as well as new emerging paradigms such as machine teaching and empirical model learning. Additionally, this paper highlights the strengths and the shortcomings of the models from a mathematical optimization perspective and discusses potential novel research directions. Following this introductory section, regression models are discussed in Section 2 while classification and clustering models are presented in Sections 3 and 4, respectively. Deep learning models are presented in Section 5 while Section 6 discusses new emerging paradigms that include machine teaching and empirical model learning. Section 7 summarizes the datasets and software systems that are commonly used in the literature. Finally, conclusions are drawn in Section 8. Regression Models Linear regression models are widely known approaches in supervised learning for predicting a quantitative response. The central assumption is that the dependence of the dependent variables (feature measurements, or predictors, or input vector ) to the independent variables (real-valued output) is representable with a linear function (regression function) with a reasonable accuracy. Linear regression models have been largely adopted since the early era of statistics, and they preserve considerable interest, given their simplicity, their extensive range of applications, and the ease of interpretability. Linear regression aims to find a regression function f that expresses the linear relation between input vector X T = (X 1 , . . . , X p ) and real-valued output Y via the regression coefficients β j such as Y = f (X) = β 0 + p j=1 X j β j .(1) In order to estimate the parameters β j , one needs training examples x 1 , . . . , x n with labels y 1 , . . . , y n , and the objective is to minimize a measure of loss due to the prediction on the training dataset. The most popular estimation is the least squared estimate, which minimizes the residual sum of squares (RSS) between the labels and the predicted outputs: RSS(β) = n i=1 (y i − β 0 − p j=1 x ij β j ) 2 .(2) The least squares estimate is known to have the smallest variance among all linear unbiased estimates, and has a closed form solution. However, this choice is not always ideal, since it can yield a model with low prediction accuracy, due to a large variance, and often leads to a large number of nonzero regression coefficients (i.e., low interpretability). Sections 2.1, 2.2 and 2.3 present some of the most important alternatives to the least squared estimate. The process of gathering input data is often affected by noise, which can impact the accuracy of statistical learning methods. A model that takes into account the adversarial noise applied to feature measurements in linear regression problems is presented in [28] which also investigates the relationship between regularization and robustness to noise. The noise is assumed to vary in an uncertainty set U ∈ R n×p , and the defender adopts the robust prospective: min β∈R p max ∆∈U g(y − (X + ∆)β)(3) where g is a convex function that measures the residuals (e.g., a norm function). The characterization of the uncertainty set U directly influences the complexity of Problem (3). Adversarial learning is extensively discussed in the context of neural networks in Section 5.2. The design of high-quality linear regression models requires several desirable properties, which are often conflicting and not simultaneously implementable. A fitting procedure based on Mixed Integer Quadratic Programming (MIQP) is presented in [31] and takes into account sparsity, joint inclusion of subset of features (called selective sparsity), robustness to noisy data, stability against outliers, modeler expertise, statistical significance, and low global multicollinearity. Mixed integer programming (MIP) models for regression and classification tasks are also investigated in [34]. The regression problem is modeled as an assignment of data points to groups with the same regression coefficients. Shrinkage methods Shrinkage methods seek to diminish the value of the regression coefficients. The aim is to obtain a more interpretable model (with less relevant features), at the price of introducing some bias in the model determination. A wellknown shrinkage method is Ridge regression, where a norm-2 penalization on the regression coefficients is added to the loss function such that RSS ridge (β) = n i=1 (y i − β 0 − p j=1 x ij β j ) 2 + λ p j=1 β 2 j(4) where λ controls the magnitude of shrinkage. Ridge regression can be equivalently expressed as min β n i=1 (y i − β 0 − p j=1 x ij β j ) 2 (5) s. t. p j=1 β 2 j ≤ t.(6) The solution of (5) -(6) can be expressed in closed form, in terms of the singular value decomposition of matrix X. Instead of the conic constraint (6), lasso regression instead penalizes the norm-1 of the regression coefficients and the following quadratic programming problem is obtained min β n i=1 (y i − β 0 − p j=1 x ij β j ) 2 (7) s. t. p j=1 \|β j \| ≤ t.(8) Ridge and lasso regression belong to a class of techniques to achieve sparse regression. As discussed in [33], the sparse regression problem can be formulated as the best subset selection problem min β∈R p 1 2γ w 2 2 + 1 2 Y − Xβ 2 2 (9) s.t β 0 ≤ k(10) where γ > 0 weights the Tikhonov regularization term and k is an upper bound on the number of predictors with a non-zero regression coefficient, i.e., the predictors to select. Problem (9)-(10) is NP-hard due to the cardinality constraint (10). Introducing the binary variables {s ∈ {0, 1} p : I T s ≤ k}, the sparse regression problem can be transformed into the following MIP min β∈R p ,x∈S p k 1 2γ β 2 2 + 1 2 Y − Xw 2 2 (11) s.t − M s j ≤ β j ≤ M s j ∀j = 1, . . . , p.(12) As shown in [33], problem (11)- (12) can be solved using a cutting plane approach. The proposed cutting plane approach is an alternative to the discrete first order algorithms of [32] which solve the MIQP (11)-(12) without regularization terms. Dimension Reduction While shrinkage methods improve model interpretability and retain the original set of p predictors, dimension reduction methods search for M < p linear combinations of the predictors such that Z m = p j=1 φ jm X j (also called projections). The M transformed predictors are then fitted in a linear regression model. Principal Components Principal Components Analysis (PCA) [120] constructs features with large variance based on the original set of features. In particular, assuming the regressors are standardized to a mean of 0 and a variance of 1, the direction of the first principal component is a unit vector φ 1 ∈ R p that is the solution of the optimization problem max φ 1 ∈R p 1 n n i=1 p j=1 φ j1 x ij 2 (13) s.t. p j=1 φ 2 j1 = 1.(14) Problem (13)- (14) is the traditional formulation of PCA and can be solved via Lagrange multipliers methods. Since the formulation is sensitive to the presence of outliers, several approaches have been proposed to improve robustness [165]. One approach is to replace the L 2 norm in (13) with the L 1 norm. As formalized by Problem (13)- (14), the first principal component Z 1 = p j=1 φ j1 X j is the projection of the original features with the largest variability. The subsequent principal components are obtained iteratively. Each principal component Z m , m = 2, . . . , M is a linear combination of the features X 1 , . . . , X n , uncorrelated with Z 1 , . . . , Z m−1 , which has the largest variance. Introducing the sample covariance matrix S of the regressors X j , the direction φ m of the m-th principal component Z m is the solution of max φm∈R p 1 n n i=1 p j=1 φ jm x ij 2 (15) s.t. p j=1 φ 2 jm = 1 (16) φ T m Sφ l = 0 l = 1, . . . , m − 1.(17) PCA can be used for several data analysis problems which benefit from reducing the problem dimension. Principal Components Regression (PCR) is a two-stage procedure that uses the first principal components as predictors for a linear regression model. PCR has the advantage of including less predictors than the original set and of retaining the variability of the dataset in the derived features. In [124], the regression loss function and the PCA objective function are combined in a one-step procedure for PCR. The one-step framework is extended in [125] to other regression models, such as logistic regression, Poisson regression and multiclass-class logistic regression. For classification models with a large number of features, PCA can be exploited to improve the interpretability of the features by performing classification on the first principal components. Since the identification of the principal components does not require any knowledge on the response y, PCA can be also adopted in unsupervised learning such as in the k-means clustering method (see section 4.1, [80]). Partial Least Squares Partial Least Squares (PLS) identifies transformed features Z 1 , . . . , Z M by taking both the predictors X 1 , . . . , X n and the response Y into account, and is an approach that is specific to regression problems. The components α j1 of the first PLS direction are found by fitting the regression with response Y and predictor X j . The approach is viable even for problems with a large number of features, because only one regressor has to be fitted in a simple regression model with one predictor. The first PLS direction points towards the variables that are more strongly related to the response. For computing the second PLS direction, the features X 1 , . . . , X p are first orthogonalized with respect to Z 1 (as per the Gram-Schmidt approach), and then individually fitted in simple regression models with response Y . The process can be iterated for M < p PLS directions. The coefficient of the simple regression of Y onto each original feature X j can also be computed as the inner product Y, X j . Analogously to PCR, PLS regression fits a linear regression model with regressors Z 1 , . . . , Z M and response Y . While the Principal Components directions maximize variance, PLS searches for directions V m = p j=1 α jm X j with both high variance and high correlation with the response. The m-th direction α m can be found by solving the optimization problem max αm Corr 2 (y, V m )Var(Xα)(18)s.t. p j=1 α 2 jm = 1 (19) α T m Sα l = 0 l = 1, . . . , m − 1(20) where Corr indicates the correlation matrix, Var the variance matrix, and S the sample covariance matrix of X j , and (20) ensures that V m is uncorrelated with the previous directions V l = p j=1 α jl X j . Non-Linear Models for Regression A natural extension of linear regression models is to consider non-linear relationship between regressors and predictors. Among the several non-linear regression models are polynomial regression, step functions, regression splines, smoothing splines and local regression. Alternatively, the Generalized Additive Models (GAMs) [108] maintains the additivity of the original predictors X 1 , . . . , X p and the relationship between each feature and the response y is expressed by non-linear functions f j (X j ) such as y = β 0 + p j=1 f j (X j ).(21) GAMs may increase the flexibility and accuracy of prediction with non-linear components, while maintaining a level of interpretability of the predictors. However, one limitation is given by the additivity of the features in the response. To further increase the model flexibility, one could include predictors of the form X i × X j , or consider non-parametric models, such as random forests and boosting. It has been empirically observed that GAMs do not represent well problems where the number of observations is much larger than the number of predictors. In [179] the Generalized Additive Model Selection is introduced to fit sparse GAMs in high dimension with a penalized likelihood approach. The penalty term is derived from the fitting criterion for smoothing splines. Alternatively, [62] proposes to fit a constrained version of GAMs by solving a conic programming problem. Classification The task of classifying data is to decide class membership y of an unknown data item x based on an input data set D = (x 1 , y 1 ), · · · , (x n , y n ) of data items x i with known class memberships y i . The x i are usually m-dimensional vectors. This section reviews the common classification approaches that include k-nearest neighbors, logistic regression, linear discriminant analysis, decision trees, and support vector machines. K-Nearest Neighbors Classification based on the k-nearest neighbor (k-NN) algorithm differs from other methods, as this approach uses the data directly, without building a model first. Unlike other supervised learning algorithms, k-NN does not learn an explicit mapping from the training data, it simply uses the training data to make predictions, thus it is called a non-parametric method. A successful application of k-NN requires a careful choice of the number of nearest neighbors k and the choice of the distance metric. While k-NN is very easy to implement, the results are highly dependent on the choice of k in some cases. By choosing small or large values of k, the model can be made more or less flexible, respectively. Given a set of training data D = (x 1 , y 1 ), · · · , (x n , y n ) with x i having m features/dimensions of the same scale and the corresponding output y i of discrete or continuous values, k-NN computes the distance from each of the training data to a test point t j . The computed distances are sorted and the k nearest neighbors are selected. The majority rule is then used for classification (output y is discrete) or the averaging is used for regression (output y is continuous). Several distance functions can be used. For real-valued features, the euclidean norm d(x i , t j ) = \|\|x − t\|\| 2 = n l=1 (x il − t jl ) 2 is commonly used. The Minkowski distance d(x i , t j ) = \|\|x − t\|\| p = n l=1 (x il − t jl ) p 1 p is a more general class of distance functions. For binary-valued features, the hamming distance is considered which counts the number of features where the two data points disagree. In binary classification problems, it is helpful to choose k to be an odd number as this avoids tied votes. Weights for each of the features can be also used when computing the distances [19]. Logistic Regression In most problem domains, there is no functional relationship y = f (x) between y and x. In this case, the relationship between x and y has to be described more generally by a probability distribution P (x, y) while assuming that the data set D contains independent samples from P . The optimal class membership decision is to choose the class label y that maximizes the posterior distribution P (y\|x). Logistic Regression provides a functional form f and a parameter vector β to express P (y\|x) as f (x, β). The parameters β are determined based on the data set D usually by maximum-likelihood estimation [82]. Generally, a logistic regression model calculates the class membership probability for one of the two categories in the data set as P (1\|x, β) = ( 1 1 + e β 0 +βx ). (22) The decision boundary between the two binary classes is formed by a hyperplane whose equation is β 0 + βx = 0. Points at this decision boundary have P (1\|x, β) = P (0\|x, β) = 0.5. The optimal parameter values β are obtained by maximizing the liklihood estimation Π n i=1 P (y i \|x i , β) which is equivalent to min − n i=1 logP (y i \|x i , β).(23) First order methods such as gradient descent as well as second order methods such as Newton's method can be applied to optimally solve problem (23). To tune the logistic regression model (22), variable selection can be performed where only the most relevant subset of the x variables are kept in the model. A forward selection approach or a backward elimination approach can be applied to add or remove variables respectively, based on the statistical significance of each of the computed coefficients. Interaction terms can be also added to (22) to further complicate the model at the risk of overfitting the training data. Variable selection can then be also applied to eliminate the non statistically significant interaction terms [91]. Linear Discriminant Analysis Linear discriminant analysis (LDA) is an approach for classification and dimensionality reduction. It is often applied to data that contains a large number of features (such as image data) where reducing the number of features is necessary to obtain robust classification. While LDA and PCA share the commonality of dimensionality reduction, LDA tends to be more robust than PCA since it takes into account the data labels in computing the optimal projection matrix [23]. Given a data set with n samples D = (x 1 , y 1 ), · · · , (x n , y n ) and K classes where x i ∈ R m and y i ∈ {0, 1} K such that if x i belongs to the k-th class then y i (k) is 1 and 0 otherwise, the input data is partitioned into K groups {π k } K k=1 where π k denotes the sample set of the k-th class which contains n k data points. LDA maps the features space [188]. The class mean of the k-th class is given by µ k = 1 n k x i ∈π k x i while the global mean in given by µ = 1 n n i=1 x i . In the projected space the class mean is given by µ k = 1 n k q i ∈π k q i while the global mean in given by µ = 1 n n i=1 q i . The within-class scatter and the between-class scatter evaluate the class separability and are defined as S w and S b respectively such that x i ∈ R m to a lower dimensional space q i ∈ R r (r < p) through a linear transformation q i = G x iS w = K k=1 x i ∈π k (x i − µ k )(x i − µ k ) (24) S b = K k=1 n k (µ k − µ)(µ k − µ) .(25) The within-class scatter evaluates the spread of the data around the class mean while the between-class scatter evaluates the spread of the class means around the global mean. For the projected data, the within-class and the between-class scatters are given by S w = K k=1 q i ∈π k (q i − µ k )(q i − µ k ) = G S w G (26) S b = K k=1 n k (µ k − µ)(µ k − µ) = G S b G.(27) The LDA optimization problem is bi-objective where the within-class should be minimized while the between-class should be maximized. Thus the optimal transformation G can be obtained by maximizing the Fisher criterion (the ratio of between-class to within-class scatters) max \|G T S b G\| \|G T S w G\| .(28) Note that since the between-class and the within-class scatters are not scalar, the determinant is used to obtain a scalar objective function. As discussed in [93], assuming that S w is invertible and non-singular, the Fisher criterion is optimized by selecting the r largest eigen values of S −1 w S B and the corre- sponding eigen vectors G * 1 , G * 2 , . . . , G * (K−1) form the optimal transformation matrix G * = [G * 1 \|G * 2 \| . . . \|G * (K−1) ]. An alternative formulation of the LDA optimization problem is provided in [58] by maximizing the minimum distance between each class center and the total class center. The proposed approach known as the large margin linear discriminant analysis requires the solution of non-convex optimization problems. A solution approach is also proposed based on solving a series of convex quadratic optimization problems. LDA can also be applied for data with multiple labels. In the multi-label case, each data point can belong to multiple classes, which is often the case in image and video data (for example an image can be labeled "ear", "dog", "animal"). In [188] the equations of the within-class and the between-class scatters are extended to incorporate the correlation between the labels. Apart from this change, the LDA optimization problem (28) remains the same. Decision Trees Decision trees are classical models for making a decision or classification using splitting rules organized into tree data structure. Tree-based methods are non-parametric models that partition the predictor space into sub-regions and then yield a prediction based on statistical indicators (e.g., median and mode) of the segmented training data. Decision trees can be used for both regression and classification problems. For regression trees, the splitting of the training dataset into distinct and non-overlapping regions, can be done using a top-down recursive binary splitting procedure. Starting from a singleregion tree, one iteratively searches for (typically univariate) cutpoint s for predictor X j such that the tree with the two splitted regions {X\|X j < s} and {X\|X j ≥ s} has the greatest possible reduction in the residual sum of squares i:x i ∈R 1 (j,s) (y i −ŷ R 1 ) 2 + i:x i ∈R 2 (j,s) (y i −ŷ R 2 ) 2 , whereŷ R denotes the mean response for the training observations in region R. A multivariate split is of the form {X\|a T X < s}, where a is a vector. Searching for a solution that minimizes the RSS often leads to overfitting. Another optimization criterion is a measure of purity [46], such as Gini's index in classification problems. To limit overfitting, it is possible to prune a decision tree so as to obtain subtrees minimizing, for example, cost complexity. For classification problems, [46] highlights that, given their greedy nature, the classical methods based on recursive splitting do not lead to the global optimality of the decision tree which limits the accuracy of decision trees. Since building optimal binary decision trees is known to be NP-hard [114], heuristic approaches based on mathematical programming paradigms, such as linear optimization [26], continuous optimization [27], dynamic programming [14,16,71,161], have been proposed. To find provably optimal decision trees, [29] propose a mixed-integer programming formulation that has an exponential complexity in the depth of the tree. Given a fixed depth D, the maximum number of nodes is T = 2 D+1 − 1 and they are indexed by t = 1, . . . , T . Following the notation of [29], the set of nodes is split into two sets, branch nodes and leaf nodes. The branch nodes T B = {1, . . . , df } apply a linear split a x < b where the left branch includes the data that satisfy this split while the right branch includes the remaining data. At the leaf nodes T F = { df + 1, . . . , T }, a class prediction is made for the data that points that are included at that node. In [29], the splits that are applied at the branch nodes are restricted to a single variable with the option of not splitting a node. These conditions are enforced through the following constraints p j=1 a jt = d t , ∀t ∈ T B 0 ≤ b t ≤ d t , ∀t ∈ T B a jt ∈ {0, 1}, ∀j = 1, . . . , p, ∀t ∈ T B d t ∈ {0, 1}, ∀t ∈ T B where d t is a binary variable that indicates if a split is performed at node t or not and a jt and b t are respectively the features coefficients and the right hand side of the split at node t. Since all the descendants of a node that does not apply a split should also not have a split, then the following constraint is applied at all the nodes apart from the root node d t ≤ d p(t) , ∀t ∈ T B \ {1} where p(t) denote the parent node of node t. To keep track of the data points, the binary variable z it indicates if data point i is assigned to leaf node t and the binary variable l t indicates if node t is used or not. The following constrains z it ≤ l t , t ∈ T L n i=1 z it ≥ N min l t , t ∈ T L indicate that data points can be assigned to a node only if that node is used and if a node is used then at least N min data points should be assigned to it. Each data point should also be assigned to exactly one leaf which is enforced by t∈T L z it = 1, i = 1, . . . , n To enforce the splitting of the data point at each of the branch nodes, the following constraints are introduced a m x i + ≤ b m + M 1 (1 − z it ), i = 1, . . . , n, ∀t ∈ T L , ∀m ∈ A L (t), a m x i ≥ b m + M 2 (1 − z it ), i = 1, . . . , n, ∀t ∈ T L , ∀m ∈ A R (t), where A L (t) is the set of ancestors of t whose left branch has been followed on the path from the root node to node t and A R (t) is the set of ancestors of t whose right branch has been followed on the path from the root node to node t. M 1 and M 2 are large numbers while is a small number to enforce the strict split a x < b at the left branch (see [29] for finding good values for M 1 , M 2 , and ). Each leaf node that is used should be assigned to a label k = 1 . . . K and thus the following constraint is added K k=1 c kt = l t , ∀t ∈ T L where c kt is a binary variable that indicates if label k is assigned to leaf node t. The misclassification loss L t at leaf node t is given by L t = N t − N kt if node t is assigned label k, where N t is the total number of data points at leaf node t and N kt is the total number of data points at node t whose true label is k. This is enforced by the constraints L t ≥ N t − N kt − n(1 − c kt ), k = 1, . . . , K, ∀t ∈ T L L t ≤ N t − N kt + nc kt , k = 1, . . . , K, ∀t ∈ T L L t ≥ 0, ∀t ∈ T L . The counting of N t and N kt is enforced by N t = n i=1 z it , ∀t ∈ T L N kt = 1 2 n i=1 (q + Y ik )z it , k = 1, . . . , K, ∀t ∈ T L where Y ik is a parameter that takes a value 1 if the true label of data point i is k and −1 otherwise. The objectives are to minimize the decision tree complexity that is given t∈T B d t and the normalized total misclassification loss 1 L t∈T L L t whereL is the baseline loss obtained by predicting the most popular class from the entire dataset. Both are included in a single objective min 1 L t∈T L L t + α t∈T B d t where α is a tuning parameter. The other hyper-parameters that need to be tuned are the minimum number of data points at each leaf node N min and the maximum tree depth D. Leveraging the hyper-parameter tuning and warm-start techniques, the optimal decision tree problem is shown to be competitive on datasets with thousands of observations and achieves a good accuracy for shallow trees compared to classical greedy approaches [29]. The proposed formulation can also be extended to multivariate splits, without increasing the computational complexity [29]. An alternative formulation to the optimal decision tree problem is provided in [104]. The main difference between the formulation of [104] and [29] is that the approach of [104] is specialized to the case where the features take categorical values. By exploiting the combinatorial structure that is present in the case of categorical variables, [104] provides a strong formulation of the optimal decision tree problem thus improving the computational performance. Furthermore the formulation of [104] is restricted to binary classification and the tree topology is fixed ,which allows the optimization problems to be solved to optimality in reasonable time. While single decision tree models are often preferred by data analysts for their high interpretability, the model accuracy can be largely improved by taking multiple decision trees into account with approaches such as bagging, random forests, and boosting. Furthermore, decision trees can also be used in a more general range of applications as algorithms for problem solving, data mining, and knowledge representation. In [15], several greedy and dynamic programming approaches are compared for building decision trees on datasets with inconsistent labels (i.e, many-valued decision approach). Many-valued decisions can be evaluated in terms of multiple cost functions in a multistage optimization [17]. Recently, [60] investigated conflicting objectives in the construction of decision trees by means of bi-criteria optimization. Since the single objectives, such as minimizing average depth or the number of terminal nodes, are known to be NP-hard, the authors propose a bi-criteria optimization approach by means of dynamic programming. Support Vector Machines Support vector machines (SVMs) are another class of supervised machine learning algorithms that are based on statistical learning and has received significant attention in the optimization literature [55,183,184]. Given a training data set of size n where each observation x has m features and a corresponding binary label y ∈ {−1, 1}, the objective of the support vector machine problem is to identify a hyperplane that separates the two classes of data points with a maximal separation margin measured as the width of the band that separates the two classes. The underlying optimization problem is a convex quadratic optimization problem. Hard Margin SVM The most basic version of SVMs is the hard margin SVM that assumes that there exists a hyperplane w x + β = 0 that geometrically separates the data points into the two classes such that no data point is misclassified [67]. The training of the SVM model involves finding the hyperplane that separates the data and whose distance to the closest data point in either of the classes, i.e. margin, is maximized. The distance of a particular data point x i to the separating hyperplane is y i (w x i + β) w where w denotes the l 2 -norm. The distance to the closest data point is normalized to 1 w . Thus the data points with labels y = −1 are on one side of the hyperplane such that w x + β ≤ 1 while the data point with labels y = 1 are on the other side w x + β ≥ 1. The optimization problem for finding the separating hyperplane is then max 1 w s.t. y i (w x i + β) ≥ 1 ∀i ∈ 1, . . . , n which is equivalent to min w 2 (29) s.t. y i (w x i + β) ≥ 1 ∀i ∈ 1, . . . , n(30) that is a convex quadratic problem. Forcing the data to be separable by a linear hyperplane is a strong condition that often does not hold in practice and thus the soft-margin SVM which relaxes the condition of perfect separability. Soft-Margin SVM When the data is not lienarly seperable, problem (29)-(30) is infeasible. Thus the soft margin SVM introduces a slack into constraints (30) which allows the data points to be on the wrong side of the hyperplane [67]. This slack is minimized as a proxy to minimizing the number of data points that are on the wrong side. The soft-margin SVM optimization problem is min w 2 + C n i i (31) s.t. y i (w x i + β) ≥ 1 − i ∀i ∈ 1, . . . , n(32)i ≥ 0 ∀i ∈ 1, . . . , n.(33) Another common alternative is to include the error term i in the objective function by using the squared hinge loss n i 2 i instead of the hinge loss n i i . Hyperparameter C is then tuned to obtain the best classifier. Besides the direct solution of problem (31)-(33) as a convex quadratic problem, replacing the l 2 -norm by the l 1 -norm leads to a linear optimization problem generally at the expense of higher misclassification rate [45]. Sparse SVM Using the l 1 -norm is also an approach to sparsify w, i.e. reduce the number of features that are involved in the classification model [45,194]. An approach known as the elastic net includes both the l 1 -norm and the l 2 -norm in the objective function and tunes the bias towards one of the norms through a hyperparameter [189,199]. The number of features can be explicitly modeled in (31)-(33) by using binary variables [56]. A constraint limiting the number of features to a maximum desired number can be enforced resulting in a mixed integer quadratic problem [94]. The Dual Problem and Kernel Tricks The data points can be mapped to a higher dimensional space through a mapping function φ(x) and then a soft margin SVM is applied such that min w 2 + C n i i (34) s.t. y i (w φ(x i ) + β) ≥ 1 − i ∀i ∈ 1, . . . , n(35)i ≥ 0 ∀i ∈ 1, . . . , n.(36) Through this mapping, the data has a linear classifier in the higher dimensional space however a non-linear separation function is obtained in the original space. To solve problem (34)-(36), the following dual problem is first obtained max α n i=1 α i − 1 2 n i,j=1 α i α j y i y j φ(x i ) φ(x j ) n i=1 α i y i = 0, ∀i = 1, . . . , n 0 ≤ α i ≤ C, ∀i = 1, . . . , n where α i are the dual variables of constraints (35). Given a kernel function K : R m × R m → R where K(x i , x j ) = φ(x i ) φ(x j ), the dual problem is max α n i=1 α i − 1 2 n i,j=1 α i α j y i y j K(x i , x j ) n i=1 α i y i = 0, ∀i = 1, . . . , n 0 ≤ α i ≤ C, ∀i = 1, . . . , n which is a convex quadratic optimization problem. Thus only the kernel function K(x i , x j ) is required while the explicit mapping φ() is not needed. The common kernel functions include polynomial K( [55,110]. x i , x j ) = (x i .x j + 1) d where d is the degree of the polynomial, radial basis function K(x i , x j ) = e − x i −x j 2 γ , and sigmoidal K(x i , x j ) = tanh(βx i x j + c) Clustering Data clustering is a class of unsupervised learning approaches that has been widely used in practice particularly in applications of data mining, pattern recognition, and information retrieval. Given The two concepts can be measured via several criteria and lead to different types of clustering algorithms. The number of clusters is typically a tuning parameter to be fixed before determining the clusters. An extensive survey on data clustering analysis is provided in [117]. In case the entities are points in a Euclidean space, the clustering problem is often modeled as a network problem and shares many similarities with classical problems in operations research, such as the p-median problem. Following the notation in [106], the following is defined X: The data matrix X contains the p characteristics of the entities of O generally modeled as a N × p matrix. D: The dissimilarities matrix D contains the pair-wise dissimilarities d kl between each pair of data points l and k and is modeled as a N × N matrix and typically assumes that d kl ≥ 0, d kl = d lk and d kk = 0. The dissimilarity d kl is commonly a distance metric. The separation of clusters can be measured by a split s(C j ) = min k:O k ∈C j ,l:O l ∈C j d kl or a cut c(C j ) = k:O k ∈C j l:O l ∈C j d kl . where C j contains the subset of the data in O that form cluster j. The following measures can also be used to evaluate homogeneity of each cluster C j Diameter: δ(C j ) = max k,l:O k ,O l ∈C j d kl Radius: r(C j ) = min k:O k ∈C j max l:O l ∈C j d kl , Star:st(C j ) = min k:O k ∈C j l:O l ∈C j d kl Clique: cl(C j ) = k,l:O k ,O l ∈C j d kl . If the observations are points of a p-dimensional Euclidean space, the homogeneity of C j can be expressed in terms of the sum-of-squares of the Euclidean distances between the elements of C j and its centroid x such that ss(C j ) = k:O k ∈C j x k −x 2 2 . The most commonly adopted types of clustering algorithms are : • Hierarchical Clustering. The clusters {C 1 , . . . , C M } are such that a hierarchy H = {P 1 , P 2 , . . . , P q } of q ≤ N partitions of O, satisfies C i ∈ P k , C l ∈ P l , k > l ⇒ C i ⊂ C j or C i ∩C j = ∅, ∀i = j, k, l = 1, 2, . . . , N. • Partitioning Clustering. The clusters {C 1 , . . . , C M } are such that: -C j = ∅, ∀j = 1, 2, . . . , M -C i ∩ C j , ∀i, j = 1, 2, . . . , M, i = j - M i=1 C j = O Hierarchical clustering methods can be divided into agglomerative, divisive, or global criterion algorithms. In the agglomerative approach, one starts with N -singleton clusters which are then iteratively merged until a local criterion is satisfied. One merging algorithm is the single-linkage where the two clusters with the smallest inter-cluster dissimilarity are merged. Alternatively, the complete-linkage algorithm merges the two clusters for which the resulting merged cluster has the smallest diameter. Divisive hierarchical clustering is done by bi-partitioning an initial cluster containing all the entities. The choice of the local criterion for dividing the clusters affects the complexity of the algorithm, and can lead to NP-hard formulations [163]. Methods searching for an optimal hierarchy overall a global criterion are still not explored. Partitioning clustering in the one-dimensional Euclidean space can be solved via Dynamic Programming. Another possibility to find partitioning solutions is to apply branch-and-bound algorithms, for criteria such as sum-of-squares , sum-of-cliques , sum-of-stars. A direct branch-and-bound approach is introduced in [81] for the problem of seeking a consensus partition, where one wants to minimize a total distance of a given set of partition. The problem is formulated as min N −1 k=1 N l=k+1 d kl y kl(37)s.t y kl + y lq − y kq ≤ 1 k = 1, 2, . . . , N − 2 (38) − y kl + y lq + y kq ≤ 1 l = k + 1, k + 2, . . . , N − 1 (39) y kl − y lq + y kq ≤ 1 q = l + 1, l + 2, . . . , N (40) y kl ∈ {0, 1} k = 1, 2, . . . , N − 1, l = k + 1, k + 2, . . . , N,(41) where y kl = 1 if O k and O l belong to the same cluster and y kl = 0 otherwise. The generic partitioning problem may be expressed as a standard set partitioning problem with columns representing all the possible subsets of the set of observations O such that min 2 N −1 t=1 f (C t )y t(42)s.t 2 N −1 i=1 a jt y t = 1 j = 1, 2, . . . , N 2 N −1 i=1 y t = M(43)y t ∈ {0, 1} t = 1, . . . , 2 N − 1(44) where f is a cost function for the clusters, a jt = 1 indicates that O j belongs to cluster C t , and y t is a decision variable associated with each cluster C t . Mathematical programming approaches can also include the decision of the cluster size such as maximizing the modularity of the associated graph [51,52]. In the following, the commonly used mathematical programming formulations that include Minimum Sum-Of-Squares Clustering, Capacitated Clustering, and k-Hyperplane Clustering are discussed. Minimum Sum-Of-Squares Clustering (a.k.a. k-Means Clustering) Minimum Sum-Of-Squares Clustering is one of the most commonly used clustering algorithms. It requires to find a number of disjoint clusters for data points A = {a 1 , . . . , a m } in R n such that the distance to the cluster centroids is minimized. Given that, typically, the number k of clusters is fixed, the problem is also referred to as k-Means Clustering. Defining the binary variables x ij = 1 if data point i belongs to cluster j 0 otherwise, the problem is formulated as the following mixed integer nonlinear program [11] min i≤N,j≤k x ij a i − µ j 2 2 s.t. j≤k x ij = 1 ∀i ≤ N µ j ∈ R n ∀j ≤ k x ij ∈ {0, 1} ∀i ≤ N, ∀j ≤ k. By adding big-M constraints, the following linearized formulation is obtained min i≤N,j≤k d ij s.t. j≤k x ij = 1 ∀i ≤ N d ij ≥ \|\|a i − µ j \|\| − M (1 − x ij ) ∀i ≤ N, ∀j ≤ k µ j ∈ R n ∀j ≤ k x ij ∈ {0, 1}, d ij ≥ 0 ∀i ≤ N, ∀j ≤ k. Solution approaches have been proposed in [18,121]. The case where the space is not Euclidean is considered in [54]. Alternatively, [171] presents the Heterogeneous Clustering Problem where the observations to cluster are associated with multiple dissimilarity matrices. The problem is formulated as a mixed-integer quadratically constrained quadratic program. Another variant is presented in [169] where the homogeneity is expressed by the minimization of the maximum diameter D max of the k clusters. The following nonconvex bilinear mixed-integer program is proposed x il = 1 ∀i = 1, . . . , n, min D max (46) s.t. D l ≥ d ij x il x jl ∀i,D max ≥ D l ∀l, l = 1, . . . , k (49) x ij ∈ {0, 1},(48) ∀i, j, i = 1, . . . , n, l = 1, . . . , k (50) D l ≥ 0 ∀l, l = 1, . . . , k. The decision variable D l expresses the diameter of cluster l and x il is activated if observation i belongs to cluster l. Capacitated Clustering The Capacitated Centered Clustering Problem (CCCP) deals with finding a set of clusters with a capacity limitation and homogeneity expressed by the similarity to the cluster centre. A mathematical formulation is given in [153] as min N i=1 M j=1 d ij x ij(52) s.t N i=1 q i x ij ≤ Q j j = 1, . . . , M(55) x ij , y j ∈ {0, 1} i = 1, . . . , N, j = 1, . . . , M, where M is an upper bound on the number of clusters, d ij is the dissimilarity measure between entities i and j, q i is the demand of entity i, Q j is the capacity of cluster j, variable x ij denotes the assignment of observation i to cluster j, and variable y j is equal to 1 if cluster j is used. If the metric d ij is a distance and the clusters are homogeneous (Q j = Q ∀j), the formulation also models the well-known facility location problem. Furthermore, when the total number of clusters is fixed to p, the resulting p-CCCP can be solved using a clustering search algorithm that is discussed in [57]. Alternative solution heuristics have also been proposed in [149] and [172] as well as a quadratic programming formulation that is presented in [138]. k-Hyperplane Clustering In the k-Hyperplane Clustering (k-HC) problem, a hyperplane, instead of a center, is associated with each cluster. This is motivated by applications where collinearity and coplanarity relations among the observations are the main interest of the unsupervised learning task, rather than the similarity. Given A = {a 1 , . . . , a N } observations in R n , the k-HC problem requires to find k clusters, and a hyperplane H j = {x ∈ R n : w T j x = γ j }, with w j ∈ R n and γ j ∈ R, for each cluster j, in order to minimize the sum of the squared 2-norm Euclidean orthogonal distances between each observation and the corresponding cluster. Given that the orthogonal distance of a i to hyperplane H j is given by \|w T j a i −γ j \| w 2 , k-HC is formulated in [12] as the following mixed integer quadratically constraint quadratic problem: min M i=1 δ 2 i (57) s.t k j=1 x ij = 1 i = 1, . . . , m(58)δ i ≥ (w T j a j − γ j ) − M (1 − x ij ) i = 1, . . . , m, j = 1, . . . , k(59)δ i ≥ (−w T j a j + γ j ) − M (1 − x ij ) i = 1, . . . , m, j = 1, . . . , k(60)w k 2 ≥ 1 j = 1, . . . , k (61) δ i ≥ 0 i = 1, . . . , m (62) w j ∈ R n , γ j ∈ R j = 1, . . . , k (63) x ij ∈ {0, 1}, i = 1, . . . , m, j = 1, . . . , k(64) Constraints (59)-(60) model the point to hyperplane distance via linear constraints. The non-convexity is due to Constraints (61). Deep Learning Deep Learning (DL) received a first momentum until the 80s due to the universal approximation results [73,112], where neural networks with a single layer with a finite number of units can represent any multivariate continuous function on a compact subset R n with arbitrary precision. However, the computational complexity required for training Deep Neural Networks (DNNs) hindered their diffusion by late 90s. Starting 2010, the empirical success of DNNs has been widely recognized, for several reasons including the development of advanced processing units, namely GPUs, the advances in the efficiency of training algorithms such as backpropagation, the establishment of proper initialization parameters, and the massive collection of data (see Section 7) enabled by new technologies in a variety of domains (e.g., healthcare, supply chain management [180], marketing, logistics [187], Internet of Things). The aim of this section is to describe the decision optimization problems associated with DNN architectures. To facilitate the presentation, the notation for the common parameters is provided in Table 1. {0, . . . , K} layers indices n k number of units, or neurons, in layer k σ element-wise activation function U (j, k) j-th unit of layer k W k ∈ R n k ×n k+1 weight matrix for layer k b k ∈ R n k bias vector for layer k x 0 training dataset y testing dataset, with observations y i , i = 1, . . . , N x 0 j j-th input feature x k output vector of layer k, k > 0 (derived feature). The output vector x K of a DNN is computed by propagating the information from the input layer to each following layer such that x k = σ(W k−1 x k−1 + b k−1 ) k = 1, . . . , K.(65) The output of the DNN is finally evaluated for regression or classification tasks. In the context of regression, the components of x K can directly represent the response values learned. For a classification problem, the vector x K corresponds to the logits of the classifier. In order to interpret x K as a vector of class probabilities, functions F such as the logistic sigmoidal or the softmax can be applied [97]. The classifier C modeled by the DNN then classifies an input x with the label correspondent to the maximum activation: C(x) = arg max i=1,...,n K F (x K i ). A data scientist may spend a considerable effort in testing configurations of parameters, such as the number of layers and their size, and the learning rates and epochs for the training algorithm. All the values that need to be determined before the training process takes place are called hyperparameters. The Hyperparameter Optimization (HPO) is typically driven by the data scientist's experience, the characteristics of the dataset, or by following heuristic rules. Alternatively HPO can be modeled in a mathematical programming framework such as the approach presented in [79] where a box-constrained model is presented and solved by a derivative-free approach. Given A the set of valid assignments of values for the hyperparameters, and f a performance metric (e.g., prediction accuracy on a test set), then the HPO requires the solution of arg max a∈A f (a, x). By bounding the number of hidden layers and their size, and relying on a surrogate model for the stochastically computable objective f , formulation (66) becomes a box-constrained problem. In terms of activation functions, the rectified linear unit ReLU : R n → R n , ReLU (z) = (max(0, z 1 ), . . . , max(0, z n )). is typically one of the preferred option. If the unit is active, the gradient is strictly positive, and this favors the adoption of gradient-based optimization algorithms for the training of DNNs. In order to learn from non-active units, some generalizations have been proposed: rectification given by the absolute value function [119], leaky ReLU [147] max(0, y) + α min(0, y) with a small α, and parametric ReLU with a learnable α for each component [109]. Other activations can also be done via the element-wise function sign : R → R, sign(z) = 1 if z ≥ 0, sign(z) = 0 if z < 0 . The task of training a DNN consists of determining the weights W k and the biases b k that make the model best fit the training data, according to a certain measure of training loss. For a regression with Q quantitative responses, a common measure of training loss L is the sum-of-squared errors on the testing dataset Q q=1 N i=1 (y iq − x K q ) 2 .(67) For classification with Q classes, the cross-entropy − Q q=1 N i=1 y iq log x K q(68) is preferred. An effective approach to minimize L is by gradient descent, called backpropagation in this setting. Typically, one is not interested in a proven local minimium of L, as this is likely to overfit the training dataset and yield a learning model with a high variance. Similar to the Ridge regression (see Section 2), the loss function can include a weight decay term such as λ K−1 k=0 n k i=1 (b k i ) 2 + K−1 k=0 n k i=1 n k+1 j=1 (W k ij ) 2(69) or alternatively a weight elimination penalty λ K−1 k=0 n k i=1 (b k i ) 2 1 + (b k i ) 2 + K−1 k=0 n k i=1 n k+1 j=1 (W k ij ) 2 1 + (W k ij ) 2 .(70) Function (70) tends to eliminate smaller weights more than (69). The aim of this section is to present the optimization models that are used in DNN. First, mixed integer programming models for DNN training are introduced in Section 5.1. Adversarial learning and data poisoning are then discussed in Sections 5.2 and 5.3, respectively. Finally ensemble approaches that train DNNs with multiple activation functions are discussed in Section 5.4. Mixed Integer Programming for DNN Architectures Motivated by the considerable improvements of Mixed-Integer Programming solvers, a natural question is how to model a DNN as a MIP. In [88], DNNs with ReLU activation x k = ReLU (W k−1 x k−1 + b k−1 ) ∀k = 1, . . . , K(71) are modeled where decision variables x k express the output vector of layer k, k > 0 and x 0 is the input vector. The following mixed integer linear problem is proposed min K k=0 n k j=1 c k j x k j + K k=1 n k j=1 γ k j z k j (72) s.t. n k−1 i=1 w k−1 ij x k−1 i + b k−1 j = x k j − s k j ∀k = 1, . . . , K, j = 1, . . . , n k (73) z k j = 1 → x k j ≤ 0 ∀k = 1, . . . , K, j = 1, . . . , n k (74) z k j = 0 → s k j ≤ 0 ∀k = 1, . . . , K, j = 1, . . . , n k (75) lb 0 j ≤ x 0 j ≤ ub 0 j ∀j = 1, . . . , n 0 (76) lb k j ≤ x k j ≤ ub k j ∀k = 1, . . . , K, j = 1, . . . , n k (77) lb k j ≤ s k j ≤ ub k j ∀k = 1, . . . , K, j = 1, . . . , n k(78) where z k j are binary activation variables and s k j are slack variables that correspond with each unit U (j, k). Depending on the application, different activation weights c k j and activation costs γ k j can also be used for each U (j, k). The proposed MIP is feasible for every input vector x 0 , since it computes the activation in the subsequent layers. Constraints (74) and (75) are indicator constraints, which are known to generate very hard optimization problems, because of their weak continuous relaxation [40]. Several optimization solvers can directly handle such kind of constraints, and the tightness of the provided bounds is crucial for their effectiveness. In [88], a bound-tightening strategy to reduce the computational times is proposed and the largest DNN tested with this approach is a 5-layer DNN with 20+20+10+10+10 internal units. Problem (72) Avg(x k ) = 1 n k n k i=1 x k i Max(x k ) = max(x k 1 , . . . , x k n k ) used for example in Convolutional Neural Networks, can be incorporated in the hidden layers. In the case of max pooling operations, additional indicator constraints are required. Adversarial learning will be discussed in more detail in Section 5.2. • Training: In this case, the weights and biases are decision variables. The resulting bilinear terms in (73) and the considerable number of decision variables in the formulation limit the applicability of (72)- (78) for DNN training. In addition, searching for proven optimal solutions is likely to lead to overfitting. Another attempt in modelling DNNs via MIPs is provided by [127], in the context of Binarized Neural Networks (BNNs). BNNs are characterized by having binary weights {−1, +1} and by using the sign function for neuron activation [70]. Given that the established search strategies for adversarial examples (e.g., fast gradient sign method, projected gradient descent (PGD) attack) rely on gradient information, the discrete and non-differentiable architecture of BNNs makes them quite robust to such attacks. In [127], a MIP is proposed for finding adversarial examples in BNNs by maximizing the difference between the activation of the targeted label l and the predicted label l of the input x 0 , in the final layer (namely, max x K l − x K l ). Contrary to [88], the MIP of [127] does not impose limitations on the search of adversarial examples, apart from the perturbation quantity. In terms of optimality criterion however, searching for the proven largest misclassified example is different from finding a targeted adversarial example. Furthermore, while there is interest in minimally perturbed adversarial examples, suboptimal solutions corresponding to adversarial examples (i.e., x K l ≥ x K l ) may have a perturbation smaller than that of the optimal solution. Besides [88], other MIP frameworks have been proposed to model certain properties of neural networks in a bounded input domain. In [59], the problem of computing maximum perturbation bounds for DNNs is formulated as a MIP, where indicator constraints and disjunctive constraints are modeled using constraints with big-M coefficients [101]. The maximum perturbation bound is a threshold such that the perturbed input may be classified correctly with a high probability. A restrictive misclassification condition is added when formulating the MIP. Hence, the infeasibility of the MIP does not certify the absence of adversarial examples. In addition to the ReLU activation, the tan −1 function is also considered by introducing quadratic constraints and several heuristics are proposed to solve the resulting problem. In [181], a model to formally measure the vulnerability to adversarial examples is proposed (the concept of vulnerability of NN is discussed in Sections (5.2.1) and (5.2.2)). A tight formulation for the resulting non-linearities and a novel presolve technique are introduced to limit the number of binary variables and improve the numerical conditioning. However, the misclassification condition is not explicitly defined but is rather left in the form "different from" and not explicitly modeled using equality/inequality constraints. In [173], the aim is to count or bound the number of linear regions that a piecewise linear classifier represented by a DNN can attain. Assuming that the input space is bounded and polyhedral, the DNN is modeled as a MIP. The contributions of adopting a MIP framework in this context are limited, especially in comparison with the results achieved in [152]. MIP frameworks can also be used to formulate the verification problem for neural networks as a satisfiability problem. In [123], a satisfiability modulo theory solver is proposed based on an extension of the simplex method to accommodate the ReLu activation functions. In [50], a branch-and-bound framework for verifying piecewise-linear neural networks is introduced. For a recent survey on the approaches for automated verification of NNs, the reader is referred to [137]. Adversarial Learning Despite the wide interest in deep learning, the integration of neural networks into safety and security related applications necessitates thorough evaluation and research. A large number of contributions in the literature showed the presence of perturbed examples, also called adversarial examples, causing classification errors [36,178]. Malicious attackers can thus exploit security falls in a general classifier. In case the attacker has a perfect knowledge of the classifier's architecture (i.e., the result of the training phase), then a white-box attack can be performed. Black-box attacks are instead performed without full information of the classifier. The attention on adversarial examples is also motivated by the transferability of the attacks to different trained models [132,182]. Adversarial learning then emerges as a framework to devise vulnerability attacks for classification models [144]. From a mathematical perspective, such security issues have been formerly expressed via min-max approaches where the learner's and the attacker's loss functions are antagonistic [74,95,133]. Non-antagonistic losses are formulated as a Stackelberg equilibrium problem involving a bi-level optimization formulation [48], or in a Nash equilibrium approach [47]. These theoretical frameworks rely on the assumption of expressing the actual problem constraints in a game-theory setting, which is often not a viable option for reallife applications. Attacks for linear and non-linear binary classifiers with a differentiable discriminant function are investigated in [35]. Assuming {+1} is the malicious class associated with sample x having discriminant g(x) ≥ 0, the aim of the attacker is to find an example x close to x, according to a metric d, such that the discriminant function g of the classifier is minimized, i.e. min x ∈χ g(x )(79)s.t. d(x , x) ≤ δ.(80) To achieve the evasion of the classifier, an estimation g of the discriminant function may also be adopted in place of g in (79). The non-linear optimization model can be solved via gradient descent or quadratic techniques, such as Newton's method, BFGS, or L-BFGS. As a particular case of adversarial learning, [178] discusses the vulnerability of neural networks to adversarial examples. Such inputs are obtained by small perturbations on the original training dataset and yield selected adversary output. In the context of image classifications, imperceptible modifications of the input images can lead to severe misclassifications. The linearity of the neural network can increase the vulnerability [115]. As discussed in the next section, evasion attacks on the test set can be conducted in a targeted or untargeted fashion [53]. In the targeted setup, the attacker aims to achieve a classification with a chosen class target, while the untargeted misclassification is not constrained to a specific output class. Targeted attacks Given a neural network classifier f : χ ∈ R n 0 → Υ = {1, . . . , n K } and a target label l ∈ Υ, a relevant problem in the framework of targeted attacks is that of finding a minimum perturbation r of a given input x, such that f (x + r) = l. This corresponds to find an input "close" to x, which is misclassified by f . Clearly, if the target l coincides with the label l x that classifies x according to f , the problem has the trivial solution r = 0 and no misclassification takes place. In [178], the minimum adversarial problem for targeted attacks is formulated as a box-constrained problem min r∈R n 0 r 2 (81) s.t. f (x + r) = l (82) x + r ∈ [0, 1] n 0 .(83) The condition (83) ensures that the perturbed example x + r belongs to the set χ of admissible inputs, in case of normalized images with pixel values ranging from 0 to 1. Assuming l = l x , the difficulty of solving Problem (81)- (83) to optimality depends on the complexity of the classifier f . Denoting by L f → χ × Υ → R + the loss function for training f , [178] approximates the problem with a box-constrained L-BFGS min r∈R n 0 c\|r\| + L f (x + r, l) (84) x + r ∈ [0, 1] n 0 .(85) The approximation is exact for convex loss functions, and can be solved via a line search algorithm on c > 0. In [102], c is fixed such that the perturbation is minimized on a sufficiently large subset ψ, and the mean prediction error rate of f (x i + r i ) i ∈ ψ is greater than a threshold τ . In [53], the L 2 distance metric of formulation (81) x + r ∈ [0, 1] n 0 , where γ is a constant that can be determined by binary search such that the solution r * satisfies the condition F(x + r * ) ≤ 0. The authors propose strategies for applying optimization algorithms (such as Adam [129]) that do not support the box constraints (87) natively. Novel classes of attacks are found for the considered metrics. Untargeted attacks In untargeted attacks, one searches for adversarial examples x close to the original input x for which l x = l x , without targeting a specific label for x . Given that the only aim is misclassification, untargeted attacks are deemed less powerful than the targeted counterpart, and received less attention in the literature. A mathematical formulation to find minimum adversarial distortion for untargeted attacks is proposed in [181]. Assuming that the classifier f is expressed by the set of functions f i associated with each label {1, . . . , n K }, and given a distance metric d, then a perturbation r for an untargeted attack is found by solving min r d(r)(88)s.t. arg max i∈Υ {f i (x + r)} = l x (89) x + r ∈ χ.(90) This formulation can easily accommodate targeted attacks in a set T by replacing (89) with arg max i {f i (x + r)} ∈ T . The most commonly adopted metrics in literature are the L 1 , L 2 , and L ∞ norms which as shown in [181], can all be expressed with continuous variables. The L 2 norm makes the objective function of the outer-level optimization problem quadratic. Problem (88)- (90) can also be expressed as the bi-level optimization problem min r,z d(r)(91)s.t. z − l x ≤ − + M y (92) z − l x ≥ − (1 − y)M (93) z ∈ arg max i∈Υ {f i (x + r)}(94) x + r ∈ χ. Constraints (91)-(92) express the condition of misclassification z = l x using constraint with big-M coefficients. The complexity of the inner-level optimization problem is dependent on the activation functions. Given that the upper-level feasibility set χ is continuous and the lower-level variable i ranges on a discrete set, the problem is in fact a continuous discrete linear bilevel programming problem [186] which are arguably the hardest class of bilevel linear programming formulations. Reformulations or approximations have been proposed in [75,84,103,170,193]. Recalling the definition given above, untargeted attacks finds a targeted attack which is 'close enough' to the original input x. A trivial way to obtain an untargeted adversarial attack is then to solve Problem (81)-(83) for every l = l x and then select the solution x closest to x, according to a given metric. This approach would however go beyond the purpose of untargeted evasion, since it would perform a targeted attack for every possible incorrect label l = l x . We introduce an alternative mathematical formulation for finding untargeted adversarial examples which generalizes condition (82). Given an untargeted example x = x + r with l x = l x and the scoring functions f i , i ∈ Υ, the condition is equivalent to f * (x) = arg max i∈Υ f i (x ), which can be then expressed as ∃ i ∈ Υ s.t. f i (x + r) > f lx (x).(96) Condition (96) is an existence condition, which can be formalized by adding the functions σ i (r) = Relu(f i (x + r) − f lx (x)), i ∈ Υ, with a composition of the scoring functions f i and the ReLU functions i∈Υ σ i (r) > ,(97) where > 0 enforces that at least one σ i function has to be activated. Therefore, untargeted adversarial examples can be found by modifying formulation (81)-(83) by adding m functions σ i (r) and replacing condition (82) with the linear condition (96). The complexity of this approach depends on the scoring functions f i . The extra ReLu functions can be expressed as a mixed integer problem as reviewed in Section 5.1. Models resistant to adversarial attacks Another interesting line of research motivated by adversarial learning deals with adversarial training, which consists of techniques to make a neural network robust to adversarial attacks. A widely known defense technique is to augment the training data with adversarial examples; this however does not offer robustness guarantees on novel kinds of attacks. The problem of measuring robustness of a neural network is considered in [22]. Three robustness metrics are proposed. The pointwise robustness evaluates if the classification of x is robust for "small" perturbations , the adversarial frequency measures the portion of training inputs for which f fails to be (x, )-robust and finally the adversarial severity measures the extent by which f fails to be robust at x, conditioned on f not being (x, )−robust. Formally, f is said to be (x, )− robust if: l x = l x , ∀x \| x − x ∞ ≤(98) The pointwise robustness ρ(f, x) is the minimum for which f fails to be (x, )-robust ρ(f, x) = inf{ ≥ 0\| f is not (x, ) -robust}.(99) As detailed in [22], ρ is computed by expressing (99) as a constraint satisfiability problem. The concept of pointwise robustness is more general than the perturbation bound defined in [59] since no restrictions are imposed on the activation of each layer. The adversarial training of neural network by a robust optimization (RO) approach is investigated in [148]. In the RO framework, a decision maker searches for solution policies that avoid a worst-case scenario [24]. In this setting, the goal is to train a neural network to be resistant to all the attacks belonging to a certain class of perturbed inputs. Particularly, the adversarial robustness with a saddle point (min-max) formulation is studied in [148] which is obtained by augmenting the Empirical Risk Minimization paradigm. Given D the data distribution of the training samples ξ ∈ D, and the corresponding labels l ∈ Υ, let θ ∈ R p be the set of model parameters to be learned. Let L(θ, ξ, l) be the loss function considered in the training phase (e.g., the cross-entropy loss) and S be the set of allowed perturbations (e.g., an L ∞ ball). The aim is to minimize the worst adversarial loss on the set of inputs perturbed by S min θ E (ξ,l) max r∈S L(θ, ξ + r, l)(100) The saddle point problem (100) is viewed as the composition of an inner maximization and an outer minimization problem. The inner problem corresponds to attacking a trained neural network by means of the perturbations S. The outer problem deals with the training of the classifier in a robust manner. The importance of formulation (100) stems both from the formalization of adversarial training and from the quantification of the robustness given by the objective function value on the chosen class of perturbations. The value ρ(θ) = E (ξ,l) max r∈S L(θ, ξ + r, θ) quantifies the magnitude of the loss. To find solutions to (100) in a reasonable time, the structure of the local minima of the loss function can be explored. A similar robust approach is investigated in [174]. A perturbation set S i is adopted for each training example ξ i . The aim is then to optimize min θ n 0 i=1 max r i ∈S i L(θ, ξ i + r i , l i )(101) As detailed in [174], an alternating ascent and descent steps procedure can be used to solve (101) with the loss function approximated by the firstorder Taylor expansion around the training points. The methodology can also handle another loss function, the so-called Manifold Tangent Classifier (MTC) [166]. The MTC adopts the following loss function J(θ, ξ, l) = L(θ, ξ, l) + β u∈B ξ ( u, ∇ ξ f (ξ) ) 2 ,(102) where B ξ is a basis of the hyperplane that is tangent to the data manifold at ξ where β is a weight parameter and f (ξ) is the label for ξ classified by f . Adversarial training is also modeled in [139] with the objective function αL(θ, ξ, l) + (1 − α)L(θ,ξ, l) where L(θ, ξ, l) is the loss function used to generate adversarial examples, ξ is the example with label l,ξ is an adversarial perturbation of ξ, and α is a constant used to control the strength of the adversary. In the fast gradient sign method, perturbationξ for example ξ with label l is computed in the direction of the gradientξ = ξ + · sign(∇ ξ L(θ,ξ, l)) In order to improve the robustness of the neural network, a distortion term L−1 i=1 β i · n i j=1 d ij can be added to (103) where L − 1 is the number of the normalization layers of the trained network, n i is the number of features in the i-th layer, d ij is the distortion of the value of the j-th normalized feature in the i-th layer, and β i is a balancing constant. Data Poisoning Another class of attacks is that of Data Poisoning. In this context, the attacker hides corrupted, altered or noisy data in the training dataset. This causes failures in the accuracy of the training process for classification purposes. Data Poisoning was first studied by [37] to degrade the accuracy of SVMs. In [177], the upper bounds on the efficacy of a class of causative data poisoning attacks are studied. The causative attacks [21] proceed as follow: • n 0 data points are drawn by the data-generating distribution p * , producing a clean dataset D C , • the attacker adds malicious examples D P in the training dataset, • the defender learns modelθ from the full dataset D = D C ∪ D P , reporting a test loss L(θ). As discussed in [177], data sanitation defenses to limit the increase of test loss include two steps: (i) data cleaning (e.g., removing outliers which are likely to be poisoned examples) producing a feasible set F, and (ii) minimizing a margin-based loss on the cleaned dataset. The learned model is thenθ = arg min θ∈Θ L(θ; (D C ∪ D P ) ∩ F). Data poisoning can be viewed as games between the attacker and the defender players where the game is based on the conflicting objectives of the two players on the test loss. At each step, the attacker is required to maximize the test loss over (ξ, l) ∈ F. In the task of binary classification for SVM, the maximization of hinge loss for each y ∈ {+1, −1} is expressed by the quadratic programming problem min ξ∈R d y d i=1 θ i ξ i (104) s.t. ξ − µ +y 2 2 ≤ r 2 y (105) \| ξ − µ +y , µ +y − µ −y \| ≤ s y .(106) Constraint (105) removes the data points that are outside a sphere with centroid µ y and radius r y (sphere defense). Constraint (106) eliminates the points that are further than a distance tolerance s y from the line between the centroids µ +y = E[ξ\|y = +1] and µ −y = E[ξ\|y = −1] (slab defense). Poisoning attacks can also be performed in semi-online or online fashion, where training data is processed in a streaming manner, and not in batches. Compared to the offline case, the attacker has the knowledge of the order in which the training data is obtained. In the semi-online context, the attacker can modify part of the training data stream so as to maximize the classification loss, and the evaluation of the objective (loss) is done only at the end of the training. Instead, in the online modality, the classifier is updated and evaluated during the training process. In [190], a white-box attacker's behavior in online learning for a linear classifier w T x with binary labels y ∈ {−1, +1} is formulated. The data stream S arrives in T instants (S = {S 1 , . . . , S T }) and the classification weights are updated using an online gradient descent algorithm [198] such that w t+1 = w t − η t (∇L(w t , (x t , y t ))) + ∇Ω(w t ), where Ω is a regularizer, η t is the step length of the iterate update, and L is a convex loss function. The semi-online attacker can be formulated as max S∈F T g(w T )(107)s.t.\|{S \ S train }\| ≤ K,(108)w t = w 0 − t−1 τ =0 η τ (∇L(ω τ , S τ ) + ∇L(w τ )), 1 ≤ t ≤ T(109) where F T is the cleaned dataset at time T , g is the attacker's objective (e.g. classification error on the test set), \| · \| denotes the cardinality of a set, S train is an input data stream, and K is a defined upper bound on the number of changed examples in S train . With respect to the offline case, the estimated weight vector w t is a complex function of the data stream S, which makes the gradient computation more challenging and the KKT conditions do not hold. The optimization problem is simplified by considering a convex surrogate for the objective function, given by the logistic loss. In addition, the expectation is conducted over a separate validation data set and a label inversion procedure is implemented to cope with the multiple local maxima of the classifier function. The fully-online case can also be addressed by replacing the objective with t t=1 g(w t ) Activation Ensembles Another important research direction in neural network architectures investigates the possibility of adopting multiple activation functions inside the layers of a neural network. Some examples in this framework are given by the maxout units [98], returning the maximum of multiple linear affine functions, and the network-in-network paradigm [140] where the classical ReLU function is replaced by fully connected network. In [10], adaptive piecewise linear activation functions are learned during training. Specifically, for each unit i and value z, activation h i (z) is considered as h i (z) = max(0, z) + S s=1 a s i max(0, −z + b s i ), where the number of hinges S is a hyperparameter to be fixed in advance, while the variables a s i , b s i have to be learned. For large enough S, h i (x) can approximate a class of continuous piecewise-linear functions [10]. In a more general perspective, Ensemble Layers are proposed in [107] to consider multiple activation functions in a neural network. The idea is to embed a family of activation functions {H 1 , . . . , H m } and let the network itself choose the magnitude of the activations during the training. To promote relatively equal contribution to learning, the activation functions need to be scaled to the interval [0, 1]. To measure the impact of the activation in the neural network, each function H j is associated with a continuous variable α j . The resulting activation for neuron i is then given by the weighted sum of the scaled functions h j i (z) = H j (z) − min ξ∈D (H j (z ξ,i )) max ξ∈D (H j (z ξ,i )) − min ξ∈D (H j (z ξ,i )) +(110)H i (z) = m j=1 α j i h j i (z)(111) where z ξ,i is the i-th output associated with training example ξ, D is the training dataset, and is a small tolerance. To achieve the desired scaling, it is sufficient to let ξ vary over a minibatch in x [107]. Equation (111) can be modified so as to allow the network to leave the activations to the original state which is achieved by adding the parameters η and δ in H i (z) = 6 Emerging Paradigms Machine Teaching An important aspect of machine learning is the size of the training set to successfully learn a model. As such, the Teaching Dimension problem identifies the minimum size of a training set to correctly teach a model [96,175]. The teaching dimension of linear learners, such as ridge regression, SVM, and logistic regression has been recently presented in [141]. With the intent to generalize the teaching dimension problem to a variety of teaching tasks, [196] and [197] provide the Machine Teaching framework. Machine Teaching is essentially an inverse problem to Machine Learning. While, in a learning task, the training dataset x is given and the model parameters θ = θ * have to be determined, the role of a teacher is to let a learner approximately learn a model θ * by providing a proper set of training examples (also called teaching dataset in this context). A Machine Teaching task requires the selection of 3 components: • A Teaching Risk associated with the approximation error of the learner. • A Teaching Cost expressing the convenience of the teaching dataset, from the prospective of the teacher, weighted by a regularization term η. • A learner L. Formally, machine teaching can be casted as a bilevel optimization problem min x,θ TR(θ) + ηTC(x)(115)s.t.θ = L(x),(116) where the upper optimization is the teacher's problem and the lower optimization L(x) is the learner's machine learning problem. TR(θ) and TC(x) are the teaching risk and the teaching cost, respectively. The teacher is aware of the learning algorithm, which could be a classifier (such as those of Section 3) or a deep neural network. Machine teaching encompasses a wide variety of applications, such as data poisoning attacks, computer tutoring systems, and the two-players games, where the teacher is an encoder of the teaching set, and the learning identifies a modelθ. The optimization problem is NP-hard due to its combinatorial and bilevel nature. The teacher is typically optimizing over a discrete space of teaching sets, hence for some problem instances, the submodularity properties of the problem may be of interest. For problems with a small teaching set, it is possible to formulate the teaching problem as a mixed integer nonlinear program. The computation of the optimal training set remains, in general, an open problem, and is especially challenging in the case where the learning algorithm does not have a closed-form solution with respect to the training set [196]. Alternatively, the single level formulation of the problem minimizes the teaching cost and allows for either approximate or exact teaching min x,θ TC(x)(117)s.t. TR(θ) ≤ (118) θ = L(x).(119) Given a teaching budget, an alternative single level formulation is then min x,θ TR(θ)(120)s.t. TC(x) ≤ B (121) θ = L(x).(122) For the teaching dimension problem, the teaching cost is the cardinality of the teaching dataset, namely its norm 0. If the empirical minimization loss L is guiding the learning process, and λ is the regularization weight, then teaching dimension can be formulated as We note that the order of the training items matters in the sequential learning context [159] while it is not relevant for batch learners. Sequential teaching can be considered in online and adaptive frameworks such as stochastic gradient descent algorithms and reinforcement learners. The machine teaching framework can also address the case of multiple learners λ taught by the same teacher. The challenge is to determine a common teaching set for all learners. Several mathematical formulations can be devised, depending on the optimization criterion chosen by the teacher that include • Minimax risk, which optimizes for the worst learner min x,θ λ max λ TR(θ λ ) + ηTC(x) (126) s.t. θ λ = L λ (x). • Bayes risk, which optimizes for the average learner min x,θ λ λ f (λ)TR(θ λ )dλ + ηTC(x) s.t. θ λ = L λ (x),(128) where f (λ) is a known probability distribution of the learner. Machine teaching approaches tailored to specific learners have also been explored in the literature. In [195], a method is proposed for the Bayesian learners, while [160] focuses on Generalized Context Model learners. In [151], the bilevel optimization of machine teaching is explored to devise optimal data poisoning attacks for a broad family of learners (i.e., SVM, logistic regression, linear regression). The attacker seeks the minimum training set poisoning to attack the learned model. By using the KKT conditions of the learner's problem, the bilevel formulation is turned into a single level optimization problem, and solved using a gradient approach. Empirical Model Learning Empirical model learning (EML) aims to integrate machine learning models in combinatorial optimization in order to support decision-making in highcomplexity systems through prescriptive analytics. This goes beyond the traditional what-if approaches where a predictive model (e.g., a simulation model) is used to estimate the parameters of an optimization model. A general framework for an EML approach is provided in [143] and requires the following: • A vector x of decision variables with x i feasible over the domain D i . • A mathematical encoding h of the Machine Learning model. • A vector z of observables obtained from h. • Logical predicates g j (x, z) such as mathematical programming inequalities or combinatorial restrictions in constraint programming. • A cost function f (x, z). EML then solves the following optimization problem min f (x, z) (130) s.t. g j (x, z) ∀j ∈ J (131) z = h(x) (132) x i ∈ D i ∀x i ∈ x.(133) The combinatorial structure of the problem is defined by (130), (131), and (133) while (132) embeds the empirical machine learning model in the combinatorial problem. Embedding techniques for neural networks and decision trees are presented in [143] using four combinatorial optimization approaches that include mixed integer non-linear programming, constraint programming, and SAT Modulo Theories, and local search. Useful Resources This section provides a summary of resources of interest for research in machine learning. Table 2 lists the datasets used by the papers that are discussed in this review article. The details that are provided include the name, a brief description, the number of instances, the default task that can be accomplished, and the year of appearance. For each dataset, Table 3 also provides a bibliographic reference, a url, and the list of papers in this review article that use it. MNIST and CIFAR-10 are by far the most commonly used datasets. However, as pointed out by several researchers, classical machine learning tools can achieve 97.5% accuracy on MNIST and therefore more complex datasets should be used as benchmark. Image datasets similar to MNIST but more complex include Fashion-MNIST and EMNIST. Also a large repository is [78] which currently maintains 442 datasets. Another important list of resources are the libraries and frameworks to perform machine learning and deep learning tasks. These are provided in Table 4. MXnet is known to be one of the most scalable frameworks. Conclusion Mathematical programming constitutes a fundamental aspect of many machine learning models where the training of these models is a large scale optimization problem. This paper surveyed a wide range of machine learning models namely regression, classification, clustering, and deep learning as well as the new emerging paradigms of machine teaching and empirical model learning. The important mathematical optimization models for training these machine learning models are presented and discussed. Exploiting the large scale optimization formulations and devising model specific solution approaches is an important line of research particularly benefiting from the maturity of commercial optimization software. The nonlinearity of the models, the associated uncertainty of the data, as well as the scale of the problems however represent some of the very important and compelling challenges to the mathematical optimization community. Furthermore, bilevel formulations play a big role in adversarial learning tasks [105], including adversarial training, data poisoning and neural network robustness. While this survey does not discuss numerical optimization techniques since they were recently surveyed in [43,72,191], we note the fundamental role of the stochastic gradient algorithm [167] and the alternating direction method of multipliers [44] on large scale machine learning. We also highlight as well the potential impact of machine learning on advancing the solution approaches of mathematical programming [25]. Neighbors . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Linear Discriminant Analysis . . . . . . . . . . . . . . . . . . 12 3.4 Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 Support Vector Machines . . . . . . . . . . . . . . . . . . . . . 17 3.5.1 Hard Margin SVM . . . . . . . . . . . . . . . . . . . . 17 3.5.2 Soft-Margin SVM . . . . . . . . . . . . . . . . . . . . . 18 3.5.3 Sparse SVM . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5.4 The Dual Problem and Kernel Tricks . . . . . . . . . . 19 4 Clustering 20 4.1 Minimum Sum-Of-Squares Clustering (a.k.a. k-Means Clustering) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Capacitated Clustering . . . . . . . . . . . . . . . . . . . . . . 24 4.3 k-Hyperplane Clustering . . . . . . . . . . . . . . . . . . . . . 25 5 Deep Learning 25 5.1 Mixed Integer Programming for DNN Architectures . . . . . . 28 5.2 Adversarial Learning . . . . . . . . . . . . . . . . . . . . . . . 31 5.2.1 Targeted attacks . . . . . . . . . . . . . . . . . . . . . 32 5.2.2 Untargeted attacks . . . . . . . . . . . . . . . . . . . . 33 5.2.3 Models resistant to adversarial attacks . . . . . . . . . 35 5.3 Data Poisoning . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.4 Activation Ensembles . . . . . . . . . . . . . . . . . . . . . . . 39 6 Emerging Paradigms 40 6.1 Machine Teaching . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Empirical Model Learning . . . . . . . . . . . . . . . . . . . . 43 N unlabeled observations O = {O 1 , . . . , O N }, Cluster Analysis aims at finding M subsets {C 1 , . . . , C M }, called clusters, which are homogeneous and well separated. Homogeneity indicates the similarity of the observations within the same cluster, while the separability accounts for the differences between entities of different clusters. j, l, i = 1, . . . , N, j = 1, . . . , N, l = 1, . . . , k, M j=1 x j=1ij = 1 i = 1, . . . , N, j = 1, . . . ≤ y j i = 1, . . . , N, j = 1, . . . , M -(78) can model several tasks in DL. These include • Pooling operations: The average and the maximum operators • Maximizing the unit activation: By maximizing the objective function (72), one can find input examples x 0 that maximize the activation of the units. This may be of interest in applications such as the visualization of image features. • Building crafted adversarial examples: Given an input vector x 0 labeled as l by the DNN, the search for perturbations of x 0 that are classified as l = l (adversarial examples), can be conducted by adding conditions on the activation of the final layer K and minimizing the perturbation. -(83) is generalized to the norm-L p with p ∈ {0, 2, ∞} and an alternative formulation considers objective functions F satisfying f (x + r) = l if and only if F(x + r) ≤ 0 are introduced. The equivalent formulation is then min r∈R n 0 r p + γF(x + r, l) η j h j i (z) + δ j ). The magnitude of the weights α j is then limited in a projection subproblem, where for each neuron the network should choose an activation function and therefore all the weights should sum to 1. Ifα j are the weight values obtained by gradient descent while training, then the projected weights are found by ≥ 0, j = 1, . . . , m. ξ, θ) + λ θ 2 . 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[ "The Microquasar XTЕ J1807-294 -Mass Evaluation by Means of the Relativistic Precession Model", "The Microquasar XTЕ J1807-294 -Mass Evaluation by Means of the Relativistic Precession Model", "The Microquasar XTЕ J1807-294 -Mass Evaluation by Means of the Relativistic Precession Model", "The Microquasar XTЕ J1807-294 -Mass Evaluation by Means of the Relativistic Precession Model" ]	[ "Radostina P Tasheva \nDepartment of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000\n", "Ivan Zh Stefanov \nDepartment of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000\n", "Radostina P Tasheva \nDepartment of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000\n", "Ivan Zh Stefanov \nDepartment of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000\n" ]	[ "Department of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000", "Department of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000", "Department of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000", "Department of Applied Physics\nTechnical University of Sofia\n8 St. Kliment Ohridski BlvdBG-1000" ]	[]	The high frequency quasiperiodic oscillations (QPOs) in the X-ray spectra of the millisecond pulsar XTE J1807-294 are under consideration. By application of the relativistic precession (RP) model an attempt is made to approximate the observed frequencies as well as to assess the mass and the angular momentum of the object. The obtained mass is too high for a neutron star.	null	[ "https://arxiv.org/pdf/1811.05223v1.pdf" ]	119,083,853	1811.05223	8ad1432eaaf158e3d043ba409eb7746aabc962c4	The Microquasar XTЕ J1807-294 -Mass Evaluation by Means of the Relativistic Precession Model Radostina P Tasheva Department of Applied Physics Technical University of Sofia 8 St. Kliment Ohridski BlvdBG-1000 Ivan Zh Stefanov Department of Applied Physics Technical University of Sofia 8 St. Kliment Ohridski BlvdBG-1000 The Microquasar XTЕ J1807-294 -Mass Evaluation by Means of the Relativistic Precession Model The high frequency quasiperiodic oscillations (QPOs) in the X-ray spectra of the millisecond pulsar XTE J1807-294 are under consideration. By application of the relativistic precession (RP) model an attempt is made to approximate the observed frequencies as well as to assess the mass and the angular momentum of the object. The obtained mass is too high for a neutron star. INTRODUCTION In the eighties physical society was introduced to an interesting phenomena taking place in both neutron stars (NS) and black holes candidates (BH) existing in low mass binary systems. In their low frequent (LF, 1 -100 Hz) X-ray spectra appeared nearly periodical spikes called quasiperiodic oscillations (QPOs, see Ref. [1]). About 20 years later the data from the Rossi X-ray Timing Explorer (RXTE) showed QPOs in the kHz range (HF)from 0.2 to 1.25 kHz. Persistent repetition of both of these oscillations provide the scientists with the opportunity to study phenomena that happen in the innermost part of accretion disks surrounding the central objects where they are believed to originate i.e. in the conditions of extremely strong gravitational fields. The behavior of the two QPO groups has proved to be completely differentwhile the LF QPOs are strong, persistent and tend to drift in frequency HF QPOs are transient and weak but do not shift their frequencies significantly. Following these data a hypothesis can be suggested -they are probably created in different parts of the accretion disk. For some of the low mass X-ray binaries containing NS or BH the lower (L) and the upper (U) HF QPOs exist in pairs. As Kotrlova et al. [2] has demonstrated the simultaneous appearance of L and U HF QPOs allows the massangular momentum relation (a-M) to be determined. If for example the mass is obtained using photometric data then angular momentum can be evaluated. In some cases twin kHz QPO are observed in more than one pair. These results allow to construct different mass-angular momentum relations. The extent of their agreement can be used as a testing ground for the model applied to explain the objects` HF QPOs. The discovery of twin kHz QPOs in the X ray flux of accretion millisecond pulsar XTE J1807-294 is firstly reported in 2005 by Linares et al. [3]. They observed eight different pairs of simultaneous kHz QPOs with ratio νU/νL varying from 3:2 to approximately 3:1. In the present paper we aim by applying the relativistic precession (RP) model to this object and solving the correspondent analytic mass-angular momentum equation to estimate the optimal mass Mopt and optimal angular momentum aopt of the object according to the χ 2 test. The comparison between the so obtained values and the results received using other methods (photometrical, spectral etc.) can serve as an assessment of the validity of the model we apply. The paper is organized as follows. The observations of the twin kHz QPOs in XTE J1807-294 are briefly presented in Section 2. Section 3 deals with the essence of the RP model. Section 4 describes the application of the χ 2 test to the object. Section 5 discusses the agreement between the observational frequencies and results of the application of the RP model. The last section is the conclusion. In this paper the masses are given in solar masses, the radii x are scaled with the gravitational radius rg ≡GM/c 2 , the specific angular momentum is a ≡cJ/GM 2 and is accepted that G = 1 = c where G is the universal gravitational constant and c is the speed of light. OBSERVATIONAL DATA XTE J1807-294 is the fourth-discovered accreting millisecond pulsar ( after SAX J1808.4-3658 , XTE J1751-305, and XTE J0929-314). The first detection of the source has been on 13th of February, 2003, during the periodic scans of the Galactic bulge region by means of RXTE. Linares et al. [3] reported that using the Proportional Counter Array (PCA), an instrument on board of RXTE on 21 st of February 2003 190. 6 Hz pulsations has been discovered thus confirming that the source is an accreting millisecond pulsar (see Ref. [4]). The data this investigation consists of are taken from 27 th of February to 16 th of March 2003. In order to fit the power spectrum of each of the eight groups available a multi-Lorentzian function is implemented. Every multi-Lorenzian is assembled by adding several Lorentzians. Each one of the Lorentzians corresponds to a recognizable component in the power density spectrum (PDS). The eight datasets and the best-fit frequencies of the QPOs are listed in Table 1.The set A according to Linares et al. (2005) do not to show simultaneous kHz QPOs and has to be treated with caution. THE RELATIVISTIC PRECESSION MODEL Neutron stars as well as black holes are final stages of evolution of stars with masses bigger than solar and much denser. If such an object exists in a binary system then accretion disc forms around it. The matter rotating around the central object and falling toward it consists of charged particles attracted by the strong gravitational field. Let a test particle is assumed orbiting along circular geodesics in the innermost part of the accretion disc i.e. close to the innermost stable circular orbit (ISCO). The values of the high frequencies are close to those of the fundamental frequencies of motionorbital frequency νϕ, radial νr and vertical νθ of the test particle. This is the reason why the models constructed to explain the creation of HF QPO often exploit simple combinations of fundamental frequencies of particle motion. The relativistic precession model proposed by Stella and Vietri [5] is the first model that relates the frequency of the LF QPOs, namely νLF to the to frame dragging off equatorial orbits and Lense-Thirring precession. The frame dragging is one of the relativistic effects of the "gravitomagnetic" field associated with rotating bodies. It coerces a test particle moving in a non-equatorial plane to start precession around the angular momentum axis of the rotating object. The value for the low frequency is νLF=νLT = \|νϕ− νθ\|. The RP model accepts that all three QPOsthe low LF and the both highlower νL and upper νU are created at the same orbit. The higher frequency νU is a direct result of modulation of the X-ray flux by the orbital frequency and νU = νϕ The lower frequency νL is ascribed to the periastron precession of the relativistic orbit planes i.e νper = νϕ−νr. The correlation between νLF and νL that has been observed for a large sample of neutron stars and black hole candidates has found its satisfactory explanation through application of the RP model (see Ref. [6]). Χ 2 TEST APPLICATION The RP model is expected to explain successfully the observational frequencies of the QPOs. It also provides a mass-angular momentum relation. The information about the main parameters of the central objectits mass and angular momentum can be retrieved (without reference to external sources) by checking the datasets for consistency. The χ 2 test can be implemented in order to show if the chosen model is applicable for investigated object. As it concerns the reliability of the so obtained results comparison with photometric or spectral data will show where our results are in conflict with other investigations. The object selected by us -XTE J1807-294 shows commensurability between the lower νL and the higher νU frequency that varies from 3:1 to 3:2. Generally we can express one of the frequencies, for example νL as a function of value νU or vice versa in the following way: = ( ) = , = ( ) = .(1) Both frequencies depend on the free parameters -the mass M and the specific angular momentum a of the neutron star. We use the experimental values { , , , } of the frequencies, i=1, 2, 3…N in order to find the optimal values Mopt and aopt for the free parameters in the chosen model. If the χ 2 test is applied to either of the above equations the solutions for Mopt and aopt will allow us to draw the line of best fit for the both relations. This will be done by minimization of the functions: . (2) In the first of Eq. (2) νL is the dependent variable and νUthe independent one, in the next equationvice versa. If the expression contains N=8 pairs of frequencies and M=2 free parameters (a, M) i.e. N-M=6 degrees of freedom i.e. the acceptable values corresponding to 90% confidence level for χ 2 are 0 ≤ χ 2 ≤ 10.6 . For N=7 pairs i.e.5 degrees of freedom 0 ≤ χ 2 ≤ 9.24. According to the RP model the both high frequencies originate at the same radius xL=xU and the observational frequencies depend in the same way on x. Then Eq. (1) can be written in the following form: ( ) = ( ( )) , ( ) = ( ( )). A substantial problem for the application of the χ 2 causes the fact that only the standard error of the dependent variable is considered during the calculation while the standard error of the independent variable is neglected. RESULTS AND DISSCUSION The χ 2 test is applied firstly to eight groups of data according to Linares et.al. (2005) and afterward the test is repeated with seven groups of data. The results of the both cases define the lines of best fitfirstly for νL=f(νU) and then for νU=f(νL) in order to get more reliable and independent of the intrinsic uncertainties results. Dataset A is excluded in the second test because according to the same paper the twin high frequencies in set A might not arise simultaneously. The dependence of the lower frequency νL as a function of the upper frequency νU is given in the Figure 1 The received value for aopt in the cases where the all datasets are included and when the A data set is excluded are aopt=0.94±0.01 and aopt=0.97±0.01 respectively. The mass estimates corresponding to both cases are Mopt=(10.0±0.4) Mʘ and Mopt=(10.9±0.4) Mʘ respectively. The optimal mass Mopt and angular momentum aopt when eight groups of data are used are defined by χ 2 min=4.2.When the number of datasets is seven, significant reduction of the χ 2 is achieved -χ 2 min=1.5 i.e. dataset A could be anomalous one. Both results are more than agreeablethe reference values are χ 2 ≤ 10.6 and χ 2 ≤ 9.24 respectively. The dependence of the upper frequency νU as a function of the upper frequency νL is given in the Figure 2 The minimum value for χ 2 min=85.57 in the first case and is again significant reduced to χ 2 min=33.33 in the second case. The line that depicts the expression νU=f(νL) is determined with a much bigger χ 2 min than is calculated for the expression νL=f(νU). The discrepancy may result from the fact that when the calculations are made the standard error for the independent variable is neglected. In the second case νL happens to be independent variable. If we accept coarsely that if the correspondent frequencies are independent variables their absolute uncertainties are respectively ΔνL,ave= (ΔνL,max + ΔνL,min)/2=17.0 and ΔνU,ave== (ΔνU,max + ΔνU,min)/2=6.0 then ΔνL,ave / ΔνU,ave ≈ 3 which may cause a significant difference between χ 2 results in both cases. Our calculations suggest that the mass of XTE J1807-294 is too big for a neutron star. One possible explanation seems to be the fact that during the retrieval of the data through the χ 2 the uncertainties of the independent variable have to be neglected. Another source of error is non-simultaneous creation of some of the twin QPO. The applicability of the RP model to this object may also be carefully reconsidered. The orbital motion frequencies near the inner edge of an accretion disc are highly susceptible to even the smallest radial perturbations which can have a strong effect over radial oscillation. Therefore some non-geodesic amendments can be made in order to adjust the models to the observational data. (see Ref. [7]). The resonant switch model by Stuchlık et al. (see Ref. [8]) for example suggests that more than one model can be applied if the frequency ratio νU / νL changes. In our case νU / νL>2 for A and B datasets and νU / νL<2 for groups C-H i.e. the next step in further investigation could be to consider a suitable resonant switch model. CONCLUSION We applied the RP model to the low mass X-ray binary XTE J1807-294. Using the data of Linares et al. (2005) for the twin QPOs we constructed graphs that represent dependence between the lower νL and the upper νU HF QPOs. Firstly νU is the independent variable and νL is the dependent i.e νL=f(νU) and secondlyvice versa i.e. νU=f(νL). For both cases we implemented the χ 2 test in order to obtain the optimal mass Mopt and optimal angular momentum aopt -the ones for which χ 2 = χ 2 min. The results from the χ 2 are more than agreeable for the first caseχ 2 min=1.5 and worse for the second -χ 2 min=33.33 (reference value χ 2 ≤ 9.24). The obtained masses -Mopt=(10.9±0.4) Mʘ in the first case and Mopt=(11.2±0.1) Mʘ in second (for seven datasets) are though too big for neutron stars and the reasons could be multiple -neglected uncertainties of the independent variable, non-simultaneous HF QPOs, inappropriate choice of model. For further investigations an amendment that includes the neglected uncertainties can be used. The switch resonant model also seems to be a viable option to have in mind. TABLE 1 . 1Twin kHz QPO frequencies with their uncertainties Group νL(Hz) νU(Hz) FIGURE 1 .FIGURE 2 . 12. The received values for aopt in the cases where the all datasets are included and when the A data set is excluded are aopt=0.95±0.01 and aopt=0.98±0.01respectively. The mass estimates corresponding to both cases are Mopt=(10.4±0.1) Mʘ and Mopt=(11.2±0.1) Mʘ respectively. The dependence of lower νL versus upper νU frequency according to RP model. The representation is a dashed line, . The positions for the experimental frequencies coming from the different groups are given with their uncertainties. (a)from A to H; (b)from B to H. The dependence of upper νU versus lower νL frequency according to RP model. The representation is a dashed line . The positions for the experimental frequencies coming from the different groups are given with their uncertainties. (a)from A to H; (b)from B to H. ACKNOWLEDGMENTSThis research is partially supported by the Bulgarian National Science Fund under Grant No N 12/11 from 20 December 2017. Rapid X-ray variability. M Van Der Klis, X-ray binaries. W. H. G. Lewin, J. van Paradijs & E. P. J.Van den HeuvelCambridgeCambridge University PressM. van der Klis, "Rapid X-ray variability",in X-ray binaries, edited by W. H. G. Lewin, J. van Paradijs & E. P. J.Van den Heuvel (Cambridge University Press, Cambridge), pp. 1-28 (1995). . A Kotrlova, G Torok, E Sramkova&, Z Stuchlık, A&A. 572A. Kotrlova, G. Torok, E. Sramkova& Z. Stuchlık, A&A,572, pp. 6-40 (2014). . M Linares, M Van Der Klis, D Altamirano, C B Markwardt, arXivastro-ph/0509011M. Linares, M. van der Klis, D. Altamirano, C. B. Markwardt, in : arXivastro-ph/0509011, pp. 18-37 (2005) . C B Markwardt, E Smith, J H Swank, Circ, 8080C. B. Markwardt, E. Smith, J. H. Swank., IAU Circ., 8080, pp. 2-25 (2003). . L Stella, M Vietri, ApJ Lett. 49259L. Stella, M. Vietri, ApJ Lett., 492, L59 (1998). Fast variability from black hole binaries. M Van Der Klis, W. H. G. Lewin & M. van der KlisCambridge University PressCambridgeCompact Stellar X-Ray SourcesM. van der Klis, "Fast variability from black hole binaries" , in "Compact Stellar X-Ray Sources", edited by W. H. G. Lewin & M. van der Klis (Cambridge University Press, Cambridge), pp. 39-45 (2006). . Z Stuchlık, A Kotrlova, & G Torok, arXiv:1010.1951Astronomy and Astrophysics. Z. Stuchlık, A. Kotrlova & G. Torok, Astronomy and Astrophysics, in : arXiv:1010.1951, pp. 1-28 (2010). . Z Stuchlık, A Kotrlova, & G Torok, Acta Astron. 62Z. Stuchlık, A. Kotrlova & G. Torok, Acta Astron., 62, pp. 389-413 (2012).	[]
[ "Separation of Moving Sound Sources Using Multichannel NMF and Acoustic Tracking", "Separation of Moving Sound Sources Using Multichannel NMF and Acoustic Tracking" ]	[ "Joonas Nikunen ", "Aleksandr Diment ", "Senior Member, IEEETuomas Virtanen " ]	[]	[]	In this paper we propose a method for separation of moving sound sources. The method is based on first tracking the sources and then estimation of source spectrograms using multichannel non-negative matrix factorization (NMF) and extracting the sources from the mixture by single-channel Wiener filtering. We propose a novel multichannel NMF model with time-varying mixing of the sources denoted by spatial covariance matrices (SCM) and provide update equations for optimizing model parameters minimizing squared Frobenius norm. The SCMs of the model are obtained based on estimated directions of arrival of tracked sources at each time frame. The evaluation is based on established objective separation criteria and using real recordings of two and three simultaneous moving sound sources. The compared methods include conventional beamforming and ideal ratio mask separation. The proposed method is shown to exceed the separation quality of other evaluated blind approaches according to all measured quantities. Additionally, we evaluate the method's susceptibility towards tracking errors by comparing the separation quality achieved using annotated ground truth source trajectories.Index TermsSound source separation, moving sources, time-varying mixing model, microphone arrays, acoustic source tracking J. Nikunen, A. Diment and T. Virtanen are with the	10.1109/taslp.2017.2774925	[ "https://arxiv.org/pdf/1710.10005v1.pdf" ]	10,765,208	1710.10005	c1b6e02db5c0bb41b5fab3fd4a3860cae16bddb2	Separation of Moving Sound Sources Using Multichannel NMF and Acoustic Tracking Joonas Nikunen Aleksandr Diment Senior Member, IEEETuomas Virtanen Separation of Moving Sound Sources Using Multichannel NMF and Acoustic Tracking 1Index Terms Sound source separationmoving sourcestime-varying mixing modelmicrophone arraysacoustic source tracking In this paper we propose a method for separation of moving sound sources. The method is based on first tracking the sources and then estimation of source spectrograms using multichannel non-negative matrix factorization (NMF) and extracting the sources from the mixture by single-channel Wiener filtering. We propose a novel multichannel NMF model with time-varying mixing of the sources denoted by spatial covariance matrices (SCM) and provide update equations for optimizing model parameters minimizing squared Frobenius norm. The SCMs of the model are obtained based on estimated directions of arrival of tracked sources at each time frame. The evaluation is based on established objective separation criteria and using real recordings of two and three simultaneous moving sound sources. The compared methods include conventional beamforming and ideal ratio mask separation. The proposed method is shown to exceed the separation quality of other evaluated blind approaches according to all measured quantities. Additionally, we evaluate the method's susceptibility towards tracking errors by comparing the separation quality achieved using annotated ground truth source trajectories.Index TermsSound source separation, moving sources, time-varying mixing model, microphone arrays, acoustic source tracking J. Nikunen, A. Diment and T. Virtanen are with the I. INTRODUCTION Separation of sound sources with time-varying mixing properties, caused by the movement of the sources, is a relevant research problem for enabling intelligent audio applications in realistic operation conditions. These applications include, for example, speech enhancement and separation for automatic speech recognition [1] especially when using voice commanded smart devices from afar [2]. Another emerging application field includes immersive audio for augmented reality [3] which requires modification of the observed sound scene for example by removing sound sources and replacing them with augmented content. Separation of non-speech sources can be also used to improve sound event detection in multi-source noisy environment [4]. Most existing works related to sound separation are assuming stationary sources, and not many blind methods have targeted the problem of moving sound sources despite its high relevance in realistic conditions. The problem of sound source separation either from single or multi-channel recordings has been tackled with various methods over the years. The methods maximizing statistical independence of non-Gaussian sources, such as the independent component analysis (ICA) have been used for unmixing the sources in frequency domain [5], [6]. The concept of binary clustering of timefrequency blocks based on inter-channel cues, namely the level and the time difference, has resulted into class of separation methods based on time-frequency masking [7], [8]. Use of binary masks requires assuming that sound sources occupy disjoint time-frequency blocks [9]. More recently, single-channel speech enhancement and separation has been performed with the aid of machine learning and specifically by using deep neural networks (DNNs) [10], [11] for predicting the time-frequency masks for separation. Combining prediction of source spectrogram using DNNs and spatial information in the form of source covariance matrices for audio source separation has been proposed in [12], [13]. Another machine learning tool for masking based source separation is the spectrogram factorization by non-negative matrix factorization (NMF) and non-negative matrix deconvolution (NMD) which both have been widely utilized for speech separation and enhancement [14], [15]. The NMF model decomposes mixture magnitude spectrogram into spectral templates and their time-dependent activations. In the case of single channel mixtures the separation is achieved by learning of noise or speaker dependent spectral templates from isolated sources in a training stage. The NMF model can be extended for multichannel mixtures by incorporating spatial covariance matrix (SCM) estimation for the NMF components as in [16], [17], [18], [19], [20]. The analysis and introduction of spatial properties for NMF components allows separation based on spatial information, i.e., NMF components with similar spatial properties are considered to originate from the same sound source. These models require operation with complex-valued data and hereafter we refer to these extensions as multichannel NMF. Recording of a realistic auditory scene often consists of sound sources which are moving with respect to the recording device, and conventional separation approaches assuming time-invariant mixing are not suitable for such a task. However, moving sound sources can be considered stationary within a short time block where the mixing can be assumed to be timeinvariant. Using the block-wise approach for separation of moving sound sources requires merging the separated sources across individually processed blocks. For example, in block-wise ICA [21], [22], [23] this is done by propagating the mixing matrix from the previous block and thus slowly adapting the mixing and preserving the source ordering in consecutive blocks. A recent generalization of multichannel NMF model [16] for time-varying mixing and separation of moving sound sources was proposed in [24]. The reported results are promising, however the proposed algorithm requires using other state-of-the art source separation method in a blind setting for initialization. Alternatively, separation of moving sources can be achieved by tracking the spatial position or direction of arrival (DOA) of the sources and using spatial filtering (beamforming or separation mask) for extracting the signal originating from the estimated position or direction in each time instance. In [25] the problem of DOA tracking and separation mask estimation is formulated jointly, however in this paper we consider a two stage approach where the acoustic tracking is done first and the separation masks are estimated in a separate (offline) stage. Also the separation masks are binary in [25] which will lead to compromised subjective separation quality even if oracle masks are used. Acoustic localization with microphone arrays can be achieved by transforming the time-difference of arrival (TDOA) obtained using generalized cross-correlation (GCC) into source position estimates [26]. Methods for estimation of trajectories of moving sound sources are based on Kalman filtering and its non-linear extensions [27], [28] for estimating the underlying state (position of the sound source) from the TDOA measurements. Alternatively, sequential Monte Carlo methods, i.e., particle filtering [29], [30] have been applied for tracking the position of the source based on TDOA measurements. For even more difficult case of multiple target tracking with data association problem, a Rao-Blackwellised particle filtering (RBPF) was proposed in [31], [32] and applied for acoustic tracking of multiple speakers in [33]. Additionally, the use of directional statistics and quantities wrapped on a unit circle or a sphere, such as the interchannel phase difference, has been recently considered for speaker tracking [34], [35]. In this paper we propose a separation method for moving sound sources based on acoustic tracking and estimation of source spectrogram from the tracked directions using the multichannel NMF with time-varying SCM model. The main contributions of this paper include: formulation of multichannel NMF model for time-varying mixing (moving sound sources), integration of the spatial model with acoustic tracking to define spatial properties of sources in form of SCMs in each time frame and finally presenting the update equations for optimizing the multichannel NMF model parameters minimizing squared Frobenius norm. The parametrization of source DOA with directional statistics and using the tracker uncertainty for defining the SCMs of sources is a novel approach for representing the spatial location and spread of sound sources in the multichannel NMF. The acoustic tracker realization is combination of existing works on wrapped Gaussian mixture models [36] and particle filtering [37], but it will be presented in detail due to its output statistics are utilized in the proposed time-varying SCM model. The evaluation of the proposed separation algorithm is based on objective separation criteria [38], [39], [40] and testing with mixtures of two and three simultaneous moving speakers. Additionally, hand-annotated source DOA trajectories are used for evaluating the performance of the acoustic tracker realization and studying the susceptibility of the proposed separation algorithm towards tracking errors. The compared separation methods include conventional beamforming (DSB and MVDR) and upper reference is obtained by ideal ratio mask (IRM) separation [41]. The proposed method achieves superior separation performance and the use of annotated trajectories shows no significant increase in separation performance, proving the proposed method suitable for realistic operation in a blind setting. The paper is organized as follows. First the problem of separating moving sound sources and an overview of the proposed processing is given in Section II. Next we introduce directional statistics and describe the acoustic source tracker realization in Section III. In Section IV the multichannel NMF separation model for sources with time-varying mixing is proposed and the utilization of tracker output within the separation model is explained. In Section V the tracking and separation performance of the proposed algorithm is evaluated using real recorded test material captured using a compact four-element microphone array. The work is concluded in Section VII. II. PROBLEM STATEMENT AND ALGORITHM OVERVIEW A. Mixing model of moving sound sources A microphone array composed of microphones (m = 1, . . . , M ) observes a mixture of p = 1, . . . , P source signals s p (t) sampled at discrete time instances indexed by t. The sources are moving and have time-varying mixing properties, denoted by room impulse response (RIR) h pmt (τ ), for each time index t. The resulting mixture signal can be given as x m (t) = P p=1 τ s p (t − τ )h pmt (τ ).(1) In sound source separation the aim is to estimate the source signals s p and their mixing h pmt (τ ) by only observing x m (t). In this paper audio is processed in frequency domain obtained using the short time Fourier transform (STFT). The STFT of a time-domain mixture signal is calculated by dividing the signal into short overlapping frames, applying a window function and taking the discrete Fourier transform (DFT) of the windowed frame. The mixing properties denoted by the time-dependent RIRs h pmt (τ ) change slowly over time and in practice the difference between adjacent time indices t is small, thus we can consider mixing being constant within a small time window. This allows to approximate the time-dependent mixing (1) in time-frequency (TF) domain as x f n ≈ P p=1 h f n,p s f n,p = P p=1 y f n,p .(2) The STFT of the mixture signal is denoted by x f n = [x f n1 , . . . , x f nM ] T for each TF-point (f, n) of B. Overview of the processing The proposed method consists of source spectrogram estimation based on the DOA of the sources of interest in each time frame and the estimated spectrograms are used for separation mask generation by generalized Wiener filter. The processing is based on two stages: the acoustic tracker and the offline separation mask estimation by multichannel NMF. The block diagram of the method is illustrated in Figure 1. The source tracking branch operates frame-by-frame and can be though as online algorithm while the parameters of the multichannel NMF model are estimated from the entire signal at once (offline). First the STFT of the input signals is calculated. The tracking branch starts with calculating the steered response power (SRP) of the signal under analysis. SRP denotes the spatial energy as a function of DOA for each time frame. A wrapped Gaussian mixture model (WGMM) [36] of the SRP function in each time frame is estimated, which converts spatial energy histogram (i.e. the SRP) into DOA measurements. WGMM parameters are used as measurements in acoustic tracking which is implemented using particle filtering [37]. The multi-target tracker detects the births and deaths of sources, solves the dataassociations of measurement belonging to one of existing sources and predicts the source trajectories. In the second stage a spatial covariance matrix model (SCM model) [19] parameterized by DOA is defined based on the acoustic tracker output (source DOA at each time-frame). The obtained SCMs denote the spatial behavior of sources over time and a spectral model of sources originating from the tracked direction is estimated using multichannel NMF. The multichannel source signals are reconstructed using a single-channel Wiener filter based on the estimated spectrogram of each source and single-channel signals are obtained by applying the delay-and-sum beamforming to the separated multichannel signals. Finally, time-domain signals are reconstructed by applying inverse STFT and overlap-add. III. SOURCE TRAJECTORY ESTIMATION The goal of the first part of the proposed algorithm is to estimate DOA trajectories of the sound sources that are to be separated. The process consist of three consecutive steps: calculating the spatial energy emitted from all directions (Section III-A), converting the discrete spatial distribution into DOA measurements (Sections III-B and III-C) and multi-target tracking consisting of source detection, data-association and source trajectory estimation (Section III-D). A. Time-difference of arrival and steered response power Spatial signal processing with spaced microphone arrays is based on observing time delays between the array elements. In far-field propagation the wavefront direction of arrival corresponds to a set of TDOA values between each microphone pair. We start by defining a unit direction vector k ∈ R 3 , \|\|k\|\| = 1 originating from the geometric center of the array p = [0, 0, 0] T and pointing towards direction parametrized by azimuth θ ∈ [0, 2π] and elevation ϕ ∈ [0, π]. Given a microphone array consisting of two microphones m 1 and m 2 at locations m 1 ∈ R 3 , m 2 ∈ R 3 the TDOA between them for a sound source at direction k is obtained as τ (m 1 , m 2 ) = −k T (m 1 − m 2 ) v ,(3) where v is the speed of sound. The above TDOA corresponds to a phase difference of exp(−jω f τ (m 1 , m 2 )) in the frequency domain, where ω f = 2π(f − 1)F s /N (F s is the sampling frequency and N is the STFT window length). From now on we operate with a set of different directions indexed by d = 1, . . . , D and the direction vector corresponding to dth direction is defined as k d resulting to TDOA of τ d (m 1 , m 2 ). The spatial energy originating from the direction [θ d , ϕ d ] at each time frame n can be calculated using the steered response power (SRP) with PHAT weighting [42] defined as S dn = M −1 m1=1 M m2=m1+1 F f =1 x f nm1 x * f nm2 \|x f nm1 x * f nm2 \| exp(jω f τ d (m 1 , m 2 )),(4) where * denotes complex-conjugate and the term exp(jω f τ d (m 1 , m 2 )) is responsible for time-aligning the microphone signals. SRP denotes the spatial distribution of the mixture consisting of spatial evidence from multiple sources and searching for multiple local maxima of the SRP function at a single time frame n corresponds to DOA estimation of sources present in that time frame. Repeating the pick peaking for all time frames of SRP would result to DOA measurements that are permuted over time and subsequently in the tracking stage the permuted DOA measurements are associated to multiple sources over time. In a general case the directions d = 1, . . . , D would uniformly sample a unit sphere around the array, but in this paper we only consider the zero elevation plane, i.e., ϕ d = 0 ∀d. We assume that the sources of interest lie approximately on the xy-plane with respect to the microphone array and the directional statistics used in tracking of the sources simplifies to a univariate case. A sparse grid of directions vectors with spacing of adjacent azimuths by π 12 is illustrated in Figure 2 along with the array casing and microphones corresponding to the actual compact array used in the evaluations. B. Wrapped Gaussian mixture model Instead of searching peaks from the SRP (4), we propose to model the mixture spatial distribution using a wrapped Gaussian mixture model (WGMM) estimated separately for each time-frame of the SRP. The estimation of the parameters of the WGMM results to converting the discrete spatial distribution obtained by SRP into multiple DOA measurements with mean, variance and weight. The individual wrapped Gaussians model the spatial evidence caused by the actual sources while some of them may model the noise or phantom peaks in the SRP caused by sound reflecting from boundaries. The use of WGMM alleviates effect of noise when the width of each peak, denoted by variance of each wrapped Gaussian, can be used to denote measurement uncertainty in the acoustic tracking stage. The probability density function (PDF) of univariate wrapped Gaussian distribution [36], [43], [44] with mean µ and variance σ 2 can be defined as P (θ; µ, σ 2 ) = ∞ l=−∞ N (θ; µ + l2π, σ 2 ) = ∞ l=−∞ 1 √ 2πσ 2 e − (θ−µ+2πl) 2 2σ 2 ,(5) where N (θ; µ, σ 2 ) is a PDF of a regular Gaussian distribution, l is the wrapping index of 2π multiples and θ ∈ [−π, π]. The multivariate version of the wrapped Gaussian distribution is given in [36], but it is not of interest in this paper. The WGMM with weights a k for each wrapped Gaussian distribution k is defined as Input: Histogram data s d and initial values for a k , µ k and σ 2 P (θ; a, µ, σ 2 ) = K k=1 a k ∞ l=−∞ N (θ; µ k + l2π, σ 2 k ).(6)k E-STEP η dkl = N (θ d ;µ k +l2π,σ 2 k )a k K k=1 ∞ l=−∞ N (θ d ;µ k +l2π,σ 2 k )a k M-STEP µ k = D d=1 ∞ l=−∞ (θ d −2πl)η dkl s d D d=1 ∞ l=−∞ η dkl s d σ 2 k = D d=1 ∞ l=−∞ (θ d −µ k −2πl)η dkl s d D d=1 ∞ l=−∞ η dkl s d a k = 1 d s d D d=1 ∞ l=−∞ η dkl s d where K is the total number of wrapped Gaussians in the model and EM algorithm for estimating parameters {a, µ, σ 2 } that maximize the log likelihood log L = D d=1 log K k=1 a k ∞ l=−∞ N (θ d ; µ k + l2π, σ 2 k ),(7) is given in [36], [44]. The parameter θ d denotes the azimuth angles of the directions indices d = 1, . . . , D used to calculate SRP in Equation (4). The EM-algorithm for WGMM as presented in [36], [44] requires observing data points generated by the underlying distribution whereas S dn for a single frame n is effectively a histogram denoting spatial energy emitted from each scanned direction indexed by d. Estimating the WGMM parameters based on the histogram requires modification of the algorithm presented in [36], [44] to account for inputs consisting of discretely sampled mixture distribution, i.e., the histogram bin values. The modification results to Algorithm 1 where SRP of a single frame n is denoted by s d and the updates are iterated until converge of η dkl . Prior knowledge can be used to set initial values for a k , µ k and σ 2 k or they can be initialized randomly. An example of the SRP S dn of a single time-frame and three component WGMM estimated from it is illustrated in Figure 3. C. DOA measurements by WGMM Algorithm 1 is applied individually for each time frame n = 1, . . . , N of S dn . A mixture of k = 1, . . . , K wrapped Gaussians for each time frame is obtained and the resulting means µ n,k with variances σ n,k and weights a n,k are considered as permuted DOA measurements. At this point of the algorithm it is unknown which of the measurements k = 1, . . . , K in each frame n are caused by actual sources and which correspond to noise. Also the detection of sources and association of different measurements k to sources p over time is unknown, i.e. kth measurement in adjacent frames may be caused by different sources. The source detection, data-association and actual source trajectory estimation is solved using the Rao-Blackwellized particle filtering introduced in Section III-D. Random initialization of Algorithm 1 could be used in each time frame. However, initial values close to the optimal ones speed up the algorithm convergence and in practice estimates from previous frame can be used as initialization for the subsequent frame. Please note that this initialization strategy does not guarantee preserving any association between kth wrapped Gaussians in adjacent frames and ordering need to be considered as permuted. The WGMM parameters {µ n,k , σ n,k , a n,k } are hereafter referred to as DOA measurements regarding the acoustic tracking: µ n,k are the measurement means, σ n,k denote measurement reliability and a n,k are the proportional weights of the measurements. Given that a WGMM with K components for each time frame is estimated, not all WGMM components are caused by actual spatial evidence but are merely modeling the noise floor and phantom peaks in the SRP. The situation is illustrated in Figure 3, where the third WGMM component with mean µ = 71 • has a very high variance σ 2 = 117 • in comparison to the actual observable peaks. By investigating the variance and weight of each WGMM component the false measurements can be efficiently removed before applying the actual tracking algorithm. The means µ n,k for each time frame for an arbitrary test signal and after removing measurements with σ n,k > 35 • or a n,k < 0.15 are illustrated in Figure 4. The removal of false measurements reveals two distinct observable trajectories, however the data association between each frame is unknown at this stage. The thresholds for measurement removal can be set to be global (signal and capturing environment independent) and their choice is discussed in more details in Section V. D. Acoustic tracking of multiple sound sources The problem setting in tracking of multiple sound sources is as follows. Multiple DOA measurements are obtained in each time frame and the task is to decide whether the new measurement is 1) associated to an existing source 2) identified as clutter, 3) evidence of new source (birth) and finally 4) determining possible deaths of existing sources. After the data-association step the dynamic state of the active sources is updated and particularly it is required to preserve the source statistics over short inactive segments (pauses between words in speech) by prediction based on previous state of the source (location, velocity and acceleration). In the following, we shortly review the Rao-Blackwellized particle filter (RBPF) framework proposed in [31] for the problem of multi-target tracking and use its freely available implementation 1 documented in [37]. We give the state-space representation of the dynamical system that is being tracked but we do not go to any further details of RBPF. The algorithm proposed in [31] and the associated implementation has been used in [45] for tracking the azimuth angle of speakers in a similar setting. Multi-target tracking by RBPF is essentially based on dividing the entire problem into two parts, estimation of data association and tracking of single targets. This can be done with the Rao-Blackwellization procedure [32] where the estimation of the posterior distribution of the data associations is done first and then applying single target tracking sub-problem conditioned on the data associations. Adding the estimation of unknown number of targets [31] using a probabilistic model results into RBPF framework that solves the entire problem of tracking unknown number of targets. The benefit of the Rao-Blackwellization is that by conditioning the data association allows calculating the filtering equations in closed form instead of using particle filtering and data sampling based techniques for all steps, which leads to generally better results. 1) State-space model for speaker DOA tracking: In the RBPF framework the single target tracking consist of Bayesian filtering which requires defining the dynamic model and the measurement model of the problem. For the time being we omit the WGMM component index k and the source index p and define the state space model equations for the single target tracking sub-problem. The goal is to estimate the state of the dynamical system in each time instance n and the state in our case is defined as a 2-D point (x, y) at the unit circle with velocities along both axes (ẋ,ẏ) defined as s n = x n , y n ,ẋ n ,ẏ n , T . The angle of the x-y coordinate (x n , y n ) represents the DOA of the source and it avoids dealing with the 2π ambiguity of 1-D DOA variables in the dynamic model. The dynamic model that predicts the target state based on previous time step is defined as s n =A n−1 s n−1 + q n−1(9) where A n−1 is the state transition matrix and q n−1 ∼ N (0, λ 2 I) is the process noise. With the above definition for state s n the transition matrix becomes linear and is defined as, A n−1 =     1 0 ∆t 0 0 1 0 ∆t 0 0 1 0 0 0 0 1     ,(10) where ∆t is the time difference between consecutive time steps. The resulting dynamic model can be described as follows: the predicted DOA at current time step n is the DOA of the previous time step in x-y coordinates added with its velocity in previous time step multiplied with the time constant, i.e., the time between consecutive processing frames. For the measurement representation we use the rotating vector model [46] that converts the wrapped 1-D angle measurements µ ∈ [0, 2π] to a 2-D point on a unit circle, resulting in measurement vector The measurement model is defined as, m n = cos(µ), sin(µ) T .(11)m n =B n s n + r n ,(12) where B n is the measurement model matrix and r n ∼ N (0, σI) is the measurement noise. The measurement model matrix B n converts the state s n into measurement m n (x-y coordinates) simply by omitting the velocities and is defined as, B n = 1 0 0 0 0 1 0 0 .(13) The above definitions result to linear dynamic and and measurement model matrices (10) and (13) allowing use of regular Kalman filter equations to update and predict the state of the particles in RBPF framework [37]. We acknowledge that the dynamic system used here is theoretically imperfect with respect to using 2-D quantities while the state and the measurements are truly 1-D, leading to additional noise in the system as pointed out in [34]. However, during the implementation of the acoustic tracker the chosen linear models were found performing better than the non-linear alternatives. Alternatively, the problem of tracking wrapped quantities using 1-D state could be addressed via wrapped Kalman filtering as proposed in [34]. 2) Multi-target DOA tracking implementation: For the actual RBPF implementation we now reintroduce the WGMM component index k and the source index p. The state vector is defined individually for each detected source p = 1, . . . , P and is denoted hereafter as s The RBPF implementation [37] is applied to the measurements m (k) n with measurement noise r (k) n ∼ N (0, σ n,k I). The multi-target tracker detects the sources and makes the association of kth WGMM measurement belonging to one of sources p = 1, . . . , P . Alternatively, if none of the active source particle distributions indicate a probability higher than the clutter prior probability, then the current measurement is regarded as clutter. The clutter prior probability is a fixed pre-set value to validate the minimum threshold when the observed measurement is linked to existing source. The output of the tracker is the state of each source p at each time frame, denoted by s The tracking result for a one test signal is illustrated in Figure 5, where the input of the acoustic tracking are the ones depicted in the bottom panel of Figure 4. The test signal is chosen such that it shows two problematic cases, sources start from the same position and intersect at 8 seconds going to the opposite directions. The tracking result indicates that the second source is detected at 2 seconds from the start just when the sources have traveled far enough from each other, resulting to approximately one first word being missed from the second speaker. The tracker is able to maintain the source association and track correctly the trajectories of intersecting sources. IV. SEPARATION MODEL A. Mixture in spatial covariance domain For the separation part we represent the microphone array signal using mixture SCMs X f n ∈ C M ×M . We use magnitude square rooted version of the mixture STFT obtained as, x f n = [\|x f n1 \| 1/2 sign(x f n1 ), . . . , \|x f nM \| 1/2 sign(x f nM )] T(15) where sign(z) = z/\|z\| is the signum function for complex numbers. The mixture SCM is calculated as X f n =x f nx H f n for each TF-point (f, n). The diagonals of each X f n contains the magnitude spectrogram of each input channel. The argument and absolute value of [X f n ] m1,m2 (off-diagonal values) represents the phase difference and magnitude correlation, respectively, between microphones (m 1 , m 2 ) for a TF-point (f, n). The TF domain mixing in Equation (2) can be approximated using mixture SCMs as X f n ≈X f n = P p=1 H f n,pŝf n,p ,(16) whereŝ f n,p = s f n,p s * f n,p is positive real-valued magnitude spectrogram of source p and H f n,p = h f n,p h H f n,p /\|\|h f n,p h H f n,p \|\| F are the SCMs of the frequency domain RIRs h f n,p . The mixing Equation (16) is hereafter referred to as spatial covariance domain mixing. B. Multichannel NMF model with time-variant mixing The proposed algorithm uses multichannel NMF for source spectrogram estimation and it is based on alternating estimation of the source magnitude spectrogramŝ f n,p and its associated spatial properties in the form of the SCMs H f n,p . In all previous works [16], [18], [20] the problem definition has been simplified for stationary sound sources and the SCMs being fixed for all STFT frames n within the analyzed audio segment. Here we present a novel extension of the multichannel NMF model for time-variant mixing. In multichannel NMF the model for magnitude spectrogram is equivalent to conventional NMF, which is composed of fixed spectral basis and their time-dependent activations. The SCMs can be unconstrained [47] or as proposed in the earlier works of the authors [19], [20] based on a model that represents SCMs as a weighted sum of entities called "DOA kernels" each containing a phase difference caused by a single direction vector. This ensures SCMs to comply with the array geometry and match with the time-delays the chosen microphone placement allows. The NMF magnitude model for source magnitude spectrogram is given aŝ s f n,p ≈ Q q=1 b q,p t f q v qn , b q,p , t f q , v qn ≥ 0.(17) Parameters t f q over all frequency indices f = 1, . . . , F represent the magnitude spectrum of single NMF component q, and v qn denotes the component gain in each frame n. One NMF component represents a single spectrally repetitive event estimated from the mixture and one source is modeled as a sum of multiple components. Parameter b q,p ∈ [0, 1] represents a weight associating NMF component q to source p. The soft valued b q,p is motivated by different types of sound sources requiring different amount of spectral templates for accurate modeling and learning of the optimal division of components is determined through parameter updates. For example, stationary noise can be represented using only a few NMF components whereas spectrum of speech varies over time and requires many spectral templates to be modeled precisely. Similar strategy for b q,p is used for example in [18]. Typically the final values of b q,p are mostly binary and only few number of NMF components are shared among sources. The multichannel NMF model with time-variant mixing can be trivially derived from the SCM mixing Equation (16) by substituting the above defined NMF model (17) in it, resulting in X f n ≈X f n = P p=1 H f n,p Q q=1 b q,p t f q v qn ≈ŝ f n,p .(18) C. Direction of arrival -based SCM model In order to constrain the spatial behavior of sources over time using information from the acoustic tracking approach or other prior information, the SCMs need to be interpreted by spatial location of each source in each time frame. The SCM model proposed in [19] parametrizes stationary SCMs based on DOA and here we extend the model for time-variant SCMs H f n,p . Converting the TDOA in Equation (3) to a phase difference results in DOA kernels W f d ∈ C M ×M for a microphone pair (m 1 , m 2 ) defined as [W f d ] m1,m2 = exp jω f τ d (m 1 , m 2 ) ,(19) where τ d (m 1 , m 2 ) denotes the time delay caused by a source at a direction k d . A linear combination of DOA kernal gives the model for time-varying source SCMs defined as H f n,p = D d=1 W f d z nd,p .(20) The directions weights z nd,p denote the spatial location and spread of the source p at each time frame n. The direction weights z nd,p can be interpreted as probabilities of source p originating from each direction d. In anechoic conditions only one of the directions weights z nd,p in each time frame would be nonzero, i.e., the direct path explains all spatial information of the source, however in reverberant conditions several of the direction weights are active. The spatial weights corresponding to tracked sources and their DOA trajectories are set using the wrapped Gaussian distribution as z nd,p = N w (θ d ;μ n,p ,σ 2 n,p ), whereμ n,p is obtained as specified in Equation (14) and the varianceσ 2 n,p is used to control the width of the spatial window of source p in the separation model. The goal is to have a large spatial window when the tracker is certain and the output suppressed (small spatial window) when no new measurements from the predicted source position are observed and the tracker state indicates high uncertainty. This strategy is motivated by ensuring that small untracked deviations in source movement do not cause the spatial focus of the separation to veer off momentarily from the target source and lead to suppression of the desired source content. In experimental tests the source spectrogram estimation by multichannel NMF proved to be sensitive to small errors if the spatial window used was very small. The small spatial window in case of no source activity can be motivated from similar perspective, the estimated source spectrogram from very constrained spatial window is less likely to capture spectrogram details of other sources close to the target trajectory. The acoustic tracker output variance denoted as σ 2 n,p at each time step n indicates the uncertainty of the source being present at its respective predicted directionμ n,p . The above specified strategy can be obtained from the tracker output variance by specifyingσ 2 n,p = c − σ 2 n,p with a constant c = max n,p (σ 2 n,p ) + min n,p (σ 2 n,p ). The operation maps the maximum output variance to the smallest spatial window and vice versa. In practice value range of σ 2 n,p is restricted to avoid specifying extremely wide and narrow spatial windows and thus the value of constant c is set in advance. The limits for σ 2 n,p are discussed in more details in Section V-C. The direction weights for each source at each time frame are scaled to unity l 1 -norm ( D d=1 z nd,p = 1). This is done to restrict H f n,p to only model spatial behavior of sources and not affect modeling the overall energy of the sources. Additionally, when the source is considered inactive, i.e., before its birth or after its death, all the direction weights in the corresponding time frame are set to zero. The spatial weights z nd,p corresponding to the tracking result in Figure 5 are illustrated in two top panels of Figure 6. By comparing to Figure 5, it can be seen that when no new measurements are observed the tracker output state variance is high and the spatial weights are concentrated tightly around the mean, whereas in case of high certainty the spatial spread is wider. In order to model the background noise and diffuse sources, an additional background source is added with direction weights set to one in indices where p z nd,p < T and zero otherwise. The threshold T is set to allow the detected and tracked sources to capture all spatial evidence within approximately +-30 degrees from their estimated DOAs when the certainty for the given source is high. With the chosen background modeling strategy the tracked sources have exclusive prior to model signals originating from the tracked DOA, with the exception of two DOA trajectories intersecting. An example of the background spatial weights is illustrated in bottom panel of Figure 6. Note that the differently colored regions at different times are due to the scaling ( D d=1 z nd,p = 1) and different spatial window widths of the tracked sources at the corresponding time indices. D. Parameter Estimation The multichannel NMF model (18) with the time-varying DOA kernel based SCM model (20) and the spatial weights as specified in Equation (21) result to model X f n ≈X f n = P p=1 D d=1 W f d z nd,p H f n,p Q q=1 b q,p t f q v qn ≈ŝ f n,p .(22) In order to use the above model for separation, parameters b q,p , t f q and v qn defining the magnitude spectrogram modeling part need to be estimated with respect to appropriate optimization criterion. We use the squared Frobenius norm as the cost function, defined as F f =1 N n=1 \|\|X f n −X f n \|\| 2 F . Multiplicative updates for estimating the optimal parameters in an iterative manner can be obtained by partial derivation of the cost function and use of auxiliary variables as in expectation maximization algorithm [48]. The procedure for obtaining multiplicative updates for different multichannel NMF models and optimization criteria are proposed and presented in [18] and can be extended for the new proposed formulation in (22). The entire probabilistic formulation is not repeated here and can be reviewed from [18]. The update equations for the non-negative parameters are b q,p ← b q,p f,n t f q v qn tr(X f n H f n,p ) f,n t f q v qn tr(X f n H f n,p ) ,(23)t f q ← t f q n,p b q,p v qn tr(X f n H f n,p ) n,p b q,p v qn tr(X f n H f n,p ) ,(24)v qn ← v qn f,p b q,p t f q tr(X f n H f n,p ) f,p b q,p t f q tr(X f n H f n,p ) .(25) Note that in contrast to earlier works on multichannel NMF for separation of stationary sound sources [18], [19], we do not update the SCM part H f n,p . It is assumed that the acoustic source tracking and spatial weights of the DOA kernels W f d fully represent the spatial behavior of the source. This strategy is assessed in more details in discussion Section VI. E. Source separation For extracting the source signal from the mixture we use combination of single-channel Wiener filter and delay-and-sum beamforming. Separation soft mask m f n,p for extracting the source spectrogram from the mixture are obtained using the estimated real valued magnitude spectrogramŝ f n,p to formulate a generalized Wiener filter defined as y f n,p = m f n,p x f n =ŝ f n,p pŝ f n,p x f n .(26) We employ delay-and-sum beamforming to produce single channel source signal from the separated multichannel signals y f n,p (having the mixture signal phase). The final estimate of the sources are given as y f n,p = w H f n,p y f n,p ,(27) where w f n,p are the DSB weights (steering vector) towards estimated direction of source p at time frame n. Finally, the time-domain signal are reconstructed by applying inverse DFT to each frame which are further combined using overlap-add processing. V. EVALUATION In this section we present the objective separation performance and tracking performance of the proposed method using real recordings of moving sound sources. Additionally, we evaluate the separation performance of the proposed algorithm in a setting where the sources are not moving allowing comparison to conventional spatial and spectrogram factorization models assuming stationary sources. A. Datasets with moving sources The development and evaluation material was recorded using a compact microphone array consisting of four Sennheiser MKE2 omnidirectional condenser microphones placed on a diamond pattern illustrated in 2 and exact locations of microphones are documented in [19]. The recordings were conducted in an acoustically treated room with dimensions 4.53 m × 3.96 m × 2.59 m and reverberation time T 60 = 0.26 s. The microphone array was placed approximately at the center of the room. Four persons spoke phonetically balanced sentences while walking clockwise (CW) and counterclockwise (CCW) around the microphone array at approximately constant velocity and at average distance of one meter from the array center. The overall assembly of the recordings and the movement of the source is illustrated in Figure 7. Two CW and two CCW 30-second recordings with four persons were done totaling to 16 signals. All speakers started from the same position and walked on average two times around the array within the recorded 30-second segment. Recordings were done individually allowing producing mixture signals by combining the recordings from different persons. Reference speech signal was captured by a close-field microphone (AKG C520). Additionally, 16 recordings with a loudspeaker playing babble noise and music outside the recording room with the door open were done and considered as a stationary (S) sound source with highly reflected propagation path. The movement of each individual speaker was annotated by hand based on SRP. An example of annotations is illustrated in Figure 5. Note that the annotations are only plotted when the source is active (a simple 8 / 8 45 • / 135 • 90 • / 180 • = 45 • 8 / 8 0 • / 45 • 90 • / 135 • = 90 • 8 / 8 0 • / 45 • 135 • / 180 • = 135 • energy threshold from the close-field microphone signal). The VAD information is only used for evaluation purposes and not by the proposed algorithm. Three different datasets were generated, one for development and two for evaluation purposes by mixing two and three individual speaker recordings. All mixture utterances in all datasets were 10 seconds in duration. In all datasets the signals were manually cut in such way that speaker trajectories based on the annotations were no closer than 45 • when going in the same direction (CW vs. CW and CCW vs. CCW). Naturally, the trajectories can intersect in the case of opposite directions (CW vs. CCW). For the development set the first 15 seconds from each recording were used, while the remaining 15 to 30 seconds were used to generate the evaluation sets. In the development set 8 mixtures of two speakers and 16 live recordings were generated and each recording was only used once. The first evaluation dataset consists of 48 mixtures of two speakers using all possible unique combinations of the recordings with different speakers. The second evaluation dataset contains 16 mixtures of three speakers based on a subset of all possible unique combinations. The subset was chosen to represent all different source trajectory combinations (all sources moving in the same direction vs one of the sources moving in the opposite direction). The datasets are summarized in Table I. The global parameters related to tracking and separation performance of the proposed algorithm were optimized using the development dataset. The recorded signals were downsampled and processed with sampling rate of F s = 24000 Hz. B. Dataset with stationary sources In order to compare the performance of the proposed algorithm against conventional methods assuming stationary sources [18], [19], [20], we include additional evaluation dataset with completely stationary sources. We use the dataset introduced in [19] consisting of two simultaneous sound sources. In short the dataset contains speech, music and noise sources convolved with RIRs from various angles captured in a regular room (7.95 m x 4.90 m x 3.25 m ) with a reverberation time T 60 = 350 ms. The array used for recording is exactly the same as the one used in the datasets introduced in Section V-A and more details of the recordings can be found from [19]. In total the dataset contains 48 samples with 8 different source types and 6 different DOA combinations. Each sample is 10 seconds in duration. The different conditions are summarized in last tabular of Table I. C. Experimental setup For the WGMM parameter estimation the peaks in the SRP function were enhanced by exponentiation S (3/2) dn , which emphasizes high energy peaks (direct path) and low energy reflected content is decreased. This was found to improve operation in moderate reverberation. A five-component (K = 5) WGMM model (6) was estimated from the SRP and parameters from a previous frame were used as an initialization for next frame. The criteria for removing WGMM measurements were set to values of σ n,k > 0.6 rad (34 • ) and a n,k < 0.15 by visually inspecting the development set results. The acoustic tracker parameters were optimized by maximizing the development set tracking performance. The parameters were set to the following values. Average variance of the WGMM measurements was scaled to σ 2 = 0.25 for each processed signal to be in appropriate range for the particle filtering toolbox 2 . The clutter prior probability was fixed for all measurements to CP = 0.1. In the particle filtering framework the life time of the target is modeled using a gamma distribution with parameters α and β. The best tracking performance was achieved with α = 3 and β = 4. The target initial state was fixed to s (p) n = [cos(π), sin(π), 0.1, 0.1]. The pre-set prior probability of source birth was set to BP = 0.005. The parameters of the multichannel NMF algorithm were set as follows: the window length was 2048 samples with 50% overlap and 80 NMF components were used for modeling the magnitude spectrogram. The entire signals were processed as whole. Before restoring the spatial weights by Equation (21) a minimum and maximum variance for σ n,p were set to 0.025 and 0.3, respectively. This was done in order to avoid unnecessarily wide or narrow spatial window (as can be seen from Figure 6). With the chosen minimum and maximum values forσ 2 n,p , the constant in Equation (21) becomes c = max(σ 2 n,p ) + min(σ 2 n,p ) = 0.325. The background source threshold for setting the spatial weights active was set to 0.01, which corresponds to approximately ±30 • exclusive spatial window for the actual tracked sources when the tracker output state variance is at its minimum indicating high certainty of source being present at the predicted direction. D. Acoustic tracking performance The acoustic tracking performance is evaluated against the hand-annotated ground truth source trajectories by using the accuracy (mean absolute error) and recall rate as the metrics. The tracking error for each source in each time frame with 2π ambiguity is specified as e n,p =μ (ann.) n,p −μ n,p =ẽ n,p + 2πN, N ∈ Z whereμ (ann.) n,p denotes the annotated DOA of pth source in time frame n andμ n,p is obtained using Equation (14). Using the error termẽ which is wrapped to [−π, π], we specify mean-absolute error (MAE) as MAE =P p=1 1 N N n=1 \|ẽ n,p \|,(29) whereP is the number of annotated sources. The recall rate is defined as the proportion of time instances the detected source is correctly active with respect to when the source was truly active and emitting sound. The ground truth of the active time instances is obtained by voice activity detection (VAD) using the close-field signal of the source. The VAD is used in order to take into account that some utterances start 1 to 2 seconds after the beginning of the signal even though the annotations are continuous for the whole duration of recordings. Additionally, if the tracked source dies before the end of the signal during a pause of speech, the duration of the pause is not accounted for as a recall error, but the remaining missing part is. We will denote the recall rate using variable recall ∈ [0, 1]. The proposed method uses multi-target tracker that can detect arbitrary number of sources and trajectories denoted as P . For evaluation of tracking and separation performance we need to match the annotated sources 1, ...,P and detected sources 1, ..., P by searching trough all possible permutations r of the detected sources denoted as P r : {1, ..., P } → {1, ...,P }. The permutation matrix P r is applied to change the ordering in which detected sources are evaluated against the annotations. We propose to choose the permutation r for final scoring that maximizes combination of MAE and recall. First MAE is converted into a proportional measure MAER = 1 − (MAE/π) ∈ [0, 1], where 1 denotes zero absolute error and 0 denotes maximum π rad = 180 • tracking error at all times. Summing the MAER and the recall rate with permutation r applied to the estimated sources equals to F r = MAER r + recall r , which is referred to as overall accuracy. The best permutation for each signal is chosen by finding the minimum value of F r over all permutations indexed by r. The combination of both measures is used to avoid favoring permutations with very short detected trajectories with small MAE over longer trajectories with slightly larger MAE, for example accurate tracking of a single word from the entire utterance. Additionally, we do not consider and compensate for cases where one sound source is correctly tracked by two trajectories with discontinuity during the pauses in speech. The effect of this is negligible due to the short test signals used (10 seconds). The acoustic tracking performance averaged over all signals in all datasets and source detection performance is reported in Table II. The tracking error measured by MAE is below 10 degrees for datasets with two sources, which can be regarded as a good result. Noticeably the accuracy of the tracking is even better with the evaluation dataset. However the recall rate drops by 4% mostly due to late detection of sources, which displays the difficulty of setting the optimal values for parameters controlling the birth and death of sources in the particle filtering. In general, a low recall rate can be considered as conservative and only detecting and tracking dominant portions of sources. Alternatively, optimizing the parameters for 100% recall rate would lead to detection of numerous phantom sources, caused by reverberation and noise in the recordings. The recall rate and tracking accuracy are noticeably decreased for the evaluation dataset with three simultaneous sources. As indicated by the second chart in Table II, the percentage of correctly detected number of sources is approximately 80% for both two source datasets and drops down to 56% for the dataset with three sources. The errors in source detection are mostly caused by overdetection in the case of two simultaneous sources, whereas in the more difficult scenario of three sources the underdetection is also a significant cause of error. E. Source separation performance 1) Separation evaluation criteria: We evaluate the separation performance of the proposed algorithm using the following objective separation criteria with the close-field microphone signal as a reference. From the separation evaluation toolbox proposed in [38] we have included signal-to-distortion ratio (SDR) and signal-to-interference ratio (SIR) evaluated in short segments of 200 ms. The score of each segment in dB scale is converted to linear scale and averaged over all segments after which the resulting average is converted back to dB scale. The resulting metrics are abbreviated as segmental SDR (SSDR) and segmental SIR (SSIR). The use of segmental evaluation was chosen due to the operation of BSSeval, which projects the separated signal into reference signal subspace and assumes that this projection is stationary. However, in the case of moving sound sources and reference by close-field microphone the initial delay to the far-field array and the room reflections change from frame to frame which requires the projection operator to be also time-variant. This is achieved by assuming projection stationarity within each 200ms segment. Other metrics include the short-time objective intelligibility measure (STOI) [39] which is used to predict the intelligibility of the separated speech in comparison to the reference signal, and the frequencyweighted segmental signal-to-noise ratio (fwSegSNR) [40]. The latter metrics, STOI and fwSeqSNR, were calculated without segmenting. 2) Reference methods: The description of methods whose separation performance is evaluated are given in Table III and can be summarized as follows. The plain microphone signal from the array acts as a lowest performance baseline whereas the IRM indicates an upper limit. The tracking information is also used in the beamforming (DSB and MVDR) to specify the weights w H f n,p to enhance the signal at each time frame originating from the estimated DOA. The separation performance of proposed method was also evaluated using the ground truth DOA trajectories for specifying the source movement for the multichannel NMF part. This evaluation indicates the highest achievable separation performance with perfect tracking information and the results can be also used for validating the robustness of the overall proposed separation algorithm towards small tracking errors. The MVDR beamforming was implemented using the sample covariance method [42] for estimating the noise covariance matrix. The noise covariance was estimated from M = 20 previous frames with respect to each processed frame and it captures the stationary noise statistics as well as the immediate spectral details of interfering speech sources. Additionally, a diagonal loading (σ = 5) of noise covariance matrices was applied to improve robustness of the MVDR beamformer. These parameters were optimized using the development set. 3) Separation results: The separation performance measured by SSDR, SSIR, STOI and fwSegSNR is calculated with the source permutation obtained by minimizing the accuracy criterion specified in Equation (30) and the results are averaged over all sources and mixtures. The separation performance for all the tested methods with all the considered criteria are given in Figure 8 (a)-(d). Evaluation with the mixture signal (mic) indicates a baseline performance resulting to SSDR of approximately 4 dB for two simultaneous sources and 2 dB for three sources. Such high absolute performance for mixture signal is because of the segmental evaluation. In contrast, evaluating the entire signals in one segment resulted into negative average SDR for all tested methods (from -6 dB SDR for the mixture to -1 dB for the IRM) while the relative differences remained the same as reported in the Figure 8. The absolute results obtained this way did not reflect the subjective separation performance and are not reported in the paper. The overall low scores were assumed to be caused by the problems of projection operation in BSSeval toolkit, discussed in the beginning of Section V-E. The beamforming methods, DSB and MVDR, consistently improve SSDR, SSIR and STOI in comparison to microphone signal. However, the overall improvement in all datasets is relatively poor: SSDR improvement varies from 0.45 dB to 0.70 dB, and STOI barely reaches the index of 0.5, indicating low predicted intelligibility for the separated sources. Additionally, MVDR beamforming has negative effect on the fwSegSNR criterion, which may be caused by unwanted canceling of target source due to the rudimentary noise covariance estimation method employed. DSB on the other hand does not have this negative effect. With all other evaluated criteria MVDR beamforming exceeds the DSB performance with a small margin. When violating the moving sources assumption and using multichannel NMF aimed for separation of stationary sound sources [20], the performance is poor especially with STOI that decreases below the array microphone baseline. The other evaluated metrics are similar to beamforming approaches. When the source is momentarily at the target static direction (blindly estimated in [20]) the separation quality is high, but as the source moves it shifts out of spatial focus of the separation. The results regarding the proposed method can be summarized by stating that it significantly increases the separation performance over the beamforming methods DSB and MVDR: in case of two sources the SSDR improves approximately by 1.5 dB and improvement is even greater for three simultaneous sources (2 dB). SSIR improvement follows the trend of SSDR and STOI increases approximately by an index of 0.1. The use of ground truth annotations in the case of two simultaneous sources does not significantly increase the performance: SSDR increases only by 0.13 dB and 0.15 dB, SSIR is unchanged and STOI increases by 0.01. This validates the good acoustic tracking results reported in Table II. The separation performance with the three sources follows the poorer tracking performance and the use of annotations has greater impact on improving the objective criteria, SSDR is increased by 0.5 dB by use of annotations but interestingly the SSIR decreases. This may be due to the annotations being always active even if the speech starts 1 to 2 seconds from the beginning of the signal resulting to nonzero separation output for the annotations, whereas in the actual tracking implementation the source signal is truly zero [19] and [47] with two simultaneous stationary sources. until it is detected. The behavior of all methods with all criteria is consistent with all datasets. Although the absolute differences in objective intelligibility are small, the increase in STOI by the proposed method over the beamforming methods is greater than the STOI improvement by beamforming over the microphone signal. The overall difficulty of each dataset can be estimated from the IRM performance, which indicates that the actual evaluation dataset with two sources is less difficult than the development dataset. The performance gap between the proposed method and IRM separation is considerable especially in case of three simultaneous sources. This is due to the fact that IRM is not much affected by the adding of third source, since speech is relatively sparse in time frequency domain and good separation can be achieved with oracle masks even with three simultaneous speakers. Evaluation of IRM performance by SSDR and SSIR for two source dataset indicates not as big difference in comparison to the proposed method. However, the IRM performance is also limited by the fact that far-field and close-field signal spaces are extremely different and time-frequency masking cannot recover the close-field signal perfectly due to mixture phase is used. In subjective evaluation IRM preserves the intelligibility of the speech much better than any separation method which is also indicated by the good results in the objective evaluation of intelligibility, i.e. STOI for IRM is around 0.7-0.8. As a final result we provide a comparison of separation performance obtained with similar DOA-based spatial and spectrogram factorization models assuming stationary sources [18], [19], [20]. The evaluation dataset consist of all sources being stationary, see Section V-B. The proposed algorithm was run as is with the exception that source reconstruction was done without DSB, since the reference signals are reverberated source signals and evaluation in [20] is based on the spatial images of the sources [49]. The details of evaluation procedure and reference results are as presented in [20]. In theory, if the source DOA trajectory estimation would be perfect, similar results between multichannnel NMF-based methods regardless of source movement assumption should be obtained. However, methods proposed in [19], [20] also update elements of the DOA kernels (Equation (19)) whereas in this work they are fixed to analytic anechoic array responses. The SDR, SIR, SAR and ISR are given in Table IV, which shows that the SDR performance of the proposed method is lower in comparison to methods utilizing the stationary assumption, while SIR is highest among the tested methods. The average tracking error (MAE) for the dataset was 8.8 • and recall rate was 79% which are similar to the tracking performance for other 2 source datasets given in Table II. The comparison to multichannel NMF models assuming stationary source motivates the future work for reducing the performance gap while assuming moving sound sources and possible research directions are discussed in the next section. VI. DISCUSSION In this section we present a few remarks regarding the algorithm development choices and possible future work for improving and extending the method. The strategy of using the estimated source DOA trajectories for definition of the SCM model in (22) means effectively using only channel-wise time differences and assuming anechoic environment. This strategy can be questioned in comparison to also updating the channel-wise level differences as in [19], [20]. However, the difficulty of updating the level differences between input channels lies within the fact that with moving sources there may be only very few frames of data observed from each direction and investigation of the updating W f d in such setting was left for future work. The multichannel NMF model (22) would allow to use multichannel Wiener filter (MWF) for source reconstruction as in [18]. Informal experiments showed inferior performance with MWF in comparison to chosen combination of singlechannel Wiener filter and DSB. There are several possible reasons to explain the findings. The multichannel model used for representing source SCMs relies only on the anechoic responses which can be suboptimal for constructing the MVF for source reconstruction. Additionally, errors in source SCM estimation can lead to unexpectedly sharp spectral and spatial responses for source reconstruction with MVF. The strategy of single-channel Wiener filter and DSB is argued to be less destructive with respect small estimation errors. Analysis of the tracking performance indicated that fairly accurate source DOA trajectories can be estimated with existing methods in realistic capturing conditions which justifies the applicability of the proposed separation algorithm for general use. Tracking errors may be caused by erroneously representing a single source with two consecutive but separate tracks due to pauses in speech. Also in the case of intersecting DOA trajectories, the estimated tracks can switch the actual acoustic targets, i.e., source 1 continues to track acoustic evidence of source 2 and vice versa. It should be noted that we did not account for the above problems in the tracking and separation performance evaluation. The extremely good results of using of deep learning for speech separation [10], [11], [51] are quickly replacing the use factorization based models in source separation. With multichannel audio the spatial parameters being complex-valued require use of other approaches for SCM estimation, for example in [52] DNNs are used for spectrogram estimation while SCMs are estimated using a probabilistic model and EM-algorithm. The strength of the proposed method compared to DNN-based separation is that it operates on spatial information and spectral factorization of the observed data only and works relatively well in any scenario and all sound content (music, noise, everyday sounds) without any training material. VII. CONCLUSIONS In this article a separation method for moving sound sources based on acoustic tracking and separation mask estimation by multichannel non-negative matrix factorization was proposed. We analyzed the objective separation performance and the proposed method exceeded the conventional beamforming using the same tracking information by a fair margin. The comparison against ground truth source DOA trajectories indicated only minor impairment to objective separation performance. Additionally, analysis of the acoustic tracking realization showed good performance, recall rate over 80 % and absolute tracking error less than 10 degrees with two simultaneous moving sound sources. In conclusion the proposed method was shown to be robust and capable of separating at least two moving targets from mixtures recorded with a compact sized microphone array in realistic capturing conditions. Fig. 1 : 1The block diagram of the proposed processing consisting of source tracking and multichannel NMF for separation of detected and tracked sound sources. Fig. 2 : 2Illustration of a sparse grid of direction vectors by lines with dot in end and an example array enclosure and enclosed microphones (circles). Fig. 3 : 3The observed SRP S dn for a single time-frame n and the WGMM with K = 3 estimated from it.Algorithm 1 EM-algorithm for estimation of WGMM model parameters for a histogram s d (single frame from the entire SRP S dn ). Fig. 4 : 4Upper panel illustrates all estimated WGMM means µ n,k for each time-frame n. In the lower panel WGMM means after removing measurements with σ n,k > 0.6 rad (≈ 35 • ) or a n,k < 0.15 are illustrated . Similarly the multiple measurements at same time step obtained from the WGMM model are denoted by m (k) n = cos(µ n,k ), sin(µ n,k ) T Extracting the DOA from the tracked source state requires calculating the angle of the vector defined by the 2-D coordinates and thus the resulting DOA trajectories are obtained asμ n,p ← atan2(s Fig. 5 : 5The acoustic tracking result of two sources intersecting and ground truth annotations illustrated when the source is active (voice activity detection by energy thresholding using signal from close-field microphone). Fig. 6 : 6The reconstructed spatial weights as given in Equation(21)for two detected sources are illustrated in two top panels and the spatial weights corresponding to the background source are illustrated in the bottom panel. Fig. 7 : 7Illustration of the recording setup and the source movement. Fig. 8 : 8Separation performance measured using various objective separation criteria: SSDR, SSIR, STOI, fwSegSNR. each input channel (m = 1, . . . , M ). The single-channel STFT of each source p is denoted by s f n,p and their frequency domain RIRs (fixed withing each time frame n) are denoted by h f n,p = [h f n1 , . . . , h f nM ] T . The source signals convolved with the impulse responses are denoted by y f n,p . TABLE I : IDescription of datasets.Developmenet dataset Number of samples Source 1 Source 2 Details 2 CW CW > 45 • 2 CCW CCW > 45 • 4 CW CCW Sources intersect 12 CW/CCW S (babble) SNR = -5, -10 and -15 dB 4 CW/CCW S (music) SNR = -10 dB Evaluation dataset, 2 sources Number of samples Source 1 Source 2 Details 16 CW CW > 45 • 16 CCW CCW > 45 • 16 CW CCW Sources intersect Evaluation dataset, 3 sources Number of samples Source 1 Source 2 Source 3 Details 4 CW CW CW > 45 • 4 CCW CCW CCW > 45 • 4 CW CW CCW Sources intersect 4 CCW CCW CW Sources intersect Dataset with stationary sources from [19] Number of samples Source 1 Source 2 Details TABLE II : IIAcoustic tracking results.Tracking performance Dev. Eval. Eval. Criteria (2 sources) (2 sources) (3 sources) MAE 7.3 • 6.1 • 10.5 • recall 86.3% 82.2% 64.7% Source detection performance Dev. Eval. Eval. Criteria (2 sources) (2 sources) (3 sources) P ==P 79.2% 81.3% 50.0% P >P 20.8% 16.7% 25.0% P <P 0.0% 2.0% 25.0% TABLE III : IIIDescription of compared separation methods. MVDR Minimum variance distortionless beamforming. MNMF sta.Multichannel NMF assuming stationary sources[20].MNMFProposed method, i.e. multichannel NMF with timevarying SCM model. MNMF ann. Proposed method with ground truth annotations as source trajectories.IRMIdeal ratio mask separation.Abbrv. Description mic Microphone signal from the array (ch #1). DSB Delay-and-sum beamforming. mic DSB MVDR MNMF sta. MNMF MNMF ann. IRM 0 2 4 6 8 10 [dB] a) SSDR Dev. Eval. (2 sources) Eval. (3 sources) mic DSB MVDR MNMF sta. MNMF MNMF ann. IRM 0 5 10 15 20 [dB] b) SSIR Dev. Eval. (2 sources) Eval. (3 sources) mic DSB MVDR MNMF sta. MNMF MNMF ann. IRM 0 0.2 0.4 0.6 0.8 1 STOI index c) STOI Dev. Eval. (2 sources) Eval. (3 sources) mic DSB MVDR MNMF sta. MNMF MNMF ann. IRM −10 0 10 20 30 40 [dB] d) fwSegSNR Dev. Eval. 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[ "Limits on evolution of the fine-structure constant in runaway dilaton models from Sunyaev-Zeldovich Observations", "Limits on evolution of the fine-structure constant in runaway dilaton models from Sunyaev-Zeldovich Observations" ]	[ "R F L Holanda \nDepartamento de Física\nUniversidade Estadual da Paraíba\nCampina Grande -PB\n58429-500Brasil\n\nDepartamento de Física\nUniversidade Federal do Rio Grande do Norte\n59300-000Natal -RN, Brasil\n\nDepartamento de Física\nUniversidade Federal de Campina Grande\nCampina Grande -PB\n58429-900Brasil\n", "L R Colaço \nDepartamento de Física\nUniversidade Federal do Rio Grande do Norte\n59300-000Natal -RN, Brasil\n", "R S Gonçalves \nObservatório Nacional\n20921-400Rio de Janeiro -RJBrasil\n", "J S Alcaniz \nObservatório Nacional\n20921-400Rio de Janeiro -RJBrasil\n" ]	[ "Departamento de Física\nUniversidade Estadual da Paraíba\nCampina Grande -PB\n58429-500Brasil", "Departamento de Física\nUniversidade Federal do Rio Grande do Norte\n59300-000Natal -RN, Brasil", "Departamento de Física\nUniversidade Federal de Campina Grande\nCampina Grande -PB\n58429-900Brasil", "Departamento de Física\nUniversidade Federal do Rio Grande do Norte\n59300-000Natal -RN, Brasil", "Observatório Nacional\n20921-400Rio de Janeiro -RJBrasil", "Observatório Nacional\n20921-400Rio de Janeiro -RJBrasil" ]	[]	In this paper, new bounds on possible variations of the fine structure constant, α, for a class of runaway dilaton models are performed. By considering a possible evolution with redshift, z, such as ∆α α = −γ ln(1 + z), where in γ are the physical properties of the model, we constrain this parameter by using a deformed cosmic distance duality relation jointly with gas mass fraction (GMF) measurements of galaxy clusters and luminosity distances of type Ia supernovae. The GMF's used in our analyses are from cluster mass data from 82 galaxy clusters in the redshift range 0.12 < z < 1.36, detected via the Sunyaev-Zeldovich effect at 148 GHz by the Atacama Cosmology Telescope. The type Ia supernovae are from the Union2.1 compilation. We also explore the dependence of the results from four models used to describe the galaxy clusters. As a result no evidence of variation was obtained.PACS numbers: 95.36.+x, 98.80.Es	10.1016/j.physletb.2017.01.055	[ "https://arxiv.org/pdf/1701.07250v1.pdf" ]	119,376,035	1701.07250	ff43c24fdb43b112c220c29a1651a9b18a6940c1	Limits on evolution of the fine-structure constant in runaway dilaton models from Sunyaev-Zeldovich Observations 25 Jan 2017 (Dated: July 10, 2021) R F L Holanda Departamento de Física Universidade Estadual da Paraíba Campina Grande -PB 58429-500Brasil Departamento de Física Universidade Federal do Rio Grande do Norte 59300-000Natal -RN, Brasil Departamento de Física Universidade Federal de Campina Grande Campina Grande -PB 58429-900Brasil L R Colaço Departamento de Física Universidade Federal do Rio Grande do Norte 59300-000Natal -RN, Brasil R S Gonçalves Observatório Nacional 20921-400Rio de Janeiro -RJBrasil J S Alcaniz Observatório Nacional 20921-400Rio de Janeiro -RJBrasil Limits on evolution of the fine-structure constant in runaway dilaton models from Sunyaev-Zeldovich Observations 25 Jan 2017 (Dated: July 10, 2021) In this paper, new bounds on possible variations of the fine structure constant, α, for a class of runaway dilaton models are performed. By considering a possible evolution with redshift, z, such as ∆α α = −γ ln(1 + z), where in γ are the physical properties of the model, we constrain this parameter by using a deformed cosmic distance duality relation jointly with gas mass fraction (GMF) measurements of galaxy clusters and luminosity distances of type Ia supernovae. The GMF's used in our analyses are from cluster mass data from 82 galaxy clusters in the redshift range 0.12 < z < 1.36, detected via the Sunyaev-Zeldovich effect at 148 GHz by the Atacama Cosmology Telescope. The type Ia supernovae are from the Union2.1 compilation. We also explore the dependence of the results from four models used to describe the galaxy clusters. As a result no evidence of variation was obtained.PACS numbers: 95.36.+x, 98.80.Es I. INTRODUCTION The Sunyaev-Zeldovich effect (SZE) is a secondary anisotropy into the cosmic microwave background radiation (CMB) temperature [1]. It is produced by the inverse Compton scattering of the CMB photons passing through a population of hot electrons in galaxy clusters. This effect encode information about the distribution of dark matter and gas throughout the Universe, being especially important at high redshifts (z > 1) where the cosmological model and abundance of clusters are critically correlated. A remarkable feature of the distortion is a decrement in low frequency (< 218 GHz) and an increment in higher frequency (> 218 GHz) in the CMB intensity (see Refs. [2,3] for excellent reviews). In the past decade, some authors obtained results from the SZE science by using the Owens Valley Radio Observatory and the Berkeley-Illinois-Maryland Association interferometric arrays. For instance, the Refs. [4,5,6] considered a jointly analysis with X-ray surface brightness of galaxy clusters and estimated the angular diameter distance (SZE/X-ray technique) of galaxy clusters by using different assumptions to their morphology. In Ref. [7] the measurements of gas mass fraction as well as the scaling relations via SZE were explored. Cosmological parameters also were inferred by using the SZE and other cosmological data in Refs. [8,9]. However, since the signal intensity of the SZE is very thin, 10 −5 , its potential as a cosmological tool has been explored only in recent years. Currently, the South Pole Telescope * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] [10,11,12], the Atacama Cosmology Telescope [13,14] and the Planck satellite [15,16,17] have detected through the SZE about 1000 galaxy clusters including more than hundreds of new galaxy clusters previously unknown by any other observational technique and put tighter constraints on cosmological parameters. The SZE observations also allow us to test the adiabatic evolution of the temperature of the cosmic microwave background (CMB), a key prediction of standard cosmology. Actually, the SZE is redshift independent only if there is no injection of photons into CMB, i.e., if its temperature evolution law is given by T CMB (z) = T 0 (1 + z). By taking a more general relation, such as T CMB (z) = T 0 (1 + z) 1+δ , recent analyses tested the evolution of the CMB temperature through different techniques and confirmed the standard relation, δ ≈ 0 (see Refs. [18,19,20]). However, in Ref. [21], the author showed that the SZE have limited applicability in these kind of tests. On the other hand, it has been showed that the combination of X-ray surface brightness of galaxy clusters with their SZE measurements also can be used to investigate fundamental physics as well as testing results from standard cosmology. For instance, the Ref. [22] showed that the so-known technique SZE/X-ray of measuring angular diameter distance (ADD) of galaxy clusters depends on the cosmic distance duality relation (CDDR), D L (1 + z) −2 /D A = η = 1, where D L and D A are the luminosity and angular diameter distances, respectively. This relation is a fundamental one from cosmology [23,24], requiring only that source and observer are connected by null geodesics in a Riemannian spacetime and that the number of photons is conserved. This relation has been verify at least within 2σ c.l. (see table I in Ref. [25], other studies testing the CDDR can be found in Ref. [26]). More recently, the authors from Refs. [27,28] showed that SZE and X-ray observations also can be combined to investigate possible variations of the fundamental constants, specifically, the fine structure constant, α = e 2 /c , where e is the charge of electron, is the Planck constant and c is the speed of the light. Constraints on variations of α for a class of dilaton runaway models were discussed. In these models of α variation, the relevant parameter is the coupling of the dilaton field to hadronic matter. Several observational analyses have been performed to study possible variations of α and to establish bounds on such variations, namely: astronomical observations, based on mainly on the analysis of high-redshift quasar absorption systems [29]; and local methods, based on atomic clocks with different atomic numbers [30,31,32]. An interesting recent debate was done by Refs. [33,34] on a possible α variation by using Keck/HIRES and VLT/UVES observations. No important deviation was verified with these observations. It is important to stress that the X-ray surface brightness depends on the CDDR and the α while the SZE depends exclusively on α. In this way, a theoretical result from Ref. [35], η 2 (z) = ϕ(z) (where ϕ(z) − 1 = ∆α/α), was used in order to put limits on ϕ(z). Indeed, the authors of Ref. [35] showed that for a large class of theories arising from modifications of gravity via the presence of a scalar field with a multiplicative coupling to the electromagnetic Lagrangian, violations of CDDR, of CMB temperature law and variations of α are intimately and unequivocally linked. In this paper, we obtained constraints on variations of α for a class of dilaton runaway models by using galaxy cluster masses from the Atacama Cosmology Telescope (ACT) data obtained via their ESZ observations and type Ia Supernovae from Union2.1 compilation [36]. More precisely, we use measurements of gas mass fraction, f gas , obtained from 82 points of galaxy cluster mass [13]. The f gas estimated for each cluster in the sample was calculated by using a semi-empirical relation presented by Ref. [37], where the observed gas fraction in galaxy clusters with z < 0.09 was verified to be a function of the total mass, M . The masses of clusters are those corresponding to M 500 , defined as the mass measured within the radius R 500 . Since these measurements depend on the physical model of the intracluster gas, the ACT team adopted four models (see Sec. III for details). So, as an extra bonus, we also verify the dependence of our results with the methods used to infer M 500 . The paper is organized as follows. In Section II, we briefly describe the samples used in our analyses. In Section III we describe our method. In section IV, we perform the analyses. Finally, the discussions and conclusions are given in Section V. II. SAMPLES The SZE gas mass fraction data used in this paper were obtained from the cluster mass measurements of the ACT [13] in the redshift range 0.12 < z < 1.36 detected via the SZE at 148 GHz. The original sample contains 91 galaxy cluster masses. In order to estimate the galaxy cluster mass, the ACT team adopted a one-parameter family of Universal Pressure Profiles (UPP) as a baseline model for the intracluster gas pressure profile [50]. The galaxy cluster masses were measured within a characteristic radius at which the enclosed mean density is 500 times the critical density at the cluster redshift, M 500 . The ACT team also used others three scaling relations to estimate M 500 , which are based on: i) structure formation simulations [39], where the density and temperature of the intracluster are modeled as a virialized ideal gas (M B12 500 ), ii) a non-thermal pressure and adiabatic model for the gas (M non−thermal 500 ) [40] and iii) a dynamical estimate of the cluster mass using the galaxy velocity dispersions (M dyn 500 ) [43]. Finally, from the total mass it is possible to obtain f gas using the following semi-empirical relation discussed by Ref. [37]: f obs gas = 0.132 + 0.039 log M 15 ,(1) where M 15 is the cluster total mass M 500 in units of Fig.1b). This relation was obtained from dozens of clusters of galaxies in z < 0.09 (see Table 2 in [37]) with mass range of 10 14 − 10 15 h −1 M ⊙ , which clearly suggested an approximately linear trend of f obs gas with log M . The uncertainties of the coefficients are negligible if compared to uncertainties of the masses in our analyses (20-30%). We extrapolate the Eq.(1) up to z = 1.36 based on the most recent hydrodynamical simulations that show no significant gas mass fraction evolution with redshift when a r 500 is used [41,42]. It is also important to comment that this relation was obtained from X-ray surface brightness observations, therefore, in this initial approach, we neglect this bias. 10 15 h −1 M ⊙ (see We also consider a sub-sample of observational measurements of SNe Ia from the original 580 data points of Ref. [20], the so-called Union2.1 compilation. The SNe Ia points are in the redshift range 0.015 < z < 1.43. The redshifts of SNe Ia were carefully chosen to match the ones of galaxy clusters. In this way, we consider the SNe Ia Union2 compilation [20] and the galaxy clusters compiled in Ref. [13] as follows: for each galaxy cluster, we select SNe Ia with redshifts obeying the criteria \|z cluster − z SN e \| ≤ 0.005. We find 2-6 SNe Ia for each galaxy cluster. This criteria resulted in 82 galaxy clusters and 82 SNe Ia sub-samples that matched this criterion, i. e., 9 galaxy clusters were ruled out from our analyses. This criterion allows us to have some SNe Ia for each galaxy cluster and so we can perform a weighted average with them in order to minimize the scatter observed on the Hubble diagram by calculating the following weighted average: 1: In Fig. (1a) we plot the SNe Ia from Union2.1 compilation (black stars) and the points used in our analyses (red circles, see Eq.2). In Fig. (1b) we plot the 82 gas mass fractions calculated from Ref. [13]. We ruled out 9 galaxy clusters from original 91 data points due to they do not have SNe Ia pairs with ∆z ≤ 0.005. µ = (µi/σ 2 µ i ) 1/σ 2 µ i , σ 2 µ = 1 1/σ 2 µ i .(2) . As is largely known, the distance moduli, µ, of Union2.1 SNe Ia compilation are dependent on the choice of the Hubble parameter H 0 = 70km/s/Mpc as well as of the ωCDM cosmological model, leaving our results somewhat dependent of a class of cosmological model (see Fig.1a). III. METHOD Our method is based on the results from Ref. [27]. In that work, the authors showed that the gas mass fraction via SZE observations is dependent on the fine structure constant. In order to clarify the method used, we describe below some fundamental aspects from their results. The spherical β model is used in this section only for simplicity but without loss of generality for the method proposed. A. Fine structure constant and SZE observations The SZE can be expressed for a dimensionless frequency x ≡ hν(z)/k B T CMB (z) as a temperature change ∆T (z) relative to the CMB temperature T CMB (z) such as: ∆T (z) T CMB (z) = f (x, T e ) σ T n e k B T e m e c 2 dℓ,(3) where n e and T e are the electron number density and the gas temperature, respectively, k B the Boltzmann constant, σ T = 8π 2 α 2 /3m 2 e c 2 is the Thomson scattering cross-section of the electron, where is the Planck constant divided by 2π, m e is the electronic mass, c is the speed of light and the integral is along the line of sight. The function f (x, T e ) contains the frequency dependence, ν, of the SZE and it can be express as: f (x, T e ) = x e x + 1 e x − 1 − 4 (1 + δ SZE (x, T e ))(4) where δ SZE (x, T e ) is a relativistic correction, written in terms of k B T e /m e c 2 [44]. As one may see, since ν = ν 0 (1 + z) and T CMB = T 0CMB (1 + z), the SZE is redshift independent. By considering the isothermal β-model, the electron number density is given by n e (r) = n 0 1 + r 2 r 2 c −3β/2 ,(5) where r is the radius from the center of the cluster, r c is the core radius of the intracluster medium (ICM) and β is a power law index. Under Eq.(3), the SZE decrement profile takes simple analytic forms ∆T = ∆T 0 1 + θ 2 θ 2 c (1−3β)/2 ,(6) where ∆T 0 is the central thermodynamic SZE temperature decrement/increment, and θ c is the angular core radius of the cluster. In this way, the central electron density can be expressed as [45]: n 0 = ∆T 0 m e c 2 Γ( 3 2 β) f (x,Te) T CMB σ T k B T e D A π 1/2 Γ( 3 2 β − 1 2 ) θ c (7) The mass of gas, inside the radius r, is obtained by integrating the best-fit 3D gas density profile: M gas (r) = A r/DA 0 1 + θ 2 θ 2 c −3β/2 θ 2 dθ,(8) where A = 4πµ e n 0 m p D 3 A , and µ e , the mean molecular weight of the electrons. On the other hand, under the hydrostatic equilibrium assumption, isothermality and Eq. (5), M tot is given by [45] M tot (< R) = 3βk B T G µGm H R 3 (r 2 c + R 2 ) ,(9) where T G is the temperature of the intracluster medium obtained from X-ray spectrum, µ and m p are, respectively, the total mean molecular weight and the proton mass and G is the gravitational constant. Finally, by considering that the gas mass fraction is defined as [46]: f gas = M gas M tot ,(10) where M tot is the total mass and M gas is the gas mass obtained by integrating the gas density model. One may shows, by using the expression for the Thompson scattering cross section, that the current gas mass fraction measurements via SZE depend on α as (see Ref. [27] for details): f obs gas ∝ α −2 .(11) B. Modified CMB temperature law It was shown in Ref. [35] that modifications of gravity generated by a multiplicative coupling of a scalar field to the electromagnetic Lagrangian lead to a breaking of Einstein equivalence principle as well as to variations of fundamental constants. As a consequence, we can have η = 1,∆α/α = 1 and δ = 0. In this framework, the CMB temperature law has to be modified to T CMB (z) = T 0 (1 + z) 1 + 0.12 ∆α α .(12) In previous papers that used SZE observations to put limits on possible α variation [27,28], the SZE observations were performed in 30 GHz, in this band the effect on the SZE from a variation of T CMB is completely negligible. In the sample considered in the present work, the frequency used to obtain the SZE signal in galaxy clusters was 148 GHz and the effect from a variation of T CMB on the SZE have to be taking into account [49]. So, following Eq.(12), the term x in Eq.(4) have to be modified to ψ = hν 0 /(k B T 0CMB [1 + 0.12(ϕ(z) − 1)]),(13) where z in Eq.(13) corresponds to galaxy cluster redshift. Note that if ∆α/α = ϕ(z) − 1 = 0, we have f (x, T e ) = f (ψ, T e ). Therefore, by using the Eqs. (7), (10), (11) and (13), if ϕ(z) = 1, current gas mass fraction measurements via SZE have to corrected by the factor (f (x, T e )/f (ψ, T e ))ϕ(z) −2 .(14) In order to calculate this ratio we consider the relativistic corrections from Ref. [44], calculated up to the fifth order of kT e /m e c 2 . The temperature of the galaxy clusters were estimated by using the scaling relation from Ref. [50], obtained via ten relaxed galaxy clusters with z ≤ 0.15, such as h(z)M = A [kT e /5keV ] τ , where A and τ are, respectively: 4.10 ± 0.19 and 1.49 ± 0.15. The h(z) parameter corrects the evolution expected in the standard self-similar model. This parameter is between 1.05 and 1.28 for the Chandra clusters located at higher redshifts (0.1 < z < 0.46). We consider a medium value ≈ 1.20. C. Observational equation for ϕ(z) The expression of the SZE gas mass fraction used as cosmological tool is [7]: f obs gas = N D * A D A ,(15) where the symbol * denotes quantities that were obtained by using a fiducial model in the observations and the parameter N defines the astrophysical modeling of the cluster. Following Eq. (15), this relation must be corrected to: ζf obs gas ϕ(z) −2 = N D * A D A ,(16) where ζ = (f (x, T e )/f (ψ, T e )). As we aim to put limits on ϕ(z), we could consider the validity of the CDDR -so D L (1 + z) −2 /D A = η = 1and use distance moduli from a Union2.1 SNe Ia compilation to obtain D L , leading to bounds on ϕ(z). However, from Eq. (12), the CDDR has to be modified to D L (1 + z) −2 /D A = ϕ(z) 1/2 before use it. After performing simple algebraic operations one obtains: ϕ obs (z) = ζf obs gas 10μ −25 5 N D * L 2/5 ,(17) where we use D L (z) = 10μ −25 5 Mpc. In our analyses, we focus on the dilaton runaway models (see more details in Refs. [48]) where the relevant parameter for studying the variation of α is the coupling of the dilaton field to hadronic matter. We are interested in the evolution of the dilaton and a reasonable approximation in the redshift range used in our analyses is to linearize the field evolution, such as or, equivalently, ϕ(z) = 1 − γ ln(1 + z), where γ = 1 40 β had,0 φ ′ 0 with φ ′ 0 = ∂φ ∂ ln a being the scalar field at present time and β had,0 being the current value of the coupling between the dilaton and hadronic matter. ∆α α (z) ≈ − 1 40 β had,0 φ ′ 0 ln (1 + z) = −γ ln(1 + z) ,(18) IV. ANALYSES AND RESULTS We evaluate our statistical analyses by defining the likelihood distribution function L ∝ e −χ 2 /2 , where χ 2 = 82 i=1 [(1 − γ ln (1 + z)) − ϕ i,obs ] 2 σ 2 i,obs ,(19) with ϕ obs (z) = ζf obs gas 10μ and σ 2 i,obs is the uncertainty associated to observational quantities: f obs gas ,μ and kT e . The parameter N carries all the information about the matter content in the cluster, such as stellar mass fraction, non-thermal pressure and the depletion parameter, which indicates the amount of cosmic baryons that are thermalized within the cluster potential [47]. From hydrodynamical simulations this quantity does not have significant dependence on redshift. Moreover, since the most of cluster masses used in our analyses are of the same order, 10 14 M ⊙ , we take it as a nuisance parameter so that we marginalize over it. Following Ref. [20] we added a 0.15 systematic error to SNe Ia data. Constraints on the quantity γ = 1 40 β had,0 φ ′ 0 are shown in Figs. (2a) and (2b). From Fig. (2a) we obtain γ = 0.008 ± 0.035 and γ = 0.018 ± 0.032 (at 68.3% c.l.) for UPP and B12 models, respectively. From Fig. (2b) we obtain γ = 0.01 ± 0.033 and γ = 0.030 ± 0.033 (at 68.3% c.l.) for NON and DYN models, respectively. As one may see, all results are fully compatible each other and with φ(z) = 1 or, equivalently, with no variation of fine structure constant α. It is interesting to compare our bounds on γ with the limits obtained recently from galaxy clusters and SNe Ia by Refs. [27,28]. In Ref. [27] the authors showed that observations of the gas mass fraction via SZE and X-ray surface brightness of the same galaxy cluster are related by f SZE = ϕ(z)f X−ray , where ϕ(z) = α α0 . Using 29 f gas measurements they found γ = 0.065 ± 0.095 at 68.3% (C.L.), in full agreement with our results. In Ref. [28] the authors showed that measurements of the SZE combined with observations of the X-ray surface brightness of galaxy clusters for estimating the ADD of galaxy clusters depends on the fine structure constant besides η [22]. By using 25 ADD and current type Ia supernovae observations they found: γ = −0.037 ± 0.0157 (at 68.3% c.l.). In this way, no significant indication of variation of α with the present data was found. V. CONCLUSIONS Nowadays, one of the most important fields of research in Cosmology is to investigate the physical assumptions implicit in the cosmological models. In this paper we analyzed a possible variation of the fine-structure constant (ϕ(z) = α α0 ), using observations of masses from galaxy clusters, for a special class of runaway dilaton models. The measurements were obtained via the Sunyaev-Zeldovich effect and the masses were obtained for four different scaling relations, named M UP P 500 , M B12 500 , M non−thermal 500 and M dyn 500 . The gas mass fraction data were then obtained using a semi-empirical relation and combined with SNe Ia measuments. In order to perform our analysis, we use a data set of 82 points of galaxy cluster mass and SNe Ia measurements, by assuming a limit of \|z cluster − z SN e \| ≤ 0.005. The gas mass fraction and the distance moduli from SNe Ia were combined using the CDDR assuming a possible variation of the fine structure constant, thus D L (1 + z) −2 /D A = ϕ(z) 1/2 . Assuming the evolution of the dilaton as ∆α α (z) = −γ ln(1 + z), we performed statistical chi-square analyses in order to obtain the best fit value of γ for the different estimates of the galaxy clusters masses. We found results in complete agreement between each other, with γ UP P = 0.008 ± 0.035, γ B12 = 0.018 ± 0.032, γ N ON = 0.01 ± 0.033 and γ DY N = 0.030 ± 0.033, with 68.3% of confidence level. By comparing our results with others in the literature, we found that the combination of gas mass fraction via SZE and SNe Ia measurements can produce results as robust as others using different methods. FIG. 1: In Fig. (1a) we plot the SNe Ia from Union2.1 compilation (black stars) and the points used in our analyses (red circles, see Eq.2). 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[ "arXiv:hep-th/0304126v2 12 Jun 2003 Anomaly, Fluxes and (2,0) Heterotic-String Compactifications", "arXiv:hep-th/0304126v2 12 Jun 2003 Anomaly, Fluxes and (2,0) Heterotic-String Compactifications" ]	[ "J Gillard \nDepartment of Mathematics King's College London Strand\nWC2R 2LSLondon, To the memory of Sonia Stanciu\n", "G Papadopoulos \nDepartment of Mathematics King's College London Strand\nWC2R 2LSLondon, To the memory of Sonia Stanciu\n", "D Tsimpis \nDepartment of Mathematics King's College London Strand\nWC2R 2LSLondon, To the memory of Sonia Stanciu\n" ]	[ "Department of Mathematics King's College London Strand\nWC2R 2LSLondon, To the memory of Sonia Stanciu", "Department of Mathematics King's College London Strand\nWC2R 2LSLondon, To the memory of Sonia Stanciu", "Department of Mathematics King's College London Strand\nWC2R 2LSLondon, To the memory of Sonia Stanciu" ]	[]	We compute the corrections to heterotic-string backgrounds with (2,0) world-sheet supersymmetry, up to two loops in sigma-model perturbation theory. We investigate the conditions for these backgrounds to preserve spacetime supersymmetry and we find that a sufficient requirement for consistency is the applicability of the ∂∂-lemma. In particular, we investigate the α ′ corrections to (2,0) heterotic-string compactifications and we find that the Calabi-Yau geometry of the internal space is deformed to a Hermitian one. We show that at first order in α ′ , the heterotic anomaly-cancellation mechanism does not induce any lifting of moduli. We explicitly compute the corrections to the conifold and to the U (n)-invariant Calabi-Yau metric at first order in α ′ . We also find a generalization of the gauge-field equations, compatible with the Donaldson equations on conformally-balanced Hermitian manifolds.1 The ∂∂-lemma is not valid on all non-Kähler Hermitian manifolds.	10.1088/1126-6708/2003/06/035	[ "https://arxiv.org/pdf/hep-th/0304126v2.pdf" ]	10,689,217	hep-th/0304126	bd8939076cfc514038fbf7e307457ab5be0098aa	arXiv:hep-th/0304126v2 12 Jun 2003 Anomaly, Fluxes and (2,0) Heterotic-String Compactifications J Gillard Department of Mathematics King's College London Strand WC2R 2LSLondon, To the memory of Sonia Stanciu G Papadopoulos Department of Mathematics King's College London Strand WC2R 2LSLondon, To the memory of Sonia Stanciu D Tsimpis Department of Mathematics King's College London Strand WC2R 2LSLondon, To the memory of Sonia Stanciu arXiv:hep-th/0304126v2 12 Jun 2003 Anomaly, Fluxes and (2,0) Heterotic-String Compactifications We compute the corrections to heterotic-string backgrounds with (2,0) world-sheet supersymmetry, up to two loops in sigma-model perturbation theory. We investigate the conditions for these backgrounds to preserve spacetime supersymmetry and we find that a sufficient requirement for consistency is the applicability of the ∂∂-lemma. In particular, we investigate the α ′ corrections to (2,0) heterotic-string compactifications and we find that the Calabi-Yau geometry of the internal space is deformed to a Hermitian one. We show that at first order in α ′ , the heterotic anomaly-cancellation mechanism does not induce any lifting of moduli. We explicitly compute the corrections to the conifold and to the U (n)-invariant Calabi-Yau metric at first order in α ′ . We also find a generalization of the gauge-field equations, compatible with the Donaldson equations on conformally-balanced Hermitian manifolds.1 The ∂∂-lemma is not valid on all non-Kähler Hermitian manifolds. Introduction The dynamics of the massless superstring modes of the various superstring theories admit a description in terms of ten-dimensional effective supergravities. The latter include, in particular, an infinite tower of α ′ corrections. From the world-sheet perspective, the equations of the target-space fields are vanishing conditions for the sigma model beta functions, i.e. conditions for conformal invariance. In this approach, α ′ is the loopcounting parameter in the sigma-model perturbation theory. At zeroth order in α ′ , the vanishing of the beta functions is equivalent to the field equations of ordinary supergravity theories. Terms linear or higher in α ′ are associated with corrections involving quadratic or higher polynomials in the spacetime curvature and their supersymmetric completions. For generic heterotic-string backgrounds the α ′ -corrections to sigma-model couplings are not known. As a result, most configurations relevant to string theory that have appeared in the literature, like the compactifications of [1,2,3], the five-brane [4] and various five-brane intersections [5,6], are solutions of ordinary supergravity theories. In cases where there is sufficient world-volume supersymmetry, one can argue that higherorder corrections are absent, see [7] for a general argument. In general however, the investigation of stringy effects requires taking the α ′ corrections into account. One such background is the heterotic five-brane [4] which emerges at one loop in sigma-model perturbation. This and other related results have been extended to three loops in [8]. The low-energy dynamics of the heterotic string is given by N = 1 supergravity in ten dimensions. The bosonic fields of the theory are the spacetime metric G, the NS three-form field strength H, the dilaton Φ and the gauge field A. The Green-Schwarz anomaly-cancellation mechanism requires that the three-form Bianchi identity receive an α ′ correction of the form dH = − α ′ 4 (p 1 (M) − p 1 (E)) + O(α ′2 ),(1.1) where p 1 (M), p 1 (E) are the first Pontrjagin forms of spacetime M and of the vector bundle E with connection A, respectively. In the α ′ expansion for H, H = T + α ′ f + O(α ′2 ), the lowest-order term T should be a closed three-form, i.e. dT = 0. This stems from the fact that at tree-level in the sigma model, the string couples to a two-form gauge potential b, where T = db. On the other hand, if p 1 (M) = p 1 (E), then df = 0 and so dH = 0. Global anomaly cancellation requires in addition that dH be exact, i.e. that H be globally defined. A class of heterotic-string backgrounds for which the Bianchi identity of the three-form H receives a correction of the type (1.1) are those with (2,0) world-volume supersymmetry. Such models were considered in [9]. The target-space geometry of (2,0)-supersymmetric sigma models has been extensively investigated in [9,10,7]. Recently, there is revived interest in these models [11,12] as string backgrounds and in connection to heterotic-string compactifications with fluxes [13,14]. In this paper we investigate the α ′ corrections to heterotic-string backgrounds with (2,0)-world-sheet supersymmetry. We take spacetime to be M = R 10−2n ×X n and demand that the background preserve 2 1−n of spacetime supersymmetry. The n = 2 case was examined in [8]. The manifold X n is Hermitian equipped with a compatible connection ∇ (+) with skew-symmetric torsion H, i.e. X n is Kähler with torsion (KT). At first order in α ′ , the holonomy of the connection ∇ (+) is contained in SU(n) and X n is conformally balanced, see appendix A. We show that at linear order in α ′ such spacetime-supersymmetric backgrounds satisfy the anomaly-cancellation condition and the field equations. In the proof, we make use of the results of [16] summarised in appendix A of the present paper. We find that the corrections to the other fields are determined by the corrections to the metric. We stress that consistency of the anomaly cancellation condition with the field equations requires that in the latter we include the two-loop contribution. A sufficient condition for the consistency of spacetime supersymmetry with the anomaly cancellation and the field equations, is the applicability of the ∂∂-lemma 1 on X n . We also consider the Donaldson equations on a non-Kähler Hermitian manifold. These are related to the gaugino Killing-spinor equation. For generic Hermitian manifolds, it is not apparent that the Donaldson equations are associated with a second-order equation for the gauge connection, i.e. a field equation. We find that they are, however, if the underlying Hermitian manifold is conformally balanced. It has been shown in [15] that if X n is compact, nonsingular and all fields are smooth, then T vanishes and to zeroth order in the α ′ expansion X n is a Calabi-Yau n-fold. We shall take this to be the starting point of the α ′ expansion. Proceeding to first order in α ′ , X n is deformed to a conformally-balanced KT manifold with hol(∇ (+) ) ⊆ SU(n). We determine the deformations of the fields using Hodge theory. Moreover, we compute the dimension of the moduli space and we find it to be the same as that of the moduli space of the underlying Calabi-Yau manifold. We therefore conclude that at this order in α ′ , there is no lifting of moduli in (2,0) compactifications with T = 0. As particular examples of the general theory mentioned above, we compute the O(α ′ ) corrections to the conifold [17] and to the U(n)-invariant Calabi-Yau metric found in [18]. We find that the singularity of the conifold persists to first order in α ′ . For other α ′ corrections to conifold geometry see [20]. This paper is organised as follows: In section two, we establish our notation and write down the field and the Killing-spinor equations for the heterotic string, up to two loops in sigma-model perturbation theory (order O(α ′ )). In section three, we give the conditions on the deformations of the metric required by spacetime supersymmetry, and express the deformations of the NS three-form and the dilaton in terms of those of the metric. We then show that the Killing-spinor equations for the aforementioned set of fields imply the field equations at this order in α ′ , provided the anomalous Bianchi identity of the H field is satisfied. In section four, we show that the gaugino Killing-spinor equation, which is equivalent to the Donaldson equations on a Hermitian manifold, implies the field equation for the gauge connection. In section five, we compute the O(α ′ ) corrections to the fields and show that the dimension of the moduli space is the same as that of the moduli space of the Calabi-Yau space we started with at zeroth order in α ′ . In section six and seven, we compute the O(α ′ ) corrections to the conifold and to the Calabi metric, respectively. In section eight, we discuss the consequences of our results in the context of compactifications with fluxes and we comment on α ′ corrections beyond two-loops. In appendix A, we summarise some of the properties of KT geometry. In appendix B, we explain the relation between the Lichnerowicz and Laplace operators. Finally, in appendix C we give a solution to the field equations at linear order in α ′ without the use of the Killing-spinor equations. Field and Killing-spinor equations The bosonic fields of the ten-dimensional supergravity which arises as low energy effective theory of the heterotic string are the spacetime metric G, the NS three-form field strength H, the dilaton Φ and the gauge connection A. We define the connections ∇ (±) M Y N = ∇ M Y N ± 1 2 H N M R Y R , where ∇ is the Levi-Civita connection of the metric G and M, N, R = 0, 1 . . . , 9 are spacetime indices. The three form H has an expansion in α ′ of the form 2 H = T − α ′ 4 Q 3 (Γ (−) ) − Q 3 (A) + O(α ′2 ) , (2.1) where T is a closed three-form, dT = 0 and Q 3 are Chern-Simons three-forms. We have dH ≡ −α ′ P + O(α ′2 ) , P = 1 4 [tr(R (−) ∧ R (−) ) − tr(F ∧ F )] ,(2.2) where the trace on the gauge indices is taken as trF ∧ F = F a b ∧ F b a , F = dA + A 2 . Similarly for the trace of R (−) , where R (−) is the curvature of the connection ∇ (−) . The four-form P is proportional to the difference of the Pontrjagin forms of the tangent bundle of spacetime and Yang-Mills bundle of the heterotic string. The string frame field equations of the heterotic string up to two-loops [19] in sigma model perturbation theory are R M N + 1 4 H R M L H L N R + 2∇ M ∂ N Φ + α ′ 4 [R (−) M P QR R (−) N P QR − F M P ab F N P ab ] + O(α ′2 ) = 0 ∇ M (e −2Φ H M RL ) + O(α ′2 ) = 0 ∇ (+)M (e −2Φ F M N ) + O(α ′2 ) = 0 , (2.3) where we have suppressed the gauge indices. The field equation of the dilaton Φ is implied from the first two equations above. Our curvature conventions are given in appendix A. Let {Γ M ; M = 0, . . . , 9} be a basis of the Clifford algebra Cliff(R 1,9 ), i.e. Γ M Γ N + Γ N Γ M = 2G M N . Then the string frame Killing-spinor equations 3 are [10,21] ∇ (+) ǫ + O(α ′2 ) = 0 (Γ M ∂ M Φ − 1 12 H M N R Γ M N R )ǫ + O(α ′2 ) = 0 F M N Γ M N ǫ + O(α ′2 ) = 0 , (2.4) where ǫ is a section of the spin bundle S + . 4 It is clear that the first Killing spinor equation is a parallel transport equation for the connection ∇ (+) . Since the connection of the spin bundle S + is induced from the tangent bundle of spacetime, the investigation of this Killing-spinor equation is greatly simplified. The first, second and third Killingspinor equations are associated with the supersymmetry transformations of the gravitino, dilatino and gaugino, respectively. We shall use this terminology in what follows to distinguish between them. It is clear from the field equations that the two-loop contribution to the Einstein equations is at the same order as the modification of the torsion H due to the cancellation of the heterotic anomaly. Consistency then requires that both should be taken into account. The various field and Killing-spinor equations are expected to receive corrections to all orders in α ′ . Therefore a solution of the field and/or Killing-spinor equations of the effective supergravity theory can be expanded as G = g + α ′ h + O(α ′2 ) H = T + α ′ f + O(α ′2 ) Φ = ϕ + α ′ φ + O(α ′2 ) A = B + α ′ Q + O(α ′2 ) . (2.5) In this expansion, the fields (g, T, ϕ, B) solve the field and the Killing-spinor equations at zeroth-order in α ′ . We again remark that dT = 0 although dH may not vanish, dH = 0. The deformation (h, f, φ, Q) linear in α ′ is the first-order correction to the background (g, T, ϕ, B). Of course, the fields receive higher-order corrections in α ′ . In what follows, we determine the deformations (h, f, φ, Q) by requiring that (G, H, Φ, A) in (2.5) solve the field (2.3) and Killing-spinor (2.4) equations. 3 The α ′ corrections to backgrounds with torsion 3.1 World-sheet and spacetime supersymmetry We restrict our attention to heterotic string backgrounds of the form ds 2 = ds 2 (R 10−2n ) + ds 2 (X n ) T = 1 3! T ijk (y)dy i ∧ dy j ∧ dy k ϕ = ϕ(y) B = B i (y)dy i (3.6) where {y i ; i = 1, . . . , 2n} are coordinates on a manifold X n , n ≤ 4, and dT = 0 as we have explained in the introduction. In addition, we require that the background (g, T, ϕ, B) be compatible with (2,0) world-sheet supersymmetry. This means that the light-cone gauged fixed string worldsheet action is (2,0)-supersymmetric. In particular this implies that X n is a hermitian manifold, (X n , J, g), with complex structure J which is parallel with respect to ∇ (+) connection, i.e. (X n , J, g) is a KT manifold. The torsion T of KT manifolds is specified by the metric g and the complex structure J, see appendix A. The background (3.6) is expected to receive α ′ corrections because the supergravity field equations are modified by two-and higher-loop contributions in sigma model perturbation theory and in particular by the heterotic anomaly-cancellation mechanism. After these corrections are included, the background is expected to be of the form ds 2 = ds 2 (R 10−n ) + ds 2 (X n ) H = 1 3! H ijk (y)dy i ∧ dy j ∧ dy k Φ = Φ(y) A = A i (y)dy i ,(3.7) where ds 2 (X n ) = G ij (y)dy i dy j . The three-form H is not necessarily closed, because of (2.2). As we have seen, sigma model loop effects and the heterotic anomaly cancellation mechanism alter the geometry of the manifold X n . Nevertheless, it is expected that if the original manifold (X n , J, g) has a KT structure, the geometry, after the corrections are taken into account, remains KT. So the manifold (X n , J, G) has a KT structure as well but now the torsion H is not closed. This is because it is expected that there is a scheme which preserves the (2,0) world-volume supersymmetry in sigma model perturbation theory [22]. A solution (g, T, ϕ, B) of the zeroth order in α ′ field equations associated with KT manifold (X n , J, g) does not necessarily satisfy the Killing-spinor equations (2.4) of supergravity theory. The conditions for (g, T, ϕ, B) to satisfy the gravitino, dilatino and gaugino Killing-spinor equations [10,15] are hol(∇ (+) ) ⊆ SU(n) , θ = 2dϕ F (B) 2,0 = F (B) 0,2 = 0 , Ω ij F (B) ij = 0 , (3.8) where hol(∇ (+) ) is the holonomy of the connection ∇ (+) , Ω ij = g jk J k j is the Kähler form and θ is the Lee form of the Hermitian geometry. (The Lee-form has been given in appendix A). KT manifolds for which the Lee-form is exact are called conformally balanced. The conditions on the curvature F (B) of the gauge connection B required by the gaugino Killing-spinor equations imply that F is a (1,1)-form with respect to the complex structure J and its trace with Ω vanishes. I.e. considered as a two-form F (B) takes values in the Lie algebra of SU(n). These conditions are the analogue of the Donaldson equations for Hermitian manifolds. It can be shown that the backgrounds of (3.8) preserve 2 1−n of spacetime supersymmetry. Conversely if (g, T, ϕ, B) in (3.6) satisfies the Killing-spinor equations (2.4) preserving 2 1−n of spacetime supersymmetry, then X n is a conformally balanced KT manifold and the holonomy of ∇ (+) is contained in SU(n). As we have explained, the geometry of the background (G, H, Φ, A) (3.7) is expected to be KT. However, it is not apparent that if the (g, T, ϕ, B) background is spacetime supersymmetric, then (G, H, Φ, A) will also be spacetime supersymmetric. The corrections to Killing-spinor equations of supergravity (3.8) at order O(α ′ ) are determined by the corrections to the metric and the torsion but otherwise their dependence on the metric and the torsion remains the same 5 . This has the consequence that if we insist that the corrected background (G, H, Φ, A) preserve the same number of supersymmetries as (g, T, ϕ, B), then (X n , J, G) is again a KT manifold for which hol∇ (+) ⊆ SU(n), θ = 2dΦ and F (A) 2,0 = F (A) 0,2 = 0, Ω ij F ij = 0. In this case, ∇ (+) , θ and Ω are the connection, Lee form and Kähler form of the metric G, respectively. We conclude that at linear order in α ′ , the corrections to the geometry of X n are deformations which preserve the following two properties: • X n is a conformally balanced KT manifold and • the holonomy ∇ (+) is contained in SU(n). In what follows, we derive the conditions on the deformations of the geometry which preserve the above properties and we solve the field and Killing-spinor equations to first order in α ′ . We also present a similar analysis for the conditions on the gauge connection. Gravitino and dilatino Killing-spinor equations As we have mentioned, in order to solve the gravitino and dilatino Killing-spinor equations, we have to specify the deformations (G, H) = (g + α ′ h, T + α ′ f ) which preserve the properties that (X n , J, g) is conformally balanced KT manifold and hol(∇ (+) ) ⊆ SU(n). First, we consider deformations which preserve the hermiticity of the metric with respect to the complex structure J. This means that h αβ = 0, where α, β = 1, . . . , n are labels for the holomorphic coordinates on X n . It can then be easily shown that hol(∇ (+) ) ⊆ U(n) provided that the deformation for the torsion is f αβγ = −∇ α h βγ + ∇ β h αγ , (3.9) where ∇ is the Levi-Civita connection of the metric g, fᾱβ γ = (f αβγ ) * and the rest of the components vanish. The latter is required because the torsion of a KT geometry is a (2,1)-and (1,2)-form. Furthermore, the deformation of the connection of the canonical bundle 6 of X n associated with ∇ (+) is ω α (G) = ω(g) α + α ′ [2i∇ β h α β − i∇ α h β β + iTδ βγ g βγ h αδ − iT αβγ h βγ ] + O(α ′2 ) (3.10) where ω(g) is the connection of the canonical bundle associated with the connection 7 ∇ (+) (g) and ωᾱ = (ω α ) * . A necessary and sufficient condition for hol(∇ (+) ) ⊆ SU(n) is that the curvature of the canonical bundle vanishes. For the connection (3.10), a sufficient condition is 2i∇ β h α β − i∇ α h β β + iTδ βγ g βγ h αδ − iT αβγ h βγ = 0 , (3.11) where ∇ = ∇(g). It remains to find the condition required for X n to remain conformally balanced after the deformation. For this, we compute the first-order deformation of the Lee form to find θ α = θ(g) α + α ′ [−∇ α (g βγ h βγ ) − ∇ β h αγ g βγ + 1 2 T αβγ h βγ − 1 2 Hδ βγ g βγ h αδ , (3.12) where θ(g) is the Lee-form of the (g, J) geometry. Substituting (3.11) in (3.12), we find that θ α = θ(g) α + α ′ 2 ∇ α (g βγ h βγ ) (3.13) The dilatino Killing-spinor equation can be solved by setting φ = 1 4 g βγ h βγ . (3.14) Therefore the dilaton is deformed to Φ = ϕ + α ′ 4 g βγ h βγ + O(α ′2 ) . The equations that remain to be solved are those in (3.11). Observe that these equations are 2n in number, i.e. as many as the diffeomorphisms of X n . Since there is some redundancy in specifying the deformation h up to an infinitesimal diffeomorphism generated by the vector field v, i.e. h ′ αβ = h αβ +∇ α vβ +∇βv α , it is expected on physical grounds that it is always possible to choose an h so that (3.11) is satisfied. Therefore (3.11) can be viewed as a choice of gauge fixing for diffeomorphisms of X n . In the following, we provide more evidence that this is a good gauge choice. It remains to examine the conditions on the gauge connection. We postpone this for after the investigation of the field equations of the metric and the two-form gauge potential. The solutions of field equations Having derived the conditions for the deformations to satisfy the gravitino and dilatino Killing-spinor equations, we now focus on the solutions of the field equations for the metric and the NS two-form potential. In particular we show that at order α ′ both these field equations are satisfied provided that the heterotic anomaly-cancellation condition holds. Assuming that the background (g, T, ϕ) satisfies the field equations at zeroth order in α ′ , substituting (2.5) in the field equation for the metric (2.3) and collecting the terms linear in α ′ , we find ∆ L h ij − 1 4 T imn f j mn − 1 4 T jmn f i mn + 1 2 h mn g kl T imk T jnl + 2∇ i ∂ j φ − g kl (∇ i h jk + ∇ j h ik − ∇ k h ij )∂ l φ 0 + S ij = 0 , (3.15) where ∆ L is the Lichnerowicz operator with respect to the metric g (see appendix B) and S ij = 1 4 [R (−) iklm R (−) j klm − F ikab F j kab ] is the two loop contribution to the beta function. The curvature R (−) is with respect to (g, T ) and F = F (B). Clearly, (3.15) is rather involved. To proceed, we take the original background (g, T, ϕ, B) to be spacetime supersymmetric in the sense described in the previous section. In addition we assume that after the deformation the background remains supersymmetric. As we have seen this is equivalent to requiring that the KT geometry (X n , J, G) be conformally balanced and hol(∇(G) (+) ) ⊆ SU(n). Substituting the deformation (2.5) in (A.4) of appendix A and collecting the terms linear in α ′ , we find that ∆ L h ij − 1 4 T imn f j mn − 1 4 T jmn f i mn + 1 2 h mn g kl T imk T jnl + 2∇ i ∂ j φ − g kl (∇ i h jk + ∇ j h ik − ∇ k h ij )∂ l φ 0 = 1 4 J k i df kjmn Ω mn (3.16) Observe that there is no explicit contribution of the metric deformation on the righthand-side of (A.4). This is because at zeroth order the torsion is closed, dT = 0. Using (3.15) and (3.16), we find that 1 4 J k i df kjmn Ω mn + S ij = 0 . (3.17) Next consider the anomaly-cancellation condition (2.2) which to linear order in α ′ can be written as df = −P , (3.18) where P depends on (g, T, B). Since we have assumed that the background (g, T, ϕ, B) is supersymmetric, R (−) and F satisfy the conditions R (−) mn i j J m k J n l = R (−) kl i j , Ω mn R (−) mn i j = 0 F mn a b J m k J n l = F kl a b , Ω mn F mn a b = 0 . (3.19) These conditions on R (−) can be easily deduced from the fact that the holonomy of ∇ (+) is contained in SU(n) and R (−) ij,kl = R (+) kl,ij provided that the torsion is closed (dT = 0). The conditions on F can be deduced from the Killing-spinor equations of the gaugino. Contracting the anomaly-cancellation condition 8 (3.18) with the Kähler form Ω and using (3.19), it is easy to see that the field equation for the metric (3.17) is satisfied. In order to solve the field equations for the metric, it is sufficient to solve the anomalycancellation condition (3.18). Substituting (3.9) in (3.18), we find that P = −2i∂∂Y (3.20) where Y ij = h ik J k j . The global anomaly-cancellation condition requires that P be exact. Since P is an exact, real (2,2)-form, if the ∂∂-lemma is valid on X n , then there is a real (1,1)-form Y globally defined on X n such that (3.20) is satisfied. On Hermitian manifolds which are not Kähler, the ∂∂-lemma is not valid in general. Therefore a sufficient condition for the existence of spacetime supersymmetric solutions in backgrounds with non-vanishing torsion, T = 0, is the validity of the ∂∂-lemma. The solution to the anomaly-cancellation condition (3.20) is not unique. Indeed, if Y satisfies (3.20), then Y ′ = Y + ∂w +∂w (3.21) where w is a (1,0)-form, is also a solution. This gauge freedom in Y is equivalent to specifying the deformation h in the metric up to an infinitesimal diffeomorphism. This can be easily seen by setting v = −iw. Therefore having determined Y from (3.20), we still have the gauge freedom to solve the supersymmetry condition (3.11). The solution of (3.20) up to a gauge transformation (3.21) is not unique. The classes of independent solutions are described by the Aeppli group V 1,1 (X n ) defined by V 1,1 = Ker(i∂∂ : Λ 1,1 (X n ) → Λ 2,2 (X n )) ∂Λ 0,1 (X n ) +∂Λ 1,0 (X n ) , see [24] for a related discussion. The dimension of this group is the dimension of the moduli space of solutions of (3.20). However it is not apparent that all elements of this group are associated with spacetime-supersymmetric deformations. The latter should in addition satisfy (3.11). It remains to show that the field equations of the NS two-form gauge potential are satisfied as well. The proof of this is based on an identity shown in [16] (corollary 3.2) which can be stated as follows: Let (X n , J, G) be a conformally balanced KT manifold with torsion H, dH = 0, and hol(∇ (+) ) ⊆ SU(n), then ∇ i H ijk = θ i H i jk . (3.22) This statement is valid irrespectively of whether or not G is a small perturbation of another metric g. Both KT structures (X n , J, g) and (X n , G, J) satisfy the aforementioned conditions because they are supersymmetric. Therefore both satisfy (3.22) with their respective torsions and Lee forms. Using (3.22) for the α ′ corrected background (G, H, Φ, A), the field equation (2.3) for the NS two-form − 2∂ i ΦH i jk + ∇ i H ijk + O(α ′2 ) = 0 (3.23) can be written as (θ i − 2∂ i Φ)H i jk + O(α ′2 ) = 0 which vanishes identically because (X n , J, G) is conformally balanced. Since the field equations for the background (g, T, ϕ, B) are satisfied by assumption and as we have shown the field equations (3.23) for (G, H, Φ, A) are satisfied as well, the part of (3.23) linear in α ′ vanishes identically. Therefore the field equations for the NS two-form gauge potential are satisfied without additional conditions on the deformation h of the metric. The gauge field The main purpose of this section is to show that the Killing-spinor equations of the gaugino imply the field equations of the gauge field before and after the α ′ corrections are taken into account. The simplest way to show this is by investigating the properties of gauge fields on conformally balanced KT manifolds. Gauge Fields on KT conformally balanced manifolds We first describe the well-known relation between the Donaldson equations and the field equations of a gauge connection on a Kähler manifold. Let E be a vector bundle over a Kähler manifold (M, J, G) equipped with a connection A with curvature F . If A satisfies the Donaldson equations F 0,2 = F 2,0 = 0 , Ω ij F ij = 0 ,(4.24) then it is straightforward to show that A solves the field equations ∇ i F ij = 0 . (4.25) The proof makes use of the Jacobi identities for F . Donaldson has shown that if E is a stable bundle over a complex surface M, then there is a unique connection which solves (4.24). Next suppose that E is a vector bundle over a non-Kähler conformally-balanced KT manifold (M, J, G) equipped with a connection A with curvature F . Donaldson equations (4.24) can be easily generalised to KT manifolds by allowing Ω to be the Kähler form of the Hermitian metric G. We shall show that the Donaldson equations imply the field equations ∇ (+)i (e −2Φ F (A) ij ) = 0 ,(4.26) where ∇ (+) is the connection of the KT structure (M, J, G) with torsion H and Lee form θ = 2dΦ. For this we choose complex coordinates with respect to the complex structure J and rewrite (4.26) as −2∇ γ G γβ Fβ α + G γβ ∇ γ Fβ α − 1 2 Hδ γβ Fδ α G γβ − 1 2 H δ γα Fβ δ G γβ = 0 The Jacobi identities imply that ∇ γ Fβ α + ∇ α F γβ = −∇βF αγ where ∇βF αγ = 1 2 Hδβ α Fδ γ + 1 2 Hδβ γ F αδ , and ∇ is the Levi-Civita connection of the metric G. Collecting the various terms together, we find that (θ γ − 2∂ γ Φ)G γβ Fβ α = 0 which vanishes identically because (M, J, G) is conformally balanced, θ = 2dΦ. Therefore equations (4.24) together with the Jacobi equations imply the field equations (4.26). The gauge field equations We shall use the result of the previous section to show that the field equations of the gauge field are satisfied provided that the Killing-spinor equations (2.4) are satisfied. Assuming that the background (g, T, ϕ, B) and its deformation (G, H, Φ, A) satisfy the Killing-spinor equations (2.4), the KT structures (X n , J, g) and (X n , J, G) are conformally balanced and the holonomies of their ∇ (+) connections are contained in SU(n). In addition both gauge connections B and its deformation A satisfy the conditions (4.24)-the latter up to linear order in α ′ . Applying the theorem proven in the previous section, we conclude that the background (g, T, ϕ, B) and its deformation (G, H, Φ, A) satisfy the field equations (4.26)-the latter up to linear order in α ′ . Expanding (4.26) in α ′ for the background (G, H, Φ, A) and since (4.26) is satisfied at zeroth order in α ′ , the linear term in α ′ will vanish identically. Thus if the background (g, T, ϕ, B) and its deformation (G, H, Φ, A) satisfy the Killing-spinor equations, then the field equations (2.3) for the gauge potential are satisfied without additional conditions on the deformations. The conditions on the deformation Q of the gauge connection B are ∇ α Q β − ∇ β Q α = ∇ᾱQβ − ∇βQᾱ = 0 h αβ F (B) αβ + g αβ ∇ α Qβ − ∇βQ α = 0 ,(4.27) where the covariant derivative ∇ is with respect to the gauge connection B. We have derived these by substituting the deformation (G, H, Φ, A) of (g, T, ϕ, B) in the Killingspinor equations (4.24) and by collecting the terms linear in α ′ . It remains to find whether the Killing-spinor equations for the gaugino or equivalently (4.24) have solutions on a general conformally balanced KT manifold. It is easy to investigate the case where A is an abelian connection. However, the non-abelian case is more involved and so we shall not pursue this further. (2,0) heterotic compactifications The compactification ansätze for the heterotic string which preserve (2,0) world-volume supersymmetry and are spacetime supersymmetric, are given in (3.6) with the additional assumption that the internal space X n is compact 9 . As we have discussed, such backgrounds are expected to receive α ′ corrections. Using the machinery developed in the previous sections, we shall investigate the deformations of these backgrounds due to α ′ corrections. The zeroth-order solution As we have explained the requirement for a compactification to preserve (2,0) world-sheet supersymmetry and 2 1−n of spacetime supersymmetry in (10 − 2n)-dimensions leads to an internal manifold X n with a conformally balanced KT structure and hol(∇ (+) ) ⊆ SU(n). The additional requirement that X n is compact 10 , the implicit assumption that all the fields (g, T, ϕ) are smooth on X n and the fact that at zeroth order in α ′ dT = 0, impose strong restrictions on the geometry of X n . It has been shown in [15] under the above assumptions 11 that X n is Calabi-Yau , T = 0 and the dilaton ϕ is constant. In addition at zeroth order in α ′ the gauge connection B satisfies the Donaldson equations (4.24) on the Calabi-Yau manifold X n . These data are the starting point of our α ′ expansion. The Calabi-Yau background (g, ϕ, B) receives α ′ corrections. The deformation (G, H, Φ, A) of (g, ϕ, B) has non-vanishing torsion H which is not closed, dH = 0, as required by the anomaly-cancellation mechanism. Note that the zeroth-order term in H vanishes and so H is purely first-order in α ′ , H = α ′ f + O(α ′2 ). The first-order solution To find the correction (G, H, Φ, A) to the Calabi-Yau geometry (g, ϕ, B), we have to solve equations (3.11), (4.27) for the first-order deformations (h, f, φ, Q). The deformation φ to the dilaton is given in (3.14), φ = 1 4 g βγ h βγ . Similarly, the deformation f of the torsion is given in (3.9). The remaining field and Killing-spinor equations are satisfied without further conditions. Since for this background the torsion vanishes at zeroth order in α ′ , condition (3.11) arising from the requirement that hol(∇ (+) ) ⊂ SU(n), can be rewritten as ∇βh αβ − 1 2 ∇ α (g γβ h γβ ) = 0 . (5.28) The above equation can be thought of as a gauge-fixing condition for the deformations associated with infinitesimal diffeomorphisms of the underlying manifold. Condition (5.28) can always be attained. To see this, first write (5.28) in real coordinates ∇ j h ji − 1 4 ∇ i h j j = 0 . (5.29) Suppose that h does not solve (5.29). We shall show that there is a v such that h ′ given by h ′ ij = h ij + ∇ i v j + ∇ j v i satisfies (5.29). v is determined by ∇ k ∇ k v i + 1 2 ∇ i ∇ k v k = ∇ j h ji − 1 4 ∇ i h j j . (5.30) First note that the Kernel of the operator on the left-hand-side of the equation above is zero on an irreducible Calabi-Yau manifold. Indeed let v be in the Kernel. Then Xn (v i ∇ k ∇ k v i + 1 2 v i ∇ i ∇ k v k ) dvol = − M ∇ k v i ∇ k v i + 1 2 (∇ k v k ) 2 dvol = 0 . Thus v is parallel with respect to the Levi-Civita connection. Since X n is irreducible, there are no parallel one-forms on X n and so the Kernel vanishes. Therefore equation (5.30) can be solved for v since the right-hand-side does not vanish, by assumption. We can also determine the metric moduli of (2,0) compactifications. Since the ∂∂lemma applies for Calabi-Yau manifolds, the Aeppli group V 1,1 (X n ) is isomorphic to the the Hodge group H 1,1 (X n ). Thus the dimension of the moduli space is the Hodge number h 1,1 which is the same as the number of metric moduli of Calabi-Yau manifolds. Of course one can also take into account the moduli associated with including in the theory NS twoform gauge potentials. The latter are harmonic (1,1)-forms. This leads to a complex moduli space of real dimension 2h 1,1 . One concludes that the metric moduli of Calabi-Yau (2,0)-compactifications, which are associated with NS fluxes induced by the heterotic anomaly, are not lifted. An effect of the heterotic anomaly-cancellation mechanism is a shift in the origin of the moduli space. The moduli of the theory associated with deformations of the complex structure of the underlying Calabi-Yau manifold is not lifted either. The analysis that we have done using the complex structure J can be repeated with any other complex structure on the Calabi-Yau manifold. Therefore, we conclude that the presence of NS flux associated with the heterotic anomaly cancellation mechanism does not lift the Calabi-Yau moduli at this order in α ′ perturbation theory. The α ′ -corrected conifold The conifold is a singular non-compact Calabi-Yau threefold [17]. Here we apply the machinery of the previous sections to compute the α ′ corrections explicitly. The conifold can be thought of as a Ricci-flat cone over a T 1,1 space, where the latter is a particular U(1) fibration over S 2 × S 2 [25]. Let 0 ≤ φ i ≤ 2π, 0 ≤ θ i ≤ π, i = 1, 2 be angular coordinates parametrising the two spheres S 2 , 0 ≤ ψ ≤ 4π be the coordinate on the U(1) fibre and ρ ≥ 0 be the radial coordinate. The line element of the conifold is ds 2 = g mn dx m dx n = dρ 2 + ρ 2 9 (dψ + cosθ 1 dφ 1 + cosθ 2 dφ 2 ) 2 + ρ 2 6 (sin 2 θ 1 dφ 2 1 + dθ 2 1 ) + ρ 2 6 (sin 2 θ 2 dφ 2 2 + dθ 2 2 ) . (6.31) The second term on the right-hand-side is the vertical displacement along the U(1) fibre whereas the last two terms represent the line element of S 2 × S 2 . There is a conical singularity at ρ = 0, where the T 1,1 base of the cone shrinks to zero size. It is useful to note that T 1,1 is topologically S 2 × S 3 . Let us now consider a deformation g mn → g mn + α ′ h mn , where h mn is hermitian, h kl J κ m J l n = h mn . (6.32) The complex structure J of the conifold is J θ 1 φ 1 = sinθ 1 , J ρ φ 1 = − ρ 3 cosθ 1 , J θ 2 φ 2 = sinθ 2 , J ρ φ 2 = − ρ 3 cosθ 2 , J φ 1 θ 1 = − 1 sinθ 1 , J ψ θ 1 = cotθ 1 , J φ 2 θ 2 = − 1 sinθ 2 , J ψ θ 2 = cotθ 2 , J ρ ψ = − ρ 3 , J ψ ρ = 3 ρ , and the rest of the components vanish. Condition (6.32) implies that h mn is of the form, h mn dx m dx n = D[dρ 2 + ρ 2 9 (dψ + cosθ 1 dφ 1 + cosθ 2 dφ 2 ) 2 ] + A(sin 2 θ 1 dφ 2 1 + dθ 2 1 ) + C(sin 2 θ 2 dφ 2 2 + dθ 2 2 ) + 2Bsinψ(sinθ 1 dφ 1 dθ 2 − sinθ 2 dφ 2 dθ 1 ) + 2Bcosψ(dθ 1 dθ 2 + sinθ 1 sinθ 2 dφ 1 dφ 2 ) . (6.33) We take A, B, C, D to be functions of the radial coordinate alone. With this assumption, it can be seen from the form of most general T 1,1 metric given in [26], that (6.33) is a foliation of T 1,1 spaces. The gauge-fixing condition (5.29) is equivalent to 0 = A + C + 3 2 ρ(A + C) ′ − 1 12 ρ 3 − 2 3 ρ 2 F 0 = B (6.34) In order to solve the Einstein equation, we need to find S mn := 1 4 R m kls R nkls . After some computation, we get S φ 1 φ 1 = S φ 2 φ 2 = sin 2 θ ρ 2 , S θ 1 θ 1 = S θ 2 θ 2 = 1 ρ 2 . (6.35) The Einstein equation (3.15) (or its gauged-fixed version (C.4) of appendix C) is then equivalent to the following set of conditions, 0 = B 0 = A + C + ρ 2 24 (−8D + 5ρD ′ + ρ 2 D ′′ ) 0 = 6ρA ′ + 6ρ 2 A ′′ + ρ 2 (−4D + 5ρD ′ + ρ 2 D ′′ ) − 3 0 = A ′ + ρA ′′ − C ′ − ρC ′′ (6.36) Demanding that the total metric be asymptotic to the conifold in the region ρ/ √ α ′ → ∞, the general solution to order O(α ′ ) reads A = − 1 16 ( 3 2 − c 1 − c 2 log ρ ρ 0 ) B = 0 C = − 1 16 ( 3 2 + c 1 + c 2 log ρ ρ 0 ) D = − 3 8ρ 2 (6.37) where ρ 0 is a dimensionful constant introduced to make the constants c 1 , c 2 dimensionless. Note that the gauge-fixing condition (6.34) is automatically satisfied by the solution (6.37). Therefore the solution is spacetime supersymmetric. To summarise, the O(α ′ )-perturbed metric is ds 2 = (1 − 3α ′ 8ρ 2 )[dρ 2 + ρ 2 9 (dψ + cosθ 1 dφ 1 + cosθ 2 dφ 2 ) 2 ] + ρ 2 6 [1 − 3α ′ 8ρ 2 ( 3 2 − c 1 − c 2 log ρ ρ 0 )](sin 2 θ 1 dφ 2 1 + dθ 2 1 ) + ρ 2 6 [1 − 3α ′ 8ρ 2 ( 3 2 + c 1 + c 2 log ρ ρ 0 )](sin 2 θ 2 dφ 2 2 + dθ 2 2 ). (6.38) The perturbation is valid in the regime ρ/ √ α ′ >> 1. In order to be able to extract any information about the behavior of the metric near the apex (ρ = 0), the full tower of α ′ corrections would have to be taken into account as these become important in the sub-stringy regime ρ/ √ α ′ ≤ 1. This is clearly beyond the validity of our analysis. Nevertheless, we can attempt to extrapolate our result to the region ρ/ √ α ′ ∼ 1. An analysis then reveals that at a radial distance of the order of √ α ′ , an S 2 in the T 1,1 base shrinks to zero volume. We therefore find that the singularity of the conifold persists, although it becomes milder in the sense that not the entire base shrinks to zero. 7 The α ′ -corrected U (n)-invariant Calabi-Yau metric To describe the U(n)-invariant Calabi-Yau metric [18], we introduce complex coordinates {z α ; α = 1, . . . , n} and consider the U(n) invariant Hermitian metric ds 2 = A(r 2 )dz · dz + B(r 2 )z · dz z · dz , where r 2 = δ αβ z α zβ, dz · dz = δ αβ dz α dzβ and similarly for the rest. The metric is Kähler if B = A ′ where the prime denotes differentiation with respect to r 2 . The Levi-Civita connection one-form is Γ α β = A −1 A ′ (δ α βz · dz +z β dz α ) + A ′′ − 2A −1 (A ′ ) 2 A + r 2 A ′ z αz βz · dz The holonomy of the above connection is contained U(n). The metric is Calabi-Yau if and only if the holonomy of the above connection is in SU(n). The connection of canonical bundle is ω α = i∂ α lndet(g αβ ) , where det(g αβ ) = A n−1 (A + r 2 A ′ ). The curvature of canonical bundle vanishes iff A n + r 2 n (A n ) ′ = λ , where λ is a constant. The most general solution of this equation is A n = λ + c r 2n , where c is constant. This metric is the Calabi-Yau metric on the orbifold C n /Z n after resolving the singularity at the origin by replacing with a CP 2 . Next we compute P = 1 4 trR 2 to find P = 1 4 C(dz ∧ dz) ∧ (dz ∧ dz) + 1 4 C ′z · dz ∧ z · dz ∧ (dz ∧ dz) , where C = (n + 1)A −2 (A ′ ) 2 + 2A −1 A ′ r 2 A ′′ − 2A −1 (A ′ ) 2 A + r 2 A ′ + r 4 A ′′ − 2A −1 (A ′ ) 2 A + r 2 A ′ 2 and (dz ∧ dz) = δ αβ dz α ∧ dzβ. After some computation, we find that C = n(n + 1) c λr 2n+2 + cr 2 2 In addition setting h = D(r 2 )dz · dz + E(r 2 )z · dz z · dz , and Y αβ = −ih αβ , we find that P = −2i∂∂Y implies D ′ − E = − 1 8 C . (7.39) Condition (5.28) for h, gives 2(n − 1)A −1 E − (n − 1)A −1 D ′ − (n − 1)A −2 A ′ D + 1 λ [A n−1 (D + r 2 E)] ′ = 0 . (7.40) The supersymmetric deformation can be computed by solving (7.39) and (7.40). Substituting for E in (7.40), we find after some algebra that 0 = − c 2 n(n + 1) 8λr 4n+4 3λ + nc r 2n (λ + c r 2n ) 2 − c 2 (n − 1) λr 4n+2 1 (λ + c r 2n ) D + [ n − 1 λ (λ − c r 2n ) + 2(λ + c r 2n )]D ′ + r 2 λ (λ + c r 2n )D ′′ . (7.41) We have not been able to obtain a closed form for D. But we can solve the equation perturbatively in a large distance expansion. The result reads, E = n(c/λ) 2 8r 4n+4 (n − 2 + −5n 2 + 8n + 1 2n + 1 (c/λ) r 2n + . . . ) D = n(c/λ) 2 8r 4n+2 ( 3 2n + 1 + n 2 − 14n − 3 (2n + 1)(3n + 1) (c/λ) r 2n + . . . ) (7.42) Note that the leading correction to the metric behaves like r −4n−2 . Concluding Remarks Compactifications with fluxes lead to lower-dimensional effective theories which exhibit potentials lifting some of the moduli. The relevant flux for (2,0) heterotic-string compactifications is the NS three-form H. There are two possibilities. One is to allow for a non-vanishing flux at zeroth order in α ′ . Then the internal manifold is either noncompact or/and some of the fields are singular. We have extensively investigated the case of Calabi-Yau compactifications where the NS flux vanishes at zeroth order in α ′ . As we have shown such compactifications develop a non-vanishing flux H and a non-constant dilaton at first order in α ′ . In addition, at this order in α ′ , the compactifications have the same dimension of moduli space as that of the moduli space of the Calabi-Yau manifolds at zeroth order in α ′ and so there is no lifting of moduli. The effective lower-dimensional theories which arise in standard Calabi-Yau compactifications, i.e. without NS flux, may be different from those with flux. If there is no NS flux induced by the heterotic anomaly-cancellation mechanism, then it is consistent to take the lower-dimensional effective action to be the standard ten-dimensional N=1 supergravity action. Of course there will be higher-order α ′ corrections. Nevertheless it is consistent to consider only the zeroth order in α ′ . On the other hand, if one wishes to consider the NS fluxes induced by the heterotic anomaly-cancellation mechanism, consistency requires that curvature square terms in the field equations and their supersymmetric completion should be considered as well. These will contribute to the lower-dimensional effective theories. The other possibility is to introduce a non-vanishing H flux at zeroth order in α ′ and to allow for some fields to be singular or/and for the internal manifold to be noncompact. In the case that the manifold or the fields are singular, one may expect that these singularities can be resolved by taking into account α ′ corrections. We have seen, for example, that after taking into account the linear order α ′ correction to the conifold, the metric is less singular but the singularity is not completely resolved. In a scenario where the singularities are removed and the internal manifold remains compact, some moduli may be lifted and potentials of the type given in [28] may be generated in the effective theory in lower dimensions. Similar conclusions can be drawn for compactifications of other theories with fluxes in the presence of an anomaly-cancellation mechanism. Such an example is M-theory [27]. To investigate the consistency of the anomaly cancellation with spacetime supersymmetry as we have done for the heterotic string, one needs to know the higher-order derivative corrections to D = 11 supergravity and their supersymmetric completions. It is of interest to speculate on the geometry of the background (3.6) after all α ′ corrections are taken into account. It is expected that such a background is a Hermitian manifold, based on the assumption that (2,0) world-sheet supersymmetry is preserved in sigma-model perturbation theory. It is unlikely that the supergravity Killing-spinor equations will remain of the form (2.4). The dependence on the metric and the torsion will change and higher-curvature terms are expected to appear [23]. So the conditions for preserving spacetime supersymmetry will not be simply expressed as conditions on the holonomy of the ∇ (+) connection and on the Lee form θ of the Hermitian geometry. However, some properties of the underlying manifold may be preserved. It is plausible to assume that the manifold satisfies the ∂∂-lemma and the canonical bundle is holomorphically trivial. It has been shown in [24] that Moishezon manifolds with these properties admit a connection with skew-symmetric torsion and holonomy contained in SU(n). However, it is not apparent that the associated metric is that which arises after taking into account all α ′ corrections. In addition, on such non-Kähler Moishezon manifolds there is no KT structure for which the associated three-form field strength is closed [24]. Therefore such manifolds cannot be used as the starting point of the sigma-model perturbation theory. So ∆ L is the first-order deformation of the Ricci tensor. A calculation reveals that ∆ L h ij = − 1 2 ∇ 2 h ij − R ikjl h kl + 1 2 ∇ i ∇ k h kj + 1 2 ∇ j ∇ k h ki − 1 2 ∇ i ∇ j h k k + 1 2 R ki h k j + 1 2 R kj h k i (B.1) The Laplacian operator ∆ on a two form Y is ∆Y ij = − 1 2 ∇ k ∇ k Y ij − R ikjℓ Y kℓ + 1 2 R ik Y k j − 1 2 R jk Y k i . On Ricci-flat Kähler manifolds, we can relate the Lichnerowicz operator to the Laplace operator on two-forms by choosing the gauge fixing condition ∇ j h ji − 1 2 ∇ i h j j = 0 for infinitesimal diffeomorphisms and by using the relation Y ij = h ik J k i between symmetric (1,1)-tensors and (1,1)-forms. The Hermiticity condition for the metric implies that the deformations h of the metric are (1,1) tensors. Note that the above gauge can always be attained. Suppose that h does not satisfy the gauge. Assume that there is an infinitesimal diffeomorphism v such that h ′ ij = h ij + ∇ i v j + ∇ j v i satisfies the gauge. Then v is determined by the equation ∇ k ∇ k v i = ∇ j h ji − 1 2 ∇ i h j j . To show that there is always such a v, we must invert the above equation. This can be achieved iff ∇ j h ji − 1 2 ∇ i h k k is orthogonal to the kernel of the elliptic operator ∇ k ∇ k . We rewrite the above equation as (dδ + δd)V = −∇ j Y ji + 1 2 J k i ∇ k h j j , where V i = v k J k i and δ is the adjoint of d. Of course this equation can be inverted iff the right-hand-side is orthogonal to harmonic one-forms. Indeed let Z be a harmonic one-form, then M Z i −∇ j Y ji + 1 2 Ω ki ∇ k h j j dvol = − M ∇ i Z j Y ij + 1 2 ∇ i Z j Ω ij h k k dvol = 0 because Z is closed. Appendix C Another derivation of the field equations for (2,0) compactifications The first-order corrections (h, f, φ) to the background of a Calabi-Yau compactification (g, ϕ), T = 0, can be determined by the field equations without using the conditions for spacetime supersymmetry. To show this, we substitute (2.5) in the field equation (2.3) and collect the linear terms in α ′ . Using the fact that at zeroth order in α ′ the geometry is Calabi-Yau, the equation for the metric gives ∆ L h ij + S ij + 2∇ i ∂ j φ = 0 , (C.1) where ∆ L is the Lichnerowicz operator (see appendix B). In this case, we have ∆ L h ij = − 1 2 ∇ 2 h ij − R ikjl h kl + 1 2 ∇ i ∇ k h kj + 1 2 ∇ j ∇ k h ki − 1 2 ∇ i ∇ j h k k (C.2) because X n is Calabi-Yau and R ij = 0, and S ij = 1 4 [R iklm R j klm − F ikab F j kab ] (C.3) The covariant derivative ∇ in (C.1) and (C.2) and the curvature R are with respect to the Calabi-Yau metric g. To solve equation (C.1) with respect to h, we shall exploit the well-known relation between the Lichnerowicz and Laplace operators on Calabi-Yau manifolds. There are two ways to do this. One is to use the scheme dependence of the two-loop beta function and the other is to impose a gauge fixing condition on the deformations h of the metric g. The latter is common in moduli problems in order for the deformation h to be orthogonal to the orbits of infinitesimal diffeomorphisms. Using the scheme dependence of the two-loop beta function, we can arrange so that the term 1 2 ∇ i ∇ k h kj + 1 2 ∇ j ∇ k h ki + 1 2 r∇ i ∂ j h k k is cancelled by a wave function renormalization, where r is a real number. Alternatively, one can impose the gauge fixing condition ∇ k h ki + r 2 ∇ i h k k = 0 . Provided that we set for the deformation of the dilaton φ = 1 4 (r + 1)h k k , the remaining equation is − 1 2 ∇ 2 h ij − R ikjl h kl + S ij = 0 . (C.4) To solve this equation observe that S is a (1,1) symmetric tensor with respect to the complex structure. This allows us to consider deformations of the metric which are (1,1) as well, i.e. to take h to be a (1,1) symmetric tensor. It is well known that on Kähler manifolds there is a 1-1 correspondence between symmetric (1,1) tensors and (1,1) twoforms. Indeed define the forms Y ij = h ik J k j and Z ij = S ik J k j associated to T and S. Equation (C.4) becomes ∆Y + Z = 0 (C.5) where ∆Y ij = − 1 2 ∇ k ∇ k Y ij − R ikjℓ Y kℓ is the standard Laplace operator on Y (see appendix B). We have taken into account that Calabi-Yau manifolds are Ricci-flat, R ij = 0. So it remains to invert the Laplace operator to determine Y in terms of Z. To solve the equation (C.5) in terms of Y , we have to show that Z is orthogonal to the harmonic two-forms of X n in the Hodge decomposition with respect to the Calabi-Yau metric g. This can be achieved by relating the two-loop contribution Z to the beta function of the metric, to the heterotic anomaly P . Using Ω ij R ijkl = Ω ij F ijab = 0 , we can show that Z ij = 1 4 P ijmn Ω mn . Now suppose that n = 3. After some computation, we find that Z ij = − 1 2 * P ij + 1 16 Ω ij P mnpq Ω mn Ω pq . The cancellation of the global anomaly implies that P is exact. As a consequence the dual two-form * P is co-exact and so it is orthogonal to harmonic two-forms. It remains to show that P mnpq Ω mn Ω pq is not harmonic. Observe that a harmonic function on X 3 is a non-vanishing constant. Integrating the identity P ∧ Ω = 1 8 P mnpq Ω mn Ω pq dvol over the compact manifold X 3 and using that P is exact, it is easy to see that P mnpq Ω mn Ω pq is not harmonic. This result can be easily extended to four-and eight-dimensional internal manifolds. The n = 2 case has been investigated in [8]. For n = 4, the computation is similar to n = 3. The only difference is the relation between the two-loop counterterm and the anomaly. Next we investigate the field equation of the three-form field strength. Again substituting (2.5) in (2.3) and collecting the terms linear in α ′ , we find ∇ i f ijk = 0 or equivalently d † f = 0 , where d † is the adjoint operator of d. To derive this, we have used the fact that H vanishes at zeroth order in α ′ . In addition, we have that df = −P ; f is well-defined because P is exact. Using the Hodge decomposition, we can write f = f h + dX + d † W where f h is harmonic and W is a four-form. Adding the counterterm b = −α ′ X , the exact three-form dX can be eliminated as follows H = −α ′ dX + α ′ f = α ′ f h + α ′ d † W . Thus the field equation is satisfied and dH = −P . Alternatively, one can choose f such that f = f h + d † W . Our form conventions are ω k = 1 k! ω i1,...,i k dx i1 ∧ . . . ∧ dx i k . We have used the notation Γ M1...M k = Γ [M1 . . . Γ M k ] .4 The spin group Spin(1,9) has two inequivalent irreducible sixteen-dimensional spinor representations and S ± are the associated bundles. It is not expected that this property of the Killing spinor equations persists to all orders in sigma model perturbation theory. The dependence of the Killing-spinor equations on the metric and the torsion will change at higher orders[23]. This is the diagonal U (1) part of a U (n) connection on the tangent bundle.7 To avoid confusion, sometimes we use the notation ∇ (+) (g) to denote the connection ∇ (+) with respect to the metric g. This derivation of (3.17) from (3.18) is sensitive to the relative numerical coefficient of the two-loop contribution to the metric field equation and that of the anomaly-cancellation condition. The ansatz (3.6) was considered in[10] where it was shown that no warp factor is allowed for the non-compact part of the metric. Therefore, as is easy to see using sigma-model perturbation theory, there can be no such warp factor in (3.7) either.10 This assumption is sufficient for the spectrum in (10 − 2n) dimensions to be discrete.11 In fact in[15] it was shown that X n is Calabi-Yau even if dT = 0 provided a certain inequality holds. AcknowledgementsWe would like to thank P. Howe for stimulating discussions. This work was partially supported by PPARC grants PPA/G/S/1998/00613 and PPA/G/O/2000/00451 and by EU grant HPRN-2000-00122.Appendix A Useful Formulae for KT GeometryLet (X n , J, G) be a KT manifold, i.e. X n is a hermitian manifold of complex dimension n with metric G and complex structure J such that ∇ (+) J = 0, where ∇ (+) has skewsymmetric torsion H. In the mathematics literature ∇ (+) is called the Bismut connection. The holonomy of ∇ (+) is contained in U(n). In complex coordinates, the holonomy condition requires Γ (+) i αβ = 0 which in turn givesThe rest of the components of the torsion are determined by complex conjugation. The (3,0) and (0,3) components of H vanish as it can be seen from the integrability of the complex structure. So H is determined uniquely in terms of the metric and complex structure of (X n , J, G). The Lee form of the KT geometry iswhere Ω ij = G ik J k j is the Kähler form. In complex coordinates, the Lee form can be written asand θᾱ = (θ α ) * . The connection of the canonical bundle induced by ∇ (+) isand ωᾱ = (ω α ) * . Let ρ = dω be the curvature of the U(1) connection ω. The holonomy of the connection ∇ (+) is contained in SU(n), iff ρ = 0. A KT manifold is conformally balanced iff there is a function Φ on X n such that θ = 2dΦ, i.e. the Lee form is exact. 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[ ". Introduction Oxygen False Positives on Habitable Zone Planets Around Sun-Like Stars", ". Introduction Oxygen False Positives on Habitable Zone Planets Around Sun-Like Stars" ]	[ "Joshua Krissansen-Totton \nDepartment of Astronomy and Astrophysics\nUniversity of California\nSanta Cruz, Santa CruzCAUSA\n\nNASA Nexus for Exoplanet System Science\nVirtual Planetary Laboratory Team\nUniversity of Washington\nSeattleWAUSA\n\nNASA Sagan Fellow\n\n\nDepartment of Astronomy and Astrophysics\nUniversity of California\nSanta CruzCA\n\nVirtual Planetary Laboratory 3 NASA Sagan Fellow\n\n", "Jonathan J Fortney \nDepartment of Astronomy and Astrophysics\nUniversity of California\nSanta Cruz, Santa CruzCAUSA\n\nDepartment of Astronomy and Astrophysics\nUniversity of California\nSanta CruzCA\n", "Francis Nimmo \nDepartment of Earth and Planetary Sciences\nUniversity of California\nSanta Cruz, Santa CruzCAUSA\n\nDepartment of Earth and Planetary Sciences\nUniversity of California\nSanta CruzCA\n", "Nicholas Wogan \nNASA Nexus for Exoplanet System Science\nVirtual Planetary Laboratory Team\nUniversity of Washington\nSeattleWAUSA\n\nDepartment of Earth and Space Sciences\nUniversity of Washington\nSeattleWAUSA\n\nAGU Advances\n\n\nVirtual Planetary Laboratory 3 NASA Sagan Fellow\n\n\nDepartment of Earth and Space Sciences\nUniversity of Washington\nSeattleWA\n" ]	[ "Department of Astronomy and Astrophysics\nUniversity of California\nSanta Cruz, Santa CruzCAUSA", "NASA Nexus for Exoplanet System Science\nVirtual Planetary Laboratory Team\nUniversity of Washington\nSeattleWAUSA", "NASA Sagan Fellow\n", "Department of Astronomy and Astrophysics\nUniversity of California\nSanta CruzCA", "Virtual Planetary Laboratory 3 NASA Sagan Fellow\n", "Department of Astronomy and Astrophysics\nUniversity of California\nSanta Cruz, Santa CruzCAUSA", "Department of Astronomy and Astrophysics\nUniversity of California\nSanta CruzCA", "Department of Earth and Planetary Sciences\nUniversity of California\nSanta Cruz, Santa CruzCAUSA", "Department of Earth and Planetary Sciences\nUniversity of California\nSanta CruzCA", "NASA Nexus for Exoplanet System Science\nVirtual Planetary Laboratory Team\nUniversity of Washington\nSeattleWAUSA", "Department of Earth and Space Sciences\nUniversity of Washington\nSeattleWAUSA", "AGU Advances\n", "Virtual Planetary Laboratory 3 NASA Sagan Fellow\n", "Department of Earth and Space Sciences\nUniversity of Washington\nSeattleWA" ]	[]	The search for life beyond Earth is a key motivator for exoplanet astronomy. Among the various life detection approaches that have been proposed, atmospheric oxygen is arguably the most promising biosignature. This is because any organism that adapts to exploit free energy from starlight will have a competitive advantage over organisms that are limited by geochemical sources of free energy. The earliest incontrovertible evidence for life on Earth around 3.5 Ga coincides with fossilized photosynthetic stromatolites (Buick, 2008). While it is unknown whether the evolution of oxygenic photosynthesis is contingent-the metabolism emerged only once in Earth's history(Mulkidjanian et al., 2006)-oxygenic photosynthesis confers a unique evolutionary advantage over other forms of photosynthesis since the required substrates, carbon dioxide Abstract Oxygen is a promising exoplanet biosignature due to the evolutionary advantage conferred by harnessing starlight for photosynthesis, and the apparent low likelihood of maintaining oxygenrich atmospheres without life. Hypothetical scenarios have been proposed for non-biological oxygen accumulation on planets around late M-dwarfs, where the extended pre-main sequence may favor abiotic O 2 accumulation. In contrast, abiotic oxygen accumulation on planets around F, G, and K-type stars is seemingly less likely, provided they possess substantial non-condensable gas inventories. The comparative robustness of oxygen biosignatures around larger stars has motivated plans for next-generation telescopes capable of oxygen detection on planets around sun-like stars. However, the general tendency of terrestrial planets to develop oxygen-rich atmospheres across a broad range of initial conditions and evolutionary scenarios has not been explored. Here, we use a coupled thermal-geochemical-climate model of terrestrial planet evolution to illustrate three scenarios whereby significant abiotic oxygen can accumulate around sun-like stars, even when significant noncondensable gas inventories are present. For Earth-mass planets, we find abiotic oxygen can accumulate to modern levels if (1) the CO 2 :H 2 O ratio of the initial volatile inventory is high, (2) the initial water inventory exceeds ∼50 Earth oceans, or (3) the initial water inventory is very low (<0.3 Earth oceans). Fortunately, these three abiotic oxygen scenarios could be distinguished from biological oxygen with observations of other atmospheric constituents or characterizing the planetary surface. This highlights the need for broadly capable next-generation telescopes that are equipped to constrain surface water inventories via time-resolved photometry and search for temporal biosignatures or disequilibrium combination biosignatures to assess whether oxygen is biogenic.Plain Language Summary Next-generation telescopes will search for life on exoplanets by looking for the spectral signatures of biogenic gases. Oxygen has been considered a reliable biosignature gas, especially for planets around sun-like stars where non-biological, photochemical production is unlikely. This motivates plans for future telescopes specifically designed for oxygen detection. Here, we develop a coupled model of the atmosphere-interior evolution of terrestrial planets to show that lifeless planets in the habitable zone could develop oxygen-rich atmospheres relatively easily. These false positives for biological oxygen could be distinguished from inhabited planets using other contextual clues, but their existence implies next-generation telescopes need to be capable of characterizing planetary environments and searching for multiple lines of evidence for life, not merely oxygen.KRISSANSEN-TOTTON ET AL.	10.1029/2020av000294	[ "https://arxiv.org/pdf/2104.06463v1.pdf" ]	233,231,686	2104.06463	adecef5c0fe7c9a0f41cbf120a17f8e88d4e2214	. Introduction Oxygen False Positives on Habitable Zone Planets Around Sun-Like Stars Joshua Krissansen-Totton Department of Astronomy and Astrophysics University of California Santa Cruz, Santa CruzCAUSA NASA Nexus for Exoplanet System Science Virtual Planetary Laboratory Team University of Washington SeattleWAUSA NASA Sagan Fellow Department of Astronomy and Astrophysics University of California Santa CruzCA Virtual Planetary Laboratory 3 NASA Sagan Fellow Jonathan J Fortney Department of Astronomy and Astrophysics University of California Santa Cruz, Santa CruzCAUSA Department of Astronomy and Astrophysics University of California Santa CruzCA Francis Nimmo Department of Earth and Planetary Sciences University of California Santa Cruz, Santa CruzCAUSA Department of Earth and Planetary Sciences University of California Santa CruzCA Nicholas Wogan NASA Nexus for Exoplanet System Science Virtual Planetary Laboratory Team University of Washington SeattleWAUSA Department of Earth and Space Sciences University of Washington SeattleWAUSA AGU Advances Virtual Planetary Laboratory 3 NASA Sagan Fellow Department of Earth and Space Sciences University of Washington SeattleWA . Introduction Oxygen False Positives on Habitable Zone Planets Around Sun-Like Stars Key Points: The search for life beyond Earth is a key motivator for exoplanet astronomy. Among the various life detection approaches that have been proposed, atmospheric oxygen is arguably the most promising biosignature. This is because any organism that adapts to exploit free energy from starlight will have a competitive advantage over organisms that are limited by geochemical sources of free energy. The earliest incontrovertible evidence for life on Earth around 3.5 Ga coincides with fossilized photosynthetic stromatolites (Buick, 2008). While it is unknown whether the evolution of oxygenic photosynthesis is contingent-the metabolism emerged only once in Earth's history(Mulkidjanian et al., 2006)-oxygenic photosynthesis confers a unique evolutionary advantage over other forms of photosynthesis since the required substrates, carbon dioxide Abstract Oxygen is a promising exoplanet biosignature due to the evolutionary advantage conferred by harnessing starlight for photosynthesis, and the apparent low likelihood of maintaining oxygenrich atmospheres without life. Hypothetical scenarios have been proposed for non-biological oxygen accumulation on planets around late M-dwarfs, where the extended pre-main sequence may favor abiotic O 2 accumulation. In contrast, abiotic oxygen accumulation on planets around F, G, and K-type stars is seemingly less likely, provided they possess substantial non-condensable gas inventories. The comparative robustness of oxygen biosignatures around larger stars has motivated plans for next-generation telescopes capable of oxygen detection on planets around sun-like stars. However, the general tendency of terrestrial planets to develop oxygen-rich atmospheres across a broad range of initial conditions and evolutionary scenarios has not been explored. Here, we use a coupled thermal-geochemical-climate model of terrestrial planet evolution to illustrate three scenarios whereby significant abiotic oxygen can accumulate around sun-like stars, even when significant noncondensable gas inventories are present. For Earth-mass planets, we find abiotic oxygen can accumulate to modern levels if (1) the CO 2 :H 2 O ratio of the initial volatile inventory is high, (2) the initial water inventory exceeds ∼50 Earth oceans, or (3) the initial water inventory is very low (<0.3 Earth oceans). Fortunately, these three abiotic oxygen scenarios could be distinguished from biological oxygen with observations of other atmospheric constituents or characterizing the planetary surface. This highlights the need for broadly capable next-generation telescopes that are equipped to constrain surface water inventories via time-resolved photometry and search for temporal biosignatures or disequilibrium combination biosignatures to assess whether oxygen is biogenic.Plain Language Summary Next-generation telescopes will search for life on exoplanets by looking for the spectral signatures of biogenic gases. Oxygen has been considered a reliable biosignature gas, especially for planets around sun-like stars where non-biological, photochemical production is unlikely. This motivates plans for future telescopes specifically designed for oxygen detection. Here, we develop a coupled model of the atmosphere-interior evolution of terrestrial planets to show that lifeless planets in the habitable zone could develop oxygen-rich atmospheres relatively easily. These false positives for biological oxygen could be distinguished from inhabited planets using other contextual clues, but their existence implies next-generation telescopes need to be capable of characterizing planetary environments and searching for multiple lines of evidence for life, not merely oxygen.KRISSANSEN-TOTTON ET AL. Planetary evolution is divided into an initial magma ocean phase, and a subsequent solid-mantle phase, as shown in Figure 1, although a planet may transition between magma ocean and solid mantle multiple times. The model is initialized with a fully molten mantle and some endowment of volatiles, radiogenic inventory, and an initial mantle oxygen fugacity (i.e., after core formation, Figure 1, left). Magma ocean solidification follows previous models such as Lebrun et al. (2013) and Schaefer et al. (2016): the magma ocean freezes from the core, upwards, as governed by the following equation:         Here, V mantle is the volume of the molten mantle, ρ m is the average density of the mantle, Q radioactive is radiogenic heat production per unit mass, r p is planetary radius, H fusion is the latent heat of fusion of silicates, r s is the solidification radius, c p is the specific heat of silicates, Q core is the heatflow from the metallic core, and T p is mantle potential temperature. The heatflow from the interior, q m , is calculated using 1-D convective parameterization, with temperature-dependent magma ocean viscosity, ν(T p ), (see Figure S3 and supporting information Section A.3):                  1/3 4 surf p m q m p T T q C T(2) Here, T surf is mean surface temperature, and C qm is a constant that depends on thermal conductivity, thermal diffusivity, critical Rayleigh number, gravity, and thermal expansivity. Supporting information Section A.1 describes this convection parameterization in more detail. Equation 1 continues to govern the thermal evolution of the mantle after the magma ocean has solidified with dr s /dt = 0.0. During magma ocean solidification H, C, and O are partitioned between dissolved melt phases, crystalline phases, and the atmosphere by assuming chemical equilibrium (supporting information Section A.7). The rate at which the mantle freezes is controlled by outgoing longwave radiative (OLR), which is balanced by heat from interior and absorbed shortwave radiation (ASR) from the host star at every timestep (see climate model description below):   ASR OLR m q(3) During the magma ocean phase, planetary oxidation may occur from the loss of hydrogen to space (less oxygen drag):     2 2 2 H O H space 0.5O(4) Free oxygen produced via H escape is dissolved in the melt and may be transferred to the solid mantle as the magma ocean solidifies (Schaefer et al., 2016). We parameterized atmospheric escape as either diffusion limited or XUV-limited, depending on the composition of the stratosphere and the stellar XUV flux (supporting information Section A.9). During XUV-driven escape of a steam-dominated atmosphere, the hydrodynamic escape of H may drag along O (and even CO 2 ) following Odert et al. (2018). In contrast, if the stratosphere is mostly dry, then the escape of H will be limited by the rate at which H 2 O can diffuse through the cold trap (Wordsworth & Pierrehumbert, 2013), and nothing heavier than H escapes. Standard parameterizations of solar bolometric luminosity (Baraffe et al., 1998(Baraffe et al., , 2002 and XUV luminosity evolution are adopted (Tu et al., 2015), as described in supporting information Section A.4. A radiative-convective climate model is used to self-consistently calculate surface temperature, OLR, ASR, the water vapor profile, and surface liquid water inventory (if any) during both the magma ocean phase and subsequent temperate evolution. OLR is a function of the surface H 2 O and CO 2 inventories, and is calculated using the publicly available correlated-k radiative transfer code of Marcq et al. (2017). To obtain KRISSANSEN-TOTTON ET AL. 10.1029/2020AV000294 4 of 20 OLR in the presence of condensable water vapor, a dry adiabat to moist adiabat to isothermal atmospheric structure is assumed (Kasting, 1988). To calculate ASR across a wide range of temperatures, we adapted the albedo parameterization described in Pluriel et al. (2019). Refer to supporting information Section A.5 for full details of radiative transfer calculations along with example outputs. When heat from accretion and short-lived radiogenics is sufficiently dissipated-the timescale for which is controlled by insolation and greenhouse warming from outgassed volatiles-a planet's surface temperature may drop below the solidus and the magma ocean phase is over. At this point, the model transitions to solid-state mantle convection and temperate geochemical cycling (Figure 1, right). The redox budget during solid-state evolution is modeled as follows: the only source of oxygen is still atmospheric escape using the same parameterization as described above. However, there are now numerous crustal sinks for oxygen including (i) subaerial and submarine outgassing of reduced species (e.g., H 2 , CO, and CH 4 ), (ii) water-rock serpentinizing reactions that generate H 2 (the "wet crustal" sink), and (iii) direct oxidation of surface crust by atmospheric oxygen (the "dry crustal" sink):                   The sizes of these three oxygen sinks are self-consistently calculated from the planetary interior evolution and mantle volatile content. Outgassing fluxes are calculated using the melt-gas equilibrium outgassing model of Wogan et al. (2020); outgassing fluxes depend on mantle oxygen fugacity, degassing overburden pressure, the volatile content of the mantle, specifically H 2 O and CO 2 content, and the rate at which melt (new crust) is produced (see supporting information Section A.10). The possible influence of pressure overburden on redox evolution has been discussed previously (Wordsworth et al., 2018), and is quantifiable within our outgassing model framework. Dry and wet crustal sinks for oxygen similarly depend on crustal production rates, and are described in full in supporting information Sections A.12 and A.13. Crustal production is calculated from interior heatflow, which is modulated by temperature-dependent mantle viscosity and radiogenic heat production (supporting information Section A.10). We assume plate tectonics when calculating crustal production rates to maximize crustal sinks of oxygen. Weathering processes (Krissansen-Totton, Arney, et al., 2018) and the deep hydrological cycle (Schaefer & Sasselov, 2015) are explicitly modeled because climate and surface volatile inventories control crustal oxygen sinks and atmospheric escape processes. Our model incorporates a rudimentary carbon cycle, which is a simplified version of that described in Krissansen-Totton, Arney, et al. (2018). Carbon is added to the atmosphere via magmatic outgassing (described above), and returned via continental weathering and seafloor weathering, whose relative contributions depend on climate and the total surface water inventory. Our carbon cycle parameterization is described in supporting information Section A.11 and the deep hydrological cycle in supporting information Section A.12. Self-consistent climate modeling enables an assessment of whether abiotic oxygen can coexist with a habitable surface climate, which would be especially problematic for unambiguous biosignature gas interpretations. The time evolution of volatile reservoirs during both the magma ocean phase and the solid state evolution is governed by the following equations (see supporting information Section A.6 for details):  Here, Λ i represents a generic volatile species (e.g., H 2 O, CO 2 , free O). The first term on the right hand side represents the transfer of volatiles from the fluid phases (magma ocean + atmosphere) to the solid mantle as the magma ocean solidifies: Λ i k is the melt-solid partition coefficient for species Λ i , and Esc . Because we are assuming a plate tectonics regime, the model does not separately track volatile reservoirs in the crust and mantle. Instead, we assume carbon, oxygen, and water added to crust is immediately subducted into the mantle; a single, wellmixed "interior" reservoir is used to represent storage of volatiles in solid silicates. The extremely broad range of crustal hydration and crustal oxidation efficiency factors sampled (see below) can accommodate differing subduction and arc volcanism efficiencies. Note that the model only tracks C, H, and O-bearing species, as well as Fe 2+ /Fe 3+ speciation in the interior. Nitrogen fluxes are not modeled, and we instead assumed a 1 bar N 2 background partial pressure in all model runs. This conservative assumption ensures that, for temperature surface conditions, there are always sufficient non-condensables to maintain a cold trap and prevent excessive water loss; any oxygen accumulation that results is due to other processes. The model does not include any explicit photochemistry; it tracks fluxes of oxygen into/out of the combined atmosphere-ocean reservoir, and all outgassed reductants are assumed to instantaneously deplete atmospheric oxygen. This simplification is adequate for estimating oxygen accumulation because if oxygen sources exceed oxygen sinks, then oxidant build-up will occur; neither photolysis reactions nor spontaneous reactions can add net reducing power the atmosphere-ocean system. Our model cannot predict low steady state oxygen abundances in predominantly anoxic atmospheres, however; atmospheric O 2 is truncated at a lower limit of 0.1 Pa for numerical efficiency. Importantly, our modeling approach is agnostic on the plausibility of the various photochemical scenarios that have been proposed for abiotic O 2 atmosphere, such as O 2 -and CO-rich atmospheres maintained by continuous CO 2 photodissociation (Gao et al., 2015;Hu et al., 2020). Studies of these scenarios enforce global redox balance at the boundaries of the atmosphere-ocean system and determine whether appreciable atmospheric oxygen exists in the resultant photochemical steady state (e.g., Harman et al., 2015). Here, we are instead using a time-dependent model to investigate whether slight imbalances in atmosphere-ocean boundary fluxes can result in atmospheric oxygen accumulation on long timescales (cf., Luger & Barnes, 2015;Schaefer et al., 2016;Wordsworth et al., 2018). There are many uncertain parameterizations and parameter values in our model, and so all results are presented as Monte Carlo ensembles that randomly sample a wide range of uncertain parameter values. Parameter ranges and their justifications are described in full in the supporting information (Table S1). We sampled a range of temperature-dependent mantle viscosities, efficiencies of XUV-driven escape, uncertain early sun XUV fluxes, carbon cycle feedbacks, deep hydrological cycle dependencies, and albedo parameterizations. Unknown parameters that are particularly important for oxygen false positives include the dry crustal oxidation efficiency, f dry−oxid , which is the fraction of Fe 2+ in newly produced crust that is oxidized to Fe 3+ in the presence of an oxidizing atmosphere via non-aqueous reactions. This parameter is sampled uniformly in log space from 10 −4 to 10% (Gillmann et al., 2009). Another important parameter is the XUV-driven escape efficiency,  lowXUV, which is the fraction of stellar XUV energy that drives H-escape. This is sampled uniformly from 0.01 to 0.3, and the portion of energy that goes into escape once the XUV flux exceeds what is required for O-drag is an additional free parameter. Model Validation To validate the model, we first show that it can successfully reproduce the atmospheric evolution of Earth and Venus. Venus results are described in detail in supporting information Section C, and here we summarize key results for Earth. Figure 2 shows Monte Carlo model outputs over a range of Earth-like volatile inventories, specifically an initial water content of 1-10 Earth oceans, an initial CO 2 content of 20-2,000 bar. Moreover, only initial CO 2 inventories less than the initial water inventory by mass are permitted, and an initial (post core-formation) mantle redox state around the Quartz-Fayalite-Magnetite (QFM) buffer is assumed. There is evidence for more reducing Hadean continental crust (Yang et al., 2014), and other terrestrial planets such as Mars likely have reducing mantles (Wadhwa, 2001). While a rapidly oxidized mantle (e.g., Zahnle et al., 2010) is assumed in all nominal calculations, the sensitivity of our results to initial mantle redox is explored in supporting information Section G. KRISSANSEN-TOTTON ET AL. 10.1029/2020AV000294 6 of 20 Figure 2a shows the time-evolution of mantle potential temperature and surface temperature, Figure 2b shows the solidification of the magma ocean from core to surface, which takes several Myr, Figure 2c shows the evolution of atmospheric volatile inventories, Figure 2d shows the globally averaged depth of liquid water oceans. The inflection in temperature evolution and solidification radius around 10 4 years reflects the transition from a low viscosity, rapidly convecting magma ocean, to more solid-like magma mush convection (Lebrun et al., 2013). Note the transfer of water from steam atmosphere ( Figure 2c) to liquid water ocean (Figure 2d) following magma ocean solidification at around 10 6 years. Figure 2e shows the planetary energy budget, Figure 2f shows carbon outgassing and weathering fluxes, Figure 2g shows total crustal production, Figure 2h shows the evolution of (solid) mantle oxygen fugacity relative to the QFM buffer, and Figure 2i shows oxygen fluxes into/out of the atmosphere, excluding loss of oxygen to the magma ocean, which is modeled as instantaneous melt-atmosphere equilibration rather than a continuous sink flux; this temperature-dependent equilibrium partitioning controls atmospheric pO 2 for the first few million years ( Figure 2c). Our modeled early Earth atmosphere-thermal-climate evolution is broadly consistent with semiquantitative reconstructions of Hadean atmospheric evolution (Zahnle et al., 2007(Zahnle et al., , 2010. We also find that in virtually every model run, after 4.5 Gyr of atmospheric evolution, the atmosphere is anoxic (Figure 2c). This is unsurprising. Hydrogen escape during the initial ∼ Myr magma ocean does not add free oxygen to the atmosphere but instead oxidizes the interior, as has been described previously (Hamano et al., 2013). In some cases, small amounts of abiotic oxygen are produced in the post magma-ocean steam atmosphere, but KRISSANSEN-TOTTON ET AL. Earth's coupled redox-thermal-climate evolution (without life). The model is applied to the Earth from magma ocean to present with initial water inventories ranging from 1 to 10 Earth oceans, and initial CO 2 inventories ranging from 20 to 2,000 bar. Additionally, we only plot model runs where the initial water inventory exceeds the initial CO 2 inventory. The lines are median values and shaded regions denote 95% confidence intervals across 3,000 model runs. In the absence of life, Earth's atmosphere after 4.5 Ga is always anoxic (c) because outgassing and crustal hydration sinks overwhelm oxygen production via photolysis and diffusion-limited hydrogen escape (i). (a, b) The magma ocean persists for a few million years, consistent with previous studies. (e) The magma ocean ends when the planet's interior cools such that heatflow from the interior drops below the runaway greenhouse limit. (d) When this occurs, liquid water oceans condense onto the surface, (f) a temperate carbon cycle commences. (c) There is sometimes a brief spike in atmospheric oxygen following magma ocean solidification due to the persistence of a steam atmosphere and hydrogen escape, (i) but this oxygen is rapidly drawn down by geological sinks. (g) Volatile cycling is controlled by the rate at which fresh crust is produced. Mantle redox evolution is plotted (h) alongside proxy estimates (O'Neill et al., 2018;Trail et al., 2011). this atmospheric oxygen is rapidly overwhelmed by outgassing and other crustal sinks. Subsequent oxygen production via diffusion-limited escape is small, and so there are no further opportunities for abiotic accumulation so long as the planet remains geologically active. For Venus, the model can recover current atmospheric conditions assuming the initial water inventory is small, and that crustal sinks of oxygen are efficient (see supporting information Section C). Venusian histories in which the surface was never habitable and in which the surface was habitable for several billion years can both be reconciled with the current atmosphere, which is broadly consistent with previous modeling of Venus ' atmospheric evolution (Chassefière et al., 2012;Kasting & Pollack, 1983;Way et al., 2016). Results If Earth's initial volatile inventories are varied, then oxygen-rich atmospheres may be possible. Here, we outline three scenarios whereby an abiotic Earth could have accumulated an oxygen-rich atmosphere after 4.5 Gyr. None of these scenarios guarantee an oxygen-rich atmosphere; instead, oxygenated atmospheres are a possible outcome that is dependent on the efficiency of oxygen crustal sinks and atmospheric escape. Scenario 1: High CO 2 :H 2 O Initial Inventory Leading to Perpetual Runaway Greenhouse Figure 3 shows selected model outputs for planets with initial CO 2 :H 2 O volatile inventories greater than one by mass and atmospheric O 2 > 10 17 kg at present (P O2 > ∼0.02 bar). For these planets, the greenhouse warming from a dense CO 2 atmosphere ensures that the surface temperature is above the critical point of water; liquid water never condenses on the surface at 1 AU, except briefly, and in small amounts, during KRISSANSEN-TOTTON ET AL. . Oxygen false positives from high initial CO 2 :H 2 O inventories (Scenario 1). The model is applied to the Earth from magma ocean to present with randomly sampled initial water inventories ranging from ∼0.1 to 10 Earth oceans, and initial CO 2 inventories ranging from ∼20 to 2,000 bar (implying CO 2 :H 2 O ranging from 0.01 to 100 by mass). Only model outputs with modern day atmospheric oxygen exceeding 10 17 kg (>∼0.02 bar) are plotted. Subplots are the same as in Figure 2, and shaded regions denote 95% confidence intervals. (a) High atmospheric CO 2 ensures the surface temperature always exceeds the critical point of water after the pre-main sequence, and (d) thus permanent liquid water oceans do not condense. The lack of surface water, low volatile content of the mantle, and high surface pressure increasing volatile solubility in partial melts all limits oxygen sinks. (a, i) The largest atmospheric sink is dry crustal oxidation, which diminishes with time as the interior cools. (c) Atmospheric oxygen produced via H escape may start to accumulate after several Gyr of evolution. early solar evolution (Figure 3d). These high CO 2 :H 2 O perpetual runaway atmospheres have been described previously Salvador et al., 2017). The lack of liquid surface water precludes CO 2 -drawdown via silicate weathering (Figure 3f). Reactions between supercritical water and silicates will be severely kinetically limited by sluggish solid state diffusion, and are therefore assumed to be negligible (Zolotov et al., 1997). Consequently, a dense CO 2 atmosphere and supercritical surface temperature persist indefinitely (Figure 3a), despite the planet residing in the habitable zone. Moreover, there is sufficient steam in the atmosphere to ensure diffusion-limited hydrogen escape provides an appreciable source flux of oxygen ( Figure 3i). Oxygen accumulation also requires limited oxygen sinks, and this may occur on high CO 2 :H 2 O worlds for two reasons. First, since permanent oceans do not condense, it is difficult to sequester outgassed volatiles left over from the magma ocean in the interior; hydration reactions and carbonatization reactions do not occur without liquid surface water. Limited mantle regassing after magma ocean outgassing implies low mantle volatile content, which inhibits the capacity of outgassed reductants to draw down oxygen. Second, the high pressure from the dense CO 2 atmosphere, while not enough to significantly increase the silicate solidus, does increases the solubility of volatiles in partial melts. In combination with low mantle volatile content, this ensures limited outgassing of reducing species (Figures 3f and 3i, cf., ). However, even with limited outgassing, new crust is still being produced that may be directly oxidized by gaseous O 2 (Figure 3g). Figure 4a shows atmospheric oxygen abundances after 4.5 Gyr as a function of dry crustal oxidation efficiency, f dry−oxid , for a large number of model runs sampling 10 20 -10 22 kg initial CO 2 and H 2 O (or ∼20-2,000 bar and 0.1-10 Earth oceans). The efficiency parameter is necessarily quite low (<0.1%) in all the model runs that produce significant oxygen. The plausibility of such inefficient crustal oxidation is explored in the discussion. Figure 4b Gyr is plotted as a function of (a) dry crustal oxidation efficiency, and (b) the initial CO 2 :H 2 O inventory by mass. Note that Scenario 1 oxygen accumulation (high CO 2 , perpetual runaway greenhouse atmospheres) requires both an initial CO 2 :H 2 O ratio >1 (green box) and for dry crustal oxidation to be relatively inefficient, with <0.1% of Fe 2+ in newly produced crust oxidized. The observed range in carbonaceous chondrite CO 2 :H 2 O ratios (purple interval) is shown in (b) as a rough proxy for Earth's initial volatile inventory. The outliers with high oxygen (red box) are Scenario 3 (desertworld) false positives, which are examined in Section 3.3. Anoxic atmospheres truncate at ∼10 −6 bar for numerical efficiency (see supporting information); these model runs represent outcomes with essentially no atmospheric oxygen. AGU Advances Scenario 2: Waterworlds Figure 5 shows selected model outputs for planets with H 2 O volatile inventories between 10 and 230 Earth oceans. For these planets, a liquid water ocean condenses out of the atmosphere after a few million years and any oxygen left over from the post-formation steam-dominated atmosphere is typically removed by oxygen sinks (Figure 5i), just as in the nominal Earth model ( Figure 2). However, the pressure overburden from the large surface water inventory dramatically increases the solidus of the silicate mantle. This effect, which has been described previously (Kite & Ford, 2018;Noack et al., 2016), causes fresh crustal production to cease completely after a few billion years when the mantle potential temperature drops below the solidus ( Figure 5g). The cessation of crustal production suppresses all geological oxygen sinks; hydration reactions stop as there is no melt production or new crust to oxidize (Figure 5i). The source flux of oxygen is low in this scenario. An effective cold trap ensures oxygen production rates of ∼0.01 Tmol/yr via diffusion-limited escape. However, since oxygen sinks are negligible, this small source flux is sufficient to accumulate modern Earth-like oxygen abundances over several Gyr (Figure 5c). There are also a small number of model runs where oxygen persists from the magma ocean, since equilibrium oxygen fugacities are high at the elevated solidus temperature under high pressures (Figure 5c). Figure 6 shows the oxygen abundance after 4.5 Gyr for individual model runs as a function of the initial water inventory. Waterworld oxygen false positives only begin to become likely when the initial water inventory exceeds around 50 Earth oceans or 10 23 kg (for Earth-sized planets). There is significant scatter in results due to uncertainty in the temperature-dependent mantle viscosity, which controls the duration of tectonics. . The model is applied to the Earth from magma ocean to present with randomly sampled initial water inventories ranging from 10 to 230 Earth oceans, and initial CO 2 inventories ranging from ∼20 to 6,000 bar. Only model outputs with modern day atmospheric oxygen exceeding 10 17 kg (>∼0.02 bar) are plotted. Subplots are the same as in Figure 2, and shaded regions denote 95% confidence intervals. (d) The large surface volatile inventory increases the (g) mantle solidus such that melt production and tectonics shut off shortly after formation. (i) This shuts down all oxygen sinks and (c) allows for the gradual accumulation of oxygen via diffusion-limited H escape over several Gyr. Scenario 3: Desertworlds The final scenario whereby abiotic oxygen could accumulate on habitable zone planets around Sun-like stars occurs for planets with extremely small initial volatile inventories (initial water inventory < ∼0.3 Earth oceans). Figure 7 shows selected model outputs representing this desertworld false positive. The required sequence of events are as follows: the low volatile inventory ensures that the magma ocean freezes quickly (typically ∼10 5 years, Figure 7b), even though the planet is still in a runaway greenhouse state due to high heatflow from the interior (Figure 7e). A steam-dominated atmosphere can therefore persist for a few million years, and oxygen may accumulate during this time because there is no surface magma ocean to dissolve the oxygen. Dry crustal oxidation will remove some oxygen during this steam atmosphere phase, but oxidation will be limited by the rate at which oxygen can diffuse into extrusive lava flows (Figures 7i and S1d). When a shallow ocean does eventually condense out as heatflow from the interior drops below the runaway greenhouse limit (Figures 7d and 7e), oxygen may persist for billions of years if oxygen sinks are small ( Figure 7c). Outgassing sinks are limited by the low volatile inventory of the planet, but inefficient dry crustal oxidation is also required post-magma ocean for the oxygen to persist for 4.5 Gyr; the efficiency parameter, f dry−oxid , must be <0.1% ( Figure S1). This habitable scenario is qualitatively different to the uninhabitable Scenario 1 false positives: in the former, atmospheric oxygen accumulates early and gradually declines due to crustal sinks, whereas in the latter, oxygen accumulation takes several Gyrs. Desertworld false positives also require efficient XUV-driven hydrodynamic escape (>10%) during the steam atmosphere phase to produce large atmospheric oxygen abundances after the magma ocean has solidified ( Figure S1). Discussion The modeling approach adopted in this paper has several important caveats and limitations. First, we consider the reasons why abiotic oxygen accumulation could be underestimated in our model. Assumptions That May Underestimate Abiotic Oxygen Accumulation As noted above, the model does not track nitrogen fluxes and instead assumes 1 bar N 2 partial pressure throughout. This limits oxygen accumulation by providing a non-condensable background gas to throttle hydrogen escape at the cold trap (Kleinböhl et al., 2018;Wordsworth & Pierrehumbert, 2014). Nitrogen atmospheric evolution for terrestrial planets is highly uncertain, and the evolution of Earth's atmospheric N 2 inventory is poorly constrained (Johnson & Goldblatt, 2018;Stüeken et al., 2016). However, for terrestrial planets that form with low nitrogen inventories, or with most of their nitrogen sequestered in the interior (Wordsworth, 2016), then oxygen accumulation on temperate planets could be a more common outcome than our modeling suggests. Our nominal outgassing model may overestimate fluxes of reduced gases per unit mass partial melt. This is because we do not account for graphite saturation and redox-dependent partitioning of carbon-bearing species between crystalline and melt phases; we instead assume a constant partition coefficient for relating solid mantle CO 2 content and total melt plus gas phase concentrations (e.g., Lebrun et al., 2013). Models of Martian outgassing (Grott et al., 2011), and more generalized terrestrial outgassing models (Ortenzi et al., 2020) both show that reducing mantles tend to outgas fewer volatiles by mass than more oxidized mantles for the same amount of crustal production. Our model does, however, account for the greater reducing power of volcanic outgassing on planets with lower mantle oxygen fugacities Wogan et al., 2020). Consequently, our conservative approach maximizes fluxes of outgassed reductants, and may KRISSANSEN-TOTTON ET AL. Gyr is plotted as a function of the initial planetary water inventory. Waterworld oxygen false positives are unlikely unless the initial water inventory exceeds 10 23 kg (∼50 Earth oceans). There is a high probability of abiotic oxygen accumulation for water inventories exceeding 100 Earth oceans. Clustering around 0.1 bar occurs because this is the amount of oxygen accumulated after 4.5 Gyr for temperate surface conditions and Earth-like stratospheric water abundances. Warmer surface oceans (∼400-600 K) result in more stratospheric water vapor and thus enable higher oxygen levels. underestimate oxygen accumulation. Sensitivity tests which consider more reducing mantles and graphite saturated melts are described below. Finally, our model may underestimate the duration of steam atmospheres following magma ocean solidification, and therefore underestimate oxygen accumulation prior to ocean condensation. Based on the time required to precipitate an Earth ocean and the apparent absence of stable climate states at the runaway greenhouse limit ( Figure S4), it is typically assumed that the time required to transition from steam atmosphere to surface water ocean is ∼10 3 years (Abe, 1993;Zahnle et al., 2007). In our model, the atmosphere-ocean system is assumed to be in radiative equilibrium with a negligible heat capacity (Lebrun et al., 2013). However, for waterworlds with hundreds of Earth oceans, the time required for a steam atmosphere to condense is long enough for abiotic oxygen accumulation to be significant. Moreover, stable climate states may exist in between a molten surface and a temperate surface ocean, especially if cloudswhich we ignore-are included in radiative transfer calculations Figure 6). Finally, in our model, oxygen is partitioned between the magma ocean and the atmosphere assuming chemical equilibrium. This is a reasonable assumption for high temperature/low-viscosity magma oceans with very short mixing times, but as the surface temperature approaches the solidus, the "magma ocean mush" will behave more like a solid (Lebrun et al., 2013;Salvador et al., 2017), and oxygen produced during the final stages of the magma ocean may not be efficiently sequestered in the melt. Assumptions That May Overestimate Abiotic Oxygen Accumulation Next, we consider model assumptions that might cause abiotic oxygen accumulation to be overestimated. . The model is applied to the Earth from magma ocean to present with randomly sampled initial water inventories ranging from ∼0.05 to 0.35 Earth oceans, and initial CO 2 inventories ranging from ∼6 to 60 bar. Only model outputs with modern day atmospheric oxygen exceeding 10 17 kg (>∼0.02 bar) are plotted. Subplots are the same as in Figure 2, and shaded regions denote 95% confidence intervals. (b) The low initial volatile inventory ensures the magma ocean solidifies before the runaway greenhouse is over, (c) allowing for significant oxygen accumulation in the ∼Myr steam atmosphere that (e) persists until the heatflow from the interior drops below the runaway greenhouse limit. If subsequent oxygen sinks are low, then the few bar oxygen that accumulate early on may persist for billions of years. (Krissansen-Totton et al., 2016), and the reaction may occur via lightning. Indeed NO-formation via lightning has previously been established as an important mechanism for removing photochemically produced atmospheric oxygen and inhibiting oxygen false positives . However, nitrogen fixation via lighting is unlikely to prevent abiotic oxygen accumulation on waterworlds. In the absence of any other sources or sinks nitrate-formation via lightning could draw down the modern Earth's atmospheric oxygen reservoir in 20-200 Myr (Krissansen-Totton et al., 2016) and could therefore remove 1 bar of nitrogen every 0.17-1.7 Gyr. Once most atmospheric nitrogen is converted to nitrate in the ocean, oxygen may accumulate rapidly due to the lack of a non-condensible cold trap (Wordsworth & Pierrehumbert, 2014), and nitrogen will not be significantly replenished by outgassing on waterworlds due to the pressure overburden effect precluding new crustal production. Waterworlds with large initial nitrogen atmospheric inventories (10 s of bar) could avoid abiotic oxygen accumulation if oxygen drawdown via lightning exceeds oxygen production via diffusion-limited hydrogen escape. However, it is debated whether nitrate is the kinetically stable form of nitrogen on habitable worlds (Hu & Diaz, 2019;Ranjan et al., 2019;Wong et al., 2017). Efficient conversion of nitrate back to molecular nitrogen via abiotic chemodentrification might return oxygen to the atmosphere and prevent its accumulation in the ocean, regardless of the initial N 2 volatile inventory. In summary, it is unlikely that nitrate sinks will always preclude abiotic oxygen accumulation on waterworlds, but better constraints on aqueous nitrogen chemistry would improve this assessment. Another important limitation of our model is that it ignores infrared cooling of the upper atmosphere and the throttling of hydrogen escape that results from a cool stratosphere. In our nominal model, an isothermal 210 K stratosphere is assumed. However, CO 2 -rich atmospheres may efficiently radiate in the IR, cooling the stratosphere and enhancing the water cold trap (Wordsworth & Pierrehumbert, 2013). To test the sensitivity of our results to stratospheric temperature, we repeated our calculations and introduced an additional stratospheric temperature variable, which was randomly sampled from 150 to 250 K. The full results of these calculations are shown in Figure S13. To summarize, neglecting the stratospheric radiation budget does not affect the viability of Scenarios 2 (waterworlds) or 3 (desertworlds) because, in the former, the oxygen source is diffusion-limited escape through a N 2 -dominated atmosphere at modern Earth-like rates, whereas for the later, oxygen accumulation occurs predominantly during the early, steam-dominated atmosphere, and so stratospheric temperature does not have a strong influence on escape rates (Figures S13b and S13c). However, oxygen accumulation via Scenario 1 (high CO 2 :H 2 O perpetual runaway greenhouse) is unlikely if the stratosphere is cooler than 200 K ( Figure S13a). Photochemically produced ozone or hazes may offset the cooling effect of CO 2 , and so a full assessment of Scenario 1 requires more detailed radiative-photochemical modeling. One additional caveat is that our model assumes an anhydrous solidus. This simplification is probably reasonable for post-magma ocean mantle conditions (Kite & Ford, 2018), but it is possible to imagine a scenario whereby waterworld mantles become increasingly hydrated via subduction after the magma ocean phase, and that this hydration offsets the pressure overburden effect to maintain geologic activity, and therefore oxygen sinks, for much longer than our nominal model suggests. To test this possibility, we conducted a sensitivity test where we accounted for mantle hydration decreasing the solidus (Katz et al., 2003). The results of this sensitivity test are discussed in detail in supporting information Section D, but in summary, mantle hydration does not have a large effect on oxygen accumulation on waterworlds; it merely shifts the ocean mass threshold for oxygen accumulation. Sensitivity tests were also conducted to assess whether the delivery of reducing material such as metallic iron and FeO via impacts (Zahnle et al., 2020), a more reducing initial mantle, or larger planet-star separations could inhibit abiotic oxygen accumulation. These sensitivity test results are described in full in supporting information Sections E, G, and H, respectively. In summary, we find that high impactor fluxes could preclude desertworld oxygen accumulation assuming all impactor material is completely oxidized (supporting information Section E). Impactors may thus prevent Scenario 3 (desertworld) false positives in some cases, although this is not guaranteed because it is possible to imagine planetary formation pathways with smaller impactor fluxes and/or where the majority of impactor material is buried or lost to space as suggested by some impact simulations (Marchi et al., 2018). Scenarios 1 (high CO 2 :H 2 O perpetual runaway greenhouse) and 2 (waterworlds) are viable under a more reducing (iron-wüstite buffer) initial mantle (supporting information Section G). This counterintuitive result occurs because, even though degassed KRISSANSEN-TOTTON ET AL. 10.1029/2020AV000294 13 of 20 volatiles are likely to be more reducing, total volatile concentrations in the melt phase are typically lower due to graphite saturation (Grott et al., 2011;Ortenzi et al., 2020). Moreover, crustal sinks are precluded by high overburden pressure, regardless of the redox state of the crustal material. Although Scenario 3 (desertworlds) is seemingly not excluded by a more reducing mantle, evaluating this would require more complete radiative transfer and photochemical modeling of CO-H 2 dominated atmospheres. Finally, when nominal calculations are repeated at 1.3 AU, both Scenario 2 and Scenario 3 oxygen false positives still occur frequently (supporting information Section H). Scenario 1 false positives do not occur at large planet-star separations because a high CO 2 :H 2 O atmosphere cannot maintain a perpetual runaway greenhouse state after magma ocean solidification. For Scenario 1 and 3 to be viable, the dry crustal oxidation parameter, f dry−oxid , must be relatively small (<0.1%). This contrasts with Venus where f dry−oxid probably needs to exceed >0.1% to remove virtually all O 2 from the atmosphere (see Venus validation in supporting information). This parameter is challenging to definitively constrain because it represents a range of physical processes including the diffusion of oxygen into extrusive lava flows (Gillmann et al., 2009), direct oxidation of small grain erosion products (Arvidson et al., 1992), and various other gas-solid redox reactions (Zolotov, 2019). Even if the oxidation of fresh crust is typically efficient, low dry crustal oxidation efficiencies cannot be ruled out because tectonic regimes where most magmatic activity is intrusive and isolated from the atmosphere are possible. In any case, the uncertainties in crustal oxidation processes highlights need for future missions to Venus to better constrain its redox evolution. Mars' crust is more oxidized than its upper mantle, but this oxidation cannot necessarily be used to constrain dry crustal oxidation efficiency since it might be attributable to early hydrous alteration (Herd et al., 2002;Wadhwa, 2001). The presence of gray, reduced sediments mere centimeters below more oxidized Martian regolith, as revealed by Curiosity, argues against efficient post depositional gas-solid oxidation under oxic conditions (Ming et al., 2014). Finally, we note that in some cases abiotic oxygen accumulation is contingent on highly uncertain atmospheric escape physics. This uncertainty does not affect the viability of the waterworld (Scenario 2) false positives because the required H escape flux is comparable to the modern Earth's diffusion-limited escape flux (Catling & Kasting, 2017, p. 148). Oxygen accumulation only occurs in this case because of the pressure overburden suppression of oxygen sinks. Scenario 1 (high CO 2 :H 2 O perpetual runaway greenhouse) is similarly unaffected. However, for Scenario 3 (desertworlds), oxygen accumulation only occurs because of efficient XUV-driven escape of hydrogen,   lowXUV 0.1. If H-escape is photochemically limited (e.g., Wordsworth et al., 2018) then oxygen accumulation may be limited by the photochemical dissociation of water by UV photons, and the rate and which H 2 O recombination reactions occur. However, the oxygen source fluxes required in our desertworld scenario (∼500 Tmol O 2 /yr, Figure 7i) are comparable to the water loss rates inferred for a steam-only early Earth atmosphere calculation using a photochemical model (Wordsworth et al., 2018, their Figure 9). Future work ought to couple the geochemical evolution model here to a photochemical model that includes C-bearing species to better assess the potential for oxygen accumulation on desertworlds. Finally, it is possible that non-thermal O loss (Airapetian et al., 2017) or photochemically modulated stoichiometric escape of H and O (McElroy, 1972) could lessen oxygen accumulation. Upcoming JWST observations of highly irradiated terrestrial planets may constrain escape processes and improve predictions of oxygen accumulation on more temperate planets. Sulfur outgassing and burial may have played an important role in the oxygenation of Earth's atmosphere (Gaillard et al., 2011;Olson et al., 2019). Sulfur species are ignored in our nominal model because their bulk abundances are probably too small to qualitatively change our oxygenation scenarios (Wordsworth et al., 2018). Supporting information Section I explores the consequences of adding reduced sulfur species to our outgassing model, and confirms that, for Earth-like sulfur mantle abundances, total oxygen sinks are comparable to when sulfur is neglected. With that said, mantle sulfur abundances are contingent on formation processes (e.g., Grewal et al., 2019) and could be highly variable. Incorporating a complete sulfur cycle into a redox evolution model to investigate the sensitivity of oxygenation to initial sulfur content is an opportunity for future research. Note, however, that mantle sulfur abundances are irrelevant for waterworlds (Scenario 2) where all crustal production is suppressed. In summary, there are several unknowns that preclude definitive predictions of how frequently the three scenarios outlined in this study might occur, but none can be ruled-out with current knowledge. KRISSANSEN-TOTTON ET AL. 10.1029/2020AV000294 14 of 20 AGU Advances Implications for Future Observations How might future observations discriminate between the three abiotic oxygen scenarios described above and oxygen produced by a biosphere? In principle, high CO 2 :H 2 O atmospheres should be possible to diagnose via direct imaging spectral observations because they are not habitable. A clear atmosphere is likely since the coexistence of atmospheric H 2 O, O 2 , O 3 , and abundant OH radicals may preclude the accumulation of photochemical hazes. Strong CO 2 absorption features ought to be visible, as should pressure-sensitive CIA features from the high pressures; more detailed photochemical and spectroscopic simulations will be required to determine the best false positive discriminants for these worlds. Because waterworlds and desertworlds are habitable, they may be more challenging to discriminate from inhabited terrestrial planets. Crucially, waterworld false positives would be ruled out by a detection of subaerial land because, for Earth-like gravity, the presence of emerged continents limits the maximum ocean depth to around 10 km (Cowan & Abbot, 2014), or equivalently, a few Earth oceans by mass. This limit arises because silicates cannot support their own weight with greater topography. Consequently, the detection of an ocean-continent dichotomy using time-resolved photometric mapping (Cowan et al., 2009;Farr et al., 2018;Fujii et al., 2010;Kawahara & Fujii, 2010;Lustig-Yaeger et al., 2018) could rule out a waterworld false positive, assuming alternative explanations for dichotomies in surface maps could be excluded. This highlights the need for large aperture direct imaging mission to ensure sufficient time-resolution to map the surface over a planet's rotation. Alternatively, independent mass and radius constraints from radial velocity observations and thermal infrared direct imaging (Quanz et al., 2019), respectively, could also help rule out large (few wt.%) water inventories based on bulk density. Desertworlds are likely the most challenging scenario to disambiguate from biological oxygen. Time-resolved photometric surface maps and/or the lack of ocean glint could help evaluate the surface water inventory and might be suggestive of a small water inventory (Lustig-Yaeger et al., 2018;Robinson et al., 2010). There are potentially other diagnostic spectral signatures of desertworlds such as spatial variation in atmospheric water vapor and photochemistry that could be tested using general circulation models and photochemical models. The presence of long-lived sulfuric acid hazes (Loftus et al., 2019) has been proposed as putting an upper bound on surface water abundances, but the desertworlds considered here likely have larger surface water inventories than this threshold. Broadly speaking, the scenarios outlined in this study emphasize that no single observation, including oxygen detection on habitable zone planets around sun-like stars, will be uniquely diagnostic of life. It will be necessary to design future telescopes that are capable of both constraining the full planetary/stellar context and identifying multiple lines of evidence for life (Catling et al., 2018;Walker et al., 2018). For example, oxygen detection on an ostensibly habitable terrestrial planet would be persuasive if accompanied by surface biosignature detections , temporal biosignatures (Olson et al., 2018), or co-existing reducing gases in atmospheric disequilibrium (Krissansen-Totton et al., 2016). The coexistence of oxygen and methane remains an excellent biosignature and would not be expected for any of the oxygen false-positive scenarios described above. Indeed, it is difficult to produce large methane abundances in habitable planet atmospheres without life, even in anoxic atmospheres (Krissansen-Totton, Olson, et al., 2018;Wogan et al., 2020). It should also be noted that the scenarios in this study were illustrated for habitable zone planets around sun-like stars, but they may also be applicable to habitable zone planets around M-dwarfs. Conclusions The redox evolution of habitable zone terrestrial planets is strongly dependent on initial volatile inventories and the efficiency of crustal sinks. Uninhabited, Earth-sized planets within the habitable zone of G-type stars are very unlikely to accumulate abiotic oxygen if their initial volatile inventories are Earthlike. However, if initial volatile inventories differ dramatically from that of the Earth, then non-biological oxygen accumulation is possible, even when atmospheric noncondensable inventories are large. This may occur when either (i) the initial CO 2 :H 2 O ratio exceeds one, which suppresses oxygen sinks due to the low mantle volatile content and because surface conditions are too hot for aqueous reactions, or (ii) the initial H 2 O inventory is very large, thereby halting crustal production after a few billion years and shutting off all KRISSANSEN-TOTTON ET AL. 10.1029/2020AV000294 15 of 20 oxygen sinks, or (iii) the planet is very volatile-poor, in which case oxygen may accumulate during the steam atmosphere that persists after magma ocean solidification. Inefficient dry crustal oxidation is required for scenarios (i) and (iii) to yield large oxygen abundances, and scenario (i) is sensitive to stratospheric temperature. Fortunately, observational discriminants exist for all three of these scenarios; scenario (i) planets are uninhabitable, whereas the ability to constrain surface water inventories using time-resolved photometry would be useful for ruling out scenarios (ii) and (iii). More generally, the possible existence of these oxygen false positive scenarios highlights the need for a systems approach to biosignature assessment where biogenicity is judged not by the presence or absence of a single biosignature gas, but by multiple lines of evidence from both spectrally resolved and temporally resolved observations. Conflict of Interest The authors declare no conflicts of interest relevant to this study. Data Availability Statement The a) b) c) d) e) f) g) h) i) Text A. Model description A.1) Thermal evolution: Planetary thermal evolution is specified by energy budget and temperature-dependent viscosity. The time-evolution of mantle potential temperature, p T (K), is determined by the following equations, representing the magma ocean and solid-state convection phases, respectively: Note that we do not account for uncertainty in core heatflow since we are already sampling an order of magnitude range in radiogenic inventories (see above). Moreover, by choosing a core heatflow history at the higher end of literature estimates (Nimmo 2007;O'Rourke & Stevenson 2016), we are effectively maximizing crustal recycling and subsequent oxygen sinks. Tidal heating is ignored; if Earth-moon system tidal heating were included then the duration of the magma ocean could be extended by a few million years, potentially providing more time for oxidation via H escape (Zahnle et al. 2015). Equation (1) governs thermal evolution except in rare cases where transition to runaway greenhouse causes surface temperature to increase above mantle potential temperature, in which case a conduction regime is adopted (see below). The heatflow from the convecting interior, m q (W/m 2 ), is parameterized as follows: Here,  =2×10 -5 K -1 is the thermal expansion coefficient for silicates, g (m/s 2 ) is surface gravity, crit Ra = 1100 is the critical Rayleigh number, k = 4.2 W/m/K is the thermal conductivity of silicates, 1 / 3  = and  =10 -6 m 2 /s is thermal diffusivity (Lebrun et al. 2013;Schaefer et al. 2016). Heatflow is dependent on the kinematic viscosity,  , which is a function of mantle potential temperature (see below). This parameterization is adopted for both solid state and magma ocean phases. A.2) Solidus parameterization and magma ocean freezing: The solidus controls both the freezing of the magma ocean and the production of partial melt (and outgassing) during temperate geochemical evolution. The adiabatic mantle temperature profile, solidus, and liquidus are parameterized as follows. Here, 1 T is a linear fit to the solidus for low pressure dry peridotite, and 2 T is a linear fit to the solidus for the high pressure lower mantle (Hirschmann 2000;Schaefer et al. 2016). A smooth function between them is assumed for solidus T so that an analytic derivative exists at all radii (see below). Following Schaefer et al. (2016), the liquidus is assumed to be 600 K warmer than the solidus at all pressures. We also allow for modulation of the solidus and liquidus by the pressure overburden of surface volatiles. Here, overburden P is the pressure from all H2O (liquid and gaseous), CO2, and O2 at the surface. The pressure overburden is only accounted for after the magma ocean has solidified and after the mantle has degassed. The time evolution of the solidification radius is determined by a similar method to Schaefer et al. (2016). The rate of change in the solidification radius can be obtained by noting that the time derivative of solidus evolution and the time derivative of the adiabatic temperature profile must be equal at the solidification radius:  − − − − − − − −− − + + − − − − −− − + + − − − + + − +  − − +  − −  =+ ( )5 00000 Note that the solidification radius must remain constant when the core-mantle boundary temperature exceeds the solidus temperature and when the surface temperature drops below the solidus to ensure the solidification radius and mantle potential temperature begin evolving together when ( ) ( ) This approach is more computationally expensive but yields an identical solidification radius evolution to the analytic expression eq. (7). A.3) Mantle Viscosity Parameterization: Our viscosity parameterization needs to have several properties. First, it must successfully reproduce the modern Earth's heatflow, melt production, and plate velocity. Second, it needs to transition smoothly from low viscosity magma ocean, to magma mush, to solid state convection. Our parameterization is a variation of those assumed in other magma ocean-to-solid interior evolution models (Lebrun et al. 2013;Salvador et al. 2017;Schaefer et al. 2016). However, the parameterizations in these studies needed modification because they predict a low viscosity magma-ocean or mush at the modern Earth's potential temperature (~1620 K). Noting that there is a large uncertainty in the critical melt fraction that controls the transition from solid-like to fluid-like convection (Costa et al. 2009), we adopted a parameterization that ensures this transition occurs at a temperature that exceeds the modern Earth's potential temperature. This is illustrated in Fig. S3, which compares our viscosity parameterization to others in the literature. T T T T T T T T T T T TT TT V        − +   −     − +  −   =        = ( )7 350000 Here, coef V = 10 1 to 10 3 Pa s is a randomly sampled parameter that accounts for uncertainty in solid-state viscosity. A.4) Stellar evolution We assume solar bolometric luminosity evolution, () Lt (W), for all model runs (Baraffe et al. 1998;Baraffe et al. 2002). For the evolution of stellar XUV luminosity, we follow the empirical fit developed in Tu et al. (2015). The early sun's rotation rate, 0  , is an unknown parameter sampled uniformly from 1.8 to 45 (relative to modern) in log space. From the early sun's rotation rate, the time for the early sun to fall out of saturation, sat t (Myr), is given by: 1.14 0 2.9 sat t =(10) For the chosen range of rotation rates, sampled saturation times range from 6 to 226 Myrs. We assume that at saturation the sun's XUV luminosity is 10 -3.13 () Lt . To retrieve the modern XUV flux, we define the exponent, L t t t Lt t t L t t t  − −    =     (11) The flux received at each planet's orbital distance, planet star D − (m), is calculated using Earth-sun and Venus-sun separations. We begin our model runs at t = 10 Myrs in the stellar evolution model, but results are insensitive to the choice of zero point. A.5) Surface Energy Budget: At each time-step in the model, the surface temperature is solved numerically by finding the surface temperature that ensures heatflow from the interior plus absorbed shortwave radiation (ASR) is exactly balanced by outgoing longwave radiation (OLR). This assumption ignores the intrinsic heat capacity of the ocean, which is reasonable for Earth-like oceans, but for waterworlds this radiative equilibrium approach may underestimate the transition time from runaway greenhouse to surface ocean (see Discussion). Surface temperature is found by solving the following equation for Here, m q is specified by equation (4). Absorbed shortwave radiation is a function of the planetary albedo and stellar luminosity: A . The "hot" and "cold" states do not refer to non-glaciated and glaciated states, which we do not consider in our model. Instead, they allow for a transition from low albedo cloud free runaway greenhouse atmospheres at high (>~ 1000 K) temperatures, to a range of cloudy and non-cloudy states under more temperate conditions (Pluriel et al. 2019). This distinction is important for modeling Venus, where the "cold" state albedo is ~0.7, but the albedo during the initial runaway greenhouse phase was potentially much lower. Albedos for hot and cold states are sampled uniformly form 0-0.3 and 0.25-0.35, respectively for Earth and 0-0.3 and 0.2-0.7 for Venus. The albedo of the hot state must always be equal to or less than that of the cold state. ( ) 2 ( ) 1 ( ,( )) To calculate the outgoing longwave radiation (OLR), we used the publicly available code from Marcq et al. (2017). This code uses DISORT (Stamnes et al. 1988) with four stream longwave radiative transfer. The radiative transfer model only considers opacity due to water vapor and carbon dioxide. Rock vapor opacities are ignored since the time spent at rock-vaporizing temperatures is very short and unlikely to affect long term redox evolution. Correlated k coefficients are calculated from the high resolution molecular absorption spectra computed with kspectrum (Eymet et al. 2016), H2O-H2O continuum absorption is taken from Clough et al. (2005), and CO2-CO2 continuum absorption from fits to Venus observations (Bézard et al. 2011). H2O-CO2 continuum opacity is not considered, and is likely negligible compared to H2O-H2O and CO2-CO2 continuum absorption (Ma & Tipping 1992). The runaway greenhouse limit calculated using the code of Marcq et al. (2017) closely agrees with line-by-line calculations in Goldblatt et al. (2013). The atmosphere model calculates atmospheric structure and abundance profiles using the expressions for dry and moist adiabats in Kasting (1988), and the thermodynamic properties of water are taken from steam tables (Haar et al. 1984). Given an assumed surface temperature and total water inventory, the code calculates the atmospheric water vapor profile assuming a dry convective regime (partial pressure of water vapor less than saturation) to moist convective regime (partial pressure of water vapor equals saturation) to isothermal temperature structure, where the isotherm temperature is the planetary skin temperature. The dry convective regime may not be present if water vapor is saturated at the surface. Once the water vapor profile has been calculated, the remainder of the surface water inventory (if any) resides in a surface water ocean. If a portion of the surface water resides in an ocean, then the partitioning of carbon between the atmosphere and ocean also determines the OLR. Thus, OLR is a function of dissolved carbonate concentrations. This is calculated as follows: 2 3 2 () sp surf KT CO Ca − +   =   (15) Here, rather than explicitly track the ocean alkalinity budget, we follow the approach of Schwieterman et al. (2019) and assume that carbonate precipitation will ensure the longterm carbonate saturation state of the ocean,  , is constant. Values for  are sampled randomly from 1 to 10 to allow for abiotic supersaturation. Similarly, rather than explicitly track cation weathering budgets, we assumed constant dissolved calcium abundances and sample uniformly in log space from 10 -4 to 3×10 -1 mol/kg. Dissolved calcium concentrations in Earth oceans have varied from 10 -2 to 3×10 -1 mol/kg over Earth history (Halevy & Bachan 2017), but we adopt a broader range to account for different crustal compositions and ocean volumes. For example, Kite and Ford (2018) consider waterworld scenarios with essentially zero Ca 2+ up to 0.25 mol/kg based on thermodynamic models of basalt-water interaction. Explicitly tracking the ocean alkalinity budget is an opportunity for future research. The temperature dependent solubility product, sp K , is the same as in Krissansen-Totton et al. (2018). Once the total carbon reservoir, the ocean size, and the dissolved carbonate concentrations are known, the entire carbonate equilibrium system of equations can be solved to determine atmospheric pCO2 (Krissansen-Totton et al. 2018). Rather than call the atmospheric radiative transfer code in real time, we precomputed a grid of OLR values as a function of surface temperature (250-4000 K), surface water (10 Pa -1 GPa), surface carbon dioxide (10 Pa to 0.1 GPa), and planetary effective temperature (150 -350 K). Within the grid we linearly interpolate between grid points, and on the rare occasion when the model moves beyond the grid, linear extrapolation is adopted. A 1 bar partial pressure of N2 is assumed at every grid point. Since the atmospheric model calculates atmospheric structure, stratospheric mixing ratios are obtained, which are used to determine atmospheric escape rates (see below).   −− − −− +− = − − − +(16) Here, zero during solid-state convection and are described in detail in their corresponding sections below. Total fluid volatile masses are converted to partial pressures using atmospheric mean molecular weight (no magma ocean) or using the melt solubility relationships described below (magma ocean). Fig. S2 shows illustrative outputs from a single model run. Subplots Fig. S2a, S2b, and S2c show the evolution free oxygen, water, and carbon dioxide reservoirs, respectively, as governed by equation (16). A.7) Magma-ocean evolution: Whilst the magma ocean exists, volatiles in fluid phases are partitioned between the melt, melt crystals, and the atmosphere. For water, this partitioning is described by the following equation (Schaefer et al. 2016 Here, we use the solubility relationship from Pan et al. (1991). For oxygen, equilibrium partitioning is more complicated because both Fe 2+ and Fe 3+ melt phases must be included. We adopt the experimental fit in Kress and Carmichael (1991) Equation (19) is also used to calculate mantle oxygen fugacity for the purposes of outgassing calculations by substituting the solid mantle molar fractions of oxidized and reduced iron. A.8) Transition from magma ocean to solid mantle convection: The model switches between magma ocean and solid-state convection freely, as dictated by the radiation and interior heating budget. At each time step, surface temperature is compared to the solidus. For as long as the surface temperature exceeds the solidus, the magma ocean model is adopted (equation (1)), and the solidification radius evolves with time according to equation (7). However, once the surface temperature drops below the solidus, the solidification radius is set to the planetary radius, and solid state interior evolution is dictated by equation (2). Volatiles are instantaneously exchanged between the magma ocean and the solid interior at this transition. The model assumes that when the surface freezes, any volatiles still dissolved in the magma mush remain in the interior (e.g. as basaltic glass, or gas bubbles in melt inclusions), which is reasonable given the short timescale for magma ocean solidification and the high viscosity of the late-stage magma mush. This assumption also maximizes the mantle's capacity for subsequent outgassing of reduced products that remove oxygen from the atmosphere. During the transition from magma ocean to solid-state convection, volatile inventories undergo the following one-off adjustment: Fig. S5 shows the fraction of total CO2 and H2O that reside in the solid mantle immediately after magma ocean solidification for outputs from Fig. 2 in the main text. These mantle fractions are determined by the partitioning of volatiles in the magma ocean as described in Supplementary Section A.7, and the instantaneous retention of leftover melt when surface temperature drops below the solidus, as described in this section. While we do not explicitly model mechanisms of volatile retention in the magma ocean, such as compaction within the moving freezing front, our spread of final mantle volatile fractions is comparable to that of more detailed models (e.g. Hier-Majumder & Hirschmann 2017). ( )   ( ) F M M M M F  − − − − − − − − − − −− = + − + − = − − + − =+ ( ) ( ) ( ) ( )M M M M M M M M M M − − − − − − =− =+(23) Fig. S5 : Volatile fraction in the solid mantle immediately after magma ocean solidification for nominal model calculations (Fig. 2). The left subplot shows the carbon dioxide mantle fraction, whereas the right subplot shows the water mantle fraction. In a few rare cases, the transition back from solid to magma ocean causes the surface temperature to exceed the mantle potential temperature. When this occurs, we modify the energy budget as follows: 22 2 3 ( , ( )) ( , ,, ,[ ])ASR L t OLR T T M M CO q T T  − −− =+(24) Here, c q (W/m 2 ) is the conduction of heat from the surface to the interior and is approximated by the diffusion equation: ( ) ( , ) surf p c p surf pc TT q T T k rr − = −(25) The time evolution of volatile reservoirs is also modified in this regime to account for the fact that the radius of solidification is moving downward towards the core (see source code for full details). This lasts until surface temperature is less than mantle potential temperature, and the model switches back to a convective mantle. A.9) Atmospheric Escape Parameterization: Atmospheric escape controls the source flux of abiotic oxygen. Escape rates are determined by the composition and temperature of the stratosphere, and by the stellar XUV flux. For low stratospheric water abundances, escape is limited by the diffusion of water through background gases, or by the XUV flux from the star (whatever is smaller). As the water content in the upper atmosphere increases, the escape regime transitions to XUV-limited because, for steam-dominated atmosphere, there is no cold trap limiting the supply of water to the upper atmosphere. Our approach does not rigorously capture the complexities of atmospheric escape physics (e.g. Owen 2019), but instead uses plausible parameterizations that incorporate broad parameter ranges to cover a wide range of uncertain physical processes. Moreover, our parameterization collapses to wellestablished solutions (e.g. the diffusion limit) for end-member cases. Following Wordsworth and Pierrehumbert (2013), the diffusion of water through noncondensible background gases is given by To calculate XUV-driven hydrodynamic escape of H, and associated O and CO2 drag, we follow Odert et al. (2018) and Zahnle and Kasting (1986). The XUV-energy mass loss rate, XUV  (kg/m 2 /s) is specified by the following equation: ( ,, , ) 4 XUVO lowXUV XUV p XUV p F X F r GM    =(27) Here, XUV F is the XUV flux (W/m 2 ) received from the star (see stellar parameterization). The efficiency of hydrodynamic escape,  , is a function of atmospheric composition and XUV stellar flux, as described below. In general, the XUV-driven mass flux will be partitioned between H loss, O drag and, very under high XUV fluxes, CO2 drag. The hydrogen escape flux, H  (molecules H/m 2 /s), can be obtained by analytically solving equations (4), (5), and (6) m X g m m b m X g m m b k T X k T b X b m X b g m m m X b X b b X b k T X b X b mX m m X b X b − − −− − − − − − − − − − −  + + ++ − −+ + + + =  + + + ( ) 2 2 2 2 2 2 2 / 1/ CO CO H CO O O CO O CO H CO O O CO m X b X b b X b −− − − −   +  + (28) Here, i m (kg) is the mass of the i-th species, i X is the stratospheric mixing ratio of the ith species, where we conservatively assume CO2 is not dissociated to minimize the drag of carbon, T (K) is stratospheric temperature, B k is Boltzmann's constant, and ij b − is the binary diffusion coefficient of i through j. We refer the reader to the original paper for the details. Crucially, once the loss of hydrogen is known, the oxygen fractionation factor, O  , can be obtained: () 1 (1 ) O H H O O H B O g m m b k T X  − − =− +(29) If 0 O   , then the hydrodynamic wind drags oxygen. The carbon dioxide fractionation factor can then be similarly calculated: It is convenient to convert these molecular escape rates to molar escape rates: ( ) 2 2 2 2 2 2 22 1 ( ) (1 ) / 1/ CO H H CO H B H CO O O H CO O O O CO CO H CO O O CO g m m b k T b X b X b b X b   − − − − −− − −  + − + = +(30) These weighting functions ensure diffusion-limited H escape for low stratospheric abundances, and a smoothly transition to XUV-driven escape as the upper atmosphere becomes steam dominated. The precise transition abundance is unknown and will, in general, depend on conductive and radiative cooling of the upper atmosphere as well as downward diffusive transport. Here, it is represented by the free parameter, tra  , which ranges from 10 -2 to 10 2 and is sampled uniformly in log space. The efficiency of hydrodynamic escape is parameterized by loosely following the approach of Wordsworth et al. (2018). If the XUV-stellar flux is insufficient to drag oxygen, then the efficiency is equal to a constant, lowXUV  , which is randomly sampled from 1% to 30%. Alternatively, if the XUV-stellar flux exceeds what is required to drag O, then some portion of the excess energy,  , goes into driving further escape, whereas the rest, 1  − , is assumed to be efficiently radiated away. The efficiency factor,  , is randomly sampled from 0-100% for complete generality. This leads to the following function for the efficiency of hydrodynamic escape: () , 4 (1 ) () ( ,, , ) (1 )( ) (1 ) , 4 (1 ) 4 Fig. S6: Illustrative examples of the escape parameterization. Each line denotes a different calculation, where uncertain escape parameters lowXUV  and tra  have been randomly sampled. Black lines denote the cold trap diffusion limit (eq. (26)), red lines denote XUVdriven hydrodynamic escape (eq. (35)), and green lines show the weighted combination that is the escape parameterization in our model (eq. (37)). On the left-hand side, the stellar XUV flux is held constant, and escape fluxes are plotted as a function of the stratospheric H mixing ratio (for an atmosphere without CO2). On the right-hand side, the composition of the upper atmosphere is held constant, and escape fluxes are plotted as a function of the stellar XUV flux received by the planet. A.10) Solid-state evolution: Outgassing and crustal production The outgassing model follows that described in Wogan et al. (2020) where we calculate redox-dependent speciation of volatiles between melt and gas phase. Given mantle concentrations of H2O and CO2 (by mass), corresponding melt fractions can be calculated assuming accumulated fractional melting: Here  is the average melt fraction over the portion of the mantle where melting occurs (see below). These melt fractions, along with mantle oxygen fugacity (eq. (19)) and magma chamber pressure-temperature conditions are used as inputs to the outgassing calculations. For subaerial outgassing, the outgassing pressure is the pressure overburden of the atmospheric inventory, whereas for submarine outgassing, the pressure overburden is the atmospheric + ocean inventory. All outgassing is assumed to occur at the solidus temperature. Given these inputs, the outgassing thermochemical equilibrium model ) outputs gaseous mixing ratios, i f , for outgassed CO2, H2O, H2, CO, and CH4, as well as the moles of gas per total moles of melt plus gaseous species, G  . Note that the outgassing model does not consider the evolution of melt composition and oxygen fugacity along a degassing path; instead, we assume that the melt oxygen fugacity is buffered to that of the source rock, and that outgassed volatiles are determined by the equilibrium gas phase mixing ratios at surface pressure. Moreover, the molecular oxygen fraction of the degassed mixed is conservatively assumed to be negligible; the only possible source of atmospheric oxygen in the model is H escape. Gaseous mixing ratios can be convolved with melt production, MP , (m 3 /s, described below) to calculate outgassing fluxes, i V (mol/s), for each species: 1 i G m im G f V MP    =   −(42) Here, m  = 15.5 mol magma/kg magma is the inverse molar mass of magma, which is assumed to be constant. The overall O2-consumption sink from outgassed volatiles is the summation of reducing species: 2 2 4 0.5 0.5 2 O sink H CO CH V V V V − = + +(43) Overall outgassing fluxes are the combination of subaerial and submarine contributions, weighted by the surface land fraction, LF (see below): (1 ) subaerial submarine i i i V V LF V LF = + −(44) To obtain mass fluxes, these molar outgassing fluxes must be weighted by their respective molecular masses: Here, the melt fraction at any given radius is given by the following expression: 0, ( ) ( , ) ( ( ), , ) 1, ( )( , ) ( ) ( , ) , ( , ) ( , Clearly these are crude approximations, but they capture the fact the planets with a few Earth oceans ought to have some subaerial land, but that for large water inventories all crust is submerged. In any case, total land fraction does not have a large impact on weathering feedbacks (Abbot et al. 2012 = 90 kJ/mol, and pH-dependence of seafloor weathering are assumed to be known constants. More sophisticated weathering parameterizations that account for kinetic dependencies, thermodynamic solute concentration limits, and a precipitationlimited runoff dependence have been proposed (Graham & Pierrehumbert 2020), but given the uncertainties in geological parameters that feed into such models, our simple CO2 and temperature dependent formulation is adequate for providing a crude thermostat. Total weathering will be the summation of continental and seafloor weathering, weighted by the fraction of liquid water at the surface, 2 (1 ) atmo H O fr − − . This ensures that weathering tends to zero as oceans evaporate. We also include a possible supply limit to weathering as an unknown variable that could potentially limit the rate at which dense pCO2 atmospheres are sequestered if the supply of erodible rock is low. This supply limit, sup lim W − = 10 5 to 10 7 (kg/s), is not coupled to crustal production since not all newly produced crust will necessarily be delivered to the surface to be weathered. The overall expression for CO2-removal via weathering is therefore: Because we are modeling a plate tectonics regime, we assume that all carbon dioxide removed by weathering is returned to the mantle. This can be seen in equation (16) (16). Note that these loss and gain quantities are not necessarily equal in because, as discussed above, H2 produced by serpentinization may be lost to space, thereby permanently removing water from the planet. A.13) Solid-state evolution: Planetary redox budget The interior may become oxidized via outgassed of reduced species (discussed above), dry oxidation, and wet oxidation. First, we consider dry (direct) crustal oxidation, which can be represented by the following reaction: 2 1.5 2FeO+0.5O 2FeO →(65) The flux of this dry crustal sink is parameterized by the following equation: ( ) M F RLF f x MP MM    − −− −−  =−   + (66) Here, there is a land fraction dependence to ensure that no dry oxidation of the crust occurs when the surface is covered in water. The unknown efficiency parameter dry oxid f − (10 -4 to 10%) is discussed in the main text, and is the fraction of reduced iron in newly produced crust that is oxidized. We also modify the melt production term to represent the fact that there is some maximum amount of surface melt accessible to oxidation via diffusion through extrusive lava flows. The diffusivity of oxygen in silicate melts is ox D ≈ 10 -7 cm 2 /s (Canil & Muehlenbachs 1990), which implies a downward diffusion depth of ~o Here, the efficiency factor, lava f , is the average fraction of the planetary surface that is continuously molten due to extrusive volcanism after the magma ocean has solidified. Because thermal diffusivity exceeds chemical diffusivity-and because mean surface temperature is below the solidus by definition-extrusive magmas will form a low permeability solid crust as they cool, precluding continuous diffusion of oxygen. Thus, even for extreme rates of resurfacing due to high heatflow, lava f is likely low. For example, on Io, where average internal heatflow is 1-3 W/m 2 (Veeder et al. 2012), only a few km 2 of the surface is estimated to be molten at any given moment (Mura et al. 2020). We uniformly sample lava f = 10 -4 to 1 in log space for full generality. Given this upper limit on melt oxidation, the amount of oxidizable melt is given by: does not affect any of our model scenarios except for Scenario 3, where values exceeding 0.01 are required for oxygen accumulation (Fig. S1d). The wet oxidation of the interior from hydration reactions is already the appropriately weighted serpentinization flux: 2 3 O wet oxid H O serp FeO FF   −− =(69) Recall the term oxid fluid F − represents the total flux of free oxygen lost from the atmosphere-ocean reservoir to the interior, whereas the term oxid fluid F − represents the total flux of free oxygen gained by the interior. Oxidized crust is assumed to be mixed back into the mantle on long timescales (equation (16)) via subduction, or slab delamination. We also assume that the outgassing of reduced species must ultimately be balanced by oxidation of the crust, and so the so net oxidation of the interior equals the reduction of the fluid reservoir: 22 oxid fluid dry oxid wet oxid O O sink oxid solid F F F V F  − − − − − = + + =(70) The only exception to this equality is when atmospheric oxygen levels are very low, and fluxes are modified for numerical reasons (see below). Text B. Numerical approach All code is written in python and available open source ([DOI_TBD]). The system of differential equations was solved explicitly with either RK45 or RK23 using the solve_ivp module in scipy. The maximum timestep was shorter for the magma ocean phase (10000 years) compared to the temperate evolution phase (10 6 years). To avoid sawtoothing and excessive computation time at low reservoir abundances, various adjustments were made to the differential equations to ensure adequate performance. First, if atmospheric CO2 dropped below 50 Pa and if weathering exceeds outgassing, then the time-derivative of surface carbon dioxide was set equal to zero. This may mean climate evolution is slightly inaccurate at low pCO2, but the effects are minor. Second, atmospheric oxygen is similarly prevented from dropping below 0.1 Pa. If atmospheric oxygen is below 0.1 Pa and if oxygen production via escape is less than oxygen consumption, then the following adjustments are made. The atmosphere is assumed to be in an anoxic steady state and so oxygen atmospheric sinks are set equal to the escape source. However, since oxygen production via escape is less than oxygen sinks (outgassing and H2, wet and dry oxidation reactions), then the interior is assumed to be oxidized by the difference as the excess reductants (H2) escape to space. This might slightly overestimate mantle oxidation because some reductants (e.g. CO) will get photochemically oxidized rather than escape, but the redox budget of the atmosphere is not directly affected, and the effects on mantle redox evolution are negligible. Text C. Venus Model validation: To further validate the model, we demonstrate that it can broadly reproduce the known atmospheric evolution of Venus. To model Venus, all parameters are kept the same as for Earth except for planet radius, mass, planet-star separation, and albedo parameterization (see Table S2). Fig. S7 shows all model outputs that reproduce modern Venus, which is defined to be atmospheric oxygen < 10 15 kg (~0.2 mbar), atmospheric CO2 exceeding 2×10 20 kg (40 bar), no surface water ocean, and atmospheric H2O < 2×10 16 kg (~3 mbar). Fig. S7 : Model runs that reproduce modern Venus conditions. Note that there are two qualitatively different histories that can reproduce modern Venus (top left). Either Venus was always in a runway greenhouse phase and never condensed a surface ocean, or Venus maintained a temperate surface for several Gyr before transitioning back to runaway greenhouse as solar insolation increased. Note that our Venus model somewhat overpredicts modern day Venusian heat flow and melt production because we assume a plate tectonics model (Nimmo & McKenzie 1998). We save a more detailed comparative study of solar system planets with stagnant lid tectonics and resurfacing events for future study. In Fig. S8 we plot key parameter values for model runs that successfully reproduce modern Venus. Initial volatile inventories are likely small (Fig. S8a) and dry crustal oxidation must be relatively efficient (Fig. S8c). Text D. Hydrous Mantle Sensitivity Test Here, we test whether modifying the solidus for hydrated mantles affects oxygen accumulation in the waterworlds scenario. Following Katz et al. (2003) we modify our expressions for the solidus and liquidus as follows: ( ) ( ) ( ) ( ) ( ) ( ) oxygen accumulation becomes increasingly likely for initial water inventories exceeding 100 Earth oceans (Fig. S10). Fig. 6 in the main text except the solidus decreases with mantle hydration. Abiotic oxygen accumulation is somewhat less frequent, and occurs at higher initial water inventories, but results are qualitatively the same. Text E. Impact Ejecta O2-sinks Here, we test whether the delivery of reducing materials from impactors could potentially draw down oxygen produced in desertworld scenarios. We introduce an impactor flux, imp F (kg/yr), that diminishes exponentially with time: ( ) 0 exp imp imp decay F F t t =(72) Here, the coefficient 0 imp F is randomly sampled (in log space) from 10 11 to 10 14.5 kg/yr and the decay time, decay t (Gyr), is randomly sampled from 0.06 to 0.14 Gyr. These ranges are adopted because they approximately reproduce plausible estimates for impactor fluxes in the Hadean and early Archean, both with and without a late heavy bombardment (Kadoya et al. 2020). Additionally, we assume that impactors are 30% metallic iron by mass, and that 100% of this iron is completely oxidized to ferric iron instantaneously upon impact, depleting atmospheric oxygen. Fig. S11 shows our desertworld calculations repeated with this impactor flux. We find oxygen accumulation and retention for several Gyr is still possible, although only when the total impactor flux is low. Fig. S12 shows oxygen accumulation after 4.5 Gyr as a function of total impactor flux. Abiotic oxygen may accumulation for impactor mass fluxes < 10 20 kg. Each dot is a model run representing an oxygen false positive. For the first scenario (Fig. S13a), abiotic oxygen only occurs when stratospheric temperature exceeds ~200 K. This is because, at lower temperatures, the cold trap becomes more effective and H escape (and therefore O accumulation) is throttled. For the waterworld scenario (Fig. S13b) oxygen accumulation may occur at any stratospheric temperature. However, this is more probable-and abiotic oxygen abundances are greater-at higher stratospheric temperatures. On waterworlds, cold stratospheres are not necessarily expected because an N2-dominated atmosphere with low CO2 is a probable outcome (Fig. 5), especially if continuous CO2-drawdown via weathering occurs (Nakayama et al. 2019). Moreover, modest oxygen accumulation would result in ozone formation, that would further warm the stratosphere, potentially resulting in a positive O2-accumulation feedback that is not considered here. Note that there are two subclasses of oxygen false positives in Fig. S13b, denoted by red and blue dots. The blue dots show model runs where oxygen accumulated during the initial magma ocean is completely sequestered in the mantle upon magma ocean solidification, whereas red dots denote scenarios whereby appreciable oxygen is left over after magma ocean solidification due to the high pressure-temperature conditions of the overburden suppressed solidus. The third, desertworld scenario ( Fig. S13c) is largely insensitive to stratospheric temperature. This is because water loss and oxygen accumulation occur immediately after magma ocean solidification while the steam-dominated atmosphere persists. There is no effective cold trap in the steam-dominated atmosphere and so escape fluxes are insensitive to stratospheric temperature. Text G. Reducing Mantle Sensitivity Test The nominal model assumes Earth-sized planets undergo rapid core formation with mantles that quickly approach ~FMQ±2. While this is a common assumption when modeling magma ocean evolution of the early Earth (e.g. Zahnle et al. 2007;Zahnle et al. 2010) and of terrestrial exoplanets (e.g. Schaefer et al. 2016), this may not be true for all terrestrial planets. To investigate the effects of a more reduced initial mantle, sensitivity tests were performed whereby the initial magma ocean was endowed with a smaller amount of free O (0.5×10 21 to 1.5×10 21 kg), such that after 4.5 Gyrs of evolution, mantle oxygen fugacity is closer to the iron-wüstite buffer than FMQ. Additionally, following Ortenzi et al. (2020), we modified the outgassing model such that the melt-solid partitioning of carbon is controlled by graphite saturation. = −  +  + − = − − − −(74) To calculate melt concentrations for outgassing calculations, we take the minimum of the concentrations in equations (41) and (73) Taking the minimum ensures that graphite saturation does not overestimate dissolved carbon concentrations under oxidizing conditions and when the total carbon content in the mantle is low. Finally, while we do not explicitly account for graphite precipitation during magma ocean solidification, we set 2 CO k =1.0 in equation (16) to allow for greater retention of carbon in the mantle. It should be emphasized that these modifications do not constitute a fully consistent model of reduced mantle planetary evolution because the radiative transfer model does not allow for CO and H2 dominated atmospheres. However, for post magma ocean evolution they are adequate approximations. Fig. S14 is identical to Fig. 2 in the main text except for the reducing mantle initial conditions and other assumptions described above. Once again, an anoxic atmosphere is assured after 4.5 Gyrs of evolution because crustal oxygen sinks overwhelm oxygen sources. Fig. S15 is the reduced mantle equivalent of Fig. 3 in the main text showing high CO2:H2O oxygen false positives. This scenario is largely unchanged by a reducing mantle; magmatic outgassing does not occur due to the high pressures and low mantle volatile concentrations following magma ocean solidification. Gradual oxygen accumulation may occur after several Gyrs of H loss to space. Fig. S16 is the reduced mantle equivalent of Fig. 5 in the main text showing waterworld oxygen false positives. The pressure overburden of a large surface ocean once again shuts down crustal production after ~1 Gyr, thereby removing all crustal oxygen sinks and permitting atmospheric oxygen to accumulate. Fig. S17 is the reduced mantle equivalent of Fig. 7 in the main text showing desertworld oxygen false positives. Although Scenario 3 is apparently unchanged by having a lower mantle redox, a fully self-consistent model that accounted for the high CO-H2 content of the originally degassed atmosphere would likely preclude early O2 accumulation, in practice. Fig. S14: Nominal Earth evolution with a more reduced initial mantle. This is identical to Fig. 2 in the main text except (i) the initial free oxygen of the mantle is lower (0.5-1.5×10 21 kg), (ii) graphite saturation in reduced melts is accounted for, (iii) and carbon is partitioned into the solid phase during magma ocean solidification. Fig. S15: Scenario 1 oxygen false positives with a more reduced initial mantle. This is identical to Fig. 3 in the main text except (i) the initial free oxygen of the mantle is lower (0.5-1.5×10 21 kg), (ii) graphite saturation in reduced melts is accounted for, (iii) and carbon is partitioned into the solid phase during magma ocean solidification. The terminal magma ocean becomes more oxidized than the solid mantle as H escape oxidizes the combined melt-volatile reservoir faster than solidification transfers oxidized material to the mantle. Oxygen accumulation may occur after several Gyr because outgassing of C-bearing volatiles is negligible from the graphite-saturated mantle. Fig. S16: Scenario 2 oxygen false positives with a more reduced initial mantle. This is identical to Fig. 5 in the main text except (i) the initial free oxygen of the mantle is lower (0.5-1.5×10 21 kg), (ii) graphite saturation in reduced melts is accounted for, (iii) and carbon is partitioned into the solid phase during magma ocean solidification. Oxygen sinks are suppressed by the large pressure overburden of the surface ocean, the same as in the nominal oxidized-mantle calculations. Fig. S17: Scenario 3 oxygen false positives with a more reduced initial mantle. This is identical to Fig. 7 in the main text except (i) the initial free oxygen of the mantle is lower (0.5-1.5×10 21 kg), (ii) graphite saturation in reduced melts is accounted for, (iii) and carbon is partitioned into the solid phase during magma ocean solidification. The terminal magma ocean becomes more oxidized than the solid mantle as H escape oxidizes the combined melt-volatile reservoir faster than solidification transfers oxidized material to the mantle. While the persistence of oxygen is permitted in these calculations, in practice, the early atmosphere is likely too reducing to permit such oxygen accumulation. Text H. Stellar Separation Sensitivity Test While this study is not an exhaustive exploration of the oxygen false positive parameter space, nominal model calculations were repeated at 1.3 AU to show that oxygen accumulation is not dependent on being close to the inner edge of the habitable zone. Fig. S18 shows all Scenario 2 false positives at 1.3 AU. The increased stellar separation results in lower H escape fluxes, but oxygen accumulation may still occur if crustal sinks are suppressed by pressure overburden. Similarly, Fig. S19 shows all Scenario 3 false positives at 1.3 AU. Oxygen accumulation on desertworlds can similarly occur at larger stellar separations because oxygen accumulation occurs early during the steamdominated atmosphere. Scenario 1 false positives do not occur at large stellar separations because even under a high CO2 atmosphere, the runaway greenhouse state is not maintained after magma ocean solidification. Fig. 7 in the main text except the assumed planet-star separation is 1.3 AU. Abiotic oxygen accumulation is still permitted due to early oxygen accumulation during the steam-atmosphere phase. Text I. The Effect of Sulfur Outgassing Sulfur outgassing and burial may have played an important role in the oxygenation of Earth's atmosphere (e.g. Gaillard et al. 2011;Olson et al. 2019). While including a complete model of sulfur cycling is beyond the scope of this study, we present calculations showing that Earth-like sulfur mantle abundances are unlikely to qualitatively change our conclusions. Following Gaillard and Scaillet (2014), sulfur speciation is added to our outgassing model by adding the following system of equations to those already described in Wogan et al. shows total oxygen sinks with (blue) and without (red) sulfur-bearing volatiles. While the inclusion of sulfur-bearing species may result in a slightly larger oxygen sink, for Earthlike mantle concentrations the effect of sulfur outgassing is minimal. Fig. S21: Comparison of oxygen sinks with and without sulfur-bearing outgassed volatiles. Subplot (a) shows outgassing sinks for Scenario 1 false positives from Fig. 3 (red) compared to outgassing sinks if sulfur-bearing species are included (blue) and a constant 300 ppm mantle sulfur concentration is assumed (von Gehlen 1992). Subplot (b) shows total oxygen sinks with (blue) and without (red) sulfur-bearing volatiles. Including sulfur-bearing species has a negligible effect on outgassing sinks because the pressure of the dense CO2 atmosphere inhibits exsolution of sulfur gases. * We apply the additional constraint that the initial water inventory is greater than the initial carbon dioxide inventory, in accordance with typical carbonaceous chondrite abundances. * We apply the additional constraint that the initial water inventory is greater than the initial carbon dioxide inventory, in accordance with typical carbonaceous chondrite abundances. ** High CO2:H2O runs are a subset of this range with initial CO2/H2O>1 (by mass). Figure 2 . 2Figure 2. Earth's coupled redox-thermal-climate evolution (without life). The model is applied to the Earth from magma ocean to present with initial water inventories ranging from 1 to 10 Earth oceans, and initial CO 2 inventories ranging from 20 to 2,000 bar. Additionally, we only plot model runs where the initial water inventory exceeds the initial CO 2 inventory. The lines are median values and shaded regions denote 95% confidence intervals across 3,000 model runs. In the absence of life, Earth's atmosphere after 4.5 Ga is always anoxic (c) because outgassing and crustal hydration sinks overwhelm oxygen production via photolysis and diffusion-limited hydrogen escape (i). (a, b) The magma ocean persists for a few million years, consistent with previous studies. (e) The magma ocean ends when the planet's interior cools such that heatflow from the interior drops below the runaway greenhouse limit. (d) When this occurs, liquid water oceans condense onto the surface, (f) a temperate carbon cycle commences. (c) There is sometimes a brief spike in atmospheric oxygen following magma ocean solidification due to the persistence of a steam atmosphere and hydrogen escape, (i) but this oxygen is rapidly drawn down by geological sinks. (g) Volatile cycling is controlled by the rate at which fresh crust is produced. Mantle redox evolution is plotted (h) alongside proxy estimates (O'Neill et al., 2018; Trail et al., 2011). Figure 3 3Figure 3. Oxygen false positives from high initial CO 2 :H 2 O inventories (Scenario 1). The model is applied to the Earth from magma ocean to present with randomly sampled initial water inventories ranging from ∼0.1 to 10 Earth oceans, and initial CO 2 inventories ranging from ∼20 to 2,000 bar (implying CO 2 :H 2 O ranging from 0.01 to 100 by mass). Only model outputs with modern day atmospheric oxygen exceeding 10 17 kg (>∼0.02 bar) are plotted. Subplots are the same as in Figure 2, and shaded regions denote 95% confidence intervals. (a) High atmospheric CO 2 ensures the surface temperature always exceeds the critical point of water after the pre-main sequence, and (d) thus permanent liquid water oceans do not condense. The lack of surface water, low volatile content of the mantle, and high surface pressure increasing volatile solubility in partial melts all limits oxygen sinks. (a, i) The largest atmospheric sink is dry crustal oxidation, which diminishes with time as the interior cools. (c) Atmospheric oxygen produced via H escape may start to accumulate after several Gyr of evolution. shows 4.5 Gyr atmospheric oxygen as a function of the initial CO 2 :H 2 O ratio, and confirms that high oxygen accumulation only occurs when CO 2 :H 2 O exceeds unity. Figure 4 . 4Conditions required for Scenario 1 oxygen false positive. Each dot denotes a single model run, and model runs are shown for uniformly sampled initial volatile abundances: 10 20 -10 22 kg CO 2 and 10 20 -10 22 kg H 2 O. Atmospheric oxygen at 4.5 Figure 5 . 5Oxygen false positives on waterworlds (Scenario 2) Figure 6 . 6Prevalence of Scenario 2 waterworld false positives. 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Characterizing Earth analogs in reflected light: Atmospheric retrieval studies for future space telescopes. The Astronomical Journal, 155(5), 200. https://doi.org/10.3847/1538-3881/aab95c Fischer, D., Bradley, P., Bean, J., Calzetti, D., Dawson, R., Dressing, C., et al. (2019). The LUVOIR Mission Concept study final report. arXiv preprint. arXiv:1912.06219. Fujii, Y., Kawahara, H., Suto, Y., Taruya, A., Fukuda, S., Nakajima, T., & Turner, E. L. (2010). Colors of a second Earth: Estimating the fractional areas of ocean, land, and vegetation of Earth-like exoplanets. The Astrophysical Journal, 715(2), 866. https://doi. org/10.1088/0004-637x/715/2/866 KRISSANSEN-TOTTON ET AL. Fig. S1 .Fig. S2 : S1S2Conditions required for Scenario 3 oxygen false positive. Each dot denotes a single model run, and model runs are shown for all model runs inFig. 7. Atmospheric oxygen at 4.5 Gyrs is plotted as a function of (a) dry oxidation efficiency, and (b) XUVdriven escape efficiency. Subplot (c) shows the initial water and carbon dioxide inventories that result in desertworld false positives, and (d) shows the maximum fractional planetary area that is continuously molten (available for oxidation) after the magma ocean has solidified. Illustrative model outputs. Selected model outputs from a single, representative model run taken fromFig. 2in the main text, the nominal Earth model. Subplots illustrate (a) free oxygen reservoirs, (b) water reservoirs, (c) carbon dioxide reservoirs, (d) free oxygen fluxes, (e) atmosphere-ocean partitioning of water, (f) carbon outgassing and weathering fluxes, (g) iron oxidation in the solid mantle, (h) surface and mantle potential temperature, and (i) the solidification radius. Early fluctuations in surface temperature and volatile fluxes are due to non-monotonic evolution of solar luminosity during the pre-main sequence. Total free oxygen (a) increases and total water decreases (b) as H is preferentially lost to space, whereas total CO2 remains constant (c). ( J/kg/s) is the heat production via radiogenic isotopes. Earth's radiogenic inventories are taken fromLebrun et al. (2013), and a range of 0.33-3.0 times this nominal, Earth-like inventory is sampled in Monte Carlo calculations. Additionally, m  = 4000 kg/m 3 is the assumed average density of mantle material, p r is the total radius of the planet, c r is the radius of the core, s r is the radius of solidification, kg is the latent heat of fusion of silicates, p c = 1200 J/kg/K is the heat capacity of silicates. We adopt a simple exponential decay for core heatflow ( be expanded using the chain rule and rearranged to obtain an expression for the solidus evolution in terms of the potential temperature time stop evolving when the magma ocean freezes. This procedure was checked against simply numerically solving for the solidification radius at every timestep: Fig. S3 : S3Assumed viscosity parameterizations compared to other parameterizations from the literature (solid lines). The dashed-cyan lines represent the mantle potential temperature and viscosity required to reproduce the modern Earth's melt production and plate velocity. Fig. S4 : S4Illustrative figure showing calculations from OLR grid. Top plots show OLR as a function of surface temperature for varying surface CO2 pressures (log10(Pa)). Here, was have assumed [CO3 2-] = 10 -3 mol/kg. Middle plots show the amount of water in the atmosphere as a fraction of the total surface water inventory, 2 atmo H O fr − . Bottom plotsshow the stratospheric water mixing ratio for different CO2 inventories. On the left-hand side 1 Earth ocean is assumed, whereas on the right-hand side 0.1 Earth oceans are assumed.A.6) Volatile reservoirs and planetary redox budget: The time evolution of planetary volatile budgets and redox states is determined by the following system of equations: M. − (kg) and Fluid i M − (kg) represent the masses of the i-th species in the solid interior and the fluid magma ocean plus surface reservoirs, respectively. Water, free oxygen, solid FeO and FeO1.5 in the mantle, and carbon dioxide are separately tracked in the model. The variables i fr represents the mass fraction of the i-th species in the magma ocean partial melt. These are calculated at every timestep using the equilibrium relations described below. Key constants include the density of mantle material, m  = 4000 kg/m 3 , molecular masses of key species, i  (kg/mol), and the assumed partition coefficients for CO2 and H2O, . See the Discussion section in the main text for possible consequences of assuming constant partition coefficients. Escape fluxes for atomic hydrogen, H E (mol/s 2 ), atomic oxygen O E (mol/s 2 ), and carbon dioxide, 2 CO E (mol/s 2 ) are parameterized below. The remaining fluxes (kg/s) are the ingassing of surface water into the interior serpentinizing reactions can remove water from the surface that never reaches the interior because the hydrogen produced is lost to space), oxidation of the interior, oxid solid F − , corresponding loss of oxidants from surface reservoirs, oxid fluid F − (typically oxid solid oxid fluid FF −− = except for anoxic atmospheres where reductants are lost to space), outgassing of carbon to the surface, Fluxes of ferrous and ferric iron are equal to oxid solid F − scaled by appropriate molecular masses. All fluxes denoted i F are only non-  (mol H2O/m 2 /s). The diffusion-limited flux should arguably be set by the diffusion of atomic hydrogen through background gases since eddy diffusion dominates vertical transport at altitudes where molecular water is more abundant than atomic H and O (Catling & Kasting 2017), but our conservative approach minimizes oxygen accumulation from H escape.  then carbon dioxide is dragged along in the hydrodynamic wind and the corresponding escape fluxes (mol/m 2 /s) of O and CO2 are given by the following expressions::  , then oxygen is dragged but not carbon dioxide, and the corresponding escape fluxes are as follows: , then then the XUV-driven escape flux is too small to drag oxygen. Carbon dioxide fractionation must be recalculated with diffusion limited escape flux exceeds the XUV-driven H loss, then the diffusion limited flux is adjusted downwards: we combine our expressions for diffusion-limit hydrogen escape and XUV-limited escape fluxes to obtain general expressions for the escape fluxes of hydrogen, given time, the average melt fraction of freshly produced crust is given by integrating the melt fraction from the radius at which mantle temperature equals the solidus to the surface: F −− , represents the flux of water lost from the surface reservoir, whereas the latter, 2 ingas H O gain F −− , represents the flux of water into the interior via subduction of hydrated crust. This transfer of water can also be seen in equation unrealistic oxygen sinks during the transition from magma mush to solid mantle where melt volumes are extremely high, but the melt accessible to atmospheric oxygen via diffusion through extrusive magmas limited. The value for lava f Fig. S8 : S8Parameter values for individual model runs inFig. S7. These shows the require initial volatile inventories (bottom left) and dry crustal oxidation efficiency (top left) required to reproduce modern Venus. A broad range of XUV escape efficiencies (top right) and fractional molten areas (bottom right) are permissible. Fig. S9 : S9Same as Fig. 5 in the main text except the solidus decreases with mantle hydration. Early crustal production is elevated (bottom left), but outcomes are qualitatively similar. Oxygen sinks are shut down by the pressure overburden after a few billion years and oxygen accumulates. Fig. S10 : S10Same as Fig. S11 : S11Same as Fig. 7 in the main text except impact ejecta sinks for oxygen have been added. Retention of abiotic oxygen is still possible if impactor fluxes are low. Fig. S12 : S12Scenario 3 abiotic oxygen accumulation as a function of total impactor flux. Large impactor fluxes preclude the retention of abiotic oxidation if all impactor material is efficiently oxidized.34 Text F. Stratospheric Temperature Sensitivity Test Fig. S13: Sensitivity of false positive results to stratospheric temperature.Calculations in main text were repeated, but stratospheric temperature is a free variable that is randomly sampled from 150 K to 250 K. Final oxygen accumulation after 4.5 Gyrs is plotted as a function of stratospheric temperature. Subplot (a) shows all Scenario 1 model runs, whereas (b) shows all Scenario 2 false positives, and (c) shows all Scenario 3 false positives. In (b) red dots denote leftover oxygen after the magma ocean, whereas blue dots show model runs where all oxygen produced during the magma ocean phase is sequestered in the mantle, and oxygen builds up subsequently, as described in the main text. Fig. S13 shows the sensitivity of each false positive scenario to stratospheric temperature. Fig. S18 : S18Identical to Fig. 5 in the main text except the assumed planet-star separation is 1.3 AU. Abiotic oxygen accumulation is still permitted due to overburden pressure suppressing oxygen sinks. Fig. S19 : S19Identical to Fig. S20 : S20Comparison of oxygen sinks with and without sulfur-bearing outgassed volatiles. Subplot (a) shows outgassing sinks in the nominal model fromFig. 2(red) compared to outgassing sinks if sulfur-bearing species are included (blue) and a constant 300 ppm mantle sulfur concentration is assumed (von Gehlen 1992). Subplot (b) AGU Advancesthe mass fraction of the volatile species in the magma. The remaining fluxes are subaerial plus submarine outgassing from the mantle to the atmosphere,Λ i fr represents KRISSANSEN-TOTTON ET AL. 10.1029/2020AV000294 5 of 20  outgas Λ i F , ingassing from the atmosphere to the mantle (e.g., crustal oxidation or hydration),  ingas Λ i F , and escape to space Λ i .rock p solidus magma liquidus p solidus p liquidus p solidus liquidus p rock p solidus magma p liquidus rock coef 16 bond bond planet star Lt ASR L t D    − − = (13) Here, planet star D − (m) is the distance between Earth (or Venus) and the sun. Following Pluriel et al. (2019) we assumed the following parameterization for albedo: ( ) ( ) 2 5 2 10 2 5 5 2 1000.0 200 log , 10 10 1000.0, 10 0.5 tanh 0.5 400 transition transition surf bond cold hot cold hot pH O pH O Pa T pH O Pa TT A A A A    +     =      −  = − + −   (14) Here, transition T (K) is the albedo transition temperature that controls the transition from a hot-state albedo, hot A to a cold-state albedo, cold The molecular mass of the atmosphere is given by  (mol/kg). The third term on the left-hand side represents the mass of water in the atmosphere, where we adopt the solubility relationship fromPapale (1997). An analogous expression can be used to calculate the partitioning of carbon dioxide:): ( ) 22 2 2 2 2 1 0.74 2 8 4 3.44 10 H O H O p H O H O crystal liquid crystal H O fluid H O fr r k fr M M M fr M g    − −  + − + =    (17) Here, ( ) 33 43 liquid m p s M r r  =− (kg) is the mass of the magma ocean, (1 ) crystal liq MM  =− is the crystal mass fraction in the magma ocean, which depends on melt fraction,  (see below). ( ) 22 2 2 2 2 2 12 4 4.4 10 CO CO p CO CO crystal CO liquid crystal fluid CO x r k x M x M M M g    − −  + − + =    FeOx . This can be done by expressing the partial pressure of oxygen in terms of the total fluid free oxygen minus the free oxygen dissolved in the magma ocean, and by noting that Here, the average molecular weight of silicates is taken to be:: ( ) ( ) 23 2 2 3 22 7 10 11492 ln 1.828 0.196ln 6.675 2.243 1673 3.201 5.854 6.215 3.36 1 ln 7.01 10 1673 1673 1.54 10 3.85 10 Fe O Fe O Al O FeO surf surf surf CaO Na O K O surf surf surf surf surf x x P x xT TP x x x TT PT T − −  = − + + − −     + + + − − − −       − −  +  2 17 surf surf P T − (19) Here, 23 Fe O x and FeO x are the mole fractions of iron-bearing species in the melt, whereas 23 2 Fe Fe O FeO x x x =+ is the total mole fraction of all iron-bearing species in the mantle (constant). The molar abundances of other species, i x , are assumed to be represent Bulk Silicate Earth (White 2013). We calculate oxygen solubility using surface temperature and pressure conditions because the interface controls volatile partitioning. In equation (19) there are three unknowns ( 2 O P , FeO x , and 23 Fe O x ) and so it is necessary to make appropriate substitutions to solve for 23 2 FeO Fe Fe O x x x =− : 23 23 2 1.5 23 23 2 3 2 2 2 0.5 ln 1.828 0.196ln 24 11492 6.675 2.243 3.201 5.854 6.215 16 3.36 1 Fe O O fluid O liquid Fe O O FeO sil Fe O Fe Fe Fe O p Al O CaO Na O K O surf M M x x x x x r g x x x x T        −      −        = − +    −      + − − + + + −− ( ) 7 10 2 17 1673 73 ln 7.01 10 1.54 10 1673 3.85 10 surf surf surf surf surf surf surf surf surf PT TP T T T P T −− − −   − −  −       + (20) 2 2 2 3 2 3 2 3 2 3 sil MgO MgO SiO SiO Al O Al O CaO CaO Fe O Fe O FeO FeO x x x x x x        = + + + + + (21) Equation (20) is solved at every timestep in the model, and from the values for melt concentrations of iron-bearing species, we can retrieve the melt fractions (by mass) that dictate the time-evolution of volatile reservoirs: 23 1.5 2 3 23 23 2 1.5 2 2 0.5 4 FeO FeO FeO sil Fe O FeO Fe O sil Fe O O fluid O liquid Fe O O FeO sil O p fr x fr x M M x P rg            − = =    −      = Note that the water transferred to the mantle cannot exceed the maximum water content of the mantle. Transition from solid state convection to magma ocean, which is typically only relevant for Venus model runs, is similarly calculated.:1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 33 33 33 33 2 4 32 4 3 4 3 O ps FeO O fluid O fluid O m FeO p s FeO Solid FeO Solid FeO m FeO p s Solid FeO Solid FeO m FeO p s Solid CO Solid CO CO CO crystal CO liq cr rr M M F r r M M F r r M M F r r M M k F M F M M        −− −− −− −− − = − − = + − = + − = + + − ( ) ( ) 2 2 2 2 2 ystal Fluid CO Fluid CO CO CO crystal CO liq crystal M M k F M F M M −− = − − − Solid i Solid i liq Solid i mantle fluid i fluid i liq Solid i mantle )solidus overburden solidus liquidus liquidus overburden solidus overburden liquidus overburden solidus overburden T r T P r T r T T T r T P r T r T P r otherwise T P r T P r       =   −   −  ( )   ( ) 0.25 max 0.25 max 0,1 / 1 2.5 / 11.4 ocean ocean LF d d LF RLF − =− = − ).Continental weathering fluxes, cont W (kg/s), and seafloor weathering fluxes, SF W (kg/s), are given by expressions similar to those described in Krissansen-Totton et al. (2018): ( ) ( ) 0.3 7.7 11 10 exp 4 3 / 8.314 285 pH coef plate SF SF deep W E W cm yr T  −−    =  − −         (55) ( ) 2 285 exp 350 surf cont coef efold T pCO W W RLF ppm T   −   =   −     (56) Here, the multiplicative factor coef W = 4000 kg/s is chosen to approximately reproduce modern Earth fluxes (or rather is the weathering flux required to balance mantle-derived CO2 outgassing). Unknown, randomly sampled parameters include the temperature dependence of continental weathering, efold T = 5-30 K and the CO2 dependence of continental weathering,  = 0.1-0.5. The temperature dependence of seafloor weathering, SF E The concentration of carbon dissolved in a graphite-saturated melt in redox state dependent: fO is mantle oxygen fugacity, and we are converting between dissolved carbonate and carbon dioxide concentrations. The temperature and pressure-dependent equilibrium constants are defined as follows:( ) ( ) ( ) ( ) ( ) 3 2 3 3 , 1, 2, 2 1, 2, 2 , , , /1 44 / 36.594 / 1 1 44 / 36.594 CO grahite sat graphite graphite graphite graphite CO grahite sat CO grahite sat CO grahite sat X K K fO K K fO X X X − − − − =+ = − − (73) Here, 2 ( ) ( ) 2 6 2 10 1, 10 2, ( ) 40.07639 2.53932 10 T 5.27096 10 T 0.0267 P 1 / T ( ) 6.24763 282.56 / T 0.119242 P 1000 / T graphite graphite log K log K −− Table S1 . S1Monte Carlo analysis and uncertain parameter ranges (for nominal Earth model). Tu et al. 2015) Implies XUV saturation time of 6 -226 Myrs. Escape efficiency at low XUV flux, lowXUV  0.01-0.3 See escape section.Transition parameter for coldtrap diffusion limited to XUVlimited escape, 10 1 -10 3 Pa s Fit to modern heatflow and melt production (seeFig. S3)Nominal range References/Notes Initial conditions Water 10 21 -10 22 kg* Approximately 1 -10 Earth oceans Carbon dioxide 10 20 -10 22 kg* Approximately 20-2000 bar (if no other atmospheric constituents). Radiogenic inventory (relative Earth) 0.33-3.0 Scalar multiplication of inventories in Lebrun et al. (2013) Mantle free oxygen 2×10 21 - 6×10 21 (kg) This ensures a post- solidification mantle redox around Quartz-Fayalite- Magnetite buffer. Solar evolution and escape parameters Early sun rotation rate (relative modern) 1.8-45 (tra  10 -2 -10 2 See escape section. XUV energy that contributes to XUV escape above hydrodynamic threshold,  0-100% See escape section. Carbon cycle parameters Temperature-dependence of continental weathering, efold T 5-30 K (Krissansen-Totton et al. 2018) CO2-dependence of continental weathering,  0.1-0.5 (Krissansen-Totton et al. 2018) Weathering supply limit, sup lim W − 10 5 -10 7 kg/s (Foley 2015) Ocean calcium concentration, 2 Ca +   10 -4 -3×10 -1 mol/kg See text for explanation (Halevy & Bachan 2017; Kite & Ford 2018) Ocean carbonate saturation,  1-10 (Zeebe & Westbroek 2003) Interior evolution parameter Mantle viscosity coefficient, coef V Crustal sinks oxygen and Crustal hydration efficiency, hydr frac fr − 10 -3 to 0.03 Upper limit wt % H2O in oceanic crust. Lower limit hydration limited by cracking. hydrological cycle parameters Dry oxidation efficiency, dry oxid f − 10 -4 to 10% Plausible range of processes for Venus (Gillmann et al. 2009) Wet oxidation efficiency, wet oxid f − 10 -3 to 10 -1 Based on oxidation of Earth's oceanic crust (Lécuyer & Ricard 1999). Maximum fractional molten area, lava f 10 -4 to 1.0 Refer Supplementary Text A.13 Max mantle water content, 2 max solid H O M −− 0.5-15 Earth oceans (Cowan & Abbot 2014) Albedo parameters Hot state albedo, H A 0-0.3 See albedo parameterization. Cold state albedo, C A 0.25-0.35 See albedo parameterization. Table S2 : S2Changes in Monte Carlo parameters for different abiotic oxygen scenarios and Venus validation:Nominal rangeHigh CO2:H2O Waterworld s Desertworlds Venus Initial cond. Table S3 : S3All fixed parameters used in the model. Parameter Value Planetary iron content (silicate mole fraction), Latent heat of fusion of silicates, Thermal expansion coefficient for silicates,  2×10 -5 K -1 Critical Rayleigh number, Molecular mass of i-th species, Binary diffusion coefficient of the i-th species through the j-th species, Mass of i-th species, − 1.496×10 11 m (Earth), 1.047×10 11 m (Venus)Fe x 0.06 Average silicate density, m  4000 kg/m 3 Planetary radius, p r 6371 km (Earth) or 6052 km (Venus) Core radius, c r 3460 km (Earth) or 3230 km (Venus) Planet mass, P M 5.972×10 24 kg (Earth) or 4.867×10 24 kg (Venus) fusion H 4×10 5 J/kg Specific heat of silicates, p c 1200 J/kg/K crit Ra 1100 Thermal conductivity of silicates, k 4.2 W/m/K Thermal diffusivity of silicates,  10 -6 m 2 /s Convective heatflow exponent,  1/3 i  Various (kg/mol) Crystal-melt partition coefficient for water, 2 HO k 0.01 Crystal-melt partition coefficient for carbon dioxide, 2 CO k 2×10 -3 ij b − Various (mol/m/s) Stratospheric temperature, strat T 200 K Inverse molar mass of magma, m  15.5 mol/kg Weathering multiplicative coefficient, coef W 4000 kg/s Activation energy seafloor weathering, SF E 90 kJ/mol i m Various (kg) Planet-star separation, planet star D Here, ij b − (mol/m/s) is the binary diffusion coefficient of the i-th species through the j-th species(Marrero & Mason 1972;Zahnle & Kasting 1986). These are weighted by the stratospheric mixing ratios of each non-condensible constituent (CO2, N2, and O2), which are obtained from the atmospheric profile calculations (see above). The scale height of water,Note that we are assuming the entire planetary area is involved in plate tectonics, which might cause us to underestimate melt production slightly. Plate velocity can be estimated by assuming a plausible ridge length, 3 p r  :Plate velocity is only used for calculating seafloor weathering rates. Outgassing fluxes and crustal sinks of oxygen all depend on melt production. Note that for Venus and stagnant lid exoplanets the assumption of plate tectonics may overestimate melt production. Future versions of the model will explore a stagnant lid regime, but plate tectonics ought to maximize melt production and therefore maximize geological sinks of oxygen, which are the focus of this study.A.11) Solid-state evolution: Weathering Carbon is transferred from surface reservoirs to the interior via silicate weathering. Silicate weathering is the combination of continental weathering and seafloor weathering. To estimate this partitioning, we need to calculate the average depth of the ocean and corresponding land fraction. Following Cowan and Abbot(2014), we assume that there is a maximum ocean depth,Given these two quantities, we approximate the planetary hypsometric curve (proportion of land as a function of elevation) with a power law, and use this to calculate average land fraction, LF , and land fraction relative to the modern Earth, RLF :Here, 973 K is the maximum surface temperature for serpentine stability(Schaefer & Sasselov 2015), and k = 4.2 W/m/K is the thermal conductivity of silicates (c.f. equation(4)). Since we assume no hydration occurs below the crust, it is also helpful to define the fractional depth of hydration as the ratio of the hydration depth to the crustal depth, or 1.0 (whichever is smaller):Water loss from surface reservoirs can be conceptually partitioned into hydration reactions that add water to the solid interior but do not alter atmospheric redox state, e.g.and hydration reactions that oxidize the solid interior and result in outgassed hydrogen (i.e. hydrogen that is lost to space under anoxic conditions, or recombines with atmospheric oxygen under oxidizing conditions):Hydration reactions that add water to the interior, equation(60)Here, hydr frac fr − = 10 -3 to 0.03 (sampled uniformly in log space) is the unknown efficiency of hydration reactions. We are assuming that, at most, hydrated crust is 3% water by 26 mass(Schaefer & Sasselov 2015), but this number could be much less depending on the tectonic regime and cracking of the crust. Some unknown portion of this crustal water may be returned to the surface via arc volcanism, for instance. After considering this efficiency factor, we assumed all hydrated crust is returned to the mantle. We assume a linear dependence on ocean depth for as long as there is emerged land, then no ocean depth dependence beyond this. Additionally, we assume that the return of water to the interior tapers off as the water content of the mantle approaches its maximum value,. This unknown variable is sampled randomly from 0.5-15 Earth oceans(Komacek & Abbot 2016).Hydration reactions that oxidize the crust and remove water from surface, but do not add water to the interior (i.e. serpentinizing reactions that produce hydrogen, equation(61)) are parameterized as follows:is another unknown efficiency parameter (10 -3 to 10 -1 ) representing the fraction of crustal iron that is oxidized via hydration reactions(Lécuyer & Ricard 1999). This efficiency parameter represents the degree of serpentinization in water-rock reactions. Total ingassing contributions are calculated as follows:This modification is a crude polynomial fit toFig. 3inKatz et al. (2003). It accounts for the depression of the solidus as the water content of the mantle increases, but the solidus never drops below about 900°C, which is the approximate water saturation limit. Waterworld calculations repeated with this hydrated solidus are shown inFig. S9. Although crustal production is elevated compared to the anhydrous solidus case in the main text, outcomes are qualitatively similar toFig. 5The equations in (76) are solved simultaneously with the system of equations describing carbon, oxygen, and hydrogen gas-melt speciation ) to determine total outgassing fluxes, and the overall outgassing oxygen sink:Rather than attempt to model complete sulfur cycling, model outputs fromFig. 2were post-processed to recalculate outgassing fluxes and oxygen sinks using the above model, assuming constant 300 ppm mantle sulfur abundances (von Gehlen 1992).Fig. S20shows the results from this calculation comparing oxygen sinks with and without sulfur outgassing. The impact on total oxygen sink fluxes is relatively minor.Fig. S21shows the same calculations performed on the Scenario 1 false positive outputs fromFig. 3. 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[ "SMOOTH LATTICE ORBITS OF NILPOTENT GROUPS AND STRICT COMPARISON OF PROJECTIONS", "SMOOTH LATTICE ORBITS OF NILPOTENT GROUPS AND STRICT COMPARISON OF PROJECTIONS" ]	[ "Erik Bédos ", "ANDUlrik Enstad ", "Jordy Timo Van Velthoven " ]	[]	[]	This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the nonexistence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C * -module. An important ingredient in the approach is that twisted group C * -algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C * -algebra has strict comparison of projections.2020 Mathematics Subject Classification. 22D25, 22E27, 42C30, 42C40, 46L08, 46L35.	10.1016/j.jfa.2022.109572	[ "https://arxiv.org/pdf/2107.13850v2.pdf" ]	249,018,195	2107.13850	8ce511955efdb832bcb33c44c40a161bd32918a1	SMOOTH LATTICE ORBITS OF NILPOTENT GROUPS AND STRICT COMPARISON OF PROJECTIONS 24 May 2022 Erik Bédos ANDUlrik Enstad Jordy Timo Van Velthoven SMOOTH LATTICE ORBITS OF NILPOTENT GROUPS AND STRICT COMPARISON OF PROJECTIONS 24 May 2022arXiv:2107.13850v2 [math.FA] This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the nonexistence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C * -module. An important ingredient in the approach is that twisted group C * -algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C * -algebra has strict comparison of projections.2020 Mathematics Subject Classification. 22D25, 22E27, 42C30, 42C40, 46L08, 46L35. Introduction Let G be a nilpotent Lie group and let (π, H π ) be an irreducible, square-integrable projective representation of G. For a lattice Γ ≤ G, consider the orbit of π under a vector g ∈ H π , π(Γ)g = (π(γ)g) γ∈Γ . (1.1) The aim of this paper is to study the existence of a vector g ∈ H π such that π(Γ)g forms a frame or Riesz sequence (Riesz basis for its span) in H π , that is, π(Γ)g satisfies the frame inequalities A f 2 Hπ ≤ γ∈Γ \| f, π(γ)g \| 2 ≤ B f 2 Hπ , f ∈ H π ,(1.2) for constants 0 < A ≤ B < ∞, or the Riesz inequalities A c 2 ℓ 2 ≤ γ∈Γ c γ π(γ)g 2 Hπ ≤ B c 2 ℓ 2 , c ∈ ℓ 2 (Γ). (1.3) A particular focus will be on the existence of frames and Riesz sequences π(Γ)g for which the associated diagonal matrix coefficient function C g g : G → C, defined by C g g(x) = g, π(x)g , x ∈ G, (1.4) possesses an additional form of localization, e.g., smoothness or L 1 -integrability. Frames and Riesz sequences are classical notions in various areas of complex and harmonic analysis, and play an important role in the applications of these areas as they provide stable and unconditionally convergent Hilbert space expansions. More modern variants of these notions have also been studied in the setting of operator theory and operator algebras, most notably in Hilbert C * -modules, where they give rise to projections in associated C * -algebras. In this paper the existence of localized frames and Riesz sequences of the form (1.1) will be studied via a correspondence to projections in an associated twisted group C * -algebra. It turns out that recent results on C * -algebras (in particular, group C * -algebras) provide adequate tools that are capable of treating localization properties in the existence problem. Before formulating the main results and describing the methods used in their proof, the requisite background and context will be sketched. 1.1. Background and context. A first fundamental obstruction to the existence of frames and Riesz sequences of lattice orbits π(Γ)g is provided by the density theorem, relating the lattice co-volume vol(G/Γ) or its reciprocal (the so-called "density") and the formal dimension d π > 0 of π. Different versions of this theorem can be found in, e.g., [12,102,106]. Theorem 1.1. Let G be a nilpotent Lie group with a lattice Γ ≤ G. Let (π, H π ) be an irreducible, square-integrable projective representation of G of formal dimension d π > 0. (i) If there exists g ∈ H π such that π(Γ)g forms a frame, then vol(G/Γ)d π ≤ 1. (ii) If there exists g ∈ H π such that π(Γ)g forms a Riesz sequence, then vol(G/Γ)d π ≥ 1. (The value vol(G/Γ)d π is independent of the choice of Haar measure on G.) Theorem 1.1 provides a critical density for a lattice to admit a frame or Riesz sequence in its orbit. In particular, a lattice admitting an orthonormal basis must have the critical density vol(G/Γ) = d −1 π . Necessary conditions of this type are commonly referred to as density conditions and can also be obtained for discrete index sets that do not necessarily form a group, see, e.g., [43,82]. There is a converse to Theorem 1.1 for irreducible representations of a nilpotent Lie group N that are square-integrable modulo the center Z = Z(N ). Representations of this type can be treated as projective representations of G = N/Z, so-called projective relative discrete series representations (see Section 6.1). The following result can be derived from [12,34] by combining [34,Theorem 1.3] and the arguments underlying [12,Theorem 3]. Theorem 1.2. Let G be a connected, simply connected nilpotent Lie group with a lattice Γ ≤ G. Let (π, H π ) be a projective relative discrete series representation of G of formal dimension d π > 0. (i) If vol(G/Γ)d π ≤ 1, then there exists g ∈ H π such that π(Γ)g forms a frame. (ii) If vol(G/Γ)d π ≥ 1, then there exists g ∈ H π such that π(Γ)g forms a Riesz sequence. Together, Theorem 1.1 and Theorem 1.2 provide a dichotomy that completely describes the reproducing properties (frame and Riesz sequence) of lattice orbits of square-integrable representations in terms of the lattice co-volume or density. The existence claims in Theorem 1.2 rely on techniques for von Neumann algebras and are not accompanied by explicit constructions. For more specific representations and lattices acting via two group actions, special cases of the existence claims in Theorem 1.2 can also be obtained via tiling arguments [29,57], in which case the generating vector can be chosen to be an indicator function of a common fundamental domain. For historical expositions on the density theorem in time-frequency analysis, see [39,59]. For vectors g ∈ H π possessing certain localization properties (i.e., a smooth or integrable vector), a second obstruction to the existence of frames and Riesz sequences of the form π(Γ)g is given by the strictness of the density conditions in Theorem 1.1. For the Euclidean plane G = R 2 and its projective Schrödinger representation (π, L 2 (R)), the fundamental Balian-Low theorem in time-frequency analysis asserts that there exists no orthonormal basis (or Riesz basis) of the form π(Γ)g for a Schwartz function g ∈ S(R), [14,27]. Alternatively, for a Schwartz function, the associated density inequalities in Theorem 1.1 are strict [6,37,51]. Balian-Low type theorems for (classes of) nilpotent groups have been obtained in [35,52] and show that the inequalities in Theorem 1.1 are strict for integrable vectors. It should be mentioned that (non-localized) orthonormal bases in the orbit of a nilpotent Lie group could still exist by Theorem 1.2, and even for nilpotent Lie groups not admitting a lattice, cf. [54,91]. A key problem in time-frequency or phase-space analysis [38,46] is the existence of smooth frames (resp. Riesz sequences) π(Γ)g for a given lattice Γ ≤ G with super-critical (resp. subcritical) density. While the mere existence of such frames and Riesz sequences are well-known for lattices possessing a qualitative "covering density" [45], there are currently no quantitative results that match the necessary conditions provided by Theorem 1.1, except for the specific setting of the Heisenberg group. Indeed, for G = R 2 and its projective Schrödinger representation (π, L 2 (R)) (for which d π = 1), the density theorems for sampling and interpolation in Bargmann-Fock spaces [79,112,113] can be recast as the Gaussian Gabor system π(Γ)g with g(t) = e −πt 2 forming a frame (resp. Riesz sequence) for L 2 (R) if and only if vol(R 2 /Γ) < 1 (resp. vol(R 2 /Γ) > 1), see also [53,55,64]. Although the frame and Riesz property of a multivariate Gaussian Gabor system cannot be simply described in terms of a density condition [48,96], it is still expected [48,Remark 2] that Gabor frames (resp. Riesz sequences) π(Γ)g with arbitrary smooth window g ∈ L 2 (R d ) exist for any lattice Γ ≤ R 2d satisfying vol(R 2d /Γ) < 1 (resp. vol(R 2d /Γ) > 1), see also [59,97]. Only recently has there been a first contribution [63] to this existence problem for Gabor frames in higher dimensions, namely for so-called nonrational lattices Γ ≤ R 2d , by exploiting the structural results on (irrational) non-commutative tori [104] and its link with Gabor frames [77]; see Section 7.1 for a more detailed discussion. Main results. Our main result concerns the existence of frames and Riesz sequences generated by smooth vectors, i.e., vectors g ∈ H π for which the orbit maps x → π(x)g are smooth; in notation, g ∈ H ∞ π . The result relies on a compatibility condition between the 2cocycle σ of the projective representation π and the lattice Γ, known as "Kleppner's condition"; see [11,70,88,89,92]. A pair (Γ, σ) satisfies Kleppner's condition if, for any non-trivial γ ∈ Γ satisfying σ(γ, γ ′ ) = σ(γ ′ , γ) for all γ ′ ∈ Γ such that γ ′ γ = γγ ′ , the associated conjugacy class {(γ ′ ) −1 γγ ′ : γ ′ ∈ Γ} is infinite. The following theorem is a special case of our main theorem (Theorem 6.6). Theorem 1.3. Let (π, H π ) be a σ-projective relative discrete series representation of a connected, simply connected nilpotent Lie group G of formal dimension d π > 0. Suppose that Γ ≤ G is a lattice such that (Γ, σ) satisfies Kleppner's condition. (i) If vol(G/Γ)d π < 1, then there exists g ∈ H ∞ π such that π(Γ)g forms a frame. (ii) If vol(G/Γ)d π > 1, then there exists g ∈ H ∞ π such that π(Γ)g forms a Riesz sequence. Under Kleppner's condition, Theorem 1.3 provides a full converse to the Balian-Low type theorems [6,37,51,52] on the strictness of the density conditions (Theorem 1.1) for smooth vectors. In fact, as a direct consequence of Theorem 1.3, the smooth vectors in Theorem 1.3 could even be chosen to be analytic (cf. Corollary 6.9). A more general version of Theorem 1.3, valid for projective representations arising from genuine representations that are merely square-integrable modulo their projective kernel, is given in Theorem 6.6. It is currently not known whether the existence claims (i) and (ii) in Theorem 1.3 also hold without the assumption of Kleppner's condition. The fact that Kleppner's condition is not needed in Theorem 1.2 and in a version of Theorem 1.3 for the 3-dimensional Heisenberg group indicates that it might be superfluous for Theorem 1.3 in general, too. For applications to time-frequency analysis, we mention that the representations (π, H π ) appearing in Theorem 1.3 can, by Kirillov's orbit method, be realized to act on some L 2 (R d ), with the action of π in a coordinate parametrization given by π(x)f (t) = e iP (x,t) f (Q(x, t)), t ∈ R d , x ∈ R n , for polynomials P and Q. In such a realization, the space of smooth vectors H ∞ π is precisely the space S(R d ) of Schwartz functions, and the corresponding matrix coefficients define functions in S(R n ). Theorem 1.3 provides therefore new classes of localized frames and Riesz sequences in L 2 (R d ). A key feature of such localized systems is that, via techniques underlying the theory of localized frames [1,9,41,47,107], the reproducing properties of frames and Riesz sequences, namely f ∈ L 2 (R d ) if and only if f = γ∈Γ c γ π(γ)g for some (c γ ) γ∈Γ ∈ ℓ 2 (Γ), respectively c ∈ ℓ 2 (Γ) if and only if c γ = f, π(γ)g for some f ∈ L 2 (R d ), automatically extend to families of associated Banach spaces; in particular, so-called coorbit spaces [36]. Therefore, such localized systems provide a description and characterization of these Banach spaces and can be used for the purpose of (generalized) time-frequency analysis on R d ; see [49] for a concrete exposition associated to lower-dimensional nilpotent groups. 1.3. Methods. With notation as in Theorem 1.3, our proof method is based on the interpretation of a vector g ∈ H π defining a lattice orbit π(Γ)g as an element of a module over an associated operator algebra. The relevant operator algebras are generated by the σ-twisted left regular representation (λ σ Γ , ℓ 2 (Γ)) of Γ, determined by λ σ Γ (γ)δ γ ′ = σ(γ, γ ′ )δ γγ ′ for γ, γ ′ ∈ Γ, where {δ γ : γ ∈ Γ} is the canonical basis for ℓ 2 (Γ). The completion of the span of the collection λ σ Γ (Γ) = {λ σ Γ (γ) : γ ∈ Γ} ⊆ B(ℓ 2 (Γ) ) in the strong operator topology gives the σ-twisted group von Neumann algebra L(Γ, σ), while completion in the norm topology gives the (reduced) σ-twisted group C * -algebra C * r (Γ, σ). Since a lattice Γ ≤ G in a nilpotent Lie group G is finitely generated and nilpotent, Kleppner's condition on (Γ, σ) is equivalent to the algebra L(Γ, σ) (resp. C * r (Γ, σ)) being a factor (resp. simple), cf. [70,92]. In addition, since Γ is amenable, the reduced algebra C * r (Γ, σ) is isomorphic to the full twisted group C * -algebra C * (Γ, σ). Our approach makes a fundamental use of the algebras C * r (Γ, σ) and C * (Γ, σ) being simple. It should be mentioned that the non-twisted group C * -algebra C * (Γ) (i.e., σ being trivial) is simple if and only if Γ is trivial, so that the use of cocycles is essential for our approach. The question of the existence of a general vector g ∈ H π generating a frame π(Γ)g in H π (see Theorem 1.2) can be naturally approached using techniques for von Neumann algebras and their Hilbert modules, as shown by Bekka [12] (cf. [34] for Riesz sequences). The fundamental observation here is that there exists a frame (resp. Riesz sequence) of the form π(Γ)g for some g ∈ H π if and only if π\| Γ is a subrepresentation of λ σ Γ (resp. λ σ Γ is a subrepresentation of π\| Γ ), cf. [12,Corollary 3 and 4]. By the square-integrability of π, the restriction π\| Γ can be extended to give H π the structure of a Hilbert L(Γ, σ)-module, so that the existence of frames (resp. Riesz sequences) in H π is equivalent to H π being a submodule of ℓ 2 (Γ) (resp. ℓ 2 (Γ) is a submodule of H π ), cf. [12,Proposition 1] and [34,Theorem 5.1]. When L(Γ, σ) is a factor, such submodule inclusions are in turn equivalent to inequalities involving the associated von Neumann dimensions, which gives rise to the inequalities vol(G/Γ)d π ≤ 1 (resp. vol(G/Γ)d π ≥ 1). The basis for these results is that projections in a II 1 factor (such as L(Γ, σ)) are completely classified by their value under the canonical tracial state by the Murray-von Neumann comparison theory. For providing density conditions for the existence of a localized vector g yielding a frame or Riesz sequence of the form π(Γ)g, the above mentioned von Neumann algebra techniques do not seem to be sufficient. In contrast, we show in the present paper that the theory of C algebras and associated Hilbert C -modules do provide powerful techniques for approaching the localization problem. The following explicitly constructed Hilbert C * -module plays a central role in the proof of Theorem 1.3. Theorem 1.4. Let (π, H π ) be a σ-projective relative discrete series representation of a connected, simply connected, nilpotent Lie group G. Suppose Γ ≤ G is a lattice. Then the space H ∞ π of smooth vectors can be completed into a finitely generated left Hilbert C * r (Γ, σ)-module E π,Γ , where the left action and the C * r (Γ, σ)-valued inner product • ·, · are determined by a · f = γ∈Γ a(γ)π(γ)f for a ∈ S(Γ) and f ∈ H ∞ π , • f, g (γ) = f, π(γ)g for f, g ∈ H ∞ π and γ ∈ Γ, where S(Γ) ⊂ ℓ 1 (Γ) denotes the Schwartz space on Γ, cf. Section 6. A general method for the construction of a Hilbert C * r (Γ, σ)-module from an integrable σ-projective representation of a discrete group Γ was outlined by Rieffel [103]. However, the explicit construction of such modules was only accomplished in [104] for the projective Heisenberg representation of a locally compact abelian group of the form G × G. The modules constructed in [104] are usually called Heisenberg modules and have found numerous applications in operator algebras and noncommutative geometry, see, e.g., [20,23,30,73,78,119]. The explicit module provided by Theorem 1.4 forms a natural generalization of the Heisenberg modules to all nilpotent Lie groups and is established here via the representation theory of nilpotent Lie groups and associated coorbit space theory. It should be mentioned that the Heisenberg modules of Rieffel [104] are, in addition, also equipped with a natural right action which gives them the structure of imprimitivity bimodules. No such extra structure is present for the modules provided by Theorem 1.4. The link between the Hilbert C * -module E π,Γ and lattice orbits π(Γ)g with g ∈ H ∞ π is given by the following theorem, see Proposition 4.3 for a more general version and Section 3.3 for definitions of the terms used below. Theorem 1.5. With notation as in Theorem 1.4, let A := C * r (Γ, σ) and let g 1 , . . . , g n ∈ H ∞ π . Then the following assertions hold: (i) The set {g 1 , . . . , g n } is an algebraic generating set for E π,Γ if and only if (π(Γ)g j ) 1≤j≤n is a frame for H π . (ii) The set {g 1 , . . . , g n } is A-linearly independent and has closed A-span in E π,Γ if and only if (π(Γ)g j ) 1≤j≤n is a Riesz sequence in H π . Theorem 1.5 provides a correspondence between spanning (resp. linear independent) sets in E π,Γ and frames (resp. Riesz sequences) in H π . For the particular setting of the projective Heisenberg representation (π, L 2 (R d )) of R 2d , the correspondence for frames (part (i)) was first proved by Luef in [77]. Combined with the general fact that imprimitivity bimodules between unital C * -algebras must be finitely generated, this was used in [77] to prove the existence of a Gabor frame (π(γ)g j ) γ∈Γ,1≤j≤n for L 2 (R d ) with finitely many localized windows over any given lattice Γ in R 2d . In the present paper we prove directly the existence of such frames in the orbit of nilpotent Lie groups by exploiting classical sampling techniques [36,45]. 1 Via Theorem 1.5, the existence of a multiwindow frame with finitely many windows in H ∞ π implies that the module E π,Γ constructed in Theorem 1.4 is finitely generated, which is essential for our approach towards proving Theorem 1.3. A finitely generated Hilbert C * -module E π,Γ as constructed in Theorem 1.4 corresponds naturally to a projection in a matrix algebra over C * r (Γ, σ), which can be explicitly constructed using module frames, see, e.g., [42,105]. Using this interpretation, the existence of frames and Riesz sequences π(Γ)g generated by a single g ∈ H ∞ π can be approached via the comparison theory for projections in (matrix algebras over) the C * -algebra C * r (Γ, σ). This approach is reminiscent of the method [12] towards Theorem 1.2 using von Neumann algebras. However, in contrast to the setting of von Neumann algebras, the comparison theory for projections in C * -algebras is remarkably subtle, see, e.g., [16,115] and the references therein. Among others, this is caused by the fact that C * -algebras, in contrast to von Neumann algebras, might not possess sufficiently many projections for a satisfactory comparison theory. This has lead, among others, to the more general notion of Cuntz subequivalence of positive elements in (matrix algebras over) a C * -algebra [26], and the corresponding comparison then concerns whether Cuntz subequivalence of positive elements can be described via tracial states. A C * -algebra satisfying such a property is said to have strict comparison of positive elements and it is this notion, which is stronger than strict comparison of projections, that forms a central ingredient in the present paper (see Section 5.1). We mention that for irrational noncommutative tori, the presence of strict comparison of projections was proven in [104] (see also [16]) and (implicitly) exploited in [63] for the study of Gabor frames; see Section 7.1. A part of the (revised) Toms-Winter conjecture predicts that for any unital separable, simple, nuclear, infinite-dimensional C * -algebra, strict comparison of positive elements is equivalent to a regularity property known as finite nuclear dimension (see e.g. [115, p. 302]). The implication from finite nuclear dimension to strict comparison of positive elements is known 2 , and follows by combining results of Rørdam [108] and Winter [120]. Therefore, for establishing strict comparison of positive elements in our setting, it would suffice to prove that the twisted group C * -algebra C * r (Γ, σ) has finite nuclear dimension. We prove the even stronger property that C * r (Γ, σ) has finite decomposition rank: Theorem 1.6. Let Γ be a finitely generated, nilpotent group and let σ be a 2-cocycle on Γ. Then the twisted group C * -algebra C * r (Γ, σ) has finite decomposition rank, in particular finite nuclear dimension. Hence C * r (Γ, σ) has strict comparison of positive elements, and therefore of projections, whenever (Γ, σ) satisfies Kleppner's condition. Finite decomposition rank of group C * -algebras associated to finitely generated, nilpotent groups is due to Eckhardt, Gillaspy and McKenney [32,33]. Our proof of Theorem 1.6 relies on [32] and extends their result to the twisted case via the theory of representation groups; in particular, we use a recent result of Hatui, Narayanan and Singla [58, Theorem 3.5], cf. Section 5.3 for precise details. Theorems 1.4, 1.5 and 1.6 are the essential ingredients in our proof of Theorem 1. 3. An additional result of independent interest, at least to operator algebraists, is the following theorem, which provides new examples of classifiable C * -algebras in the sense of the classification program for simple separable nuclear C * -algebras (see e.g. [115,Chapter 18] and references therein). Theorem 1.7. Let Γ be a finitely generated, nilpotent group and let σ be a 2-cocycle on Γ such that (Γ, σ) satisfies Kleppner's condition. Then C * r (Γ, σ) is a unital, separable, simple C * -algebra with finite nuclear dimension that satisfies the UCT, hence is classifiable by the Elliott invariant. Moreover, C * r (Γ, σ) has stable rank one. 2 In fact, under various additional assumptions (which all hold in our setting), the full equivalence is known, see, e.g., [115] and our discussion after Theorem 5.1. We note that there are remarkably few examples in the literature of pairs (Γ, σ) (with Γ countable) such that C * r (Γ, σ) has stable rank one (cf. the discussion in [11, p. 293 and Appendix A]). Within the class of countable amenable groups, our result about this property in Theorem 1.7 has previously only been known for finitely generated free abelian groups (this may be deduced from [19,Theorem 1.5], which deals with simple noncommutative tori), and for some examples of finitely generated nilpotent groups (see the comment after Theorem 4.7 in [90]). For a countable nilpotent group and a 2-cocycle σ on Γ such that (Γ, σ) satisfies Kleppner's condition, Osaka and Phillips raise in [90,Problem 4.8] the question whether C * (Γ, σ) ≃ C * r (Γ, σ) has real rank zero and stable rank one, and whether the order on projections over C * (Γ, σ) is determined by traces (this is equivalent to the one we call strict comparison of projections). Our Theorems 1.6 and 1.7 provide a partial answer to their question. 1.4. Extensions. The proof of Theorem 1.3 makes a fundamental use of the presence of strict comparison of projections in twisted group C * -algebras C * (Γ, σ) of finitely generated nilpotent groups Γ, which we show by relying on the paper [32]. The results in [32] are, however, also valid for virtually nilpotent groups, and it might be that a version of Theorem 1.3 is also valid for more general (classes of) groups of polynomial growth. For such an extension, several parts in our approach, among others, the explicitly constructed Hilbert C * -modules (Theorem 1.4) would require different arguments, while there are also several ingredients that do currently not have analogues for virtual nilpotent groups, among others, the existence of suitable representation groups [58]. With an eye on such possible extensions, we prove several auxiliary results in a slightly more general setting than strictly needed for our main result Theorem 1.3, provided they do not require additional arguments. As another extension, we mention that our current approach also allows a version of Theorem 1.3 for Gabor systems on general (compactly generated) locally compact abelian groups, which was left open in [63]. For this extension, the module constructed in Theorem 1.4 could be replaced by the Heisenberg modules of [104]. 1.5. Outline. Section 2 provides preliminary results on frames, square-integrable representations and group operator algebras. Hilbert C * -modules and their generating sets are discussed in Section 3. A general construction of Hilbert C * -modules associated to projective representations of discrete groups is outlined in Section 4. Section 5 is devoted to strict comparison of positive elements in C * -algebras. In particular, the presence of strict comparison of projections in simple twisted group C * -algebras (Theorem 1.6) associated to finitely generated nilpotent groups is proven in Section 5.3, along with Theorem 1.7. In Section 6 the results obtained in previous sections are applied to the setting of nilpotent Lie groups to prove Theorem 1.3 (cf. Theorem 6.6), along with Theorem 1.4 and Theorem 1.5. Lastly, we discuss some examples in Section 7. Notation. The notation N 0 will be used for the natural numbers including zero 0. The complex numbers without zero will be denoted by A frame (g j ) j∈J is called a Parseval frame if (2.1) can be chosen to be equality. A system (g j ) j∈J is called a Bessel sequence if the associated analysis operator C : H → ℓ 2 (J) given by C × = C \ {0}. The cardinality of a set X is denoted by \|X\| ∈ [0, ∞]. For functions f 1 , f 2 : X → [0, ∞), we write f 1 ≍ f 2 if there exist constants C 1 , C 2 > 0 such that f 1 (x) ≤ C 1 f 2 (x) and f 2 (x) ≤ C 2 f 1 (x) for all x ∈ X.C f = ( f, g j ) j∈J for f ∈ H is a bounded, linear operator. Its adjoint, the synthesis operator D : ℓ 2 (J) → H, is determined by De j = g j , where e j denotes the standard basis vector of ℓ 2 (J) corresponding to an index j ∈ J. The frame operator associated to (g j ) j∈J is given by S = C * C : H → H. The system (g j ) j∈J is a frame for H if and only if S is invertible. The family (g j ) j∈J is called a Riesz sequence in H if c 2 ℓ 2 ≍ j∈J c j g j 2 for all c = (c j ) j∈J ∈ ℓ 2 (Γ). Alternatively, (g j ) j∈J is a Riesz sequence if and only if the associated Gramian operator G := D * D : ℓ 2 (J) → ℓ 2 (J) is invertible, where D is the synthesis operator of the sequence (g j ) j . For background and further results on frames and Riesz sequences, see, e.g., [22,121]. 2.2. Cocycles and projective representations. Throughout, G denotes a second countable, locally compact, unimodular group with identity element e. We assume that a Haar measure µ G on G is fixed and let L p (G) be the associated Lebesgue space for p ∈ [1, ∞]. By a cocycle on G we will mean a Borel measurable map σ : G × G → T that satisfies the identities (1) σ(x, y)σ(xy, z) = σ(x, yz)σ(y, z) for all x, y, z ∈ G, (2) σ(e, e) = 1. Such maps are frequently called normalized 2-cocycles, or multipliers, in the literature. We denote by Z 2 (G, T) the set of all such cocycles. Given a cocycle σ on G, a σ-projective unitary representation π of G on a Hilbert space H π is a map π : G → U (H π ) (where U (H π ) denotes the unitary operators on H π ) such that π(x)π(y) = σ(x, y) π(xy) for all x, y ∈ G. We will always assume that representations are measurable, i.e., x → π(x)f is a Borel measurable function on G for every f ∈ H π . A subspace of H π is said to be invariant under π if it invariant under π(x) for every x ∈ G. We say that π is irreducible if {0} and H π are the only closed subspaces of H π which are invariant under π. Given f, g ∈ H π , we can form the function C g f : G → C given by C g f (x) = f, π(x)g for x ∈ G. Such functions on G are called matrix coefficients associated to π. If f = g, then C f f is called a diagonal matrix coefficient. Matrix coefficients satisfy the relation C g (π(x)f )(y) = σ(x, x −1 y)C g f (x −1 y) for f, g ∈ H π and x, y ∈ G. (2.2) The σ-twisted left regular representation of G is the σ-projective unitary representation of G on L 2 (G) given by λ σ G (x)f (y) = σ(x, x −1 y) f (x −1 y) for x, y ∈ G, f ∈ L 2 (G). In terms of λ σ G , (2.2) can be stated as the intertwining relation C g (π(x)f ) = λ σ G (x)C g f , provided that C g f ∈ L 2 (G). An irreducible, projective unitary representation π is called square-integrable if there exist nonzero f, g ∈ H π such that G \| f, π(x)g \| 2 dµ G (x) < ∞. In that case, there exists a unique d π > 0 called the formal dimension of π (depending on the Haar measure on G) such that G f, π(x)g f ′ , π(x)g ′ dµ G (x) = 1 d π f, f ′ g, g ′ for all f, f ′ , g, g ′ ∈ H π . (2.3) Square-integrability of π implies that for each g ∈ H π , the coefficient operator C g : H π → L 2 (G), mapping each f ∈ H π to C g f , is a well-defined, bounded, linear operator. Moreover, d 1/2 π C g is an isometry which intertwines π and λ σ G , cf. (2.2), and thus realizes π as a subrepresentation of λ σ G . For more details on projective and square-integrable representations, cf. [3,93,110,118]. 2.3. Lattices. Let Γ ≤ G be a discrete subgroup of a second countable unimodular locally compact group G. A left (resp. right) fundamental domain of Γ in G is a Borel set Σ ⊂ G such that G = Γ · Σ and γΣ ∩ γ ′ Σ = ∅ (resp. G = Σ · Γ and Σγ ∩ Σγ ′ = ∅) for all γ, γ ′ ∈ Γ with γ = γ ′ . The discrete Γ ≤ G is called a lattice if it admits a left or right fundamental domain of finite measure. Alternatively, a discrete subgroup Γ ≤ G is a lattice if, and only if, the quotient G/Γ carries a finite G-invariant Radon measure. By Weil's integral formula, vol(G/Γ) = µ G (Σ) for any choice of fundamental domain Σ ⊂ G for Γ. See, e.g., [100] for more details and properties. 2.4. Twisted group operator algebras. Let Γ be a countable discrete group, and let σ be a cocycle on Γ. The σ-twisted convolution of two functions a, b : Γ → C in ℓ 1 (Γ) is defined by (a * σ b)(γ ′ ) = γ∈Γ σ(γ, γ −1 γ ′ )a(γ)b(γ −1 γ ′ ) for γ ′ ∈ Γ. We often simply write * = * σ . The σ-twisted involution of a is given by a * (γ) = σ(γ, γ −1 )a(γ −1 ) for γ ∈ Γ. The Banach space ℓ 1 (Γ) becomes a unital Banach * -algebra with respect to σ-twisted convolution and σ-twisted involution, which we denote by ℓ 1 (Γ, σ). The full σ-twisted group C * -algebra of Γ is the completion C * (Γ, σ) of ℓ 1 (Γ, σ) with respect to the universal C * -norm a u = sup π π(a) for a ∈ ℓ 1 (Γ, σ), where the supremum is taken over all nondegenerate * -representations of ℓ 1 (Γ, σ) on a Hilbert space. We will frequently consider ℓ 1 (Γ, σ) as embedded in C * (Γ, σ). There is a bijective correspondence π → π between σ-projective unitary representations of Γ and nondegenerate * -representations of C * (Γ, σ), where π is determined from π by π(a) = γ∈Γ a(γ)π(γ) for all a ∈ ℓ 1 (Γ, σ). As π is irreducible if and only if π is irreducible, one deduces from the Gelfand-Naimark theory for C * -algebras that there always exists enough σ-projective irreducible unitary representations of Γ to separate its elements. Let λ σ Γ be the σ-twisted left regular representation of Γ on ℓ 2 (Γ). The C * -algebra generated by λ σ Γ (Γ) ⊆ B(ℓ 2 (Γ)) is called the reduced σ-twisted group C * -algebra of Γ and is denoted by C * r (Γ, σ). Equivalently, we have C * r (Γ, σ) = λ σ Γ C * (Γ, σ) . Similarly, the von Neumann algebra generated by λ σ Γ (Γ) ⊆ B(ℓ 2 (Γ)) is called the σ-twisted group von Neumann algebra of Γ and is denoted by L(Γ, σ). It is equipped with a faithful, normal tracial state τ given by τ (a) = aδ e , δ e for a ∈ L(Γ, σ), where δ e ∈ ℓ 2 (Γ) denotes the characteristic function of {e} in Γ. We refer to τ as the canonical tracial state, and denote its restriction to C * r (Γ, σ) also by τ . The canonical map λ σ Γ : C * (Γ, σ) → C * r (Γ, σ) is always faithful on ℓ 1 (Γ, σ). Hence we will often consider ℓ 1 (Γ, σ) as embedded in C * r (Γ, σ) via this map. If the group Γ is amenable, then λ σ Γ is a * -isomorphism. In this case, e.g., when Γ is nilpotent, we will frequently identify C * (Γ, σ) with C * r (Γ, σ). For additional information about the operator algebras associated to (Γ, σ), the reader may consult, e.g., [10,93,122] and references therein. An element γ ∈ Γ is called σ-regular if we have σ(γ, γ ′ ) = σ(γ ′ , γ) whenever γ ′ ∈ Γ and γγ ′ = γ ′ γ. If γ is σ-regular, then every element in the conjugacy class of γ is σ-regular, hence it makes sense to talk about σ-regular conjugacy classes. The pair (Γ, σ) is said to satisfy Kleppner's condition if every nontrivial, σ-regular conjugacy class is infinite. The twisted group von Neumann algebra L(Γ, σ) is a factor (i.e., has a trivial center) if and only if (Γ, σ) satisfies Kleppner's condition, cf. [70,Theorem 2]. Kleppner's argument shows that C * r (G, σ) has a nontrivial center whenever (G, σ) does not satisfy Kleppner's condition. Hence this condition is necessary for C * r (G, σ) to be simple (i.e., to have no non-trivial ideals), but it is not always sufficient, cf. [11]. See also [88,92] for other results relying on this condition. A function a : Γ → C is said to be σ-positive definite if n i,j=1 c i c j a(γ j γ −1 i )σ(γ j γ −1 i , γ i ) ≥ 0 for all γ 1 , . . . , γ n ∈ Γ, c 1 , . . . , c n ∈ C; (2.4) see [10,72] for a slightly different definition (where a would be calledσ-positive definite). The following characterization of σ-positive functions will be convenient for our purposes. The result is part of the folklore for trivial cocycles σ ≡ 1. Proposition 2.1. Assume a ∈ ℓ 1 (Γ, σ) . Then a is σ-positive definite as a function on Γ if and only if λ σ Γ (a) is positive as an element of C * r (Γ, σ). It follows that if a ∈ ℓ 1 (Γ, σ) is a diagonal matrix coefficient associated to a σ-projective unitary representation of Γ, then λ σ Γ (a) is positive in C * r (Γ, σ). Proof. Given a finite subset F = {γ 1 , . . . , γ n } of Γ, denote by P F the orthogonal projection of ℓ 2 (Γ) onto span{δ γ : γ ∈ F }. Then it is well-known that an operator T ∈ B(ℓ 2 (Γ)) is positive if and only if P F T P F is positive for any such F . Let η ∈ ℓ 2 (Γ). Write P F η = n i=1 c i δ γ i for some scalars c 1 , . . . , c n ∈ C and note that λ σ Γ (a)δ γ i , δ γ j = γ∈Γ a(γ) λ σ Γ (γ)δ γ i , δ γ j = γ∈Γ a(γ)σ(γ, γ i ) δ γγ i , δ γ j = a(γ j γ −1 i )σ(γ j γ −1 i , γ i ). Hence P F λ σ Γ (a)P F η, η = λ σ Γ (a)P F η, P F η = n i,j=1 c i c j λ σ Γ (a)δ γ i , δ γ j = n i,j=1 c i c j a(γ j γ −1 i )σ(γ j γ −1 i , γ i ). Thus the condition λ σ Γ (a) ≥ 0 is equivalent to the condition that the above expression is nonnegative for all γ 1 , . . . γ n ∈ G and c 1 , . . . , c n ∈ C, i.e., to the σ-positive definiteness of a, as desired. Assume now that the function a ∈ ℓ 1 (Γ, σ) may be written as a(γ) = f, π(γ)f for some σ-projective unitary representation π of Γ on H π and some f ∈ H π . Then for γ 1 , . . . , γ n ∈ Γ and c 1 , . . . , c n ∈ Γ we have that n i,j=1 c i c j f, π(γ j γ −1 i )f σ(γ j γ −1 i , γ i ) = n i,j=1 c i c j f, π(γ j )π(γ −1 i )f σ(γ j γ −1 i , γ i )σ(γ j , γ −1 i ) = n i,j=1 c i c j π(γ i ) * f, π(γ −1 i )f σ(γ −1 i , γ i ) = n i=1 c i π(γ i ) * f, n j=1 c j π(γ j ) * f ≥ 0. This shows that a is σ-positive definite, hence that λ σ Γ (a) is positive in C * r (Γ, σ). Hilbert C-modules, generating sets and localization Throughout this section, A denotes a unital C -algebra with unit 1 A . 3.1. Hilbert C-modules. We follow the conventions in [71,99], except that we prefer to work with left Hilbert C -modules. Thus, by an inner product A-module we mean a complex vector space E together with a left A-module structure and a map • ·, · : E × E → A such that the following axioms are satisfied: (a1) • af + bg, h = a • f, h + b • g, h for all a, b ∈ A and f, g, h ∈ E. (a2) • f, g * = • g, f for all f, g ∈ E. (a3) • f, f ≥ 0 (as a positive element of A) and • f, f = 0 if and only if f = 0. An inner product A-module becomes a normed space with respect to f E = • f, f 1/2 for f ∈ E. If E is complete with respect to this norm, E is called a Hilbert A-module. We will often consider A itself as a Hilbert A-module with respect to the inner product • a, b = ab * , a, b ∈ A. If A 0 is a dense * -subalgebra of A, then a pre-inner product A 0 -module is a complex vector space E 0 together with a left A 0 -module structure and a map • ·, · : E 0 ×E 0 → A 0 such that the above three axioms are satisfied for a ∈ A 0 and f, g ∈ E 0 , where the positivity is interpreted in the completion A of A 0 . A pre-inner product A 0 -module E 0 can always be completed into a Hilbert A-module E, see [99,Lemma 2.16]. Given a closed A-submodule E 0 of a Hilbert A-module E, the orthogonal complement of E is the set E ⊥ 0 = {f ∈ E : • f, g = 0 for all g ∈ E 0 }. One always has E 0 ∩ E ⊥ 0 = {0} but not necessarily E 0 + E ⊥ 0 = E. If the latter is the case, E 0 is called orthogonally complementable in E. One may form the direct sum of finitely many Hilbert A-modules in the obvious way. The direct sum of n copies of a Hilbert A-module E is denoted by E n . Adjointable operators. A map T : E → F between Hilbert A-modules is called ad- jointable if there exists a (uniquely determined) map T * : F → E such that • T f, g = • f, T * g for all f ∈ E and g ∈ F. An adjointable map is automatically a bounded, A-linear operator. We say that E and F are isomorphic (as Hilbert A-modules) if there exists an adjointable map T : E → F which is a unitary, i.e., satisfies that T * T = I E and T T * = I F . We denote by L A (E, F) the set of all adjointable maps from E into F, which is a Banach space with respect to the operator norm. Furthermore, we set L A (E) := L A (E, E). The map T → T * is an involution on L A (E), and L A (E) becomes a C * -algebra with respect to the operator norm. If a map T : E → F is A-linear and isometric (as a map between the underlying Banach spaces), it need not be adjointable. However, the following conditions are equivalent: 1) T is A-linear, isometric, and Im(T ) is orthogonally complementable in E; 2) T is adjointable with T * T = I E ; 3) T is an adjointable isometry. The equivalence between 1) and 2) is [71,Proposition 3.6]. The fact that 2) implies 3) is straightforward. Finally, if 3) holds, then Im(T ) is closed, and orthogonally complementable in F by [71, Theorem 3.2], so 1) holds. An immediate consequence is that there exists an adjointable isometry E → F if and only if E is isomorphic to a closed, orthogonally comple- mentable A-submodule of F. Given g, h ∈ E, the rank-one operator Θ g,h ∈ L A (E) is given by Θ g,h f = • f, g h for f ∈ E. Spanning and independence. Let E be a Hilbert A-module. The A-span of a set S ⊆ E is the set span A S of all finite A-linear combinations n j=1 a j g j where g j ∈ S, a j ∈ A for 1 ≤ j ≤ n. A finite set S ⊆ E is called a generating set for E if span A S = E, and E is called finitely generated if it admits a finite generating set. Note that this notion is often called algebraically finitely generated in Hilbert C * -module theory to distinguish it from the weaker notion of being topologically finitely generated. Associated to a finite set {g 1 , . . . , g n } ⊆ E are the analysis operator C : E → A n and the synthesis operator D : A n → E given by C f = ( • f, g j ) n j=1 , D(a j ) n j=1 = n j=1 a j g j for f ∈ E and (a j ) j ∈ A n . Both these operators are adjointable, with C * = D. The operator S : C * C : E → E is called the frame operator and the operator G = D * D : A n → A n is called the Gramian operator. The following characterization of generating sets will be convenient for our purposes, cf. [42, Theorem 5.9]. . . , g n } ⊆ E is a generating set for E if and only if it is a frame for E, that is, there exist C 1 , C 2 > 0 such that C 1• f, f ≤ n j=1 • f, g j • f, g j * ≤ C 2• f, f for all f ∈ E, (3.1) if and only if the associated frame operator S is invertible in L A (E). Proof. Consider the set {g 1 , . . . , g n } ⊆ E. If it is generating for E, then it is a frame for E by [42,Theorem 5.9]. If it is a frame for E, satisfying (3.1), then, using [71, Lemma 4.1], we get that the positive operator S = n j=1 Θ g j ,g j satisfies that C 1 I E ≤ S ≤ C 2 I E ; since C 1 > 0, it follows that S is invertible in L A (E). Finally, if S is invertible in L A (E), and f ∈ E, then f = S S −1 f = n j=1 • S −1 f, g j g j , which shows that {g 1 , . . . , g n } is generating for E. As for Hilbert spaces, if one can choose C 1 = C 2 = 1 in (3.1), the frame is called Parseval. We call a finite set {g 1 , . . . , g n } A-linearly independent if whenever a 1 , . . . , a n ∈ A are such that n j=1 a j g j = 0, then a j = 0 for 1 ≤ j ≤ n. Note that contrary to the Hilbert space case, the A-span of a finite set might not be topologically closed. We say that a finite set S has closed A-span if span A S is topologically closed. Proof. Consider a finite set {g 1 , . . . , g n } ⊆ E. Applying [5, Proposition 2.1] to the associated analysis operator C : E → A n , and setting D = C * , we get that the following conditions are equivalent: 1) C is surjective; 2) There exists C > 0 such that C (a j ) j A n ≤ D(a j ) j E for all (a j ) j ∈ A n ; 3) There exists C ′ > 0 such that C ′ • (a j ) j , (a j ) j ≤ • D(a j ) j , D(a j ) j for all (a j ) j ∈ A n . In fact, to show these equivalences, it is shown in [5] that 1) ⇒ D * D is invertible ⇒ 3) ⇒ 2) ⇒ D is injective with closed range ⇒ 1) . This means that the associated Gramian operator G = D * D is invertible in L A (E) if and only if D is injective with closed range, which is equivalent to {g 1 , . . . , g n } being A-linearly independent with closed A-span. In [8], sets satisfying the properties of Lemma 3.2 are called module Riesz sequences. We recall that if M n (A) denotes the C * -algebra consisting of all n × n matrices over A and p ∈ M n (A) is a projection (i.e., p is self-adjoint and idempotent), then A n p is a Hilbert A-submodule of A n (we consider here elements of A n as row vectors). Furthermore, if there exists a generating set (resp. A-linearly independent set with closed Aspan) with n elements, then one can find a generating set (resp. A-linearly independent set with closed A-span) with n elements that belong to any dense subspace E 0 of E. Proof. (i) A generating set with n elements is a frame by Lemma 3.1. Hence, the corresponding frame operator S is invertible. As in the Hilbert space case, by applying S −1/2 to each element of the frame, one obtains a new frame with n elements which is Parseval. But then the associated analysis operator C : E → A n is an adjointable isometry. Next, assume that there exists an adjointable isometry C : E → A n for some n ∈ N. Then one checks readily that Im(C ) = A n p, where p is the projection in M n (A) whose i-th row vector is C C * e i , where e i = (δ i,j 1 A ) n j=1 ∈ A n . Thus, E ∼ = A n p. Finally, if there exists a projection p in M n (A) such that E ∼ = A n p, then, as A n p has clearly a generating set with n elements, this is also true for E. (ii) Assume first that g 1 , . . . , g n are A-linearly independent and that F = span A {g 1 , . . . , g n } is closed in E. Then F is a Hilbert A-module, and {g 1 , . . . , g n } is a generating set for F. Denoting by S ∈ L A (F) the corresponding frame operator, S is positive and invertible, and (g j ) n j=1 is a frame. Since S f = n j=1 • f, g j g j for every f ∈ F, we get that g i = n j=1 • S −1 g i , g j g j for every 1 ≤ i ≤ n. By A-linear independence, this forces • S −1 g i , g j = δ i,j 1 A . Hence, the set {g j : 1 ≤ j ≤ n}, whereg j := S −1/2 g j , is orthonormal in the sense that • g i ,g j = • S −1 g i , g j = δ i,j 1 A for all 1 ≤ i, j ≤ n. It follows that the associated synthesis operator D : A n → E given by D(a j ) j = n j=1 a jgj is an adjointable isometry. Conversely, suppose D : A n → E is an adjointable isometry. Set g j = D(e j ) for each 1 ≤ j ≤ n, where e j denotes the jth element of the standard basis of A n . Then {g 1 , . . . , g j } is an orthonormal set, hence A-linearly independent. This finishes the proof of (ii). Suppose now that {g 1 , . . . , g n } is a generating set for E, which by the argument for (i) above can be assumed to be a Parseval frame. Let S be the corresponding frame operator. In terms of rank-one operators, we get S = n j=1 Θ g j ,g j = I E . Now it is an easy exercise to check that Θ g,g − Θ g ′ ,g ′ ≤ ( g + g ′ ) g − g ′ for all g, g ′ ∈ E. By density of E 0 in E, we can find g ′ j ∈ E 0 such that g j − g ′ j < δ for every j = 1, . . . , n, where M := max{ g 1 , . . . , g n } and δ := min{1, ((2M + 1)n) −1 }. This gives that I E − n j=1 Θ g ′ j ,g ′ j ≤ n j=1 Θ g j ,g j − Θ g ′ j ,g ′ j ≤ (2M + 1) n j=1 g j − g ′ j < (2M + 1)nδ ≤ 1, hence that S ′ := n j=1 Θ g ′ j ,g ′ j is invertible in L A (E). Since S ′ is the frame operator associated to the family {g ′ 1 , . . . , g ′ n }, it follows from Lemma 3.1 that this family, which lies in E 0 , is a generating set for E. By considering the Gramian operator instead of the frame operator, a similar argument shows the analogous property for A-linearly independent sets with closed A-span. The first part of Proposition 3.3 is essentially known, see, e.g., [105,Section 7] and [42,Section 5] for somewhat similar statements. 3.4. Localization of Hilbert C-modules. We will repeatedly use the following simple observation, which relies on the elementary fact that cac ≤ cbc * for every c ∈ A whenever a, b ∈ A are self-adjoint and a ≤ b. Proof. Indeed, since a is positive, we have that 0 ≤ a ≤ a 1 A . Hence we get 0 ≤ b 1/2 ab 1/2 ≤ b 1/2 a 1 A b 1/2 = a b, which implies that 0 ≤ τ (b 1/2 ab 1/2 ) ≤ τ ( a b) = a τ (b). As τ is tracial, τ (b 1/2 ab 1/2 ) = τ (ab), and the result follows. We assume from now on that A has a faithful, tracial state τ . We denote by H the Hilbert space obtained from the GNS construction applied to (A, τ ). Thus H is the Hilbert space completion of A with respect to the inner product given by a, b τ = τ (ab * ) for a, b ∈ A. To avoid confusion, we write a when we view a ∈ A as an element of H. Since τ is faithful, we can view A as a C * -subalgebra of B(H), whose action on H is determined by a b = ab for a, b ∈ A. The vector f 0 := 1 A ∈ H is then cyclic and separating for A, and we have τ (a) = af 0 , f 0 τ for every a ∈ A. Let M = A ′′ ⊆ B(H) be the von Neumann algebra on H generated by A. By [116,Proposition V.3.19], the functional on M given by a → af 0 , f 0 τ is a faithful tracial normal state on M , which we also denote by τ . The GNS-space of (M, τ ), which is usually denoted by L 2 (M, τ ), can then be identified with H, and M acts also on it from the right in the obvious way. Throughout this subsection we also fix a Hilbert A-module E, where we denote the A-valued inner product by • ·, · . We define a scalar-valued inner product on E by setting f, g H τ E = τ ( • f, g ) for f, g ∈ E, and denote by H τ E the corresponding Hilbert space completion of E. (This is a special case of a procedure known as localization of Hilbert C * -modules, see [71, p. 7].) Since τ is a tracial state, the left action of A on E extends to a representation π τ E of A on H τ E . Indeed, using Lemma 3.4, we get that for all a ∈ A and f ∈ E, af 2 H τ E = τ ( • af, af ) = τ (a • f, f a * ) = τ (a * a • f, f ) ≤ a * a τ ( • f, f ) = a 2 f 2 H τ E . It follows that the linear operator f → af extends to a bounded linear operator π τ E (a) on H τ E for each a ∈ A, and one checks readily that the map a → π τ E (a) is a * -homomorphism from A into B(H τ E ). We refer to the pair (H τ E , π τ E ) as the localization of E with respect to (A, τ ). We recall the notion of a Hilbert module over a von Neumann algebra, which is different from the notion of a Hilbert C * -module over a C * -algebra. If N denotes a von Neumann algebra, a (left, normal) Hilbert N -module is a Hilbert space K together with a normal unital representation of N on K. The following proposition seems part of the folklore. As we could not find a suitable reference in the literature, we include a proof, for the ease of the reader. Proof. Given a ∈ M , we define a map φ a : E × E → C by φ a (f, g) = τ (a • f, g ) for a ∈ M and f, g ∈ E. Then φ a is linear in the first variable and conjugate-linear in the second. We also have that φ a (f, g) = τ (a • f, g ) = τ ((a * ) * • g, f * ) = τ (( • g, f a * ) * ) = τ ( • g, f a * ) = φ a * (g, f ). In particular, if a is self-adjoint, then φ a (f, g) = φ a (g, f ). Moreover, if a ≥ 0, then by Lemma 3.4 we have that φ a (f, f ) = τ (a • f, f ) ≥ 0. Thus, for fixed a ≥ 0 we have shown that φ a is a semi-inner product on E. Consequently it satisfies the Cauchy-Schwarz inequality: \|φ a (f, g)\| ≤ φ a (f, f ) 1/2 φ a (g, g) 1/2 . (3.2) From Lemma 3.4 it also follows for a ≥ 0 that φ a (f, f ) = τ (a • f, f ) ≤ a τ ( • f, f ) = a f 2 H τ E . (3.3) Combining (3.2) and (3.3), we arrive at \|φ a (f, g)\| ≤ a f H τ E g H τ E for a ≥ 0. By writing a given a ∈ M as a linear combination of positive elements in M , one easily deduces that φ a is a bounded, sesquilinear form on E for every a ∈ M , so it extends uniquely to a bounded, sesquilinear form φ a on H τ E . By Riesz' representation theorem there exists a unique bounded linear operator π τ E (a) on H τ E such that π τ E (a)f, g H τ E = φ a (f, g) for all f, g ∈ H τ E . Thus we get a map π τ E : M → B(H τ E ) . Note that for a ∈ A and f, g ∈ E we have that π τ E (a)f, g H τ E = τ (a • f, g ) = τ ( • af, g ) = af, g H τ E . It follows that π τ E extends the representation of A on H τ E . Since φ a (f, g) is linear in a for fixed f, g ∈ E, it also follows that π τ E is linear on M . Further, from what we showed earlier, we have that π τ E (a) * f, g = π τ E (a)g, f = φ a (g, f ) = φ a * (f, g) = π τ E (a * )f, g for f, g ∈ E. This implies that π τ E preserves adjoints. Next, we claim that π τ E is a positive map: Indeed, let a ∈ M be positive. Then for every f ∈ E we get from Lemma 3.4 that π τ E (a)f, f H τ E = τ (a • f, f ) ≥ 0. Writing a general f ∈ H τ E as a limit of a sequence (f n ) n∈N in E, we get that π τ E (a)f, f H τ E = lim n→∞ π τ E (a)f n , f n ≥ 0. We will now show that π τ E is normal. Since π τ E is a positive linear map, it suffices to show that for any given bounded, increasing net (a i ) i∈I of positive elements in M with s.o.t. limit a, π τ E (a) is the s.o.t. limit of (π τ E (a i )) i∈I . (By s.o.t., we mean the strong operator topology). Note that (π τ E (a i )) i∈I is a bounded, increasing net of positive operators in B(H τ E ), so it has a s.o.t. limit which we denote by T . Thus, we want to show that π τ E (a) = T . For this, let f, g ∈ E. Since a is the s.o.t. limit of (a i ) i , the net (a i• f, g ) i converges in M to a • f, g in the s.o.t. Using the s.o.t. continuity of τ , we get lim i π τ E (a i )f, g H τ E = lim i τ (a i• f, g ) = τ (a • f, g ) = π τ E (a)f, g H τ E However, since T is the s.o.t. limit of (π τ E (a i )) i , we have lim i π τ E (a i )f = T f , so lim i π τ E (a i )f, g H τ E = lim i π τ E (a i )f, g H τ E = T f, g H τ E as well. We have thus shown that π τ E (a)f, g = T f, g for all f, g ∈ E. Next, let f ∈ E and g ∈ H τ E . Let g = lim n g n where g n ∈ E for all n. Then it follows easily by the above that T f, g H τ E = lim n π τ E (a)f, g n H τ E = π τ E (a)f, g H τ E . Therefore, π τ E (a)f, g H τ E = T f, g H τ E for arbitrary g ∈ H τ E , so we can conclude that π τ E (a)f = T f for all f ∈ E. By density, it follows that π τ E (a) = T . This completes the proof of the normality of π τ E . To see that π τ E is multiplicative on M , we first observe that π τ E , being normal, is ultraweakly continuous. Moreover, π τ E is multiplicative on A, and A is ultraweakly dense in M . Since multiplication in a von Neumann algebra is separately continuous in each variable for the ultraweak topology, it is then straightforward to check that π τ E (ab) = π τ E (a)π τ E (b), first for a ∈ A and b ∈ M , and next for a, b ∈ M . Finally, since any normal representation of M is ultraweakly continuous, π τ E is clearly the only normal representation of M on H τ E extending the given representation of A on H τ E . 3.5. Localization of adjointable operators. As in Section 3.4, let τ be a faithful tracial state on A and let M be the von Neumann algebra coming from the GNS construction applied to (A, τ ). Given two Hilbert M -modules H and H ′ , we denote by B M (H, H ′ ) the bounded, M -linear operators from H into H ′ . We will make use of a procedure called localization of adjointable operators. Parts of the statements in the following result can be found in [7, p. 13] and [71, p. 58]. Lemma 3.6. Let E, F and K be Hilbert A-modules. Then the following hold: (i) Every adjointable operator T : E → F extends uniquely to a bounded, M -linear map T τ : H τ E → H τ F . The map T → T τ defines an injective, bounded, linear operator from L A (E, F) into B M (H τ E , H τ F ) (ii) Let T ∈ L A (E, F) and S ∈ L A (F, K). Then (ST ) τ = S τ T τ and (T τ ) * = (T * ) τ . Proof. (i) Let T : E → F be an adjointable map, and denote by T its operator norm as a bounded linear map between the Banach spaces E and F. Then by [71, Proposition 1.2], we have that • T f, T f ≤ T 2 • f, f for all f ∈ E. Applying τ to the above inequality, it follows that T f 2 H τ F ≤ T 2 f 2 H τ E for all f ∈ E. Hence, T extends uniquely to a bounded linear operator T τ : H τ E → H τ F , which satisfies that T τ ≤ T , where T τ denotes the operator norm of T τ as a bounded linear map from H τ E to H τ F . Thus T → T τ defines an injective, bounded linear map L A (E, F) → B(H τ E , H τ F ). For the M -linearity, let a ∈ M and let (a i ) i∈I be a net in A that converges to a in the strong operator topology. Then, for any f ∈ E, we get that T τ (af ) = T τ (lim i (a i f )) = lim i (a i T f ) = aT τ f . Hence, if f ∈ H τ E , say f = lim n f n for some sequence (f n ) n∈N in E, then T τ (af ) = T τ (lim n (af n )) = lim n T τ (af n ) = lim n aT τ f n = aT τ f , which shows that T τ is M -linear. (ii) For f ∈ E, we get that (S τ T τ )f = S τ T f = ST f since T τ (resp. S τ ) extends T (resp. S). Thus, by uniqueness of the extension (ST ) τ , it follows that (ST ) τ = S τ T τ . For f ∈ E and g ∈ F, applying τ to the equality • T f, g = • f, T * g immediately gives that T f, g H τ F = f, T * g H τ E , from which it readily follows that T τ f, g H τ F = f, (T * ) τ g H τ E for f ∈ H τ E and g ∈ H τ F . Hence, (T τ ) * = (T * ) τ . Setting E = F = K in Lemma 3.6, we get that the map T → T τ from L A (E) into B(H τ E ) is an injective * -homomorphism, hence an isometry, see, e.g., [85, Theorem 3.1.5]. Thus, L A (E) can be viewed as a unital C * -subalgebra of B(H τ E ), and spectral invariance is known to hold (see, e.g., [85, Theorem 2.1.11]), that is, if T ∈ L A (E) is invertible as an element of B(H τ E ), then its inverse is necessarily in L A (E). Hilbert C-modules from projective representations of discrete groups This section provides a general method for the construction of Hilbert C -modules from (integrable) σ-projective representations of discrete groups. Our method follows the approach outlined in [103] and [104, Section 1] closely, complementing it with statements on frames/Riesz sequences. Henceforth, Γ denotes a countable discrete group, and σ denotes a cocycle on Γ. In this section we set A equal to C * r (Γ, σ), the reduced twisted group C * -algebra of (Γ, σ), and let τ denote the canonical tracial state on A. We let C c (Γ, σ) denote the * -subalgebra of ℓ 1 (Γ, σ) consisting of all finitely supported functions on Γ. We fix a σ-projective unitary group representation π of Γ on a Hilbert space H π . In this context we make the following definition: Definition 4.1. Let A 0 be a * -subalgebra of ℓ 1 (Γ, σ) containing C c (Γ, σ), and let H 0 be a dense subspace of H π . We call the pair (A 0 , H 0 ) admissible for π if the following hold: (1) For every a ∈ A 0 and f ∈ H 0 , the vector a · f := π(a)f = γ∈Γ a(γ)π(γ)f (4.1) is an element of H 0 . (2) For every f, g ∈ H 0 , the function • f, g : Γ → C given by • f, g (γ) := f, π(γ)g Hπ for γ ∈ Γ (4.2) is an element of A 0 . 2), H 0 becomes a left pre-inner product A 0 -module, which can be completed to a Hilbert A-module E. Proof. The assumption that (A 0 , H 0 ) is an admissible pair ensures that the action of A 0 on H 0 is well-defined into H 0 and that the inner product on H 0 takes values in A 0 . Since A 0 contains C c (Γ, σ), it is dense in ℓ 1 (Γ, σ). Moreover, ℓ 1 (Γ, σ) is dense in C * r (Γ, σ), and the ℓ 1 -norm dominates the C * r -norm on ℓ 1 (Γ, σ). Hence it follows that A 0 is dense in C * r (Γ, σ). For the A 0 -linearity in the first argument of • ·, · , let f, g ∈ H 0 . Consider first a = δ γ for some γ ∈ Γ. Note that • f, g is simply the matrix coefficient C g f associated to π as in Section 2.2. Using relation (2.2), we get • a · f, g (γ ′ ) = π(γ)f, π(γ ′ )g Hπ = [C g (π(γ)f )](γ ′ ) = C g f (γ −1 γ ′ ) σ(γ, γ −1 γ ′ ) = • f, g (γ −1 γ ′ ) σ(γ, γ −1 γ ′ ) = a * σ • f, g (γ ′ ) for every γ ′ ∈ Γ, i.e., • a · f, g = a * σ • f, g . This formula clearly extends by linearity to every a ∈ C c (Γ, σ). Now assume that a ∈ A 0 and pick a sequence (a n ) ∞ n=1 in C c (Γ, σ) such that a − a n 1 → 0 as n → ∞. Using what we just have shown, we get that, for every γ ′ ∈ Γ, • a · f, g − a * σ • f, g (γ ′ ) ≤ • (a − a n ) · f, g (γ ′ ) + (a n − a) * σ • f, g (γ ′ ) ≤ (a − a n ) · f, π(γ ′ )f Hπ + (a − a n ) * σ • f, g 1 ≤ (a − a n ) · f Hπ π(γ ′ )f Hπ + a − a n 1 • f, g 1 ≤ a − a n 1 f 2 Hπ + a − a n 1 • f, g 1 → 0 as n → ∞. Hence, • a · f, g = a * σ • f, g for every a ∈ A 0 , as desired. Moreover, for each f ∈ H 0 , the function • f, f ∈ A 0 ⊆ ℓ 1 (Γ, σ) is a diagonal matrix coefficient on Γ associated to π, hence positive in C * r (Γ, σ) by Proposition 2.1. Lastly, if • f, f = 0 for some f ∈ H 0 , then f 2 Hπ = • f, f (e) = 0, hence f = 0. This proves all the properties needed for H 0 to be a pre-inner product A 0 -module as defined in Section 3.1. Let (A 0 , H 0 ) be an admissible pair for π, as in Proposition 4.2. Note that for f, g ∈ H 0 we have τ ( • f, g ) = • f, g (e) = f, π(e)g Hπ = f, g Hπ . This implies that the localization space H τ E can be naturally identified with H π . Thus, the representation π τ E of A on H τ E induces a representation π r of A = C * r (Γ, σ) on H π , which satisfies that π = π r • λ σ Γ . (This shows that π is weakly contained in λ σ Γ whenever there exists an admissible pair for π.) Similarly, for a, b ∈ ℓ 1 (Γ, σ), the localized inner product on A as a left module over itself is given by: τ ( • a, b ) = (a * b * )(e) = γ∈Γ σ(γ, γ −1 e)a(γ)b * (γ −1 e) = γ∈Γ σ(γ, γ −1 )a(γ)σ(γ −1 , γ)b(γ) = a, b ℓ 2 (Γ) . Consequently, the localization space H τ A of A, considered as a left module over itself, can be naturally identified with ℓ 2 (Γ). It is readily verified that the representation π τ A of A on H τ A corresponds then to the identity representation of A on ℓ 2 (Γ). is a frame for H π . (ii) The finite set {g 1 , . . . , g n } is an A-linearly independent set in E with closed A-span if and only if (π(γ)g j ) γ∈Γ,1≤j≤n is a Riesz sequence in H π . Proof. Denote by C : E → A n the analysis operator associated to a finite set {g 1 , . . . , g n } ⊆ E, so C f = ( • f, g j ) n j=1 for f ∈ E. It maps H 0 into ℓ 1 (Γ) n ∼ = ℓ 1 (Γ × {1, . . . , n}), and after this identification, it acts on H 0 by C f = ( f, π(γ)g j ) γ∈Γ,1≤j≤n . Thus, the action of C on H 0 coincides with the action of the analysis operator C : H π → ℓ 2 (Γ× {1, . . . , n}) associated to the system (π(γ)g j ) γ∈Γ,1≤j≤n . By density, it follows that the localized operator C τ : H τ E → H τ A n can be identified with C . Similarly, the localization of the synthesis operator D : A n → E can be identified with the synthesis operator D : ℓ 2 (Γ×{1, . . . , n}) → H π of (π(γ)g j ) γ∈Γ,1≤j≤n . Consequently, by Lemma 3.6, the same identifications hold for the frame operator S ∈ L A (E) and the Gramian operator G ∈ L A (A n ). (i) By the discussion below Lemma 3.6, the frame operator S ∈ L A (E) is invertible if and only if its localization, the frame operator S associated to (π(γ)g j ) γ∈Γ,1≤j≤n , is invertible. Invertibility of the former is equivalent to {g 1 , . . . , g n } being a generating set by Lemma 3.1, while invertibility of the latter is equivalent to (π(γ)g j ) γ∈Γ,1≤j≤n being a frame for H π . (ii) Similarly, the Gramian associated to {g 1 , . . . , g n } is invertible if and only if the Gramian G associated to (π(γ)g j ) γ∈Γ,1≤j≤n is invertible. The former invertibility is equivalent to {g 1 , . . . , g n } being A-linearly independent with closed A-span by Lemma 3.2, while the latter invertibility is equivalent to (π(γ)g j ) γ∈Γ,1≤j≤n being a Riesz sequence for H π . Strict comparison in C-algebras Throughout this section, A denotes a unital C -algebra. Strict comparison of positive elements and of projections. We first recall a few facts about the comparison of positive elements in C * -algebras, originally introduced by Cuntz [26]. We follow Rørdam [109] (see e.g. [4,108,115] for alternative presentations). Let Assume now that A has at least one tracial state. If τ is a tracial state on A, we define τ : M n (A) → C for each n ∈ N by τ ([a ij ]) := n i=1 τ (a ii ). Moreover, we define d τ : M ∞ (A) + → [0, ∞) by d τ (a) = lim k→∞ τ (a 1/k ) whenever a ∈ M n (A) + . Then we say that A has strict comparison of positive elements whenever the following implication holds for a, b ∈ M ∞ (A) + : If d τ (a) < d τ (b) for every tracial state τ on A, then a b. We note that this property implies that A has strict comparison of projections, in the following extended sense (compared to [16,FCQ2,p. 22] and [115,Definition 11.3.8 ]): If n ∈ N and p, q are projections in M n (A) such that τ (p) < τ (q) for every tracial state τ on A, then p is Murray-von Neumann subequivalent to q in M n (A), i.e., there exists some v ∈ M n (A) such that p = v * v and vv * ≤ q. Indeed, if A has strict comparison of positive elements and the projections p, q ∈ M n (A) satisfy the assumption above, then we readily get that d τ (p) = τ (p) < τ (q) = d τ (q) for every tracial state τ on A, hence that p q. This means that there exists a sequence (r k ) ∞ k=1 in M n (A) such that r * k qr k → p as k → ∞, and it is well-known that this implies that p is Murrayvon Neumann subequivalent to q in M n (A), cf. [ We next mention some conditions ensuring that strict comparison of positive elements hold whenever A belongs to a certain class of C * -algebras. As we will not work explicitly with any of the properties involved, we do not recall the lengthy definitions and simply refer the reader to Strung's book [115] for undefined terminology in the following theorem and in the comments related to it. Theorem 5 .1 ([108,109,120]). Let A be a unital, separable, simple, nuclear, infinite-dimensional C * -algebra with at least one tracial state, and let Z denote the Jiang-Su algebra [66]. Consider the following conditions: 1) A has finite decomposition rank; 2) A has finite nuclear dimension; 3) A is Z-stable; 4) A has strict comparison of positive elements. 5) A has stable rank one. Then 1) ⇒ 2) ⇒ 3) ⇒ 4). We also have 3) ⇒ 5). The first implication is by definition, the second is due to Winter, cf. [120,Corollary 7.3], and the third is due to Rørdam,cf. [109,Corollary 4.6] and [115,Theorem 15.4.6]; to be a bit more precise, this implication can be deduced from [109,Theorem 4.5] by arguing in the same way as in the proof of [108, Theorem 5.2 (a)], taking into account Blackadar and Handelman's characterization of lower semi-continuous dimension functions, cf. [18], and Haagerup's result that quasitraces on exact C * -algebras, hence on nuclear C * -algebras, are traces, cf. [56]. The implication 3) ⇒ 5) follows from [109,Theorem 6.7]. Let A be as in Theorem 5.1. For completeness, we add that if A is also assumed to have a unique tracial state (or more generally, if the extreme boundary of the tracial state space of A has a finite topological dimension), then Matui and Sato have shown in [80] that 4) implies 2), i.e., conditions 2), 3) and 4) are equivalent in this case. This means that the (revised) Toms-Winter conjecture (cf. [115, p. 302]) holds in this case. 5.2. Relation to finitely generated Hilbert C-modules. Consider the Hilbert A-modules E = A n p and F = A n q associated to some projections p, q ∈ M n (A). Then p is Murray-von Neumann subequivalent to q if and only if E is isomorphic to an orthogonally complementable submodule of F, i.e., there exists an A-submodule E ′ of F such that E ∼ = E ′ and E ′ ⊕ E ′⊥ = F. This is again equivalent to the existence of an adjointable isometry E → F. Let E be a finitely generated Hilbert A-module. If A has a tracial state τ , then we define τ (E) = τ (p) = i τ (p ii ) for any projection p = [p ij ] ∈ M n (A) such that E ∼ = A n p. (It is not difficult to check that τ (p) = τ (q) if we also have E ∼ = A k q for some projection q ∈ M k (A)). The correspondence between finitely generated Hilbert A-modules and projections in matrix algebras over A allows us to prove the following result in the presence of strict comparison of projections (in the sense defined in the previous subsection): Proposition 5.2. Suppose A has at least one tracial state and strict comparison of projections. Let E be a finitely generated Hilbert A-module and n ∈ N. Then the following hold: (i) If τ (E) < n for all tracial states τ on A, then E admits a generating set with n elements. (ii) If τ (E) > n for all tracial states τ on A, then E admits an A-linearly independent set with n elements such that its A-span is closed. Proof. (i) Assume that τ (E) < n for all tracial states τ on A. Since E is finitely generated, we can find k ∈ N and a projection p ∈ M k (A) such that E ∼ = A k p. We let I n denote the n × n identity matrix in M n (A), and 0 r denote the zero matrix in M r (A) for r ∈ N. If b ∈ M n (A) and c ∈ M r (A), we denote by b ⊕ c the matrix in M n+r (A) given by b ⊕ c : = b 0 0 c . Set p = p ⊕ 0 n−k if k < n, p if n ≤ k, , I n = I n if k ≤ n, I n ⊕ 0 k−n if n ≤ k. Further, set m = max(n, k). Then p and I n are projections in M m (A), satisfying that τ ( p) = τ (p) < n = τ (I n ) = τ ( I n ) for all tracial states τ on A. Strict comparison of projections implies that p I n . In terms of Hilbert A-modules, this means that there exists an adjointable isometry A m p → A m I n , so we get an adjointable isometry E ∼ = A k p ∼ = A m p → A m I n ∼ = A n , which by Proposition 3.3 means that E admits a generating set consisting of n elements. (ii) Arguing similarly as in (i), we now get that there exists an adjointable isometry A n → E, which by Proposition 3.3 means that E admits an A-linearly independent set with n elements that has closed A-span. Assume now that A has a faithful tracial state τ and E is a finitely generated Hilbert Amodule. Then one may use the localization procedure of Section 3.4 to express τ (E) in terms of the dimension of the Hilbert M -module H τ E , where M is the von Neumann algebra associated to (A, τ ). Indeed, by Proposition 3.3 (and its proof), we can find an adjointable isometry C : E → A n for some n ∈ N, and we then have Im(C ) = A n p for some projection p ∈ M n (A). Considering the localization of C (with respect to τ ) as in Lemma 3.6, we obtain a bounded, M -linear map C τ : H τ E → H τ A n ∼ = (H τ A ) n = L 2 (M, τ ) n . In addition, since C C = I, we obtain by Lemma 3.6 that (C τ ) * C τ = (C * C ) τ = I, hence C τ is an isometry. Moreover, we have Im(C τ ) = L 2 (M, τ ) n p, where we consider p as a projection in M n (M ). It follows that H τ E is finitely generated as a Hilbert M -module (cf. [2,Proposition 8.5.3]) and that its dimension (with respect to τ ) is given by dim (M,τ ) H τ E = τ (p), cf. [2,Definition 8.5.4]; see also [74,Definition 1.6], and [68,Chapter 2] in the case where M is a II 1 -factor. The above discussion yields the following result. Proposition 5.3. Suppose E is finitely generated Hilbert A-module and τ is a faithful tracial state on A. Then τ (E) = dim (M,τ ) H τ E . 5.3. Twisted group C-algebras of finitely generated, nilpotent groups. In this section we will extend the results from [32,33] on the finite decomposition rank (nuclear dimension) of group C -algebras associated to finitely generated nilpotent groups to twisted group algebras of such groups. By Theorem 5.1, this will imply the presence of strict comparison of projections, which will allow us to exploit Proposition 5.2 in the setting of Section 4. The following result (cf. [58, Theorem 3.5]) will be useful to us for extending the relevant results in [32,33]. Theorem 5.4 ([58]). Let Γ be a finitely generated nilpotent group. Then Γ has a representation group Γ which is finitely generated and nilpotent. In [58, Definition 1.1], a group Γ is called a representation group 3 of a discrete group Γ if there is a central extension 1 → B → Γ → Γ → 1 such that the associated transgression map from Hom(B, C × ) into the second cohomology group H 2 (Γ, C × ) is an isomorphism. As pointed out in [58,Section 3], see [15,114] for more information, Γ has always a representation group Γ, and one may alternatively say that Γ is a Schur cover of Γ (sometimes called a stem cover of Γ), meaning that there is a central extension 1 → N → Γ → Γ → 1 such that N is contained in the commutator subgroup of Γ and is isomorphic to the second homology group H 2 (Γ, Z). We let φ : Γ → Γ denote the homomorphism appearing in the sequence above. The relevance of representation groups for our purposes lies in the fact that any projective representation of Γ corresponds to a genuine representation of Γ. Indeed, let σ ∈ Z 2 (Γ, T) and let π be a σ-projective unitary representation Γ on a Hilbert space H π . Identifying the circle group T with the center T · I Hπ of the group of unitary operators U (H π ), and letting q : U (H π ) → U (H π )/T denote the quotient map, we get a homomorphism ρ π : Γ → U (H π )/T given by ρ π = q • π. Since Ext(Γ ab , T) = 0 (because T is divisible, cf. [60, Chapter III, Proposition 2.6]), we may then invoke [114,Proposition V.5.5], and deduce that there exists a homomorphism π : Γ → U (H π ), i.e., a unitary representation π of Γ on H π , satisfying that q • π = ρ π • φ. From the relation between π and π, it follows readily that for every γ ∈ Γ, there exists (a unique) µ γ ∈ T such that π(γ) = µ γ π(φ(γ)) . Note that π is irreducible if and only if π is irreducible: Indeed, as φ is surjective, we obviously have π(Γ) ′ = π( Γ) ′ , so this is a consequence of Schur's lemma. It is also immediate that π(Γ) and π( Γ) generate the same C * -algebra of operators on H π . If π is a σ-projective unitary representation of Γ on a Hilbert space H π , we will denote by C * (π(Γ)) the C * -subalgebra of B(H π ) generated by π(Γ). In other words, C * (π(Γ)) = π(C * (Γ, σ)). Since C * (Γ, σ) is nuclear whenever Γ is amenable [94,Corollary 3.9], and as any * -homomorphic image of a nuclear C * -algebra is also nuclear (see for example [17, Corollary IV.1.13]), we get in particular that C * (π(Γ)) is nuclear whenever Γ is nilpotent. Moreover, using results due to Eckhardt, Gillaspy and McKenney [32] and Eckhardt and Gillaspy [31] in the case of ordinary unitary representations, we obtain the following: Theorem 5.5. Let Γ be a finitely generated nilpotent group and σ ∈ Z 2 (Γ, T). Let π be a σ-projective unitary representation of Γ on a Hilbert space H π . Then (i) C * (π(Γ)) and C * (Γ, σ) ≃ C * r (Γ, σ) are nuclear, quasidiagonal and have finite decomposition rank ; (ii) If π is irreducible, then C * (π(Γ)) is also simple, it satisfies the universal coefficient theorem (UCT), and its decomposition rank is less or equal to 1. Proof. (i) The assertion about nuclearity follows from our comment above. Next, according to Theorem 5.4, Γ has a representation group Γ which is finitely generated and nilpotent. As explained previously, there exists a unitary representation π of Γ on H π such that C * (π(Γ)) = C * ( π( Γ)). Now, [32,Theorem 5.1] gives that C * ( Γ) has finite decomposition rank. Using [69, (3.3)], we deduce that any * -homomorphic image of C * ( Γ), in particular C * ( π( Γ)), has finite decomposition rank. Since C * r (Γ, σ) = C * (λ σ Γ (Γ)) = C * ( λ σ Γ ( Γ)), this implies that 3 There is a similar notion of representation group for certain locally compact groups which was introduced by C.C. Moore [83], see also [93,Section 3]. C * (Γ, σ) ≃ C * r (Γ, σ) has finite decomposition rank. Since any separable C * -algebra with finite decomposition rank is quasidiagonal (cf. [115,Theorem 17.4.3]), we have shown (i). (ii) Assume now that π is irreducible. Then π is irreducible too. As any primitive ideal of C * ( Γ) is maximal [98], we get that C * ( π( Γ)) is simple. Further, [31,Theorem 3.5] gives that C * ( π( Γ)) satisfies the UCT. Finally, [32,Theorem 6.2] gives that the decomposition rank of C * ( π( Γ)) is less or equal to 1. Remark 5.6. By definition, the decomposition rank of a separable C * -algebra bounds its nuclear dimension. Thus, in Theorem 5.5 we could replace finite decomposition rank with finite nuclear dimension. To show that this weaker property holds, we could then have invoked [33,Theorem 4.4] instead of quoting [32, Theorem 5.1]. Theorem 5.7. Let Γ be a finitely generated nilpotent group and σ ∈ Z 2 (Γ, T). Assume that (Γ, σ) satisfies Kleppner's condition. Then C * (Γ, σ) ≃ C * r (Γ, σ) is a (unital, separable, nuclear) simple, quasidiagonal C * -algebra with a unique tracial state, which satisfies the UCT and has decomposition rank less or equal to 1. Proof. The fact that C * (Γ, σ) ≃ C * r (Γ, σ) is simple with a unique tracial state whenever Γ is nilpotent and (Γ, σ) satisfies Kleppner's condition is due to Packer, cf. [92]. Let now π be any σ-projective irreducible unitary representation of Γ. Since C * (Γ, σ) is simple, we have C * (Γ, σ) ≃ C * (π(Γ)), which we know is quasidiagonal and has decomposition rank less or equal to 1 from Theorem 5.5. Combining Theorem 5.1 and Theorem 5.7, we finally get: Corollary 5.8. Let Γ be a finitely generated nilpotent group, σ ∈ Z 2 (Γ, T), and assume (Γ, σ) satisfies Kleppner's condition. Then C * (Γ, σ) ≃ C * r (Γ, σ) has strict comparison of positive elements and stable rank one. In particular, C * r (Γ, σ) has strict comparison of projections (in our extended sense). Note that a combination of Theorem 5.7 and Corollary 5.8 directly provides Theorem 1.6. Lastly, we mention how to obtain Theorem 1.7 from these results. Proof of Theorem 1.7. As shown by Tikuisis, White and Winter, cf. [117,Corollary D], the class C of all separable, unital, simple, infinite-dimensional C * -algebras with finite nuclear dimension and which satisfy the UCT is classified by the Elliott invariant. Now Theorem 5.7 gives that C * r (Γ, σ) belongs to C whenever Γ and σ satisfy the assumptions of this theorem (and Γ is infinite). Lattice orbits of nilpotent Lie groups Let (π, H π ) be a projective representation of a nilpotent Lie group G on H π . For a lattice Γ ≤ G and a vector g ∈ H π , we consider the system of vectors π(Γ)g = (π(γ)g) γ∈Γ . (6.1) A system (6.1) will be treated as a Γ-indexed family, possibly with repetitions. In this section the results obtained in the previous sections are applied to the restriction π\| Γ of π to Γ and an explicitly constructed associated Hilbert C * -module. 6.1. Relative discrete series and projective representations. Let N be a connected, simply connected nilpotent Lie group and let (π, H π ) be an irreducible unitary representation of N . Denote by P π = x ∈ N : π(x) ∈ C·I Hπ the projective kernel of π. Then P π ≤ N forms a connected, simply connected normal subgroup. Assume that (π, H π ) is square-integrable modulo P π , i.e., there exist f, g ∈ H π \ {0} such that N/Pπ \| f, π(x)g \| 2 dµ N/Pπ (xP π ) < ∞. (6.2) Since π(x) = χ(x)I Hπ for a character χ ∈ P π , the integrand N ∋ x → \| f, π(x)g \| ∈ [0, ∞) in (6.2) defines a function on N/P π . The orthogonality relations for (π, H π ) yields that there exists a unique d π > 0 such that N/Pπ \| f, π(x)g \| 2 dµ N/Pπ (xP π ) = d −1 π f 2 Hπ g 2 Hπ , f, g ∈ H π ; (6.3) see, e.g., [25,84,95]. An irreducible representation π that is square-integrable modulo P π is called a relative discrete series representations; this will be denoted by π ∈ SI/P π . In particular, if π is irreducible and square-integrable modulo the center Z of N , then π ∈ SI/P π , and P π = Z by [25,Theorem 3.2.3] and [25,Corollary 4.5.4]. A representation π ∈ SI/P π can be treated as a square-integrable projective representation of N/P π : Given a smooth cross-section s : N/P π → N for the quotient map p : N → N/P π , the mapping π ′ : N/P π → U (H π ), xP π → π(s(xP π )) (6.4) defines an irreducible projective unitary representation of N/P π . The assumption on π yields that (π ′ , H π ) is square-integrable on N/P π in the strict sense, i.e., g 1 , π ′ (·)g 2 ∈ L 2 (N/P π ) for all g 1 , g 2 ∈ H π . The constant d π > 0 in (2.3) coincides with the formal dimension d π ′ of π ′ normalized according to Haar measure on N/P π (see Section 2.2). A different choice of cross-section yields equivalent projective unitary representations (cf. [3] for details). A square-integrable projective representation π ′ obtained via a cross-section as in (6.4) will be referred to as a projective relative discrete series representation of the connected, simply connected nilpotent Lie group G = N/P π . For simplicity, it will often also be written π = π ′ . Notation. Throughout, unless stated otherwise, any nilpotent Lie group G is assumed to be connected and simply connected. The Lie algebra of G is denoted by g and its dimension by d. The associated exponential map is denoted by exp G : g → G and forms a global diffeomorphism. The Schwartz space on G consists of all F : G → C such that F •exp G ∈ S(g). 6.2. Smooth vectors and matrix coefficients. Let (π, H π ) be an irreducible unitary representation of a nilpotent Lie group N . The space of smooth vectors H ∞ π consists of all vectors g ∈ H π such that the orbit maps N ∋ x → π(x)g ∈ H π are smooth. The Lie algebra n acts on H ∞ π via the derived representation dπ(X)g = d dt t=0 π(exp N (tX))g, X ∈ n, g ∈ H ∞ π . For a basis {X 1 , ..., X d } for n, a family of semi-norms in H ∞ π is defined by g β := dπ(X β )g Hπ = dπ(X β 1 1 ) · · · dπ(X β d d )g Hπ , β ∈ N d 0 . The space H ∞ π is π-invariant and is norm dense in H π , see, e.g., [25,Appendix A.1]. For smooth vectors, the associated matrix coefficients of a square-integrable representation define Schwartz functions, see, e.g., [24,25,61,95] 95]). Let π ∈ SI/P π . If f, g ∈ H ∞ π , then C g f = f, π ′ (·)g ∈ S(N/P π ). Proof. The result follows from the general theorem [95,Theorem 2.6] in the following manner. Let p ⊆ n denote the Lie algebra of P π . As in [95,Remark 2.2.8], let n e ⊂ n be a subspace of even dimension, so that the orthogonal decomposition n = n e ⊕ p gives rise to the diffeomorphism φ : n e → N/P π , X → exp N (X)P π (cf. [25, Section 1.2]), with inverse φ −1 : N/P π → n e given by exp N (X)P π → X. Hence, a (smooth) cross-section s : N/P π → N for p : N → N/P π is given by s(exp N (X)P π ) = exp N (X), i.e., p • s = id N/Pπ . For this cross-section, denote by π ′ the projective representation of N/P π as defined in (6.4). If f, g ∈ H ∞ π , then [95, Theorem 2.6] yields that n e ∋ X → f, π(exp N (X))g ∈ C is in S(n e ). A direct calculation using exp N/Pπ (X + p) = p(exp N (X)) and the definition of π ′ shows f, π ′ (exp N/Pπ (X + p))g = f, π ′ (exp N (X)P π )g = f, π(exp N (X))g , X ∈ n. Therefore, X + p → f, π ′ (exp N/Pπ (X + p))g defines a Schwartz function on n/p ∼ = n e , i.e., f, π ′ (·)g ∈ S(N/P π ). Lemma 6.1 allows to prove a convenient characterization of the space H ∞ π . For this and other purposes, a family of semi-norms on S(N/P π ) defined via polynomial weights and leftinvariant differential operators will be used, cf. [62,75,111] for more details on what follows. Let U be fixed a symmetric, compact generating set for G = N/P π and define the length The space S(G) is independent of the choice of the neighborhood U and the exponent p, cf. [62,75,111]. function τ : G → [0, ∞) by τ (x) = min{n ∈ N 0 \| x ∈ U n }, with U 0 := {e}. Then τ (xy) ≤ τ (x) + τ (y), τ (x −1 ) = τ (x) and τ (e) = 0 for x, y ∈ G. Given α ∈ N 0 , let w α : G → [1, ∞) be defined as x → (1 + τ (x)) α and define L p wα (G) to be the collection of all F ∈ L p (G) such that F L p wα := w α · F L p < ∞. Proposition 6.2. Let π ∈ SI/P π . For g ∈ H ∞ π \ {0} and α ∈ N 0 , let H 1,α π := f ∈ H π : C g f L 1 wα = G \|C g f (x)\|w α (x) dµ G (x) < ∞ . Then H ∞ π = α∈N 0 H 1,α π . Proof. As in the proof of Lemma 6.1, consider the orthogonal decomposition n = n e ⊕p and the diffeomorphism φ : n e → N/P π , X → exp N (X)P π . Let π ′ denote the projective representation of G = N/P π defined via the section s : N/P π → N, s(exp N (X)P π ) = exp N (X) as in (6.4). If f ∈ H ∞ π , then C g f ∈ S(G) by Lemma 6.1. In particular, using the semi-norms (6.5) with p = 1, yields directly that C g f ∈ L 1 wα (G) for all α ∈ N 0 . For the converse, let f ∈ α∈N 0 H 1,α π and let g ∈ H ∞ π \ {0} be normalized such that f = G f, π ′ (x)g π ′ (x)g dµ G (x) = N/Pπ f, π(s(xP π ))g π(s(xP π ))g dµ N/Pπ (xP π ); (6.6) cf. the orthogonality relations (2.3) and (6.3). By [25,Theorem 1.2.10], the map φ : n e → N/P π transforms the Lebesgue measure dY on n e to Haar measure µ G on G = N/P π . Therefore, the reproducing formula (6.6) and the change-of-variables formula yields f = ne f, π(exp N (Y ))g π(exp N (Y ))g dY. Given a basis {X 1 , ..., X d } of n and a multi-index β ∈ N d 0 , a direct calculation entails then that dπ(X β )f = ne f, π(exp N (Y ))g dπ(X β )π(exp N (Y ))g dY. Since g ∈ H ∞ π , it follows as in the proof of [25,Lemma A.1 .1] that dπ(X β )π(exp N (Y ))g = π(exp N (Y )) \|β ′ \|≤\|β\| p β ′ (exp N (Y ))dπ(X β ′ )g, where p β ′ are polynomial functions on N , i.e., p β ′ • exp N is a polynomial on n. Combining these identities with norm estimates for vector-valued integrals (see, e.g., [40,Theorem A.22]) gives dπ(X β )f Hπ ≤ \|β ′ \|≤\|β\| dπ(X β ′ )g Hπ ne \| f, π(exp N (Y ))g \|\|p β ′ (exp N (Y ))\| dY = \|β ′ \|≤\|β\| dπ(X β ′ )g Hπ N/Pπ \| f, π(s(xP π ))g \|\|p β ′ (s(xP π ))\| dµ N/Pπ (xP π ). (6.7) By the assumption g ∈ H ∞ π , we have C 1 := max \|β ′ \|≤\|β\| dπ(X β ′ )g Hπ < ∞. Since p β ′ • exp N is a polynomial on n e , it follows by the identity p β ′ (exp N (Y )) = p β ′ (s(exp N/Pπ (Y + p))) that p β ′ • s is a polynomial function on G = N/P π . By [75,Section 1.5] or [76,Section 3.5], any polynomial function p on N/P π is comparable to the polynomial weight w = 1 + τ (·) in the sense that there exist C 2 , M > 0 (depending on p) such that p(x) ≤ C 2 w(x) M for x ∈ G. Choosing α > 0 sufficiently large, it follows therefore easily from this and (6.7) that there exists C > 0 such that dπ(X β )f Hπ ≤ C G \| f, π(s(x))g \|w α (x) dµ G (x) < ∞. (6.8) Since β ∈ N d 0 was chosen arbitrary, it follows that f ∈ H ∞ π . The arguments used in the proof of Proposition 6.2 are reminiscent of some arguments used in [46, Section 11.2] and [13, Section 2.2], which require different assumptions than used here. 6.3. Mapping properties for smooth vectors. Henceforth, π = π ′ will denote a projective relative discrete series representation of G = N/P π and Γ ≤ G will denote a lattice. The Schwartz space S(Γ) on the discrete subgroup Γ ≤ G is defined by S(Γ) = c ∈ C Γ : γ∈Γ \|c γ \|w α (γ) < ∞, ∀α ∈ N 0 , and equipped with the semi-norms c α := c ℓ 1 wα = w α · c ℓ 1 for α ∈ N 0 ; see [65,67,111]. The following result shows that the action of the analysis (resp. synthesis) operator associated to smooth vectors is well-defined into (resp. on) the space S(Γ). Proposition 6.3. Let (π, H π ) be a projective relative discrete series representation of a nilpotent Lie group G. Suppose that Γ ≤ G is a lattice. Then the following assertions hold: (i) For all f, g ∈ H ∞ π , the mapping Γ ∋ γ → f, π(γ)g ∈ C defines an element of S(Γ). (ii) For (c γ ) γ∈Γ ∈ S(Γ) and g ∈ H ∞ π , the series γ∈Γ c γ π(γ)g defines an element of H ∞ π . Thus (S(Γ), H ∞ π ) forms an admissible pair for π\| Γ in the sense of Definition 4.1. Proof. The space L 1 wα (G) is invariant under translations L x F := F (x −1 ·) and R x F := F (·x) for x ∈ G, with L x B(L 1 wα ) , R x B(L 1 wα ) ≤ w α (x) , see, e.g., [101,Proposition 3.7.6]. In particular, the weight w α : G → [1, ∞) is a control weight for L 1 wα (G) in the sense of [36, Section 3]. Throughout, let H 1,α π be as defined in Proposition 6.2, see [21,36] for basic properties. (i) Let α ∈ N 0 . If f, g ∈ H ∞ π , then C g f ∈ S(G) by Lemma 6.1. In particular, this implies, by using the semi-norms (6.5) with p = ∞, that, for all α ′ ∈ N 0 , there exists C α ′ > 0 such that \|C g f (x)\| ≤ C α ′ (1 + τ (x)) −α ′ for x ∈ G. Since (1 + τ (·)) −α ′ ∈ L 1 wα (G) for a sufficiently large α ′ ≥ α (cf. [111, Proposition 1.5.1]), the submultiplicativity and local boundedness of w = (1 + τ (·)) yields that G sup y∈V \|C g f (xy)\|w α (x) dµ G (x) ≤ C α ′ G sup y∈V (1 + τ (xy)) −α ′ w α (x) dµ G (x) ≤ C α ′ sup y∈V (1 + τ (y −1 )) α ′ G (1 + τ (x)) −α ′ w α (x) dµ G (x) (6.9) < ∞ for any relatively compact unit neighborhood V ⊂ G. The property (6.9) allows an application of [36,Lemma 3.8], which yields that ( f, π(γ)g ) γ∈Γ ∈ ℓ 1 wα (Γ). Since α ∈ N 0 was chosen arbitrary, it follows that ( f, π(γ)g ) γ∈Γ ∈ α∈N 0 ℓ 1 wα (Γ) = S(Γ). (ii) If (c γ ) γ∈Γ ∈ S(Γ) and g ∈ H ∞ π , then (c γ ) γ∈Γ ∈ ℓ 1 wα (Γ) and g satisfies (6.9) with the choice g = f for all α ∈ N 0 . Hence, by [36,Proposition 5.2] or [21, Theorem 6.1], the mapping (c γ ) γ∈Γ → γ∈Γ c γ π(γ)g is bounded from ℓ 1 wα (Γ) into H 1,α π for α ∈ N 0 . This shows γ∈Γ c γ π(γ)g ∈ α∈N 0 H 1,α π = H ∞ π by Proposition 6.2. For the admissibility of the pair (S(Γ), H ∞ π ) for π\| Γ , it is obvious that C c (Γ, σ) is contained in S(Γ), so it remains only to show that S(Γ) is a * -subalgebra of ℓ 1 (Γ, σ). It is straightforward to see that S(Γ) is closed under twisted involution. For the algebra property, note that S(Γ) = α∈N 0 ℓ 1 wα (Γ) and that each ℓ 1 wα (Γ), where α ∈ N 0 , is an ordinary convolution algebra. Since \|c * σ d\| ≤ \|c\| * \|d\| for c, d ∈ ℓ 1 (Γ), it follows readily that S(Γ) is also closed under twisted convolution. 6.4. Finitely generated modules associated to lattices. This section is devoted to the construction of a Hilbert C * -module from H ∞ π . The following observation will guarantee that this module is finitely generated. Proposition 6.4. Let (π, H π ) be a projective relative discrete series representation of a nilpotent Lie group G. Suppose that Γ ≤ G is a lattice. Then there exists a finite family (g j ) n j=1 of vectors g j ∈ H ∞ π such that (π(γ)g j ) γ∈Γ,1≤j≤n is a frame for H π . Proof. Let g ∈ H ∞ π \ {0}, so that C g g satifies the property (6.9) with g = f . Then, by [21,Theorem 6.4] or [45,Section 4], there exists a compact unit neighborhood U ⊂ G such that for any discrete family Λ in G satisfying G = λ∈Λ λU and sup x∈G \|Λ ∩ xU \| < ∞, f 2 Hπ ≍ λ∈Λ \| f, π(λ)g \| 2 , f ∈ H π . Since G is a nilpotent Lie group, Γ ≤ G is also co-compact, see, e.g., [25,Corollary 5.4.6]. Hence, there exists a relatively compact fundamental domain Σ ⊂ G for Γ. Let (x j U ) n j=1 be a finite cover of Σ. Then Λ ′ := {γx j : γ ∈ Γ, j = 1, . . . , n} satisfies G = λ ′ ∈Λ ′ λ ′ U and sup x∈G \|Λ ′ ∩ xU \| < ∞, so that f 2 Hπ ≍ λ∈Λ ′ \| f, π(λ ′ )g \| 2 = n j=1 γ∈Γ \| f, π(γ)π(x j )g \| 2 , f ∈ H π . Therefore, defining g j := π(x j )g ∈ H ∞ π for j = 1, . . . , n, gives the desired result. The existence of localized multi-window Gabor frames was proven in [77, Theorem 4.6] via a correspondence to projective modules over non-commutative tori. The proof of Proposition 6.4 shows that this is also a direct consequence of the classical sampling techniques [21,36]. Theorem 6.5. Let (π, H π ) be a σ-projective relative discrete series representation of a nilpotent Lie group G of formal dimension d π > 0. Suppose that Γ ≤ G is a lattice. Then (S(Γ), H ∞ π ) is an admissible pair for π\| Γ in the sense of Definition 4.1, so that H ∞ π can be completed into a Hilbert C * r (Γ, σ)-module E. The module E is finitely generated, and if τ denotes the canonical tracial state on C * r (Γ, σ), then τ (E) = vol(G/Γ)d π . (The constant vol(G/Γ)d π is independent of the choice of Haar measure on G.) Proof. Admissibility of the pair (S(Γ), H ∞ π ) was proved in Proposition 6.3. Combining Proposition 6.4 with Proposition 4.3, we get that E is finitely generated. By the discussion preceding Proposition 4.3, the localization space H τ E of E with respect to τ can be naturally identified with H π , and the representation π τ E of C * r (Γ, σ) on H τ E induces a representation π r of C * r (Γ, σ) on H π . By Proposition 3.5, this representation can be extended to give H π the structure of a Hilbert L(Γ, σ)-module, where the action is determined by λ σ Γ (γ) · f = π(γ)f for γ ∈ Γ and f ∈ H π . The dimension dim (L(Γ,σ),τ ) H π of this Hilbert L(Γ, σ)-module was computed in [34] to be vol(G/Γ)d π , see [34,Theorem 4.3]. Therefore, by Proposition 5.3, it follows that τ (E) = dim (L(Γ,σ),τ ) H π = vol(G/Γ)d π , as required. Proof of Theorem 1.4. The statement of Theorem 1.4 follows directly from Theorem 6.5 combined with the fact that π ∈ SI/P π and P π = Z, cf. Section 6.1. Proof of Theorem 1.5. Theorem 1.5 follows by applying Proposition 4.3 to the module of Theorem 6.5. 6.5. Existence of smooth frames and Riesz sequences. The following theorem is the main result of this paper. Theorem 6.6. Let (π, H π ) be a σ-projective relative discrete series representation of a nilpotent Lie group G. Suppose Γ ≤ G is a lattice such that (Γ, σ) satisfies Kleppner's condition. Then the following assertions hold: (i) If vol(G/Γ)d π < 1, then there exists g ∈ H ∞ π such that π(Γ)g is a frame for H π . (ii) If vol(G/Γ)d π > 1, then there exists g ∈ H ∞ π such that π(Γ)g is a Riesz sequence in H π . Proof. For the applicability of the results of Section 5.3, we note that a discrete Γ ≤ G is finitely generated, see, e.g., [25,Corollary 5.4.4]. (i) Suppose vol(G/Γ)d π < 1. By Theorem 6.5, it follows that τ (E) < 1 for the canonical trace τ on C * r (Γ, σ), which is the unique tracial state on C * r (Γ, σ) by [92]. Since C * r (Γ, σ) has strict comparison of projections by Corollary 5.8, it follows from Proposition 5.2 that E admits a generating set with one element. By Proposition 3.3, the generating element may be chosen to be g ∈ H ∞ π . Hence, π(Γ)g is a frame for H π by Proposition 4.3. (ii) Suppose vol(G/Γ)d π > 1. Just as in (i), we get τ (E) > 1 for the unique tracial state τ on C * r (Γ, σ), so by strict comparison of projections and Proposition 5.2, E admits an A-linearly independent set {g} which is closed in E. By Proposition 3.3, g can be chosen in H ∞ π , so π(Γ)g is a Riesz sequence by Proposition 4.3. Proof of Theorem 1.3. The statement of Theorem 1.3 follows directly from Theorem 6.6 combined with the fact that π ∈ SI/P π and P π = Z, cf. Section 6.1. Remark 6.7. Theorem 6.6 can be extended to multi-window and super systems, cf. [34] for these notions. Under Kleppner's condition, the inequality vol(G/Γ)d π < n/d (resp. vol(G/Γ)d π > n/d) implies the existence of an n-multiwindow d-super frame (resp. Riesz sequence) in H d π with windows in H ∞ π . 6.6. Special classes of smooth vectors. Theorem 6.6 can also be used to prove the existence of frames and Riesz sequences generated by smooth vectors with additional qualities, such as Gårding vectors or analytic vectors for a representation (π, H π ) of a Lie group N . For k ∈ C ∞ c (N ) and g ∈ H π , a Gårding vector is defined by π(k)g = N k(x)π(x)g dµ N (x). (6.10) The Gårding subspace H γ π ⊆ H π is the linear span of all vectors of the form (6.10). The space H γ π is π-invariant and norm dense in H π and satisfies H γ π ⊆ H ∞ π , cf. [25, Appendix A]. A vector g ∈ H π is called analytic if the orbit map x → π(x)g is real-analytic. The space of all analytic vectors is denoted by H ω π and is a π-invariant dense subspace of H π , cf. [44,86]. The following modification result can be proved in a similar manner as [52,Proposition 4.4] (cf. also [50,Proposition 1]). Its proof will be omitted here. Lemma 6.8. Let π be an irreducible, square-integrable projective representation of a nilpotent Lie group G. For g ∈ H ∞ π \ {0}, let H 1 π = {f ∈ H π : C g f ∈ L 1 (G)} be equipped with the norm f H 1 π := C g f L 1 . Suppose that V ⊂ H 1 π is a norm dense subspace. Then the following assertions hold: (i) If π(Γ)g is a frame, then there exists g ∈ V such that π(Γ) g is a frame. (ii) If π(Γ)g is a Riesz sequence, then there exists g ∈ V such that π(Γ) g is a Riesz sequence. Corollary 6.9. Under the assumptions of Theorem 6.6, the following hold: (i) If vol(G/Γ)d π < 1, there exists g ∈ H ω π (resp. g ∈ H γ π ) such that π(Γ)g is a frame. (ii) If vol(G/Γ)d π > 1, there exists g ∈ H ω π (resp. g ∈ H γ π ) such that π(Γ)g is a Riesz sequence. Proof. The result follows from Theorem 6.6 and Lemma 6.8 after showing that H ω π (resp. H γ π ) is dense in H 1 π . To show the latter, let h ∈ H ω π (resp. h ∈ H γ π ) be non-zero. By the atomic decomposition of H 1 π (cf. [21,36]), there exists a sequence (x i ) i∈N in G such that any f ∈ H 1 π can be represented as a norm convergent series f = i∈I c i π(x i )h for some (c i ) i∈I ∈ ℓ 1 (I). If f n := n i=1 c i π(x i )h for n ∈ N, then f n ∈ H ω π (resp. f n ∈ H γ π ), and f n → f in H 1 π as n → ∞. This completes the proof. Remark 6.10. An alternative argument for the existence claims (i) and (ii) in Corollary 6.9 for the Gårding space H γ π can be obtained via the Dixmier-Malliavin theorem [28], which asserts that H ∞ π = H γ π for a nilpotent Lie group. Then (i) and (ii) follow already from Theorem 6.6. Examples In this section we discuss two examples that illustrate our main result. Example 7.1 (The Heisenberg group). Let N be the 2d + 1-dimensional Heisenberg group, i.e., N = R d × R d × R with multiplication (x, ω, s)(x ′ , ω ′ , s ′ ) = (x + x ′ , ω + ω ′ , s + s ′ + x · ω ′ ). The center of N is given by Z = {0}× {0}× R ∼ = R, hence the quotient G = N/Z is isomorphic to the abelian group R d × R d . The Schrödinger representation of N on L 2 (R d ) is given by π(x, ω, s)f (t) = e 2πis e −2πiωt f (t − x). The corresponding projective representation of R 2d ∼ = R d × R d can be given by π(x, ω)f (t) = e −2πiωt f (t − x) where the associated cocycle is given by σ((x, ω), (x ′ , ω ′ )) = e −2πix·ω ′ . A lattice orbit π(Γ)g for Γ a lattice in R 2d and g ∈ L 2 (R d ) is in this context known as a Gabor system. A lattice Γ in R 2d is of the form Γ = M Z 2d for some M ∈ GL 2n (R). Viewing instead Γ as Z 2d , the cocycle is given by σ Θ (k, l) = e −2πi(Θk)·l for k, l ∈ Z 2d , where Θ = M t JM and J denotes the standard symplectic 2n × 2n matrix J = 0 I n −I n 0 . With this notation, Kleppner's condition translates into the statement that whenever k ∈ Z 2d satisfies e 2πi(Θk)·l = 1 for all l ∈ Z 2d , then k = 0. For a lattice of the form Γ = αZ d × βZ d with α, β > 0, this translates into the number αβ being irrational. Kleppner's condition implies a weaker condition, namely that Θ contains at least one irrational entry. Let us call Γ nonrational when the latter condition holds. For nonrational lattices, Rieffel proved that the non-commutative tori C * (Γ, σ) have strict comparison of projections and cancellation [104] (see also [16,Theorem 5.3.2]). A consequence of this (cf. [104,Corollary 7.10]) was used in [63,Theorem 5.4], combined with the link between Heisenberg modules over non-commutative tori and Gabor frames [77], to prove the existence of Gabor frames π(Γ)g with integrable vector g ∈ H 1 π (hence, g ∈ S(R d ) by Lemma 6.8) for nonrational lattices Γ satisfying vol(R 2d /Γ) < 1. Therefore, in this setting, our main result is already covered by the result in [63]. The following example considers the group G 5,3 from Nielsen's catalogue [87]. This example is of interest to time-frequency analysis as it leads to so-called coorbit spaces [36] that are different [49,Example 3.3] from the coorbit spaces associated to the Schrödinger representation defined in Example 7.1, so-called modulation spaces. In addition, we mention that group C algebras associated with lattices in G 5,3 have been studied in [81]. Example 7.2 (The group G 5,3 ). Consider the group G 5,3 from [87, p. 6]. This is a step 3 nilpotent Lie group with R 5 as underlying manifold. The group operation is given by (x 1 , . . . , x 5 )(y 1 , . . . , y 5 ) = (x 1 +y 1 +x 4 y 2 +x 5 y 3 +x 2 5 y 4 /2, x 2 +y 2 , x 3 +y 3 +x 5 y 4 , x 4 +y 4 , x 5 +y 5 ). The center of G 5,3 is given by R × {0} 4 . An irreducible representation (π, L 2 (R 2 )) of G 5,3 which is square-integrable modulo the center is given by π(x 1 , . . . , x 5 )f (s, t) = e 2πi(x 1 −x 2 x 4 +x 4 s−x 3 t+x 4 t 2 /2) g(s − x 2 , t − x 5 ). The formal dimension of π is d π = 1. The quotient of G 5,3 by the center is isomorphic to G := R × N , where N denotes the 3-dimensional Heisenberg group from the previous example, although the multiplication is in a different order: (x 1 , x 2 , x 3 , x 4 )(y 1 , y 2 , y 3 , y 4 ) = (x 1 + y 1 , x 2 + y 2 + x 4 y 3 , x 3 + y 3 , x 4 + y 4 ). Here we have relabeled the coordinates from x j to x j−1 for j = 2, 3, 4, 5. The Haar measure µ G on G is just the 4-dimensional Lebesgue measure. The center Z of G is R 2 × {0} 2 . The corresponding projective representation of R × N corresponding to π (which we denote also by π) is given by π(x 1 , . . . , x 4 )f (s, t) = e 2πi(x 3 s−x 2 t+x 3 t 2 /2) f (s − x 1 , t − x 4 ). The cocycle is given by σ((x 1 , x 2 , x 3 , x 4 ), (y 1 , y 2 , y 3 , y 4 )) = exp(2πi(−x 1 y 3 + x 4 y 2 + x 2 4 y 3 /2)). Let N ′ = N ∩ Z 3 denote the discrete Heisenberg group, which is a cocompact lattice in the Heisenberg group. Hence the group Γ = Z × N ′ is a lattice in G. The conjugacy class of an element (k 1 , k 2 , k 3 , k 4 ) ∈ Γ is given by {(k 1 , k 2 + pk 3 + qk 4 , k 3 , k 4 ) : p, q ∈ Z}. From this we see that the elements with finite conjugacy class in Γ are exactly elements of the center Z ∩ Γ = Z 2 × {0} 2 , and these elements have singleton conjugacy classes. Let us consider the dilation automorphisms δ α,β of G (α, β > 0) given by δ α,β (x 1 , x 2 , x 3 , x 4 ) = (αx 1 , β 2 x 2 , βx 3 , βx 4 ). Applying these to Γ ⊆ G, we get a family of lattices Γ α,β := δ α,β (Γ) in G which are isomorphic as discrete groups to Γ. We can compute the covolume of Γ α,β as Let us check Kleppner's condition for (Γ α,β , σ). We need only check the elements with finite conjugacy class, i.e., those in the center of Γ α,β . Thus, an element (αk 1 , β 2 k 2 , 0, 0) is σ-regular if and only if for all (αl 1 , β 2 l 2 , βl 3 , βl 4 ) ∈ Γ α,β we have that 1 = σ((αk 1 , β 2 k 2 , 0, 0), (αl 1 , β 2 l 2 , βl 3 , βl 4 ))σ((αl 1 , β 2 l 2 , βl 3 , βl 4 ), (αk 1 , β 2 k 2 , 0, 0)) = exp(−2πi(αβk 1 l 3 + β 3 k 2 l 4 )). For this to happen, we need αβk 1 l 3 + β 3 k 2 l 4 ∈ Z for all l 3 , l 4 ∈ Z. Hence, we see that if at least one of the numbers αβ and β 3 is rational, then nontrivial σ-regular conjugacy classes exist. On the other hand, if both αβ and β 3 are irrational, then Kleppner's condition is satisfied. Our main result now states the following: Let α, β > 0 such that both β 3 and αβ are irrational numbers. If αβ 4 < 1 (resp. αβ 4 > 1), then there exists g ∈ H ∞ π = S(R 2 ) such that π(Γ α,β )g is a frame (resp. Riesz sequence) for L 2 (R 2 ). Lemma 3.1 ([42]). A finite set {g 1 , . Lemma 3.2 ([5]). A finite set {g 1 , . . . , g n } ⊆ E is A-linearly independent with closed A-span if and only if the associated Gramian operator G is invertible in L A (E). Proposition 3 . 3 . 33Let n ∈ N. Then the following hold: (i) There exists a generating set with n elements in E if and only if there exists an adjointable isometry E → A n , if and only if there exists a projection p in M n (A) such that E ∼ = A n p. (ii) There exists an A-linearly independent set with n elements and closed A-span in E if and only if there exists an adjointable isometry A n → E. Lemma 3. 4 . 4Assume τ is a tracial state on A and let a, b ∈ A be positive. Then 0 ≤ τ (ab) ≤ a τ (b). Proposition 3. 5 . 5The representation π τ E of A on H τ E extends uniquely to a normal representation of M on H τ E . In other words, H τ E is a Hilbert M -module. Proposition 4. 2 . 2Let (A 0 , H 0 ) be an admissible pair for π. Then, with the action of A 0 on H 0 given by (4.1) and the A 0 -valued inner product on H 0 given by (4. Proposition 4. 3 . 3Let (A 0 , H 0 ) be an admissible pair for π, and E be the associated Hilbert A-module. Let g 1 , . . . , g n ∈ H 0 . Then the following hold:(i) The finite set {g 1 , . . . , g n } is an algebraic generating set for E if and only if (π(γ)g j ) γ∈Γ,1≤j≤n A + denote the cone of positive elements of A. For m, n ∈ N, we let M m,n (A) denote the space of m × n matrices over A. We let M ∞ (A) + denote the (disjoint) union n∈N M n (A) + . For a, b ∈ M ∞ (A) + , say a ∈ M n (A) + and b ∈ M m (A) + , we say that a is Cuntz subequivalent to b, and write a b, if there exists a sequence (r k ) ∞ k=1 in M m,n (A) such that r k br k − a → 0 in M n (A) as k → ∞. Let D(G) be the (unital) algebra of all left-invariant differential operators on G, i.e., all linear operators D :C ∞ (G) → C ∞ (G) of the form D = β∈N d 0 c β X β ,where all but finitely many c β ∈ C are zero and X β = X β 1 1 · · · X β d d for a basis {X 1 , ..., X d } for g. Let p ∈ [1, ∞]. 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[ "Measurement of tW differential cross-sections with ATLAS at √ s = 13 TeV", "Measurement of tW differential cross-sections with ATLAS at √ s = 13 TeV" ]	[ "\nPhysikalisches Institut\nUniversity of Bonn\nBonn, BragaGermany, Portugal\n" ]	[ "Physikalisches Institut\nUniversity of Bonn\nBonn, BragaGermany, Portugal" ]	[]	The cross-section to produce a W boson in association with a top quark is measured differentially with respect to several particle-level final-state observable quantities. The measurements are performed using 36.1 fb −1 of pp collision data at √ s = 13 TeV collected in 2015 and 2016, by the ATLAS detector at the LHC. Cross-sections are measured in a fiducial phase-space defined by the presence of two charged leptons and exactly one jet identified as containing B hadrons. Measurements are normalised to the fiducial cross-section, causing several of the main uncertainties to cancel. The results are found to be in good agreement with predictions from several Monte Carlo generators.PRESENTED AT 10 th International Workshop on Top Quark Physics	null	[ "https://arxiv.org/pdf/1809.01433v1.pdf" ]	89,611,307	1809.01433	ed3af447d0c1b033a0c3281130988e52a07846e4	Measurement of tW differential cross-sections with ATLAS at √ s = 13 TeV September 6, 2018 September 17-22, 2017 5 Sep 2018 Physikalisches Institut University of Bonn Bonn, BragaGermany, Portugal Measurement of tW differential cross-sections with ATLAS at √ s = 13 TeV September 6, 2018 September 17-22, 2017 5 Sep 2018Rui Zhang, on behalf of the ATLAS Collaboration 1 1 Copyright [2018] CERN for the benefit of the [ATLAS Collaboration]. CC-BY-4.0 license. The cross-section to produce a W boson in association with a top quark is measured differentially with respect to several particle-level final-state observable quantities. The measurements are performed using 36.1 fb −1 of pp collision data at √ s = 13 TeV collected in 2015 and 2016, by the ATLAS detector at the LHC. Cross-sections are measured in a fiducial phase-space defined by the presence of two charged leptons and exactly one jet identified as containing B hadrons. Measurements are normalised to the fiducial cross-section, causing several of the main uncertainties to cancel. The results are found to be in good agreement with predictions from several Monte Carlo generators.PRESENTED AT 10 th International Workshop on Top Quark Physics Introduction Single-top-quark production via electroweak interactions involving a W tb vertex at leading order is an excellent probe of the W tb couplings. Among all the possible mechanisms, the top-quark production in association with a W boson (tW ) is the second largest process at the LHC. This cross-section has been measured by the ATLAS [1] and CMS [2] collaborations using 13 TeV collision data. These proceedings describe differential cross-section measurements by the ATLAS collaboration in the tW dilepton final state, which explore different kinematic regimes in a more detailed way, and thus will be able to improve Monte Carlo (MC) modelling. Analysis strategy The data correspond to an integrated luminosity of 36.1 fb −1 at √ s = 13 TeV collected by ATLAS [3] in 2015 and 2016. Events are required to have exactly two oppositely charged leptons (henceforth "lepton" refers to an electron or muon) with p T > 27 GeV and p T > 20 GeV, respectively, at least one of which has to be triggered on. Additionally, events are required to have exactly one jet with p T > 25 GeV which is b-tagged (b-jet). Finally a certain amount of missing transverse momentum, E miss T , is required depending on the invariant mass of two leptons, to further reduce background from Z + jets events. A boosted decision tree (BDT) technique [4] is used to combine several observables with increased separation power into a single discriminant. The variables considered are derived from the kinematic properties of subsets of the objects involved in the final states. The BDT discriminant distributions from MC predictions and data are compared and shown in Figure 1. To select a signal-enriched portion of events in the signal region, the BDT response is required to be larger than 0.3. The value of the requirement is optimised to reduce the total uncertainty of the measurement over all bins, considering both statistical and systematic uncertainties. The remained events are corrected for detector acceptance and resolution effects and the efficiency to pass the event selection by using the iterative Bayesian unfolding technique [6] implemented in the RooUnfold software package [7]. The unfolding procedure includes bin-by-bin correction for out-of-fiducial (C oof ) events which are reconstructed but fall outside the fiducial acceptance at particle level, followed by the iterative matrix unfolding procedure M −1 , as well as another bin-by-bin correction (C eff ) to the efficiency to reconstruct a fiducial event: Uncertainty bands reflect the total systematic uncertainties. The first and last bins contain underflow and overflow events, respectively [5]. N ufd i = 1 C eff i j M −1 ij C oof j (N data j − B j ), where N ufd i represents the unfolded event yields, i (j) indicates the bin at particle (reconstruction) level, N data j is the number of events in data and B j is the sum of all background events. Unfolded event yields are converted to cross-section values as a function of an observable X using the expression: dσ i dX = N ufd i L∆ i , where L is the integrated luminosity of the data sample and ∆ i is the width of bin i of the particle-level distribution. Differential cross-sections are divided by the fiducial cross-section to create a normalised distribution. The fiducial cross-section is simply the sum of the cross-sections in each bin multiplied by the corresponding bin widths: σ fid = i dσ i dX · ∆ i = i N ufd i L . Many sources of experimental systematic uncertainties are taken into account. These include the luminosity measurement, lepton efficiency scale factors used to correct simulation to data, lepton/jet energy scale and resolution, E miss T related terms and the efficiency of b-jet. The dominant systematic uncertainties in this category are related to the measurement of the jet energy scale and resolution. Apart from the experimental systematics, uncertainties that arise due to theoretical modelling of the signal and tt background are also evaluated. The dominant uncertainties for this analysis are the next-to-leading order (NLO) matrix element generator and the parton shower and hadronisation generator. Results Differential cross-sections are measured and compared to a variety of theory predictions (see Figure 2) for the following variables: • The uncertainty on the measurements is at the 20 − 50% level. While this does not allow firm conclusions to be drawn, in general, most of the MC models show fair agreement with the measured cross-sections. Notably, for each distribution there is a substantial negative slope in the ratio of predicted to observed cross-sections, indicating there are more events with high-momentum final-state objects than several of the MC models predict. In most cases, differences between the MC predictions are smaller than the uncertainty on the data, but there are some signs that Powheg-Box+Herwig++ deviates more from the data and from the other predictions in certain bins of the E( b), m( b), and m( 1 b) distributions. The predictions of DS and DR samples * likewise give very similar results for all observables as expected from the fiducial selection. The predictions of Powheg-Box+Pythia 6 with varied initial-and final-state radiation tuning were also examined but not found to give significantly different distributions in the fiducial phase space of this analysis. , and E( b). Data points are placed at the horizontal centre of each bin, and the error bars on the data points show the statistical uncertainties. The total uncertainty in the first bin of the m( 1 b) distribution (not shown) is 140% [5]. Figure 1 : 1Comparison of data and MC predictions for the BDT response in the signal region. The tW signal is normalised with the measured fiducial cross-section. the energy of the b-jet, E(b); • the mass of the leading lepton and b-jet, m( 1 b); • the mass of the sub-leading lepton and the b-jet, m( 2 b); • the energy of the system of the two leptons and b-jet, E( b); • the transverse mass of the leptons, b-jet and neutrinos, m T ( ννb); and • the mass of the two leptons and the b-jet, m( b). They are either related to the event, top quark or W boson kinematics. Figure 2 : 2Normalised differential cross-sections unfolded from data, compared with selected MC models, with respect to E(b), m( 1 b), m( 2 b) . arXiv:1612.07231JHEP. 0163hep-exATLAS Collaboration, JHEP 01, 063 (2018), arXiv:1612.07231 [hep-ex]. . JINST. 38003ATLAS Collaboration, 2008 JINST 3, S08003 (2008). * Diagram removal (DR) and diagram subtraction (DS) are two commonly used approaches to deal with quantum interference between tW and tt processes. * Diagram removal (DR) and diagram subtraction (DS) are two commonly used approaches to deal with quantum interference between tW and tt processes. . J H Friedman, Comput. Stat. & Data Analysis. 38367J. H. Friedman, Comput. Stat. & Data Analysis 38, 367 (2002). . arXiv:1712.01602Eur. Phys. J. C. 78186hepexATLAS Collaboration, Eur. Phys. J. C 78, 186 (2018), arXiv:1712.01602 [hep- ex]. . G , Nucl. Instrum. Meth. A. 362487G. D'Agostini, Nucl. Instrum. Meth. A, 362, 487 (1995). . T Adye, arXiv:1105.1160physics.data-anT. Adye, arXiv: 1105.1160 [physics.data-an], (2011).	[]
[ "SPATIAL ANOMALY DETECTION WITH OPTIMAL TRANSPORT", "SPATIAL ANOMALY DETECTION WITH OPTIMAL TRANSPORT" ]	[ "Pranay Seshadri ", "Andrew B Duncan ", "George Thorne ", "Raúl Vázquez Díaz ", "\nImperial College London\nLondonUnited Kingdom\n", "\n‡ Rolls-Royce plc\nDerbyUnited Kingdom\n" ]	[ "Imperial College London\nLondonUnited Kingdom", "‡ Rolls-Royce plc\nDerbyUnited Kingdom" ]	[]	This manuscript outlines an automated anomaly detection framework for jet engines. It is tailored for identifying spatial anomalies in steady-state temperature measurements at various axial stations in an engine. The framework rests upon ideas from optimal transport theory for Gaussian measures which yields analytical solutions for both Wasserstein distances and barycenters. The anomaly detection framework proposed builds upon our prior efforts that view the spatial distribution of temperature as a Gaussian random field. We demonstrate the utility of our approach by training on a dataset from one engine family, and applying them across a fleet of engines-successfully detecting anomalies while avoiding both false positives and false negatives. Although the primary application considered in this paper are the temperature measurements in engines, applications to other internal flows and related thermodynamic quantities are made lucid. Copyright (c) 2022 by Rolls-Royce plc (b) respectively. Build 163's mean and standard deviation are shown in (c) and (d) respectively. Distances between the two cases d in (c); classified anomalies based on τ (NA: not anomalous; A: anomalous) in (d), and radial distribution at 142.0 • in (e). Copyright (c) 2022 by Rolls-Royce plc Page 27The results presented in this paper demonstrate the utility of the proposed framework for capturing distinct spatial anomalies. The methodology is invariant to the specific thermodynamic quantity considered, and can also be adapted to other turbomachinery applications.	null	[ "https://arxiv.org/pdf/2207.06166v1.pdf" ]	250,491,015	2207.06166	6e13971552ede2d6a19592357d163217c7b4336a	SPATIAL ANOMALY DETECTION WITH OPTIMAL TRANSPORT Pranay Seshadri Andrew B Duncan George Thorne Raúl Vázquez Díaz Imperial College London LondonUnited Kingdom ‡ Rolls-Royce plc DerbyUnited Kingdom SPATIAL ANOMALY DETECTION WITH OPTIMAL TRANSPORT This manuscript outlines an automated anomaly detection framework for jet engines. It is tailored for identifying spatial anomalies in steady-state temperature measurements at various axial stations in an engine. The framework rests upon ideas from optimal transport theory for Gaussian measures which yields analytical solutions for both Wasserstein distances and barycenters. The anomaly detection framework proposed builds upon our prior efforts that view the spatial distribution of temperature as a Gaussian random field. We demonstrate the utility of our approach by training on a dataset from one engine family, and applying them across a fleet of engines-successfully detecting anomalies while avoiding both false positives and false negatives. Although the primary application considered in this paper are the temperature measurements in engines, applications to other internal flows and related thermodynamic quantities are made lucid. Copyright (c) 2022 by Rolls-Royce plc (b) respectively. Build 163's mean and standard deviation are shown in (c) and (d) respectively. Distances between the two cases d in (c); classified anomalies based on τ (NA: not anomalous; A: anomalous) in (d), and radial distribution at 142.0 • in (e). Copyright (c) 2022 by Rolls-Royce plc Page 27The results presented in this paper demonstrate the utility of the proposed framework for capturing distinct spatial anomalies. The methodology is invariant to the specific thermodynamic quantity considered, and can also be adapted to other turbomachinery applications. Introduction To discern if an observed measurement is anomalous, one must have some baseline measurement to compare it against. This introduces two requirements. First, a metric that measures the difference-or more generally the distance-between the observed and baseline. Second, a threshold that delineates whether the distance is large enough to be classified as anomalous or not. For instance, if the absolute value of the difference between the observed measurement and the baseline is greater than 2, then it is anomalous. This delineation between an acceptable and anomalous observation is tedious to quantify when there are multiple related observations and consequently multiple related baseline measurements. Such is the case that we consider in this paper. More specifically, we wish to identify spatial anomalies in stagnation temperature sensors in an engine; applications to other thermodynamic quantities, and indeed other internal flow applications are extensions of the present work. The sensors considered in this work are positioned on both rakes and vanes, and thus at a given axial station are functions of radial and circumferential locations (r , θ). Rake placements may differ across engines, as may absolute values of their thermodynamic quantities. Thus, one cannot simply compute the distance between two sets of measurements. Moreover, in an operational engine environment, practitioners want to know if sensors are reporting anomalous values, and if so, which ones. Thus, a scalar distance between an observed and baseline set of measurements, in isolation, will fail to offer necessary information on the precise location of the anomaly and potential cause thereof. Finally, to arrive at such a delineation, data-driven anomaly detection methods [1,2,3,4,5] are seen as the way forward, with the caveat that they require large training repositories. This may be infeasible for certain applications, such as ours, where anomaly detection over a higher granularity of measurements is sought for which training data is limited by virtue of the costs of well-instrumented engine tests. To address these issues, in this paper, we consider the following ideas. 1. It may be beneficial to construct a probabilistic spatial model using each set of measurements independently. In doing so, we have at hand an annular model for observed measurements, and another annular model for the baseline measurements. At the sensor locations, we expect this probabilistic model to have a very small uncertainty-dictated by the measurement apparatus. As we move away from the sensor locations, the model uncertainty will increase based on model assumptions and data availability. 2. Prior to computing any distances, it will be important to normalise the data as different engine tests may have relatively higher or lower values and we would not want the distance to be dominated by the apparent difference in the mean. One way to do this would be to normalise by the area average. That said, this area average should not be based on the sensor positions-which would invariably introduce a bias-but rather based on spatially integrating the aforementioned probabilistic model. 3. It may be prudent to have sufficient granularity for identifying which part of the space the anomaly originates from. Therefore it makes sense to consider multiple anomaly detection tasks. To do this, a vector of distances based on the location of the sensors may be appropriate. This will yield the location of the spatial anomaly, and offer relative comparisons between neighboring measurements. 4. Finally, rather than rely on the availability of a large training repository, it may be sagacious to combine available data with synthetically generated data to boost the overall training repository size. However, the precise manner this synthetic data is to be generated must be carefully considered. We are still left with the matter of selecting a suitable distance and threshold. In terms of the distance, one point to recognise is that we are no longer comparing scalars or vectors: we are contrasting probability distributions. To motivate this departure from existing anomaly detection literature [6], consider the data shown in Figure 1. Subfigure (a) shows two sample measurements taken at distinct non-dimensional radial locations. Each measurement is denoted by a circular maker and the numerous circular overlays reflect the uncertainty in a given measurement. The interpolating curves offer plausible explanations of the data, based on any prior information. In a Bayesian context these curves are referred to as the predictive posterior distribution. In (b) we consider the inclusion of an extra sensor that is not observed in (a). It is clear that some of the interpolating curves in (a) are inconsistent with the predictive posterior distribution in (b), in the sense that they lie in the tails of that distribution. From an anomaly detection perspective, we are interested in the following: if the two sensors in (a) represent a baseline (or gold standard) in measurements, then is the measurement at a non-dimensional radial location of 0.8 in (b) anomalous? If our answer only utilised the mean curve in (a), then we would be inclined to say "no". However, we see that curves which would be plausible under the posterior in (a) would become implausible (i.e., lie in the tails of the predictive posterior distribution) for (b). Taking into account this uncertainty we would therefore consider (a) and (b) to be far away from each other. In Figure 1(c) we illustrate another possible outcome from the sensor's value at 0.8. Now if we assume (b) is the baseline, then we want the difference in the mean to be adequately accounted for. To summarise: in this paper we utilise a probabilistic paradigm for anomaly detection to appropriately account for plausible explanations of the data, which would be consistent with the baseline model and its composite measurements. Even at the extreme when comparing an observed and baseline, each endowed with only one sensor at the same location (r , θ), we are still comparing two probability distributions. This is because the sensor measurement will likely have uncertainty arising from temporal averaging, signal-to-noise filtering, and a variety of thermodynamic calibrations-for converting from volts to Kelvin (or Pascals)-yielding a probability distribution. Thus we restrict our search to distances that can be used to compare probability distributions. Classical statistical metrics for comparing distributions that may be utilised include the total variation distance, the Hellinger norm, and the L 2 (Euclidean) norm. However, these metrics do not consider the underlying space of the distributions. For instance, the distances between two uniform distributions that have similar means is identical to a uniform distribution with a very different one if they all have the same variance. The Kullback-Leibler divergence [7], is possibly a candidate, however it is not a metric in the sense that the distance between a baseline and observed is not equivalent to the distance between the observed and baseline; in other words it is asymmetric 1 . This may introduce additional issues when trying to set a threshold. What we do therefore is to exploit ideas from the field of optimal transport and Bayesian inference to address the the points raised above (point-by-point respectively). We formalise the ideas discussed above as follows. 1. To build a probabilistic spatial model, we use recent ideas in [8,9] that view the thermodynamic quantities at an engine axial plane as a Gaussian random field. In scope, this builds upon prior least-squares based methods [10,11,12]. 2. To negotiate issues pertaining to normalisation, we compute the Bayesian area average as derived in [8]. This is an analytical calculation as it is a linear operator acting over the Gaussian random field. 3. We introduce a vector of one-dimensional Wasserstein distances, where each component of the vector is the distance for a particular sensor location's (r , θ) coordinate. Our choice in adopting the Wasserstein metric is based on its symmetry, its closed-form expressions for Gaussian distributions, and its ability to factor the underlying space of the distributions. 4. For generating synthetic data, we introduce a barycentric interpolation methodology that extends standard Wasserstein geodesics-amendable with only two distributions-to a higher dimensional manifold. This permits us to generate synethetic samples that combine multiple baseline datasets. Following these introductory comments, the structure of this paper is set down. Section 2 offers a cursory overview to Gaussian processes, followed by an overview of the chosen kernel functions and the method for inference. What follows is a condensed form of the fundamentals of optimal transport in section 3, and the spatial anomaly detection framework is detailed in 4 with a specific focus on the distance metric, the threshold selection strategy, and synthetic data generation. Finally, numerical examples of the propose framework at work are given in section 5. Gaussian random field model Consider the spatial distribution of stagnation temperature at an isolated axial measurement plane. We denote this as t (x), where x = (r , θ), with r = {r : 0 ≤ r ≤ 1} and θ = {θ : 0 ≤ θ < 2π} representing the 1 Note that the Kullback-Leibler divergence can be made symmetric via the Jensen Shannon distance. non-dimensional span and circumferential location (in radians) respectively. What underpins our modelling paradigm is that the relationship between x and t has both a systematic and random component. Our overarching objective in this section is to describe the conditional distribution p (t\|x). We assume the existence of a set of m pairwise observations of the stagnation temperature D = (x i , t i ) \| M i=1 , herewith referred to as the training data. This set of data may also be written as D = (X, t) where X = (x 1 , . . . , x M ) that is X ∈ R M ×2 , and t = (t 1 , . . . , t M ) with t ∈ R M . Note that the distribution t (x) cannot be observed directly as individual measurements are corrupted by a variety of noise sources as described above. Mathematically, we assume that this corruption is Gaussian, yielding t (x) ∼ f (x) + N 0, σ 2 where σ is the standard deviation associated with an individual measurement. Extending this across the training data D we write t ∼     f (x 1 ) . . . f (x M )     + N (0, Σ)(1) where Σ = σ 2 I, where I ∈ R M ×M is the identity matrix. In instances where noise correlations between the measurements can be inferred, Σ can be appropriately altered to encode such correlations, and thus need not be restricted to the identity. Given the observed non-dimensional span and circumferential locations, the likelihood of t may be given as p (t\|f ) = N (0, Σ) .(2) We now wish to represent the conditional distribution p (f \|X), which is designed to capture the systematic component of the relationship between x and t. To this end, we define a Gaussian model prior of the form p (f \|X) = N     f     m (x 1 ) . . . m (x M )     ,     k (x 1 , x 1 ) · k (x 1 , x M ) . . . . . . . . . k (x M , x 1 ) · k (x M , x M )         ,(3)p (f \|X) = N (f \|m, K)(4) where m (x) and k (x, x) are chosen mean and covariance functions respectively. Covariance functions are typically parameterised by certain hyperparameters, i.e., k (x, x; ψ), values for which need to inferred based on both the data and the assumed likelihood (noise model). We define them to be ψ ∈ R d and express the model prior as p (f \|X, ψ). Note that these hyperparameters will also have a prior distribution p (ψ), which must be used to evaluate the posterior distribution p (f \|D, ψ). According to Bayes' rule, this is given by p (f \|D, ψ) = p (t\|f ) p (f \|X, ψ) p (D, ψ) = p (t\|f ) p (f \|X, ψ) p (ψ) p (t\|f ) p (f \|X, ψ) p (ψ) d ψ d f .(5) where the denominator is termed the evidence or marginal likelihood; it is essentially a scaling constant. As we assumed a Gaussian likelihood, for a chosen value of ψ * , inference over f the vector of latent values can be derived analytically p (f \|D, ψ) ∝ p (t\|f ) p (f \|X, ψ * ) = N (t\|f , Σ) N (f \|m, K) ∝ N f \|m + Σ -1 K -1 + Σ -1 -1 (t -m) , K -1 + Σ -1 -1 .(6) For simplicity, we set m = 0 and zero-mean the data t. Before detailing how we can use the formulations above for predicting t at testing spatial locations, a few statements on equations (5) and (6) are in order. Typically, when using Bayes' rule, we wish to identify the full posterior distribution p (f \|D, ψ), rather than just its moments or maximum value. To do so, one can utilise well-worn Markov chain Monte Carlo (MCMC) methods that generate samples from the prior distribution p (ψ) to inform the posterior distribution, based on a variety of factors including whether the chosen sample ψ * yields a higher posterior density. Note that in practice ψ, may also be a function of certain other hyerperparameters, in which case priors must be assigned and duly sampled from. As a technicality, it should be noted that each value of the hyperparameters yields a Gaussian random field. From MCMC, we obtain a distribution of values for the hyperparameters, and thus a distribution of Gaussian random fields. This therefore does not yield a posterior Gaussian distribution, but a mixture of Gaussian distributions. Rather than negotiate a mixture of Gaussians, we choose to identify the single value of the hyperparameters that maximises the likelihood, given the data and the priors. This maximum a posteriori (MAP) value yields a posterior Gaussian distribution which as we will see later results in an analytical form of the distance required for anomaly detection. It should be noted that although the MAP will likely offer a reduced estimate of the overall uncertainty, it has the advantage of delivering faster inference, which is necessary for anomaly detection. Once the posterior distribution or its mode has been computed, it can be used to predict the stagnation temperature at other locations. LetX ∈ R N ×2 withX = (x 1 , . . . ,x N ) T be a set of such locations and f ∈ R N the corresponding [unknown] predictive values, i.e.,f = (f (x 1 ) , . . . , f (x N )) T . Evaluating the covariance function at these locations, we defineK ij = k x i ,x j ; ψ and K ij = k x i ,x j ; ψ . The predictive posterior distribution can then be written as p f \|D,X, ψ = p f \|f , X,X, ψ p (f \|D, ψ) d ψ d f = N f f 0, KK K T K N (t\|f , Σ) N (f \|m, K) d ψ d f .(7) Note that the first term on the right hand side in (7) is the joint distribution of the observed f and unobserved f . The covariance matrix is partitioned into four blocks to capture the covariances between the training and testing input locations. Evaluating (7) yields p f \|D,X, ψ = N f \|K T (K + Σ) -1 t, K -K T (K + Σ) -1K ,(8) revealing the predictive posterior mean and predictive posterior covariance. The kernel function used in this paper is adapted from our prior work in [8] and [9] as it was found to capture the type of variability expected in the radial and circumferential profiles. The kernel function is expressed as a product of two kernels, one denoting the kernel in the radial direction k r r, r and another along the circumferential direction k c θ, θ k x, x = k r r, r × k c θ, θ .(9) First, we introduce their circumferential kernel, which comprises a Fourier series kernel which has the form k c θ, θ = F (θ) Λ 2 F (θ) T ,(10) where F (θ) = 1 sin (Ω 1 θ) cos (Ω 1 θ) . . . sin (Ω k θ) cos (Ω k θ) ,(11) with Ω = (Ω 1 , . . . , Ω k ) being the k wave numbers and where Λ 2 ∈ R 2K +1 is a diagonal matrix of hyperparameters, i.e., Λ 2 = diag λ 2 1 , . . . , λ 2 2k +1 , whose values need to be determined. The squared terms are used here to show that the hyperparameters represent the variances associated with either the sine or cosine of each wave number. Note that as λ 2 j → 0, for j = 1, . . . , 2k +1, implies that corresponding mode does not play an important role in the Fourier series expansion. Half normal priors are assigned for the hyperparameters (1); this distribution has the support [0, ∞), and takes in the parameter variance as the argument. Along the radial direction, we use the well-worn squared exponential kernel λ 2 j ∼ N +k r r, r = σ 2 f exp - 1 2l 2 r -r T r -r(12) which is parameterised by two hyperparameters: a kernel noise variance σ 2 f and a kernel length scale l 2 . These are both assigned half Normal priors σ 2 f ∼ N + (1) , l 2 = N + (1) ,(13) with variances set to 1. For convenience we set ψ = λ 2 1 , . . . , λ 2 2k +1 , σ f , l and thus the prior p (ψ) represents 2k + 3 independent half Normal distributions with a variance of 1. Posterior inference via MAP In MAP, the objective is to solve an optimization problem over the space of hyperparameters ψ for identifying the mode of the posterior. This is done via maximise ψ p (t\|f ) p (f \|X, ψ) p (ψ) ,(14) where the terms p (t\|f ) p (f \|X, ψ) come from (5). Whilst this optimisation problem is generally non-convex, its gradients may be computed-either analytically for via any automatic differentiation package-and used for accelerating the optimisation with a standard gradient-based optimiser. All results in this paper use the MAP for parameter inference. Bayesian area average One important consequence of interpreting the spatial distribution of temperature or pressure as a Gaussian random field over an annulus, is that one can derive analytical expressions for linear operators that act over the random field. This is precisely what we introduced in [8], and describe below for convenience. The standard area average for a thermodynamic quantity f (r , θ) at an annular axial plane is written as A (f ) = r o -r i π r 2 o -r 2 i 1 0 2π 0 f (r , θ) v (r ) dr d θ = r o -r i π r 2 o -r 2 i f (x) v (r ) d z(15) where r ∈ [0, 1], θ ∈ [0, 2π) and v (r ) = r (r o -r i ) + r i , with r i and r o being the inner and outer radii of annular section respectively. For convenience we use the coordinates x = (r , θ) as before. One can rewrite (15) as a linear operator acting upon the spatially varying quantity. Now, recall the joint distribution (7) based on an available training dataset D. We can apply the same linear operator across the posterior predictive distribution to arrive at p (A (f ) \|D, ψ) = N    w T (K + Σ) -1 t µ A(p) , ω -w T (K + Σ) -1w σ 2 A(p)     ,(16)wherew = r o -r i π r 2 o -r 2 i k (x, z) v (r ) d z, wherew ∈ R M ×1 , ω = r o -r i π r 2 o -r 2 i 2 k z, z v 2 (r ) d z d z , where ω ∈ R.(17) To clarify, for a given value of the hyperparameters, the Bayesian area average is a Gaussian distribution where the mean and variance can be calculated by plugging in the values of hyperparameters ψ into (15). The Wasserstein distance and optimal transport In this section we introduce the proposed methodology for spatial anomaly detection. The overarching idea is based on the distance between two probability distributions, one which represents a baseline while the other represents an observed sample. We can think of the baseline akin to a gold standard, as it represents an idealised distribution of a quantity-i.e., what we expect the quantity to be. In what follows we define the chosen distance metric and detail our data-driven strategy for classification. Optimal transport is the study of moving a collection of related items from one configuration into another. This movement may entail items from one discrete distribution to another discrete distribution; from one discrete distribution to one continuous distribution, or from one continuous distribution to another continuous one. These collections may include images, graphs, or more generally probability distributions-both discrete or continuous. Optimal transport has recently seen applications in myriad of diverse fields including signal processing, statistical machine learning, computer vision, and medical sciences [13]. It is extremely useful for comparing signals across different coordinate systems and signal families, and thus has naturally seen some application in anomaly detection [14]. Our exposition below closely follows the notation in [15]. To offer a deeper understanding of optimal transport, consider two measures α and β. A measure can be either a discrete or a continuous distribution and it need not integrate to unity. That said, the measure should be integrable against any continuous function and yield a real-valued output as the integral. We assume that the measures α and β are defined over spaces X and V respectively. We further assume that Figure 2: A schematic of optimal transport for two continuous distributions. these spaces are equipped with a distance metric. Mathematically, we state that over the set of Radon measures R (X ) and R (V) we have α ∈ R (X ) and β ∈ R (V). For a given X , let R (X ) + denotes the space of all the positive measures, while R (X ) 1 + denotes the space all positive measures that satisfy X d α = 1 for all α ∈ R (X ) 1 + [15]. There are two key ideas in optimal transport: conservation of mass when transporting elements of α to β, and the ability to split mass when doing so-also termed the Kantorovich relaxation which is particularly suited for discrete measures. Thus, we seek a transport map T that pushes all the mass of α towards the mass β, through which the mass itself may be split. To crystallise the relationship over the two measures, we consider couplings ϕ ∈ R 1 + (X × V) which represent all the joint distributions over the product space of the marginals X × V. The notation ϕ ∈ U [α, β], encodes the mass conservation constraint, i.e., ϕ is uniformly distributed between α and β (see Figure 2). For random samples x ∈ α and v ∈ β, the optimal transport problem is L (α, β) := minimum ϕ ∈ U [α,β] X ×Y c (x, v) d ϕ (x, v) ,(18) as the minimisation of a distance metric subject to a certain cost function c (x, v). If we consider the standard L ρ -norm distance between α and β, then the optimal minimiser, should it exist, is given by the L ρ -th Wasserstein distance W Lρ (α, β) = minimum ϕ ∈ U [α,β] X ×Y x -v ρ Lρ d ϕ (x, v) 1 ρ(19) Optimal transport with multivariate Gaussians A closed form expression for the Wasserstein metric exists when evaluating the distance between two Gaussian distributions with L ρ = 2. Let us define two Gaussian annular random fields (see Figure 3) for a thermodynamic quantity α = N (µ α , Σ α ) and β = N µ β , Σ β , where µ α ∈ R N , µ β ∈ R N , Σ α ∈ R N ×N and Σ β ∈ R N ×N . We also assume that both covariance matrices Σ α and Σ β are symmetric positive definite. By construction, the spaces X and Y are equivalent. The Wasserstein distance (see page 34 in [15]) between them using a quadratic cost, is given by W 2 2 (α, β) := µ α -µ β 2 2 + tr (Σ α ) + tr Σ β -2tr Σ 1/2 α Σ β Σ 1/2 α 1/2 ,(20) where the superscript 1/2 denotes the matrix square root, and the expression · 2 2 denotes the sum of the squares of the argument (·). Note that this is equivalent to the Bures-Wasserstein distance between covariance matrices Σ α and Σ β . One can interpret the movement of probability mass as a pseudo-temporal map g connecting α and β, where g (0) = α and g (1) = β. The map g is parameterised by a scalar time parameter {t : 0 ≤ t ≤ 1} for which g (t) returns the probability mass at time t. The resulting path from t = 0 to t = 1 is called the Wasserstein geodesic, as shown in Figure 4. For a sample ζ 0 from the distribution α, the temporal movement is given by ζ t = (1 -t) ζ 0 + tH,(21) where ζ t is the transported sample, and H is the optimal transport map. For two Gaussian distributions, the optimal transport map is given by As the displacement of each sample in (21) is affine, i.e., H = µ β + R (ζ 0 -µ α ) , with R = Σ -1/2 1 Σ 1/2 α Σ β Σ 1/2 α 1/2 Σ -1/2 α .(22)ζ t = ζ 0 (1 -t + tR) + tµ β -tRµ α ,(23) the distribution for any t is Gaussian with moments given by p (t) = N (1 -t) µ α + tµ β , ((1 -t) I + tR) T Σ α ((1 -t) I + tR) ,(24) where I ∈ R N ×N is the identity matrix. It is straightforward to show that when t = 0 the right hand side of (24) is N (µ α , Σ α ) and setting t = 1 yields N µ β , Σ β . Fusing multiple distributions via a weighted barycenter As comparisons between distributions are inherently done in pairs, it will be useful to ensure that the baseline measurement is a good representation of possibly distinct, yet completely non-anomalous measurements. We utilise ideas within optimal transport as a means to fuse multiple gold standard measurements into a single representative one. The task of computing a representative distribution from a set of distributions is analogous to the idea of computing the centroid via k-means clustering for data (see Remark 9.2 in [15]). For a collection of samples the mean or the barycenter is the minima of a weighted sum of the distances between a candidate and all the samples. To clarify, let α 1 , . . . , α k denote the set of K input distributions for which we wish to compute the barycenter. We define the weighted barycenter α as minimise α ∈ M 1 + (X ) K j =1 ϑ j L α , α j ,(25) with weights ϑ 1 , . . . , ϑ M , where ϑ j ≥ 0 and K j =1 ϑ j = 1. When the weights in (25) are equal, we say that α star is the barycentre of α 1 , . . . , α K . Additionally, note that in the case where X = R and L ρ = 2, then under certain circumstances the barycenter is unique [16]. Computing the barycenter with multivariate Gaussians As shown in Agueh and Carlier [16], for a collection of Gaussian distributions the Wasserstein barycenter is Gaussian N (µ , Σ ) with known mean µ and covariance Σ . Consider a set of K Gaussian distributions α j = N µ j , Σ j for j = 1, . . . , K . The mean and covariance of the barycenter is then given by µ = K j =1 ϑ j µ j , Σ = minimise Σ K j =1 ϑ j tr Σ + Σ j -2 Σ 1/2 Σ j Σ 1/2 1/2 .(26) The weights ϑ j ≥ 0 can be set uniform, i.e., ϑ j = 1/K or they may be chosen to appropriately weight certain distributions; we discuss this salient point in the context of generating synthetic samples later. Whilst the mean in (26) is trivially computed, the covariance requires some clarification. Whilst a closed-form analytical solution is not available, an iterative solution is at hand. Following the fixed point technique in [17], we compute Σ through iterative updates via Σ (i+1) = K j =1 ϑ j Σ (i) 1/2 Σ j Σ (i) 1/2 1/2 ,(27) where the superscript (i ) denotes the present iterate. Whilst no convergence proof for (26) exists, convergence is obtained in practice [17]. It may be useful to also compute the Bayesian area average of the barycenter, which is also a univariate Gaussian. Let N µ A(α i ) , σ 2 A(α i ) , for i = 1, . . . , K(28) denote the Bayesian area average for the K distributions above. We can then write the barycenter's area average as N   K i=1 ϑ i µ A(α i ) , K i=1 ϑ i σ 2 A(α i ) 1/2 2   ;(29) here the variance does not require a fixed point iteration and is analytically solved for. Spatial anomaly detection via optimal transport In this section we make precise our framework for spatial anomaly detection. Following our introductory remarks, this requires a distance metric and a threshold. Distance metric for anomaly detection We begin by formalising the anomaly detection problem. Given S sensors with spatial locationsX = (x 1 , . . . ,x S ) T , we wish to ascertain which sensor(s) is anomalous based on its location and thermodynamic readingsf = f 1 , . . . ,f S T . To do so, we use the Bayesian inference framework in 2 to arrive at an observed Gaussian random field β. We assume that we have also have access to the Gaussian random field arising from some baseline measurements which we hold to be representative of what one would expect. We term this Gaussian random field α. Let the computed area average means for the observed and baseline Gaussian random fields be given by µ A(β) and µ A(α) (computed via (15)). The distance metric we propose is a quintessentially a weighted 1D Wasserstein distance evaluated at the same spatial locations (r, θ) across α and β of the form d = (d 1 , . . . , d S ), where 2 values. We assume that a majority of these Q datasets are standard, however, there are a few anomalous ones included too. d j = µ α x j µ A(α) - µ β x j µ A(β) 2 + Σ α x j ,x j µ 2 A(α) + Σ β x j ,x j µ 2 A(β) - 2 Σ α x j ,x j · Σ β x j ,x j 1/2 µ A(α) µ A(β)(30) Setting the threshold Then we proceed to calculate the 95% percentile value of all the aggregated distances, and use that as our threshold value τ . In other words, if d i ≥ τ then it is likely anomalous. Synthetic data generation via manifold sampling We envisage a shortage of quality training data, and thus offer a recipe to synthetically generate more data. Recall in section 3.1, the Wasserstein geodesic was introduced as an affine transformation between two distributions α and β; parameterised by a pseudo-temporal parameter t. In section 3.2, formulas for computing the barycenter associated with multiple distributions was provided. One can generalise these ideas to a Riemannian simplex [18], which comprises K vertices given by distributions α 1 , . . . , α K and the inner geodesic convex hull, i.e., the space bounded by the simplex edges. To generate samples, we assign ϑ 1 , . . . ϑ K to be Dirchlet distributed random variables, and solve (26). The resulting distributions are guaranteed to fill the Riemannian simplex (see Figure 5 for a schematic). Demonstration on engine data Following a brief overview of the measurements used, we present results of our spatial anomaly detection framework on real engine data. We also illustrate how one can generate synthetic data using the ideas above. 2 The notation Q 2 denotes the number of combinations of Q data sets in pairs without repetitions, i.e., Q!/ (2!(Q -2)!). Figure 5: A schematic of a Riemannian simplex (shaded yellow region) with K = 3 distributions α 1 , α 2 , α 3 . Each vertex represents a Gaussian annular random field where the mean and standard deviations are shown across the annulus. The grey hemisphere is shown to contrast a standard Euclidean simplex with the Riemannian one. Each edge of the Riemannian simplex denotes the geodesic between any two vertices. Measurement and data overview All the data shown here corresponds to steady-state temperature measurements taken at a given engine thrust level. The greater the thrust, the higher the temperatures at each station. We define a single engine test as being a run up (and often down) the power curve, and therefore comprising numerous engine extracts. An extract represents data collected when the engine is effectively adiabatic, i.e., it has had time to stabilise at the required operating thrust. At a given extract, the measurements are obtained by sampling all the thermocouple voltages at 192 kHz with a rolling average running for the last 20 milliseconds. After filtering the signal to remove noise and electrical system artefacts, it is averaged over a 30 second interval at a rate of 33 Hz. Subsequently, these millivolt values are converted into Kelvin via a series of calibrations that cover static, batch-wire and recovery effects. This yields both a mean stagnation temperature for each measurement and an uncertainty. For the same thrust level, at a given measurement station, the uncertainties for all sensors are assumed to be similar. Our dataset is born from stagnation temperature measurements taken from a multitude of engines that had been allocated as development test assets, in which a high level of gas path instrumentation had been provisioned to gain insight into the engine functional behaviour. As a given engine design will have a limited number of these test assets, due to their significant cost, we make use of similarity of the various recent projects to increase our sample size. Here we denote each of these different projects as A, B, C and D. Each engine project will contain numerous of these physical test assets, run over multiple builds where some of the hardware or instrumentation is swapped out to achieve a particular test aim in each build. Each test asset build will likely undergo numerous tests for which data is collected. While the basic architecture of each test asset build, including the number of rotating and stationary components in each sub-system, are the same, there will be variations in the geometry of the constituent components beyond those implied by the manufacturing tolerances to ensure the specific test objectives of each build are achieved. It is also a fact that the exact flow conditions at which the measurements were taken on one test extract will be almost impossible to match on a subsequent test, due to the geometry changes between assets, but also other factors such as the ambient conditions. This makes the task of anomaly detection more challenging, as minor variations are permitted. As we treat each data set independently-regardless of whether they are two tests from the same build or two tests from distinct builds-we do not distinguish between test number or build in what follows. In terms of the measurements themselves, we concern ourselves with high power measurements as this represents airplane cruise conditions. For the purposes of our example, we focus on stations within the engine that were provisioned with gas path instrumentation across several projects and several test assets to provide us with data to train for a threshold and then test this on data not used in the training phase. In this paper we study three different axial stations, aptly named station 1, station 2 and station 3. Extracting thresholds from engine A As a majority of the data we have available is from engine project A, we train exclusively on it and test on the engines projects B, C, D and E. To begin, consider the subfigures in Figure 6 that chart our training workflow. Subfigure (a) shows the radial and circumferentially placed sensors, i.e., D = (x i , t i ) \| M i=1 with M = 27 across the annulus as coloured markers for the data from a test asset build (termed build 348) at station 1, where build 348 is one of the assets for engine project A. The interpolated spatial field here represents the posterior predictive mean of the Gaussian random field. The interpolated spatial field here represents the posterior predictive mean of the Gaussian random field. Subfigure (b) shows the posterior predictive standard deviation; both the mean and standard deviations are evaluated using (7). For these results, we set Ω = (1,2,3,4,5,6,7,8) and σ 2 = 0.04. Similar plots are shown in subfigure (c) and (d) for a second test asset build (termed build 565) at the same instrumentation location in the engine (station 1), which is also for an engine project A test asset using the same values of Ω and σ 2 . In subfigure (e) the values of d are shown across the different spatial locations associated with the sensor positions in (c). It should be clear that these distance values represent a continuous spectrum of possible distances that we wish to threshold via an appropriately chosen scalar parameter τ , i.e., if a given distance d i ≥ τ for any i = 1, . . . , S then it is flagged as anomalous. A similar workflow at station 3 for another pair of measurements from two test assets from engine project A is shown in Figure 7. Here we set Ω = (1,2,3,4,5,6,7,8,9,10,11) and σ 2 = 0.035. One observation we make is that across different engine stations, the distance values are distinct in magnitude warranting a bespoke τ parameter for each station. The wave numbers and noise for station 2 is set to be the same as those set for station 3. We aggregate the distance values obtained from numerous pairwise comparisons for the three axial stations, and plot them as histograms in Figure 8. The 95% percentile value associated with each of the three stations is also shown. For station 1, we set τ = 0.0103, at station 2 τ = 0.032, and for station 3 τ = 0.0184. Demonstrating anomaly detection on engines B, C, D and E Here we demonstrate the utility of our approach on a few test cases-all on different engines families. While distinct, there are similar characteristics across these engines which make them suitable candidates to test the framework, even though the different values for τ were ascertained solely from engine A. We consider two very similar test asset builds (termed build 283 and 278) in Figure 9 for station 1 from engine project B to demonstrate that the threshold chosen is sufficient for not yielding false positives, i.e., it is not overly penalising. Figure 9(d) are classified as anomalous (A), but are classified as not anomalous (NA). Note that this is a binary classification, and the apparent colour gradient in this subfigure should be ignored. The next case studied, also at station 1, is for test asset builds (termed build 270 and 208) from engine project C. Here we have two very distinct builds as is reflected in their spatial mean distributions in Figures 10(a) and (b). It is readily apparent that there is something amiss with the rake at 342 • on build 270. Our spatial anomaly detection approach registers this as an anomaly and also picks up an anomaly on the rake at 306 • . Radial profiles comparing the two predictive posterior distributions associated with the profiles highlight the extent of dissimilarity. Two more such studies are carried out on test asset builds from engine projects D and E in Figures 11 and 12 and demonstrate the ability of the framework to deal with distinct anomalies. In the case of Figures The results above are a snapshot of some of the anomaly detection test cases studied-a selection of a much larger test campaign. While in some cases it is easy to ascertain that a given station has an anomaly via inspection, in most cases it is not. Additionally, there is the time it takes to undertake a manual inspection- having an engineer plot the data for each rake across the distinct measurement stations and comparing the data from an observed engine test to a series of baseline ones-not a matter of minutes. Our framework is fully automated, and as a result drastically reduces the time it takes to identify anomalies. Additionally, it offers a more comprehensive treatment, going beyond the capabilities of a human engineer. Using the barycenter for anomaly detection For completeness, we offer a demonstration of computing the barycenter for multiple distinct distributions at the same plane. Figure 13(a-e) shows the mean (top) and standard deviation (bottom) associated with the posterior predictive distributions for five distinct builds at station 2 for engine A. We compute the barycenter (26) using the fixed point iteration in (27) setting ϑ j = 1/5 for all j . Note that in some cases, a weighted barycenter may be more appropriate, i.e., when assigning certain measurement sets more weight than others. Owing to the computational cost of storing and inverting the covariance matrices, we evaluate the barycenter on a coarser grid compared to the distributions above. The final assimilated result is shown in Figure 13(f). To demonstrate the utility of the barycenter for spatial anomaly detection, we revisit build 163, shown previously in Figure 11. Rather than contrast it with data from another build, here we evaluate our anomaly detection approach using the barycenter, with the same threshold τ determine before. We report the results in Figure 14 and demonstrate that even using the barycenter, the anomalies in build 163 are correctly captured. A valid line of inquiry here is whether a computed barycenter is robust to the inclusion of one or possibly two anomalous data sets. The rationale for this notion is that there may be instances where a given set of measurements may seem non-anomalous until a new build is tested and then compared against. To study this idea, we 1. create a new barycenter for station 2-termed barycenter II-with the five builds in Figure 13(a-e) and build 168, and 2. create another barycenter for station 2-termed barycenter III-with the five aforementioned builds and two instances of build 168. The latter is analogous (but not equivalent) to doubling the barycenter weight ν corresponding to build 168. Then we compute d between these new barycenters and build 168; the results are plotted in Figure 15. While a clear reduction in the Wasserstein distances are observed, especially for barycenter III, the threshold adequately detects the anomalies, giving us some confidence in this approach. Conclusions This manuscript presents an anomaly detection framework for identifying spatial anomalies in stagnation temperature measurements in jet engines. It builds upon prior work on interpreting stagnation temperature measurements at an isolated axial plane as a Gaussian random field. We borrow ideas from optimal transport to define a weighted 1D Wasserstein distance between the same locations across two different engine data sets. When this distance exceeds a certain data-driven threshold at an annular location, we classify the corresponding sensor as anomalous. As the definition of an anomaly rests upon what is considered a baseline or gold standard measurement, we exploit the Wasserstein barycenter to aid in assimilating multiple gold standard measurements. Figure 1 : 1Motivating a probabilistic interpretation of anomaly detection: (a) two sensor measurements; (b) inclusion of an extra sensor; (c) inclusion of an extra sensor with a reduced sensor value. Figure 3 : 3A schematic of two Gaussian annular random fields. Figure 4 : 4A schematic of showing the temporal map between two distributions α and β. for j = 1 , 1. . . , S . There are a few remarks to make regarding the proposed distance metric. First, this metric is a 1D analogue of the Wasserstein distance presented in (20), and by construction provides a distance between the distributions of α and β indexed by each sensor's location. Note that the measurements used to infer β and α can be distinct, as they are based solely on their respective predictive posteriors atX.Second, to mitigate the relatively large penalty imposed by subtracting the square of the means in (20), we normalise all the mean terms by the corresponding area average mean and normalise all the variance terms by the square of the corresponding area average mean. This in practice should facilitate comparisons between engines that have slightly different means without flagging them as anomalous. d has been computed for an observed set of data, a delineation has to be made with regards to whether any of the S sensors are yielding anomalous values. This paper adopts a relatively straightforward data-driven approach to set the threshold. Given a repository of Q data sets, D = D 1 , . . . , D Q , all of the form previously shown, we can evaluate d in (30) for each pair, yielding d 1 , . . . , d ( Q 2 ) Figures 9(a) and (b) show the posterior predictive means for two very similar measurements, with the computed values of d. As none of the distance values exceed the threshold of τ = 0.103 for this station, none of the sensor positions in 11, the anomaly was caused by unwanted coolant leakage flow in build 163; in the case of Figures 12 the culprit was a faulty sensor readings. Figure 6 : 6Computing d for a pair of builds from engine product A station 1: (a) mean of build 348; (b) standard deviation of build 348; (c) mean of build 565; (d) standard deviation of build 565; (e) distances between the two builds d evaluated at the sensor locations of the second build. Figure 7 : 7Computing d for a pair of builds from engine A station 3: (a) mean of build 145; (b) standard deviation of build 145; (c) mean of build 565; (b) standard deviation of build 565; (d) distances between the two builds d evaluated at the sensor locations of the second build. Figure 8 :Figure 9 : 89Aggregated pairwise distance values d across numerous engine data sets for engine stations: (a) 1; (b) 2; (c) 3. The number of pairwise comparisons vary as not every engine test will have instrumentation at a given station. Testing on engine B station 1: (a) mean of build 283; (b) mean of build 278; (c) distances between the two builds d evaluated at the sensor locations of the second build; (d) classified anomalies based on τ (NA: not anomalous; A: anomalous). Figure 10 :Figure 11 :Figure 12 : 101112Testing on engine C, station 1: (a) mean of build 270; (b) mean of build 208; (c) distances between the two builds d evaluated at the sensor locations of the second build; (d) classified anomalies based on τ (NA: not anomalous; A: anomalous); (e) radial distribution at 306 • , and (f) radial distribution at 342 • . Testing on engine D, station 2: (a) mean of build 163; (b) mean of build 574; (c) distances between the two builds d evaluated at the sensor locations of the second build; (d) classified anomalies based on τ (NA: not anomalous; A: anomalous); (e) circumferential distribution at a span of 0.74, and (f) circumferential distribution at a span of 0.87. Testing on engine E, station 3: (a) mean of build 391; (b) mean of build 180; (c) distances between the two builds d evaluated at the sensor locations of the second build; (d) classified anomalies based on τ (NA: not anomalous; A: anomalous); (e) radial distribution at 164.3 • . Figure 13 : 13Mean and standard deviations in the predictive posterior distribution for five different builds at station 2 in(a, b, c, d, e). The barycenter is shown in (f). Figure 14 : 14Anomaly detection with the barycenter with mean and standard deviation in (a) and (b) respectively. Build 163's mean and standard deviation are shown in (c) and (d) respectively. Distances between the two cases d in (c); classified anomalies based on τ (NA: not anomalous; A: anomalous) in (d), and radial distribution at 142.0 • in (e). Figure 15 : 15Anomaly detection with barycenters II and III, with mean and standard deviation in (a) and AcknowledgementsThe work was part funded by the Fan and Nacelle Future Aerodynamic Research (FANFARE) project under grant number 113286, which receives UK national funding through the Aerospace Technology Institute (ATI) and Innovate UK together with Rolls-Royce plc. The authors are grateful to Rolls-Royce plc for permission to publish this paper. 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[ "COVID-19'S (MIS)INFORMATION ECOSYSTEM ON TWITTER HOW PARTISANSHIP BOOSTS THE SPREAD OF CONSPIRACY NARRATIVES ON GERMAN SPEAKING TWITTER", "COVID-19'S (MIS)INFORMATION ECOSYSTEM ON TWITTER HOW PARTISANSHIP BOOSTS THE SPREAD OF CONSPIRACY NARRATIVES ON GERMAN SPEAKING TWITTER" ]	[ "Morteza Shahrezaye [email protected] \nInstitute for Media and Communications Management\nInstitute for Media and Communications Management\nUniversity of St\nGallen\n", "Miriam Meckel [email protected] \nUniversity of St\nGallen\n", "Léa Steinacker [email protected] \nInstitute for Media and Communications Management\nUniversity of St\nGallen\n", "Viktor Suter [email protected] \nUniversity of St\nGallen\n" ]	[ "Institute for Media and Communications Management\nInstitute for Media and Communications Management\nUniversity of St\nGallen", "University of St\nGallen", "Institute for Media and Communications Management\nUniversity of St\nGallen", "University of St\nGallen" ]	[]	In late 2019, the gravest pandemic in a century began spreading across the world. A state of uncertainty related to what has become known as SARS-CoV-2 has since fueled conspiracy narratives on social media about the origin, transmission and medical treatment of and vaccination against the resulting disease, COVID-19. Using social media intelligence to monitor and understand the proliferation of conspiracy narratives is one way to analyze the distribution of misinformation on the pandemic. We analyzed more than 9.5M German language tweets about COVID-19. The results show that only about 0.6% of all those tweets deal with conspiracy theory narratives. We also found that the political orientation of users correlates with the volume of content users contribute to the dissemination of conspiracy narratives, implying that partisan communicators have a higher motivation to take part in conspiratorial discussions on Twitter. Finally, we showed that contrary to other studies, automated accounts do not significantly influence the spread of misinformation in the German speaking Twitter sphere. They only represent about 1.31% of all conspiracy-related activities in our database. 2020; Frank et al.] have left many people to uncertainty and fear of further developments. Previous research has shown that lack of certainty and control often results in the emergence and circulation of conspiracy theory narratives[Whitson and Galinsky, 2008]. Popper defined conspiracy mentality as the "mistaken theory that, whatever happens in societyespecially happenings such as war, unemployment, poverty, shortages, which people as a rule dislike -is the result of direct design by some powerful individuals and groups"[Popper, 2002]. This one-sided or even pathological method of reasoning regularly facilitates coping with uncertainty and fear by making the world more understandable and providing individuals with an illusion of control[Kruglanski et al., 2006].	10.1007/978-3-030-73100-7_73	[ "https://arxiv.org/pdf/2009.12905v1.pdf" ]	221,970,124	2009.12905	0e993227ec2af143e90e6b7c75df77732105957c	COVID-19'S (MIS)INFORMATION ECOSYSTEM ON TWITTER HOW PARTISANSHIP BOOSTS THE SPREAD OF CONSPIRACY NARRATIVES ON GERMAN SPEAKING TWITTER Morteza Shahrezaye [email protected] Institute for Media and Communications Management Institute for Media and Communications Management University of St Gallen Miriam Meckel [email protected] University of St Gallen Léa Steinacker [email protected] Institute for Media and Communications Management University of St Gallen Viktor Suter [email protected] University of St Gallen COVID-19'S (MIS)INFORMATION ECOSYSTEM ON TWITTER HOW PARTISANSHIP BOOSTS THE SPREAD OF CONSPIRACY NARRATIVES ON GERMAN SPEAKING TWITTER social networks · misinformation · conspiracy theory · political polarization In late 2019, the gravest pandemic in a century began spreading across the world. A state of uncertainty related to what has become known as SARS-CoV-2 has since fueled conspiracy narratives on social media about the origin, transmission and medical treatment of and vaccination against the resulting disease, COVID-19. Using social media intelligence to monitor and understand the proliferation of conspiracy narratives is one way to analyze the distribution of misinformation on the pandemic. We analyzed more than 9.5M German language tweets about COVID-19. The results show that only about 0.6% of all those tweets deal with conspiracy theory narratives. We also found that the political orientation of users correlates with the volume of content users contribute to the dissemination of conspiracy narratives, implying that partisan communicators have a higher motivation to take part in conspiratorial discussions on Twitter. Finally, we showed that contrary to other studies, automated accounts do not significantly influence the spread of misinformation in the German speaking Twitter sphere. They only represent about 1.31% of all conspiracy-related activities in our database. 2020; Frank et al.] have left many people to uncertainty and fear of further developments. Previous research has shown that lack of certainty and control often results in the emergence and circulation of conspiracy theory narratives[Whitson and Galinsky, 2008]. Popper defined conspiracy mentality as the "mistaken theory that, whatever happens in societyespecially happenings such as war, unemployment, poverty, shortages, which people as a rule dislike -is the result of direct design by some powerful individuals and groups"[Popper, 2002]. This one-sided or even pathological method of reasoning regularly facilitates coping with uncertainty and fear by making the world more understandable and providing individuals with an illusion of control[Kruglanski et al., 2006]. Introduction In November 2019, a febrile respiratory illness caused by SARS-CoV-2 infected people in the city of Wuhan, China. On January 30th 2020, the World Health Organization (WHO) declared the spread of the virus a worldwide pandemic [BBC, 2020]. Shortly after, the WHO reported multiple COVID-19-related knowledge gaps relating to its origin, transmission, vaccinations, clinical considerations, and concerns regarding the safety of healthcare workers [WHO, 2020a]. The organization warned of an "infodemic", defined by "an overabundance of information and the rapid spread of misleading or fabricated news, images, and videos" [WHO, 2020b]. By August 2020, more than 22 million people worldwide had contracted the virus [WHO, 2020c]. The Organization for Economic Co-operation and Development (OECD) put forward estimates of negative GDP growth for all member countries in 2020 due to the crisis [OECD, 2020]. COVID-19's indomitable dissemination around the globe combined with a lack of effective medical remedies [Guo et al., 2020;Xie et al., 2020] and its psychological and economic side effects [OECD, 2020;Ho et al., 2020;Rajkumar, There are two main conditions conducive to the emergence of conspiracy narratives: individuals' psychological traits and socio-political factors. Regarding psychological traits, numerous laboratory studies demonstrate the correlation between conspiracy beliefs and psychological features like negative attitude toward authorities [Imhoff and Bruder, 2014], self-esteem [Abalakina-Paap et al., 1999], paranoia and threat [Mancosu et al., 2017], powerlessness [Abalakina-Paap et al., 1999], education, gender and age [van Prooijen, 2017], level of agreeableness [Swami et al., 2011], and death-related anxiety [Newheiser et al., 2011]. Another part of reasoning sees conspiracy mentality as a generalized political attitude [Imhoff and Bruder, 2014] and correlates conspiracy beliefs to socio-political factors like political orientation. Enders et al. showed that conspiracy beliefs can be a product of partisanship [Enders et al., 2020]. Several other studies show a quadratic correlation between partisanship and the belief in certain conspiracy theories [van Prooijen et al., 2015]. These insights imply that extremists on both sides of the political spectrum are more prone to believe in and to discuss conspiracy narratives. We define conspiracy narratives as part of the overall phenomenon of misinformation on the internet. We use misinformation as the broader concept of fake or inaccurate information that is not necessarily intentionally produced (distinguished from disinformation which is regularly based on the intention to mislead the recipients). Among all the conspiracy narratives, we are interested in those propagated in times of pandemic crises. The spread of health-related conspiracy theories is not a new phenomenon [Geissler and Sprinkle, 2013;Bogart et al., 2010;Klofstad et al., 2019] but seems to be even accelerated in world connected via social media. The COVID-19 pandemic's unknown features, its psychological and economical side effects, the ubiquitous availability of Online Social Networks (OSNs) [Pew, 2019], and high levels of political polarization in many countries [Fletcher et al., 2020;Yang et al., 2016] make this pandemic a potential breeding ground for the spread of conspiracy narratives. From the outset of the crisis, "misleading rumors and conspiracy theories about the origin circulated the globe paired with fear-mongering, racism, and the mass purchase of face masks [...]. The social media panic travelled faster than the COVID-19 spread" [Depoux et al., 2020]. Such conspiracy narratives can obstruct the efforts to properly inform the general public via medical and scientific findings [Grimes, 2016]. Therefore, investigating the origins and circulation of conspiracy narratives as well as the potential political motives supporting their spread on OSNs is of vital public relevance. With this objective, we analyzed more than 9.5M German language tweets about COVID-19 to answer the following research questions: Research Question 1: What volume of German speaking Twitter activities comprises COVID-19 conspiracy discussions and how much of this content is removed from Twitter? Research Question 2: Does the engagement with COVID-19 conspiracy narratives on Twitter correlate with political orientation of users? Research Question 3: To what degree do automated accounts contribute to the circulation of conspiracy narratives in the German speaking Twitter sphere? Data We collected the data for this study during the early phase of the crisis, namely, between March 11th, the day on which the WHO declared the spread of the SARS-CoV-2 virus a pandemic [BBC, 2020] and May 31st, 2020. The data was downloaded using the Twitter's Streaming API by looking for the following keywords: "COVID", "COVID-19", "corona", and "coronavirus". Only Tweets posted by German speaking users or with German language were included. The final dataset comprises more than 9.5M tweets from which two categories of conspiracy narratives were selected: conspiracy narratives about the origin of the COVID-19 illness (Table 1) and those about its potential treatments ( Table 2). The conspiracy narratives about the origin of the COVID-19 illness were selected based on Shahsavari et al., who automatically detected the significant circulation of the underlying conspiracy theories on Twitter using machine learning methods [Shahsavari et al., 2020]. The second group of conspiracy narratives were chosen based on the fact that they were in the center of attention in German media [Tagesschau, 2020] and thus a considerable number of tweets discussed them [Netzpolitik, 2020]. Table 3 indicates the number of tweets belonging to each conspiracy narrative and the keywords that are used to filter them out 1 . There were 68,466 tweets in total discussing the underlying conspiracy narratives. Figure 1 shows the timeline of the tweets. In addition to the six conspiracy narratives, 9000 tweets were randomly extracted from the dataset and served as a control group. To answer research question 2, a list was extracted from official party websites; this list contains members of parliament (MPs) who are active on Twitter and belong to one of the six political parties in Germany's federal legislature. Each party runs several official Twitter pages that were added to the list of Twitter pages of each political party; for example, the official Twitter page of the Social Democratic Party (SPD) in the federal state of Bavaria, called "BayernSPD", was added to the SPD list. For each twitter account in the extracted list a maximum of 4000 tweet handles were downloaded from the Twitter API. Table 4 shows the relevant statistics on the political tweets. In the next step, for each of the 68,466 users spreading conspiracy narratives ( Table 3) the lists of their tweet handles were downloaded (Table 5). Finally, for each of them we counted the number of times they retweeted one of the political tweets in table 4. Based on Boyd et al. retweets are mainly a form of endorsement [Boyd et al., 2010]. Therefore, we assume if a user collects a discernible number of retweets from members of a certain political party, this user will most likely share the corresponding political orientation. This method of inference about the political orientation of users has been applied in similar studies [Garimella et al., 2017]. Results There are multiple studies showing that exposure to misinformation can lead to persistent negative effects on citizens. The respondents in a study adjusted their judgment proportional to their cognitive ability after they realized that their initial evaluation was based on inaccurate information. In other words, respondents with lower levels of cognitive ability tend to keep biased judgments even after exposure to the truth [Keersmaecker and Roets, 2017]. In another study, Tangherlini et al. found that conspiracy narratives stabilize based on the alignment of various narratives, domains, people, and places such that the removal of one or some of these entities would cause the conspiracy narrative to quickly fall apart [Tangherlini et al., 2020]. Imhoff and Lamberty have shown that believing COVID-19 to be a hoax negatively correlated with compliance with self-reported, infection-reducing, containment-related behavior [Imhoff and Lamberty, 2020]. On that account, to assess a democratic information ecosystem that is balanced rather towards reliable information than misinformation we need to monitor and estimate if COVID-19 conspiracy theory narratives circulate significantly on Twitter. Based on a survey in mid-March 2020, about 48% of respondents stated that they have seen some pieces of likely misinformation about COVID-19[Pew, 2020]. Shahsavari et al. used automated machine learning methods to automatically detect COVID-19 conspiracy narratives on Reddit, 4Chan, and news data [Shahsavari et al., 2020]. Multiple other studies found evidence of COVID-19 misinformation spread on different OSNs [Boberg et al., 2020;Ahmed et al., 2020;Serrano et al., 2020]. To address the public concerns many of the service providers claimed that they will remove or tag this sort of content on their platforms. On March 16th 2020, Facebook, Microsoft, Google, Twitter and Reddit said they are teaming up to combat COVID-19 misinformation on their platforms [Bloomberg, 2020]. On April 22nd, Twitter stated that they have removed over 2230 tweets containing misleading and potentially harmful COVID-19-related content [Twitter, 2020]. On June 7th 2020, we examined how many of the German conspiracy-related tweets still exist on Twitter in order to understand if conspiracy-related tweets tend to exist on Twitter for a longer period of time compared to non conspiracy-related tweets. Table 6 shows the results. Research Question 1 Based on Table 6, only about 0.61% of all COVID-19 German tweets are about one of the conspiracy narratives under consideration. These German tweets are posted by more than 36,000 unique Twitter users. While 0.61% is small in magnitude, it still comprises a relevant number of citizens. It is important to note though that this finding does not imply that only about 36,000 Twitter users believe in conspiracy theories. While our data shows the spread of conspiracy narratives, they do not reveal a user's stance towards the respective content. In terms of content moderation by Twitter, on average 7.3% of conspiracy narrative tweets are deleted after a certain period of time which is significantly higher than 6% of tweets in the control group. We speculate that more of the conspiracy-related tweets are deleted because of Twitter's content moderation efforts that have been enforced due to recent public debates about misinformation on OSNs. Research Question 2 There is a long list of laboratory studies that show a correlation between conspiracy mentality and extreme political orientation [Enders et al., 2020;van Prooijen et al., 2015]. In this study we answer the slightly different question if the partisanship of Twitter users correlates with their contribution to conspiracy theory narrative discussions. Table 7 shows the distribution of the political orientations of users who discuss each of the underlying conspiracy narratives. Table 7 demonstrates that users who are likely to be supporters of AfD and SPD most actively discuss and spread COVID-related conspiracy narratives on Twitter. To check if contributions to conspiracy narratives are correlated with the political orientation of users, we ran a saturated Poisson log-linear model on the contingency Table 7. The model defines the counts as independent observations of a Poisson random variable and includes the linear combination and the interaction between conspiracy narratives and the political orientation of users [Agresti, 2003]. log(µ ij ) = λ + λ N i + λ P j + λ N P ij (1) where µ ij = E(n ij ) represents the expected counts, λs are parameters to be estimated and N and P stand for Narrative and Political Orientation. λ N P ij s corresponds to the interaction and association between conspiracy narratives and also reflects the departure from independence [Agresti, 2003]. Since we suspect that beliefs in certain mutually contradictory conspiracy theories can be positively correlated [Wood et al., 2012], we aggregated the six conspiracy theory cases to two based on to which category they belong and formed Table 8 to remove any possible correlation. Table 9 shows the ANOVA analysis of the underlying saturated Poisson log-linear model applied on Table 8. The last line of resulting p-values in Table 9 shows that in interaction parameter, µ ij = E(n ij ), is statistically significant. Therefore, we can reject the hypothesis that the contribution to conspiracy narratives is independent of the political orientation of users. The fact that there is evidence of a correlation between the contribution to conspiracy narratives and the political orientation of users, however, does not imply any causality. To further estimate the relative effect of political orientation on the contribution to conspiracy narratives on Twitter, we applied six Chi-Square goodness of fit tests on the control group and each of the other six conspiracy narratives. For all of the six tests the p-values were significantly less than 0.05, which suggests that the distributions of the contribution to the six different conspiracy narratives are statistically different compared to the control group. Figure 2 shows the distribution of the tests' residuals. The last column of Figure 2 shows that the Twitter users without a certain political orientation contributed relatively less to conspiracy narratives in comparison to the control group. In other words, compared to the control group, users with certain political orientations contributed more to the circulation of conspiracy narratives. Research Question 3 Automated accounts, or users who post programmatically, make up a significant amount of between 9% and 15% of Twitter users worldwide [Davis et al., 2016]. Multiple studies hold automated accounts responsible for political manipulation and undue influence on the political agenda [Shao et al., 2017;. However, more recent studies shed light on these previous results and showed that the influence of automated accounts is overestimated. Ferrara finds that automated accounts comprise less than 10% of users who post generally about COVID-19 [Ferrara, 2020]. There are multiple methods to automatically detect automated accounts on OSNs [Alothali et al., 2018]. For this study, we used the method developed by . They applied random forest classification trees on more than a thousand public meta-data available using the Twitter API and on other human engineered features. Table Figure 2: Distribution of residuals of Chi-Square goodness of fit tests 10 displays the percentage of automated accounts (users with Complete Automation Probability higher than 0.5) and verified users who contribute to conspiracy narratives. Based on this analysis, 1.31% of COVID-19 conspiracy narrative tweets are suspected to be posted by automated accounts. This number is significantly lower than many other studies on bot activities on Twitter. We speculate that this occurs due to three reasons. First, the importance of the topic might have captured a lot of public attention, so that significantly more users discuss COVID-19-related topics compared to usual Twitter discussions. Second, many service providers, including Twitter, have started to combat COVID-19 misinformation because of widespread warnings. Finally, we have concentrated on German tweets while the past estimates apply to tweets in English. Discussions and limitations In this study we analyzed more than 9.5M German language tweets and showed that the volume of tweets that discuss one of the six considered conspiracy narratives represents about 0.6% of all COVID-19 tweets. This translates to more than 36,000 unique German speaking Twitter users. Imhoff and Lamberty found that "believing that COVID-19 was a hoax was a strong negative prediction of containment-related behaviors like hand washing and keeping physical distance". To provide the public with accurate information about the importance of such measures, social media intelligence can help elevate potential pitfalls of the Twitter information ecosystem. Using more than 38,000 tweets and 36,000 unique Twitter users, we formed the contingency table of political orientation and of contribution to COVID-19 conspiracy narratives (Table 8). We then applied a saturated Poisson log-linear regression and showed that we cannot statistically reject independence among the underlying variables. This implies partisans have a higher motivation for taking part in COVID-19-related conspiracy discussions. This shows that politically polarized citizens increase the spread of health misinformation on Twitter. Finally, we employed an automated accounts detection tool and showed that on average about 1.31% of the users who discuss COVID-19 conspiracy narratives are potentially automated accounts or bots. This number is much lower than estimations on general bot activity on Twitter, which is assumed to be up to 15% [Davis et al., 2016;. This study holds new insights as well as some limitations: • Our results shed light on the problem of misinformation on Twitter in times of crises for a certain cultural and language context: Germany. We showed that the political orientation of politically polarized users translates to higher circulation of health-related conspiracy narratives on Twitter. Further research could compare the results of this study with other countries and language realms on Twitter. • We also offer indications between political or ideological partisanship and engagement in the dissemination of misinformation on Twitter. In this study we examined if political partisanship motivates individuals to take part in conspiracy discussions. In other words, we did not distinguish between tweets promoting the conspiracy narratives and those rejecting them. One could extend the analysis and study the effect of partisanship on promoting conspiracy theories. Further research will also need to combine quantitative data analysis and qualitative content analysis to better understand the underlying motivations for engaging in conspiracy communication on OSNs. • Finally, we offer a more nuanced view on the role of automated tweets regarding a highly emotionally-charged topic. There are numerous studies showing contradictory estimates of bot activity on OSNs. We found only about 1.31% of users who spread COVID-19 conspiracy tweets are potentially bots. This number is much lower than many of those put forward by other researchers. Further research could investigate this result in order to understand the reasons why this estimation is lower than other case studies. Figure 1 : 1Daily number of tweets for each conspiracy narrative Table 1 : 1Conspiracy narratives about the origin of COVID-19case description 5G conspiracy narrative suggesting that the 5G network activates the virus Bill Gates conspiracy narrative suggesting that Bill Gates aims to use COVID-19 to initiate a global surveillance regime Wuhan laboratory conspiracy theory narrative suggesting that the virus originates from a laboratory in Wuhan, China Table 2 : 2Conspiracy narratives about potential treatments of COVID-19 case description Ibuprofen conspiracy narrative suggesting that Ibuprofen reduces COVID-19 symptoms Homoeopathy conspiracy narrative suggesting that homeopathy medicines reduce COVID-19 symp- toms Malaria conspiracy narrative suggesting that a malaria drug is an antiviral against SARS-CoV-2 virus Table 3 : 3Number of tweets for each conspiracy narrative case number of tweets keywords 5G 5,762 5G, #5g Bill Gates 24,653 Bill Gates, #billgates Wuhan laboratory 9,366 #wuhanlab Wuhan Lab Ibuprofen 7,016 Ibuprofen, #ibuprofen Homeopathy 4,714 Homöopath, #Homöopath Malaria 7,955 Malaria, #malaria Control group 9,000 __ Table 4 : 4Number of political tweets extracted from politicians' twitter pages political party number of MPs on Twitter number of extra official Twitter pagestotal number of tweet han- dles Table 5 : 5Number of tweets extracted from users spreading conspiracy narrative tweetscase number of tweets number of downloaded tweets from the contributing users 5G 5762 10,967,158 Bill Gates 24653 35,144,536 Wuhan laboratory 9366 14,332,403 Ibuprofen 7016 14,855,267 Homoeopathy 4714 7,746,555 Malaria 7955 16,258,164 Control group 9000 12,217,082 Table 6 : 6Share of conspiracy narratives among all COVID-19 tweets case number of tweets share (among all 9.5M COVID-19 tweets) share deleted on 7th June 2020 5G 5762 0.06% 6% Bill Gates 24653 0.25% 7% Wuhan laboratory 9366 0.098% 9% Ibuprofen 7016 0.073% 14% Homoeopathy 4714 0.049% 3% Malaria 7955 0.083% 5% Control group 9000 0.094% 6% Table 7 : 7Political orientation of users discussing conspiracy narratives case AfD CDU/ CSU FDP Bündnis 90/Die Grünen Linke SPD Unknown 5G 11% 3% 3% 8% 10% 14% 51% Bill Gates 16% 3% 3% 8% 12% 14% 44% Wuhan lab- oratory 27% 3% 15% 7% 5% 8% 35% Ibuprofen 9% 3% 3% 8% 9% 16% 52% Homoeopathy 5% 4% 6% 13% 15% 24% 33% Malaria 10% 2% 2% 5% 6% 11% 64% Control group 10% 2% 2% 4% 6% 10% 66% Table 8 : 8Political orientation of users who discuss two conspiracy narratives (absolute counts)case AfD CDU/ CSU FDP Bündnis 90/Die Grünen Linke SPD Unknown Origins of COVID-19 4133 694 1396 1873 2449 3028 10296 Possible treatments of COVID- 19 1263 432 497 1149 1347 2330 7841 Table 9 : 9ANOVA of Poisson log-linear model on the contingency Table 7 Df Deviance Resid. Df Resid. Dev Pr(>Chi) NULL 13 31332.82 narrative 1 2115.49 12 29217.32 0.0000 party 6 28294.31 6 923.02 0.0000 narrative:party 6 923.02 0 0.00 0.0000 Table 10 : 10Ratio of tweets posted by automated and verified users case share of tweets posted by ver- ified users share of tweets posted by au- tomated accounts Ratio of automated accounts to verified users 5G 3.141% 1.578% 0.5 Bill Gates 1.85% 1.358% 0.73 Wuhan laboratory 9.065% 1.3% 0.14 Ibuprofen 3.349% 1.386% 0.41 Homoeopathy 1.039% 0.921% 0.89 Malaria 4.626% 1.343% 0.29 Control group 4.644% 0.89% 0.19 We used "homöopath" in order to match both German words "homöopathie" and "homöopathisch" . Oecd, Oecd Economic, Outlook, 10.1787/7969896b-enReportOECD. OECD Economic Outlook, Interim Report March 2020. 2020. doi:https://doi.org/https://doi.org/10.1787/7969896b-en. URL https://www.oecd-ilibrary.org/content/ publication/7969896b-en. 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[ "Development of a finite element based haptic interactive surgery simulation from Computer Tomography data", "Development of a finite element based haptic interactive surgery simulation from Computer Tomography data" ]	[ "Kaustav Bora [email protected] \nDepartment of Mechanical Engineering\nIIT Kharagpur\nWest BengalIndia\n", "Adarsh Mishra \nDepartment of Mechanical Engineering\nIIT Kharagpur\nWest BengalIndia\n", "C S Kumar [email protected]. \nDepartment of Mechanical Engineering\nIIT Kharagpur\nWest BengalIndia\n" ]	[ "Department of Mechanical Engineering\nIIT Kharagpur\nWest BengalIndia", "Department of Mechanical Engineering\nIIT Kharagpur\nWest BengalIndia", "Department of Mechanical Engineering\nIIT Kharagpur\nWest BengalIndia" ]	[]	Virtual models are important for training and teaching tools used in medical imaging research. We introduce a workflow that can be used to convert volumetric medical imaging data (as generated by Computer Tomography (CT)) to computer-based models where we can perform interaction of tool with the tissue. This process is broken up into two steps: image segmentation and tool-tissue interaction. We demonstrate the utility of this streamlined workflow by creating models of a liver. A FE model for probe insertion has been developed using cohesive elements to simulate the tissue rupture phenomena. FE based simulations are performed and the results are compared with that of various published papers. An analytical model that governs the reaction forces on the needle tip has also been developed. Expressions for reaction forces acting in both axial and transverse directions on a symmetric tip needle and a bevel tip needle are developed. These models consider local tissue deformation by the needle and the frictional forces generated due to the inclusion of the needle in the tissue, but do not consider the role of tissue rupture toughness parameter due to which some differences are visible in FE based simulations and these analytical results.	null	[ "https://arxiv.org/pdf/2201.12520v1.pdf" ]	246,430,370	2201.12520	e944e33f428e4cda72286a32ef3ebe8e2118ed11	Development of a finite element based haptic interactive surgery simulation from Computer Tomography data Kaustav Bora [email protected] Department of Mechanical Engineering IIT Kharagpur West BengalIndia Adarsh Mishra Department of Mechanical Engineering IIT Kharagpur West BengalIndia C S Kumar [email protected]. Department of Mechanical Engineering IIT Kharagpur West BengalIndia Development of a finite element based haptic interactive surgery simulation from Computer Tomography data 10.3390/xxxxxBioengineering 2022, 9, x. https://doi.org/10.3390/xxxxx ArticleComputer Tomographyinsertion modellingfinite element modeltissue ruptureim- age segmentation Virtual models are important for training and teaching tools used in medical imaging research. We introduce a workflow that can be used to convert volumetric medical imaging data (as generated by Computer Tomography (CT)) to computer-based models where we can perform interaction of tool with the tissue. This process is broken up into two steps: image segmentation and tool-tissue interaction. We demonstrate the utility of this streamlined workflow by creating models of a liver. A FE model for probe insertion has been developed using cohesive elements to simulate the tissue rupture phenomena. FE based simulations are performed and the results are compared with that of various published papers. An analytical model that governs the reaction forces on the needle tip has also been developed. Expressions for reaction forces acting in both axial and transverse directions on a symmetric tip needle and a bevel tip needle are developed. These models consider local tissue deformation by the needle and the frictional forces generated due to the inclusion of the needle in the tissue, but do not consider the role of tissue rupture toughness parameter due to which some differences are visible in FE based simulations and these analytical results. Introduction Majority of surgical procedures involve tissue rupture and damage by a needle, scissors or blade. Hence accurate modelling of these invasive needle-tissue interactions is very important for development of a surgical simulator. Modelling and simulation of invasive surgical procedures are complex due to tissue rupture/damage, presence of friction, and varying boundary conditions. Linear elasticity-based FE models are the most predominate technique to model these invasive surgical procedures as well. DiMaio et al. [15] developed a linear elastic FE based 2D needle insertion model, where phantom tissue was used to acquire tissue properties. They used condensed FE technique during pre-processing to achieve real-time haptic feedback. Alterovitz et al. [32] used this model to simulate needle steering and Goksel et al. [33] extended this model for 3D simulations. Due to high computation cost associated with FE based models for invasive surgical procedures, and also, the difficulties associated with the characterization of non-linear behavior tissue during rupture, only a few studies have used non-linear model for simulating soft-tissue. Nienhuys and van der Stappen [34] used the Neo-Hookean model for simulating needle insertion in a three-dimensional model, but they have not provided any comparison of the simulation results with any experimental data. Picinbono et al. [35] have used a nonlinear isotropic model to simulate cutting of liver, but no validation or comparison for the simulation results was provided. Analytical Modelling of Needle-Tissue Interactions: An ideal mechanics-based model of the forces acting on needle tip would require input information about needle geometries and tissue material properties. Such a model could be used in surgical simulators for real-time haptic feedback or forces for given needle displacement. Nienhuys et al. [34] have presented a mechanics-based computational technique for estimating the reaction forces on the needle by a tissue. Wei Dong et al. [36] has developed a mechanics-based model of needle tip when it is inserted and rotated into a soft tissue. They have attempted to develop a 3D model of needle-tissue interaction, but they have not provided any experimental results for validation of their model. Okamura et al. [19] developed a mechanics-based model to predict needle behavior using mechanical properties of tissue and geometrical properties of needle. They presented an analytical model for the load developed at needle tip based on its geometry and material properties of tissue, guided by microscopic observation of needle-tissue interactions. Also, an analytical model for calculation of deflection of needle tip in transverse direction is developed using energy-based approach. They have also validated their model with experimental observations. Materials and Methods Modelling Tissue Failure/Rupture: Within the context of finite element modelling, the 'energy of fracture' approach is a very effective method of incorporating failure of an otherwise continuous mesh. In finite element software like ABAQUS, cohesive elements implement this concept through a fracture toughness parameter which is analogous to the strain energy release rate. In 2D simulations, the cohesive elements are four-node with two 'active' faces. The traction-separation relationship between these two faces of the element determines whether or not the cohesive element is intact or, having failed, is removed from the simulation. To incorporate elements of zero thickness, constitutive thickness parameter (Tc) is included in the element formulation linking strain ( ) and separation ( ): Modelling and Simulation: To model the needle insertion phenomena, FE based 2D simulations are performed. The Poisson ratio for the tissue is kept near 0.5 to simulate the incompressible nature of tissue, and the value of the coefficient of friction between needle and tissue is kept low at 0.1, to replicate biological tissue behavior where the presence of fluid within the organs may act as a lubricant [11]. Material and Geometric Properties The geometric and material properties for tissue and cohesive elements are taken from M. Oldfield et al. [3] and shown in table 3.1. The needle is modelled as a discrete rigid body, as the deformation in needle would be very small in comparison to that in tissue, hence to reduce the computational cost a rigid part type was selected instead of a deformable part type for the needle. 3-node 2D plane strain (CPE3) elements are used to mesh the tissue parts, and the cohesive layer is mesh using 4-node 2D cohesive element (COH2D4). Performing the convergence analysis, it has been found that an element size of 1.6 mm is suitable for the tissue elements. The tissue rupture parameter for the cohesive elements is modelled using traction-separation relationship as shown in figure 3.1. The value is set close to the needle diameter, and value of is calculated from the relation between the tissue rupture toughness parameter ( ), tissue elasticity ( ), and fracture separation ( ). Three sets of simulations are performed and a unique set of values of and are taken for each simulation as shown in figure 3.5. Figure 3.3 shows the simulation setup for these FE based simulations. The tissue has been made of two parts joined together using tie constraint. The cohesive element is modelled as a thin layer of 2D cohesive elements. This cohesive layer is put between the two tissue parts and joined together using tie constraint. The needle is placed in the needle the junction of the two tissue parts, vertical overt the cohesive layer as shown in figure 3.3. To enable lateral forces across the cohesive elements two elements at the top of tissue block are removed. This enables the initial reparation across the crack surface. Simulation Setup Boundary Conditions Appropriate boundary conditions have been applied to simulate realistic modelling as shown in figure 3.3. The sides of tissue block are pinned to replicate the presence of a container. Nodes along the bottom of the tissue block are free to move in the lateral direction (x-direction) but are prevented to move in normal direction (y-direction) to replicate the presence of container bottom. An additional constraint of equal and opposite displacement is applied to the nodes on either side of the crack axis to reduce the possibility of buckling associated with small numerical errors. The needle is given a displacement boundary condition of 25 mm along the needle shaft in y-direction to press against the tissue, to imitate the tissue rupture. Results The simulations are performed in ABAQUS/CAE (version 6.14) using an explicit solver. The deformation of tissue due to needle insertion and resulting contour plot for von-mises stress in tissue is shown in figure 3.4 for 25 mm depth of insertion of needle. and . The value are kept close to the needle diameter, and value of is calculated from the relation between , , and . From the figure 3.5 is can be seen that small variation in the values of and does not make any significant change in values of reaction force and is more important parameter of needle insertion modelling. Verification of Insertion Model To verify the results obtained from the FE based simulations, the reaction forces are compared with some of the previously published research work [2] and [3], as shown in figure 3.6. The plots shown in these figures are obtained from the experimental measurements for the reaction forces acting along the needle shaft during needle-insertion in soft tissue. It can be observed that trend in the results of the simulation match closely with these experimental results. The difference in the magnitude of the reaction forces is because of the fact that each research work have used different tissues. Discussion The needle-tissue interaction forces are analyzed in both palpation and insertion phenomena. To verify the results obtained from the FE based simulations performed for the needle-tissue palpation model, they are compared with the reported experimental results. Data obtained from the palpation models are used to develop a parametric model that can be implemented in a real-time surgical simulator to provide information about the tissue deformation and the reaction forces at a very high rate. Cohesive elements are used in the insertion model implements to simulate the tissue failure and crack generation phenomena. Results from the finite element-based simulation for the insertion model are compared with the reported experimental results in various other research work and a similar trend in insertion force is observed. The presence of a peak in reaction force just before the initiation of the tissue rupture, followed by a drastic reduction in the reaction forces, indicates the occurrence of tissue relaxation phenomena. Hence the implementation of cohesive element for insertion modelling is appropriate. An analytical model is developed for the reaction forces acting on the needle during needle-insertion, the expressions for axial and transverse reaction forces in the symmetric and bevel tip needle are compared with a finite element-based simulation and both results follow the same trend. This validates the analytical model developed on the basis of local tissue deformation, compressive and frictional forces generated due to inclusion on the needle in the tissue. The constant difference in the current analytical and simulation results is due to the absence of tissue rupture toughness parameter in the analytical model. Sensitivity analyses are performed for variation in axial and transverse reaction forces with respect to tip bevel angle and tissue elasticity using FE based simulations. It is observed that the variation in reaction forces with tip bevel angle and tissue elasticity follow the same trend as the analytical model, further validating the analytical model. Supplementary Materials: The following supporting information can be downloaded at: www.mdpi.com/xxx/s1, Figure S1: title; Table S1: title; Video S1: title. Author Contributions: For research articles with several authors, a short paragraph specifying their individual contributions must be provided. The following statements should be used "Conceptualization, X.X. and Y.Y.; methodology, X.X.; software, X.X.; validation, X.X., Y.Y. and Z.Z.; formal analysis, X.X.; investigation, X.X.; resources, X.X.; data curation, X.X.; writing-original draft preparation, X.X.; writing-review and editing, X.X.; visualization, X.X.; supervision, X.X.; project administration, X.X.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript." Please turn to the CRediT taxonomy for the term explanation. Authorship must be limited to those who have contributed substantially to the work reported. Funding: This research received no external funding. Institutional Review Board Statement: In this section, you should add the Institutional Review Board Statement and approval number, if relevant to your study. You might choose to exclude this statement if the study did not require ethical approval. Please note that the Editorial Office might ask you for further information. Please add "The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of NAME OF INSTITUTE (protocol code XXX and date of approval)." for studies involving humans. OR "The animal study protocol was approved by the Institutional Review Board (or Ethics Committee) of NAME OF INSTITUTE (protocol code XXX and date of approval)." for studies involving animals. OR "Ethical review and approval were waived for this study due to REASON (please provide a detailed justification)." OR "Not applicable" for studies not involving humans or animals. Informed Consent Statement: Any research article describing a study involving humans should contain this statement. Please add "Informed consent was obtained from all subjects involved in the study." OR "Patient consent was waived due to REASON (please provide a detailed justification)." OR "Not applicable." for studies not involving humans. You might also choose to exclude this statement if the study did not involve humans. Written informed consent for publication must be obtained from participating patients who can be identified (including by the patients themselves). Please state "Written informed consent has been obtained from the patient(s) to publish this paper" if applicable. Data Availability Statement: In this section, please provide details regarding where data supporting reported results can be found, including links to publicly archived datasets analyzed or generated during the study. Please refer to suggested Data Availability Statements in section "MDPI Research Data Policies" at https://www.mdpi.com/ethics. If the study did not report any data, you might add "Not applicable" here. Acknowledgments: In this section, you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g., materials used for experiments). Conflicts of Interest: The authors declare no conflict of interest. Appendix A The appendix is an optional section that can contain details and data supplemental to the main text-for example, explanations of experimental details that would disrupt the flow of the main text but nonetheless remain crucial to understanding and reproducing the research shown; figures of replicates for experiments of which representative data is shown in the main text can be added here if brief, or as Supplementary data. Mathematical proofs of results not central to the paper can be added as an appendix. Figure 1 . 1(a) Simulation setup for needle-insertion model; (b) Boundary conditions imposed on the needle-insertion model. Figure 2 . 2Contour plot for von-mises stress over deformed tissue. Figure 3 . 3Reaction force vs. insertion depth for different values of δ0 and δy. Figure 3 3shows the reaction forces acting along the needle shaft in y-direction, for different values of Table 1 : 1Material and Geometric properties of needle, tissue and cohesive elements for needle insertion modelling[3].Needle Material Properties: Rigid Material (as the elastic deformation in needle is negligible) Geometrical Properties: Diameter : 8 mm Length : 160 mm Tip-include angle : 50 ° Tissue Material Properties: Young Modulus : 7 kPa Poisson Ratio : 0.475 Geometrical Properties: Length of each part : 80 mm Depth of each part : 86 mm Cohesive Elements Material Properties: Stiffness : 6.64 × 10 −4 / 3 Fracture Toughness : 17.43 J/mm 2 Geometrical Properties: Length : 86 mm Geometric Thickness : 0.001 mm Constitutive Thickness : 1 Appendix BAll appendix sections must be cited in the main text. 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[ "Topology for Substrate Routing in Semiconductor Package Design", "Topology for Substrate Routing in Semiconductor Package Design" ]	[ "Rak-Kyeong Seong \nAdvanced Research Lab\nSamsung SDS\nSamsung R&D Campus\nSeocho-GuSeoulAISouth Korea\n", "Jaeho Yang \nAdvanced Research Lab\nSamsung SDS\nSamsung R&D Campus\nSeocho-GuSeoulAISouth Korea\n", "Sang-Hoon Han \nAdvanced Research Lab\nSamsung SDS\nSamsung R&D Campus\nSeocho-GuSeoulAISouth Korea\n" ]	[ "Advanced Research Lab\nSamsung SDS\nSamsung R&D Campus\nSeocho-GuSeoulAISouth Korea", "Advanced Research Lab\nSamsung SDS\nSamsung R&D Campus\nSeocho-GuSeoulAISouth Korea", "Advanced Research Lab\nSamsung SDS\nSamsung R&D Campus\nSeocho-GuSeoulAISouth Korea" ]	[]	In this work, we propose a new signal routing method for solving routing problems that occur in the design process of semiconductor package substrates. Our work uses a topological transformation of the layers of the package substrate in order to simplify the routing problem into a problem of connecting points on a circle with non-intersecting straight line segments. The circle, which we call the Circular Frame, is a polygonal schema, which is originally used in topology to study the topological structure of 2manifolds. We show through experiments that our new routing method based on the Circular Frame competes with certain grid-based routing algorithms.	10.1016/j.cad.2022.103269	[ "https://arxiv.org/pdf/2105.07892v1.pdf" ]	234,741,785	2105.07892	f69eb0abb3ec2b009630796066ca499c42f42615	Topology for Substrate Routing in Semiconductor Package Design Rak-Kyeong Seong Advanced Research Lab Samsung SDS Samsung R&D Campus Seocho-GuSeoulAISouth Korea Jaeho Yang Advanced Research Lab Samsung SDS Samsung R&D Campus Seocho-GuSeoulAISouth Korea Sang-Hoon Han Advanced Research Lab Samsung SDS Samsung R&D Campus Seocho-GuSeoulAISouth Korea Topology for Substrate Routing in Semiconductor Package Design In this work, we propose a new signal routing method for solving routing problems that occur in the design process of semiconductor package substrates. Our work uses a topological transformation of the layers of the package substrate in order to simplify the routing problem into a problem of connecting points on a circle with non-intersecting straight line segments. The circle, which we call the Circular Frame, is a polygonal schema, which is originally used in topology to study the topological structure of 2manifolds. We show through experiments that our new routing method based on the Circular Frame competes with certain grid-based routing algorithms. Introduction Semiconductor devices are at the forefront of innovation in the information technology (IT) industry and play an essential role in driving innovations in areas such as consumer electronics, telecommunications, artificial intelligence or data analysis and security. Although semiconductor devices play such a pivotal role in IT innovation, the integrated circuit (IC) packaging process of semiconductor devices still heavily relies on human expertise. For substrates in, for example, chip-scale packages such as multilayered Fine Pitched Ball Grid Array (FBGA) packages as illustrated in Fig. 1, most of the design process is about finding the optimal connections between bond fingers, vias and solder balls. Given the variety of types for semiconductor packages, the problem of substrate routing is challenging. As a result, substrate routing problems are often solved with the help of routing methods that are implemented in many computer-aided design (CAD) solutions. In line with recent advances in Electronic Design Automation (EDA), in this work, we outline a new routing method for package substrate design that competes with the performance of other routing methods. The problem of finding non-intersecting paths that connect a set of start and end points on a plane is one of the oldest problems in computational geometry and graph theory. We know that Dijkstra's algorithm and the A-algorithm [Dij59,HNR68] are examples of graph traversal algorithms, which are used to solve such routing problems. However, substrate routing becomes exponentially more complicated with an increasing number of start and end point pairs. Most routing algorithms such as Dijkstra's algorithm, the A-algorithm and other grid-based Maze Router algorithms [Lee61,KC93,JKRS94,Alb01,CRN97] are known as geometrical routers. Their disadvantage is that when start and end point pairs are connected sequentially on consecutive shortest paths, it becomes increasingly more likely that there will be not enough clearance left for consecutive connections between pairs of points. This problem with geometrical routers is illustrated in Fig. 2. In this work, we are interested in a different class of routers known as topological routers [DKJS90]. In order to connect fully all points, topological routers aim to find the topological class of the connections first, i.e. the relative positions of paths. After the topological class of the connecting paths is found, with a choice of representation scheme, absolute coordinates are assigned to represent the routing result in real space. This avoids situations where there is a lack of clearance as it is often the case for geometrical routers. For topological routers, paths can always be inserted between already routed paths in order to solve the connection problem. Fig. 2 illustrates this difference between geometrical and topological routers. Our work is based on the concept of topological routers and proposes a novel topological representation and routing algorithm for substrate routing that competes with the performance of conventional geometrical routers. We make use of topology, more specifically the study of 2-manifolds and polygonal schema [Ful13,Pap96,EKL06,EN11] in mathematics in order to topologically transform the package substrate into a simpler abstract environment where routing design can be performed more straightforwardly. In an earlier work [SMH * 21], we outlined the general principle of our new method for general routing problems. In the current work, we extend our proposal with a focus on the problem of substrate routing in semiconductor chip package design. In particular, we apply our substrate routing method to an explicit example of a Fine Pitch Ball Grid Array (FBGA) package. Note that our work concentrates on a substrate routing method that finds a fully connected routing solution and does not take into account other metrics such as the wire length or optimal placement of via points. 1 Our work also concentrates on signal routing in substrates where the routing problem involves connections between a single start and a single corresponding end point. 2 We test our routing method's performance against geometrical routers and conclude with a summary of results and an actual FBGA package substrate design that was completed using our new routing method. In this work, we propose based on our earlier work in [SMH * 21] a new method of solving routing problems that occur during the package substrate design process by using topology. The idea of making use of concepts in topology for designing circuits is not new as shown by the works on rubber-band routing in [DKJS90,DDS91]. Background These works discuss how certain design features in circuit design can be altered without changing the connections between points, i.e. the topology of the paths, as illustrated in Fig. 3. Moreover, they give an insight into how paths can be bent and moved in such a way that problems of clearance occurring with traditional geometrical routers can be avoided. Several routing algorithms have been proposed for EDA since the 1990s [KC93, JKRS94, DDS91, CRN97], which are based on the idea of grid-dependent geometrical routers. Moreover, more recent work in EDA considers applications and improvements on geometrical routers in areas such as length matching routing [CWC19], escape routing [CKK19, AZN17, BHH16, WHJ * 20], routing with obstacle avoidance [MCS19] and pin assignment and placement algorithms [HXF * 19]. In comparison, topological routers have been studied less extensively [DKJS90,DDS91] although, as mentioned above, they have considerable advantages over geometrical routers. In contrast to the developments made in geometrical routing, our work tries to push forward the development of topological routing. In particular, our work proposes the use of a novel topological transformation to completely transform the substrate routing environment into a topologically equivalent environment. This is a completely new approach for routing in package substrates. Our proposed transformation maps the routing problem to a topologically simpler space where the problem can be solved more straightforwardly. This is the case when in the new environment only relative positions are preserved under the transformation. Given that the transformation is reversible, after all nets are connected, the space with the routing result is transformed back to its original substrate environment. Such topological transformations and representations that preserve relative posi-tions rather than absolute positions occur extensively in the study of compact 2-manifolds through polygonal schema [Ful13]. These were introduced in mathematics to study the topology of compact 2-manifolds and are particularly useful in representing the homotopy of paths on these manifolds [EKL06]. As a result, polygonal schema appeared also extensively in relation to so-called non-crossing walk problems on compact 2manifolds [Pap96,EN11]. Let us illustrate briefly the concept behind polygonal schema using one of the simplest compact 2-manifolds, the Riemann surface of genus 1, which is also known as a torus or doughnut. The torus can be represented by a rectangle when opposite sides of the rectangle are identified with each other. Any such simple convex polygon together with a boundary gluing pattern shown in Fig. 4 is known as a polygonal schema of the represented 2-manifold. Using the example of the 2-torus, we learn that a rectangle with its opposite boundary sides identified with each other is topologically equivalent to a torus. We can see from this example that even though a torus is 3-dimensional, it can be much more straightforwardly represented by its 2-dimensional polygonal schema. Figure 5: Routing Problem in a multi-layered FBGA Package Substrate. Each layer of the package substrate has its own set of start and end points. After solving the routing problem on each substrate layer, the layers can be connected again along the vias. We claim that a semiconductor package substrate, which usually contains multiple interconnected layers, can be described topologically in terms of polygonal schema. Substrate layers, which are connected by vias, can be separated and individually represented by polygonal schema. Because we split the layers for the topological transformation, each layer has its layer-specific start and end points corresponding to either pins, solder balls or vias. We keep track of which via connects which layers together so that when we reverse the topological transformation, we are able to sew back together the vias between each pair of layers to form the original multi-layered package substrate as shown in Fig. 5. Note that the locations of the via points plays an important role in the overall global routing solution. Since we focus on the problem of finding a fully connected routing solution and consider no other optimization metrics, we refer to future work on optimizing the routing solution using our method. In the following section, we describe how we make use of the topological transformation specific to our problem and describe a method of how to complete the routing in the topologically transformed routing environment. Circular Frame Let there be a set S of start points s i and a set T of end points t i with pairwise identification s i → t i . For our routing problem, we call such a pair a net. These points are on a plane bounded by B as shown in Fig. 6 (a). In order to transform this environment, we introduce trees R consisting of a set of edges r i such that these edges have at their ends either s i ∈ S, t i ∈ T or b i ∈ B. All points in S and T are each connected to a single tree. Note that a tree R is always connected by exactly one edge with the boundary B at a point b i as shown in Fig. 6 (b). These trees R can be found using a minimum spanning tree algorithm such as Kruskal's algorithm. 3 Such an algorithm needs to be generalized such that each tree R gets connected to the boundary B at a point b i by a single edge r i . The start and end points do not need to be connected to B by a single tree R. Each point can be connected to the boundary B by separate trees where each tree is separately connected to B. Our proposed topological transformation cuts the plane along all the edges r i such that all points in S and T are now placed on a new boundary that includes the cut-lines along r i as shown in Fig. 7. The cutting process splits some of the points s i and t i to multiple copies if the original points are connected to more than one tree edge r i . The boundary points at which trees are attached to the original boundary B are always separated into a pair b i and b i . We also notice that during the cutting process the edges r i separate into pairs r i and r i . We pinch the edges r i and r i originating from the trees R in such a way that they are also represented by points on the new boundary H as shown in Fig. 8. As a The combined boundary H can be deformed into a circle as illustrated in Fig. 9. We call this representation of the original substrate layer the Circular Frame. The order in which the points appear along the circle is the same as they appear when one traverses H in a given orientation as shown in Fig. 9. The Circular Frame is topologically equivalent to the original substrate layer where the routing is taking place. The advantage of using the Circular Frame representation of the routing problem is that paths connecting pairs of points are represented as straight line segments connecting points on the boundary of the Circular Frame. These points are either start or end points of the original path, points representing r i or r i , or points on the original boundary B. When a path is connected to r i or r i in the Circular Frame, it corresponds in the substrate layer to a path that crosses the associated tree edge r i as illustrated in Fig. 10. A further advantage of the Circular Frame is that line intersections can be easily detected by going through the ordering of line ends on the boundary of the Circular Frame. The fact that the topological transformation is reversible enables us to solve the routing problem in the simpler Circular Frame environment and then transform the routing solution back to the original substrate layer environment. This is done by reversing the transformation as illustrated in Fig. 10. Within the Circular Frame, the routing problem is simply a problem of connecting points on the boundary of a circle with non-intersecting straight line segments as illustrated in Fig. 10 (a). Routing Method In this section, we outline a method of connecting the nets in the Circular Frame. As noted in the section above, although the Circular Frame is topologically equivalent to the original planar substrate layer bounded by B, it simplifies the routing problem to a problem of connecting points on a circle with straight line segments that do not intersect in the interior of the circle. The following section outlines how the Circular Frame simplifies the routing problem. Starting from a Circular Frame with no points connected, as illustrated in Fig. 9 (b), we can choose to connect the first net, i.e. s 1 with t 1 . Due to the cutting process of the original routing plane, as shown in Fig. 8, the end point t 1 is split into 3 copies in the Circular Frame, i.e. t 1 , t 1 and t i . We note that in the Circular Frame, connecting s 1 to either t 1 , t 1 or t 1 is possible. In the actual routing plane, the choice will determine in which direction the connecting path is going to enter the end point t 1 in the original substrate layer environment. For the moment, without loss of generality, let us assume that we connect in the Circular Frame s 1 with t 1 as illustrated in Fig. 11. Note that any connection between two points in the Circular Frame can be realized in terms of straight line segments that do not intersect in the interior of the Circular Frame. Due to the line segment connecting s 1 with t 1 , the Circular Frame gets divided into two sections, which we call slices. Fig. 11 shows the two slices σ 1 and σ 2 . Each slice has its own boundary with a subset of points from the boundary of the Circular Frame. For our example in Fig. 11, the two slices σ 1 and σ 2 have the points {s 1 , t 1 , r 1 , b 1 , b 1 , r 1 , t 1 , r 2 , s 2 , r 3 } and {s 1 , r 3 , s 2 , r 2 , t 1 , r 4 , t 2 , r 4 , t 1 } each on their respective boundaries. Note that the points that we connected, s 1 and t 1 , are both shared by the boundary of the two slices. The line segment, which connects s 1 with t 1 , is precisely the overlap of the two boundaries. As shown in Fig. 12, the two slices σ 1 and σ 2 are not completely disconnected. We recall that the points r i and r i that represent tree edges in the Circular Frame always come in pairs as explained in Section 3. r i and r i precisely identify the tree edges along which the original substrate layer was cut in order to obtain the Circular Frame as illustrated in Fig. 7. Accordingly, they represent points that need to be pairwise glued together when the Circular Frame is transformed back to the original substrate layer environment. Fig. 12 shows these pairwise connections as dotted lines. The two slices σ 1 and σ 2 in Fig. 12 are connected by the pairs (r 2 , r 2 ) and (r 3 , r 3 ). When we now attempt to connect start point s 2 , which is on the boundary of σ 1 , with its corresponding end point t 2 , which is on the boundary of σ 2 , we have to move between the two slices σ 1 and σ 2 . As we noted above, the two slices are connected by the point pairs (r 2 , r 2 ) and (r 3 , r 3 ). Without loss of generality, by choosing point pair (r 2 , r 2 ), s 2 is connected with r 2 in σ 1 , and then its partner r 2 is connected with t 2 in σ 2 as illustrated in Fig. 13. Note that by connecting s 2 to t 2 through the point pair (r 2 , r 2 ), the original slices σ 1 and σ 2 are each divided into two slices by the two line segments connecting s 2 with r 2 and r 2 with t 2 . As a result, we end up with a total of four slices. There is also the possibility that more than one path goes through a point pair (r 1 , r 1 ) as shown in Fig. 14. In the example in Fig. 14, both (s 1 , t 1 ) and (s 2 , t 2 ) are connected through the point pair (r 1 , r 1 ). In such a situation, one has to make sure that the slice containing the origin point and the slice containing the destination point are in the same order. Let us define the order o(σ, r i ) of σ with respect to the point r i as the segment number of σ attached to r i in the Circular Frame when one counts anti-clockwise around r i starting from the boundary of the Circular Frame. In analogy, let us define the order o(σ, r i ) of σ with respect to the point r i as the segment number of σ attached to r i in the Circular Frame when one counts clockwise around r i starting from the boundary of the Circular Frame. For example, in Fig. 14, we note that o(σ 1 , r 1 ) = o(σ 5 , r 1 ) = 1, o(σ 2 , r 1 ) = o(σ 4 , r 1 ) = 2 and o(σ 3 , r 1 ) = o(σ 3 , r 1 ) = 3. Accordingly, anything starting in slice σ 1 can go through (r 1 , r 1 ) to slice σ 5 , not any other slice. Similarly, we have o(σ 2 , r 1 ) = o(σ 4 , r 1 ) and o(σ 3 , r 1 ) = o(σ 3 , r 1 ), meaning that anything in slice σ 2 can be connected to σ 4 and anything in slice σ 3 can be connected to σ 4 via the edge point pair (r i , r i ). Note that slice ordering is essential to make sure that when the edge points are glued together, the correct slices recombine with each other to give the original substrate layer as shown in Fig. 14 (c). Following these rules on connecting points in the Circular Frame, we outline a basic algorithm for connecting all points in Fig. 15. In line 12 of the algorithm in Fig. 15, the closest r k to a given point p in a slice π is identified by the smallest number of points one needs to pass in order to go from p to r k along the boundary of π. We note that the algorithm in Fig. 15 is one example amongst many possible connection algorithms that one can formulate with the help of the Circular Frame. We plan to present variations of this algorithm in future work. For now, the algorithm in Fig. 15 does not have the aim to find the shortest possible paths between start and end point pairs. Instead, the algorithm in Fig. 15 simply has the aim to achieve full connection for all start and end point pairs. In fact, with the Circular Frame and the algorithm in Fig. 15, complete connection is always guaranteed. This is because the Circular Frame is only encoding the topology of the routing problem as illustrated in Fig. 2 and as a result there is no problem of clearance as it is the case for geometrical routers. Furthermore, in signal routing, paths always connect a single start point s i with a single end point t i and hence there is no possibility that a path completely encircles points that need to be connected by other paths in the Circular Frame. 4 Fig. 16 illustrates an example where the algorithm in Fig. 15 is applied to solve the connection problem in the Circular Frame. Embedding We call the process of transforming the routing result in the Circular Frame back to the original substrate layer the embedding process. As discussed in the sections above, the Circular Frame can be transformed back to the original substrate layer by glueing together the point pairs (r i , r i ) for all i. Under this reverse transformation, the topology of the routing result, i.e. the identified paths connecting start points with corresponding end points, is preserved. The information about which points correspond to which slices in the Circular Frame and the information about which slices are adjacent to each other is called the topological class T Ful13] of a topological class is a specific embedding of the connecting paths in the routing solution. An example of such a topological class T i (P i , W i , H i ) for a path ρ i is the following information: • The set of points P i = {p (i) k . If p (i) k = s i or t i then h (i) k = 0. Note that T i (P i , W i , H i ) for any path ρ i can be obtained from the Circular Frame of the routing solution. 5 In the example in Fig. 16, for path ρ 4 connecting s 4 with t 4 , T 4 is given by P 4 = (s 4 , t 3 , t 2 , t 4 ), W 4 = (0, +1, +1, 0) and H 4 = (0, 1, 1, 0). We make use of the rubber-band sketch from [DKJS90,DDS91] in order to represent the topological class of the routing result from the Circular Frame on the original 5 We note that one can introduce several other topological classes that encapsulate the routing result in the Circular Frame, for instance including the edges r i . We hope to present further versions in future work. By cutting the substrate layer along the tree edges r i , we obtain (b) the corresponding Circular Frame. We select s 1 and t 1 as the first pair to be connected. (c) Since both s 1 and t 1 are in slice σ 1 , we connect them. We select s 2 and t 2 as the second pair to be connected. (d) Because s 2 and t 2 are both in slice σ 1 , we connect them. We select s 3 and t 3 as the next pair to be connected. (e) Because s 3 and t 3 are both in slice σ 1 , we connect them. The final pair to be connected is selected as s 4 and t 4 . Here, s 4 is in σ 4 and t 4 is in σ 1 . (f) In σ 4 , the closest tunnelling point to σ 4 is r 3 and we connect σ 4 with r 3 . The partner of r 3 , r 3 , is in σ 2 . (g) Since in σ 2 , we still have not t 4 , we look for the closest tunnelling point to r 3 . We identify r 4 and connect r 3 with r 4 . (h) The partner of r 4 , r 4 , is in slice σ 1 where we also have our destination point t 4 . We connect r 4 with t 4 and hence have connected using tunnelling points s 4 with t 4 . (i) We reverse transform the Circular Frame with the routing result back to the original substrate layer by glueing together the edge point pairs (r i , r i ). planar environment. Fig. 3 (a) shows the rubber-band sketch of the same topological class represented in Fig. 3 (b). A characteristic feature of the rubber-band sketch is that paths are represented as line segments that can have any angle and the line segments are connected by arcs whenever the path passes a point. Fig. 17 illustrates an example of a topological class and its corresponding rubber-band representation. For the purpose of this work, which is to present a new topological routing method that results in a topological class of a fully-connected routing result via the Circular Frame, we keep the review on topological classes and the rubber-band sketch representation short and refer to the works in [DKJS90,DDS91]. Experiment Let us design experiments to compare the performance of the proposed routing algorithm based on the Circular Frame (CF) with variations of the A-algorithm (AS). General Setup Let For each n, we generate N = 1000 end ball sets T whose coordinates are generated randomly within the boundary of the planar environment. The randomly generated end points t i have a minimum center-to-center separation d min (t i , t j ) = 11 to other end points t j as well as a minimum center-to-center separation d min (t i , s j ) = 11 to start points s j . The randomly generated end points also satisfy a minimum distance d min (t i , b j ) = 3 to any boundary point b j ∈ B of the plane. We call each generated set (S, T ) a routing environment E h=1...N . Fig. 18 shows an environment with n = 2, where all N = 1000 randomly generated end points for s 1 are illustrated simultaneously in order to illustrate that the randomly generated points are evenly distributed on the bounded plane. Figure 18: Random Positions for End Points. N = 1000 randomly generated end points t 1 for s 1 in an environment with n = 2 start and end point pairs. We have shown all end points t 1 for s 1 (red) at once to illustrate the random distribution of the points within the boundary of the environment. Measurements For each environment E h , the routing problem is to connect all s i with the corresponding t i with non-intersecting paths. We run different routing algorithms for each environment E h and measure the time t h that the algorithms take to complete the routing for all nets. Note that all algorithms are run on a laptop with CPU at 1.80 GHz (Intel i7-8550U) and 8 GB memory. If any of the nets are left disconnected, we label the routing result as incomplete. For completed routing environments, we also measure for each connecting path between s i and t i the Euclidean path length l h i . The mean path length l h and the corresponding standard deviation σ h for all connecting paths in E h are also obtained. In addition, we also measure the Manhattan distance between the start node s i and corresponding end node t i , d h i (s i , t i ) = \|x(s i ) − x(t i )\| + \|y(s i ) − y(t i )\| ,(1) and compare it to the Euclidean path length l h i of the path that was found by the chosen routing algorithm. In particular, we calculate the ratio r h i (s i , t i ) = l h i (s i , t i )/d h i (s i , t i ) . The Manhattan distance is the shortest path length between start and end points on a square grid and is a measure of how direct a path has been taken between a start point and its corresponding end point. Accordingly, a smaller r h i indicates that the path is closer to the shortest path on a square grid. For all our measurements, we have two different types of means. A measurement X h i corresponding to (s i , t i ) in environment E h can be averaged over all paths in E h to give X h = 1 n i X h i and then further averaged over all environments E h to give X = 1 N h X h . The corresponding standard deviation of sample means is denoted as σ X . We are going to use this notation when we summarize our experimental results in Section 7. A-Algorithm Let us give a brief overview of the A-algorithm used in this work for the purpose of benchmarking our new routing algorithm based on the Circular Frame. The reader is referred to previous work for a more extended overview of the A-algorithm [HNR68]. The A-algorithm is a graph traverser algorithm, which at each iteration of the algorithm extends a tree of candidate paths originating from the start node s i until one of the branches of the tree reaches the end node t i . The incremental extension is made at a given node p of the graph if a cost function f (p) is minimized by the extension. The cost function is defined as f (p) = g(p) + h(p), where g(p) is the cost of the path from the start node to p and h(p) is the heuristic function that estimates the cost of the cheapest path from p to the end node. Without loss of generality we define g(p) as the Euclidean path length from the start point to p unless the path intersects with another path in which case its value is set to infinity. For our experiments, we consider two different implementations of the A-algorithm. The first implementation (AS1) uses as the graph the integer square grid of the plane bounded by B with the neighbourhood of a given node defined by the 4 direction vectors (±1, 0) and (0, ±1). The corresponding heuristic uses the Manhattan distance between a given node p and the corresponding destination t i , where here we set D = 1, and dx = \|x(p)−x(t i )\| and dy = \|y(p)−y(t i )\|. The second implementation (AS2) defines the neighbourhood of a node in the integer square grid by the 8 direction vectors (±1, 0), (0, ±1) and (±1, ±1). We use here the Chebyshev distance heuristic, h AS1 (p) = D(dx + dy) ,(2)h AS2 (p) = D 1 (dx + dy) + (D 2 − 2D 1 ) min(dx, dy) ,(3) where we set D 1 = 1 and D 2 = 1. Fig. 19 illustrates the difference between the two implementations of the A-algorithm that we use in this work. For our experiments we use Python implementations of the above A-algorithms and a Python implementation of the Circular Frame algorithm described in Section 4. Results and Discussions Let us summarize the results of the experiments in the following section. Table 1 shows the number of successfully completed routing problems under the Circular Frame algorithm (N C F ), the A-algorithm under the Manhattan distance heuristic (N AS1 ) and the A-algorithm under Chebyshev distance heuristic (N AS2 ). Originally N = 1000 routing environments were generated as outlined in Section 6. We can see that for all number of start points n, the Circular Frame algorithm consistently completes the routing for all generated environments, whereas the number of routing failures increases with increasing n for the two implementations of the Aalgorithm. Reliability and Performance Table 1 also shows the number N C of environments where the routing was completed by all 3 tested routing algorithms. For these completed environments and for each n, we measure the mean routing times t with the corresponding standard deviations σ t . Fig. 20 shows that the average routing time for the Circular Frame algorithm stays consistently below the average routing times for the two implementations of the A-algorithm. From the routing completion numbers in Table 1 and the average routing times illustrated in Fig. 20, we conclude for the test environments that the Circular Frame algorithm is more reliable and faster than the two implementations of the A-algorithm. This is not a surprising result since the number of points that needs to be traversed on the boundary of the Circular Frame is far less than the grid points used for grid-based geometrical routers. Moreover, as a topological router, the Circular Frame algorithm does not suffer from clearance problems as the A-algorithms do as illustrated in Fig. 2. Table 3 shows the grand mean of the path lengths l with the corresponding standard deviation of the mean σ l for the N C completed routing environments under the 3 tested algorithms. We observe that paths connecting nets under the Circular Frame algorithms tend to be shorter than for the two implementations of the A-algorithm for all n. This is not surprising since the rubber-band sketch representation of the resulting topological class uses arcs and any-angle straight lines for paths, making the overall routing more compact than grid-based representations. We also calculated the ratio r h i between the Euclidean path length l h i and the corresponding Manhattan distance d h i between the connected start and end points (s i , t i ) for all completed E h . The grand mean of the ratio r with σ r is shown in Table 3. As noted in Section 6, a smaller ratio r indicates that the connection is closer to the shortest path on a square grid. As we can see in Table 3, the Circular Frame algorithm at n = 2 has a mean ratio r < 1, indicating that for some routing results, the Circular Frame identified connecting paths that are even shorter than the Manhattan distance d. Moreover, consistently the Circular Frame algorithm achieved on average a smaller value of r than the two implementations of the A-algorithm, indicating that overall the Circular Frame algorithm found more direct and hence more accurate connections. Fig. 21 shows the completed routing results under the Circular Frame algorithm and the two implementations of the A-algorithm for a given routing environment. We note that our observations here are as expected since the Circular Frame algorithm avoids as a topological router problems caused by a lack of clearance as discussed in Fig. 2. Furthermore, the rubber-band sketch representation optimizes the length of the connecting paths in comparison to grid-based geometrical routers. Routing Accuracy Conclusions In this paper, we have proposed a new method based on our earlier work in [SMH 21] for solving substrate routing problems using topology. Our proposed topological transformation of the original routing environment into the Circular Frame has accelerated in experiments the substrate routing process significantly in comparison to grid-based geometrical routers such as the A-algorithm. Moreover, the Circular Frame representation guarantees for substrate routing problems with start and end point pairs full connection as a topological router. In addition, Fig. 22 shows a 2-layered substrate for a FBGA package with 200 solder balls and a completed routing design that was obtained using our new Circular Frame algorithm. Our experiments and the positive routing results on real semiconductor package substrates are a clear indication that our new Circular Frame routing method has the potential to significantly improve and at the end fully automate the package substrate routing process. We note that the Circular Frame routing algorithm can lead to different routing solutions given by topological classes depending on via placement, spanning tree generation and even net ordering during the routing process in the Circular Frame. Moreover, in our work we have given only a single basic routing method based on the Circular Frame representation and depending on other routing algorithms based on the Circular Frame representation, the routing result can differ significantly. Finding the most optimal routing solution based on the Circular Frame representation depends on metrics such as wire length or wire widths and is an optimization problem that we hope to cover in future work. We are currently testing the Circular Frame routing algorithm on larger FBGA packages and other package designs. Moreover, beyond semiconductor package design, we are applying our routing method on problems related to the design of printed circuit boards (PCB) and the logistics and manufacturing industry. We hope to report on our progress in these areas in future work. Figure 22: FBGA Routing Sample. 2-layered FBGA package substrate that has been connected using the Circular Frame routing algorithm. Layer 1 consists of fingers (red and blue) and net connections (gray) while layer 2 consists of solder balls (blue) and net connections (green). The two layers are connected by vias (white circles). Figure 1 : 1Fine Pitch Ball Grid Array (FBGA) Package Substrate Layout. (a) An illustration of a 3layered FBGA package substrate with vias connecting different substrate layers. (b) Each individual layer (here layer 2) has its own set of start and end points that need to be connected with non-intersecting paths. Figure 2 : 2Geometrical and Topological Routers. (a) In geometrical routers, start (s i ) and end (t i ) points are sequentially connected with shortest paths, which can result in a lack of clearance for any following pairs, in this case s 3 and t 3 . (b) In topological routers, the connection problem only deals with relative positions, avoiding problems of clearance. Figure 3 : 3Preserving Routing Topology. (a) Rubber-band sketch representation of a connected set of start and end points, (b) compared to a rectilinear representation of the same connected solution with the same routing topology. Figure 4 : 4Polygonal Schema. (a) A torus with its corresponding polygonal schema, which is a rectangle with opposite edges identified with each other. (b) A path on the torus can be represented as a path on the corresponding polygonal schema. Figure 6 : 6Start Points, End Points and Trees. (a) Start points s i ∈ S and end points t i ∈ T on a plane bounded by B. (b) Trees R made of edges r i connect all s i and t i to points b j on the boundary B. Figure 7 : 7Cutting along Trees. (a) Cutting the plane along the tree edges r i ∈ R (b) splits the points connected to the edges. Figure 8 : 8Tree Lines as Points. (a) The cutting process splits the edges into pairs r i , r i . (b) Each of the edges can be represented as points on the combined boundary H. All points are now on H.result, the start points s i , end points t i , the tree edges r i , boundary points b i and their corresponding partners generated by the cutting process are all represented as points on a single combined boundary H as shown inFig. 8. Figure 9 : 9Circular Frame. (a) The combined boundary H can be deformed to form (b) a circle. The interior of the circle represents the original substrate layer that was cut, and the start, end, boundary and tree edge points are all on the circle. We call this representation of the original substrate layer the Circular Frame. Figure 10 : 10Routing Representation in the Circular Frame. (a) Paths connecting start and end points in the Circular Frame via the point pairs (r i , r i ) (b) are combined by glueing together r i with r i to form (c) the original substrate layer with the complete routing solution. Figure 11 : 11Slices in the Circular Frame. (a) A path connecting two points on the boundary of the Circular Frame (b) divides the Circular Frame into 2 slices σ 1 (green) and σ 2 (blue). Figure 12 : 12Moving Between Slices. (a) Points r i and r i , which always appear in pairs, correspond to the tree edges along which the original substrate layer was cut to give the Circular Frame. (b) The pairs can be pulled together along the dotted lines to give (c) the original substrate layer. We can consider these pairs as 'tunnels' along which a connecting path can move between different slices of the Circular Frame. Figure 13 : 13Multiple Slices. (a) By connecting s 2 to t 2 through (r 2 , r 2 ), the original slices σ 1 and σ 2 are each divided into two slices giving a total of four slices. (b) By glueing together r i with r i , we obtain (c) the original substrate layer. Figure 14 : 14Slice Ordering. (a) Both (s 1 , t 1 ) and (s 2 , t 2 ) are connected through the point pair (r 1 , r 1 ). (b) In order to make sure that when the points r 1 and r 1 are glued together the correct slices recombine with each other, (c) we need o(σ 1 , r 1 ) = o(σ 5 , r 1 ), o(σ 2 , r 1 ) = o(σ 4 , r 1 ) and o(σ 3 , r 1 ) = o(σ 3 , r 1 ). [Ful13] of the routing solution. The geometric sketch[DKJS90, 1: for each si ∈ Figure 15 : 15τ containing r k with o(π, r k ) = o(τ, r k ) A Basic Circular Frame Routing Algorithm. Pseudocode for connecting nets in the Circular Frame. k•k } a path ρ i passes starting from s i and ending at t i . • The set of orientations W i = {w (i) k } a path ρ i takes when it passes points {p The set of heights H i = {h (i) k } a path ρ i has when it passes points {p = m indicates that between ρ i and p (i) k there are m − 1 other paths passing p Figure 16 : 16Routing Algorithm Example. (a) The original substrate layer consists of 4 start points s i placed on the boundary B of the plane and 4 corresponding end points t i . Figure 17 : 17Topological Class and Rubber-Band Sketch. (a) A topological class T (P, W, H) for a 3path routing solution with (b) the corresponding rubber-band sketch representation. Note that the separation between paths passing t 1 is given by the height interval ∆h = 0.5. The paths are made of line segments (blue) and concentric arcs (red). us first construct a planar environment with a boundary B having −50.0 ≤ x ≤ 50.0 and −50.0 ≤ y ≤ 50.0. The start points S = {s 1 , . . . , s n } and end points T = {t 1 , . . . , t n } are represented as circles with radius r = 0.5 and their positions are given by the coordinates of their centers. For our experiment, we vary n by setting it to n = 2, 4, 6, 8, 10. 6 The number of start points S increases by adding consecutively (50.0, ±4.0), (50.0, ±12.0), (50.0, ±20.0), (50.0, ±28.0) and (50.0, ±36.0). Figure 19 : 19A-Algorithm. Implementations of the A-algorithm using a square grid with (a) 4 directions of movement and a Manhattan distance heuristic (AS1), and (b) 8 directions of movement and a Chebyshev distance heuristic (AS2). Figure 20 : 20Average Routing Times. Average routing times t to complete the routing problem at given n for the Circular Frame algorithm (CF) and A-algorithms (AS1 and AS2). The error bars show the standard deviation σ t of t. Figure 21 : 21Routing Sample. Complete routing result for the same routing environment under (a) the Circular Frame algorithm, (b) the A-algorithm with Manhattan distance heuristic (AS1) and (c) the Aalgorithm with the Chebyshev distance heuristic (AS2). Table 1 : 1Routing Completion Results.n 2 4 6 8 10 N 1000 1000 1000 1000 1000 N CF 1000 1000 1000 1000 1000 N AS1 1000 931 742 424 209 N AS2 934 913 749 496 220 N C 934 868 628 320 116 Table 2 : 2Average Routing Completion Times.n 2 4 6 8 10 N C 934 868 628 320 116 CF t 0.0041 0.0122 0.0269 0.0494 0.0846 σ t 0.0009 0.0032 0.0088 0.0163 0.0312 AS1 t 1.94452 8.25649 18.35662 38.50646 46.85451 σ t 1.0035 4.0399 9.1618 16.3210 14.5456 AS2 t 1.04229 4.12858 9.16055 18.71960 22.36284 σ t 0.4988 2.0021 4.8156 8.7838 7.8897 Table 3 : 3Routing Path Length Results.n 2 4 6 8 10 N C 934 868 628 320 116 Manhattan d 73.926 76.836 77.789 78.220 79.055 σ d 29.004 29.090 30.362 30.964 30.755 CF l 62.787 76.221 85.638 92.658 103.115 σ l 18.662 19.451 24.711 26.031 34.285 r 0.951 1.151 1.308 1.439 1.591 σ r 0.386 0.574 0.743 0.862 1.084 AS1 l 87.565 115.357 126.014 130.207 128.716 σ l 32.862 35.974 35.765 32.015 25.660 r 1.327 1.754 1.940 2.034 2.005 σ r 0.663 0.990 1.180 1.244 1.301 AS2 l 83.903 108.047 116.246 120.616 119.095 σ l 32.070 31.939 30.476 27.257 20.304 r 1.277 1.645 1.791 1.889 1.857 σ r 0.669 0.927 1.058 1.153 1.181 Our routing method can be adjusted to take into account an optimization metric and this will be the subject of upcoming work.2 Multi-pin routing that occurs in power and ground routing or plating lines can be covered in a generalized version of our method, which we plan to cover in future works. Note that the choice of method for finding the spanning trees may lead to a single tree. Furthermore, the choice will impact the routing result and leads to questions about optimization that will be studied in future work [CKT * 13]. Note that there is an optimization problem in terms of the choice of geometric sketch one uses to represent the routing solution given by its corresponding topological class. The problem of optimization will be the subject of future work. Note that in many substrate designs, nets can be grouped into independent substrate segments containing on average around 10 start and end points in signal routing. AcknowledgementsThe authors would like to thank Minsoo Kim for suggesting the problem and Seungjai Min and Youngjae Gwon at Samsung SDS for helpful discussions. The authors are also grateful to Joung Oh Yun and Minkyu Jung for helpful guidance during the project, and Chanho Min for collaborating on an earlier project. 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[ "Scaling limits of Maximal Entropy Random Walks on a non-negative integer lattice perturbed at the origin", "Scaling limits of Maximal Entropy Random Walks on a non-negative integer lattice perturbed at the origin" ]	[ "Duboux Thibaut \nInstitut de Mathématiques de Bourgogne (IMB)\nUMR CNRS 5584\nUniversité de Bourgogne Franche-Comté\n21000DijonFrance\n", "Offret Yoann \nInstitut de Mathématiques de Bourgogne (IMB)\nUMR CNRS 5584\nUniversité de Bourgogne Franche-Comté\n21000DijonFrance\n" ]	[ "Institut de Mathématiques de Bourgogne (IMB)\nUMR CNRS 5584\nUniversité de Bourgogne Franche-Comté\n21000DijonFrance", "Institut de Mathématiques de Bourgogne (IMB)\nUMR CNRS 5584\nUniversité de Bourgogne Franche-Comté\n21000DijonFrance" ]	[]	We investigate random walks on the integer lattice perturbed at the origin which maximize the entropy along the path or equivalently the entropy rate. Compared to usual simple random walks which maximize the entropy locally, those are a complete paradigm shift. For finite graph, they have been introduced as such by Physicists in [BDLW09] and they enjoy strong localization properties in heterogeneous environments. We first give an extended definition of such random walks when the graph is infinite. They are not always uniquely defined contrary to the finite situation. Then we introduce our model and we show there is a phase transition phenomenon according with the magnitude of the perturbation. We obtain either Bessel-like or asymmetric random walks and we study there scaling limits. In the subcritical situation, we get the classical three-dimensional Bessel process whereas in the supercritical one we get a recurrent reflected drifted Brownian motion. We produce a unify proof relying on the equivalence between submartingale problems and reflected stochastic differential equations obtained in[KR17]. Finally, we remark that applying our results to a couple of particles subject to the famous Pauli exclusion principle, we can recover the electrostatic force assuming only the maximal entropy principle. Also, we briefly explain how to retrieve those results in the macroscopic situation without go through the microscopic one by using the Kullback-Leibler divergence and the Girsanov theorem.	null	[ "https://arxiv.org/pdf/2203.05274v1.pdf" ]	247,363,004	2203.05274	1c326c36c6d568838b00acb9f118e492315eb16b	Scaling limits of Maximal Entropy Random Walks on a non-negative integer lattice perturbed at the origin March 11, 2022 Duboux Thibaut Institut de Mathématiques de Bourgogne (IMB) UMR CNRS 5584 Université de Bourgogne Franche-Comté 21000DijonFrance Offret Yoann Institut de Mathématiques de Bourgogne (IMB) UMR CNRS 5584 Université de Bourgogne Franche-Comté 21000DijonFrance Scaling limits of Maximal Entropy Random Walks on a non-negative integer lattice perturbed at the origin March 11, 2022Random walks Maximum entropy principle Functional scaling limits Reflected diffusions Submartinagle problem Mathematics Subject Classification We investigate random walks on the integer lattice perturbed at the origin which maximize the entropy along the path or equivalently the entropy rate. Compared to usual simple random walks which maximize the entropy locally, those are a complete paradigm shift. For finite graph, they have been introduced as such by Physicists in [BDLW09] and they enjoy strong localization properties in heterogeneous environments. We first give an extended definition of such random walks when the graph is infinite. They are not always uniquely defined contrary to the finite situation. Then we introduce our model and we show there is a phase transition phenomenon according with the magnitude of the perturbation. We obtain either Bessel-like or asymmetric random walks and we study there scaling limits. In the subcritical situation, we get the classical three-dimensional Bessel process whereas in the supercritical one we get a recurrent reflected drifted Brownian motion. We produce a unify proof relying on the equivalence between submartingale problems and reflected stochastic differential equations obtained in[KR17]. Finally, we remark that applying our results to a couple of particles subject to the famous Pauli exclusion principle, we can recover the electrostatic force assuming only the maximal entropy principle. Also, we briefly explain how to retrieve those results in the macroscopic situation without go through the microscopic one by using the Kullback-Leibler divergence and the Girsanov theorem. Introduction The most popular way to randomly explore a given locally finite graph G, without any further information, is to assume that the walker sitting at a node of out degree k jumps onto any neighboring node with uniform probability 1/k, and that independently at every time .This Markov process is called a Generic Random Walk (GRW) and this choice among all the random walks we can consider is that maximizing the entropy production at each step. The concept of entropy introduced by Ludwig Boltzmann 1870s is fundamental in the fields of Statistical Physics and Thermodynamics but also in the field of Information Theory developed by Claude Shannon in the 1940s. We refer to their groundbreaking papers [SM15,Sha48]. Here, all we need to known is that the entropy of a distribution µ on a countable set E is defined by H(µ) = − x∈E µ(x) ln(µ(x)) ∈ [0, ∞]. (1.1) When X is a random variable on E, the quantity H(X) has to be understood as the entropy of the distribution of X. Besides, if N = card(E) is finite, the maximum of H(µ) is achieved when µ is the uniform probability measure on E and it is equal to ln(N ). Regarding the Markov chain theory on a countable state space E, a quantity of particular interest is the entropy rate h = lim n→∞ H(X 0 , · · · , X n ) n = lim n→∞ 1 n c∈Cn µ n (c) ln(µ n (c)), (1.2) where µ n denotes the distribution of the Markov chain trajectory (X 0 , · · · , X n ) ∈ C n . When the latter is irreducible and positive recurrent, h does not depend on the distribution of X 0 and it can be written from its invariant probability measure π and its transition kernel (p(x, y)) x,y∈E as h = − x,y∈E π(x)p(x, y) ln(p(x, y)). (1.3) When the stochastic process is stationary, that is when X 0 is distributed as π, it has to be noted that the sequence in the right-hand-side of (1.2) is constant. Concerning Markov chain and entropy we refer to [Khi57,EC93] for instance. Besides, since for an irreducible GRW on a finite lattice, the probability of finding the particle at node x is proportional to the outer degree d(x) in the stationary state, it comes h GRW = x∈E d(x) ln(d(x)) x∈E d(x) . (1.4) The Maximum Entropy Random Walks (MERWs) are somehow a paradigm shift, from local to global. Roughly speaking, we are looking for irreducible random walks on a graph maximizing the path entropy or, equivalently, the entropy rate. This formulation has been introduced recently in [Dud12, BDLW10,BDLW09]. Among others results, the authors highlight strong localization phenomenon of MERWs for slightly disordered environments, which is particularly relevant in Quantum Mechanics when we study the phenomenon of Anderson localization. We refer for instance to [K16] for a mathematical survey. Also, this notion of MERW is very close to that of Parry measure for sub-shifts of finite type defined in [Par64]. Also, it has a hand with the iteration of matrices, it appears between the lines in [Het84,AGD94], and it could be useful to study complex networks as in [SGGnL + 11,DL11,DM05]. Indeed, when the graph is finite and irreducible, the Perron-Frobenius theorem insures that such random walk exists and is unique. Its Markov kernel and invariant probability measure is given by p(x, y) = A(x, y) ψ(y) ρψ(x) and π(x) = ψ 2 (x), (1.5) where A is the adjacency matrix of G, ρ its spectral radius and ψ the associated 2 (G)-normalized positive eigenfunction. In that case the entropy rate is nothing but h M ERW = ln(ρ). Surprisingly, or not, all the trajectories of length n between two given points x and y are of the probability given by ψ(y)/ρ n ψ(x). Even if the path distributions are not uniform, conditionally on the their length and the extreme stated they are. Hence we can glimpse all the combinatorial features that can result from MERWs. Besides, the kernel in (1.5) reminds the well-known Doob h-transform appearing in particular when stochastic processes are conditioned to stay in a given domain. We can refer to [Doo01] and [KS10]. The positive eigenfunction ψ is well-known when we classify the influence of nodes in complex networks, it is used in the eigenvector centrality (see [ASGD] for instance). For the Physicists, it can be viewed as a wave function and, more precisely, it is the ground state of a discrete Schrödinger equation whose potential V is given by the opposite of the degree, that is − ∆ψ(x) + V (x)ψ(x) = −ρ ψ(x), (1.6) where ∆ is the discrete Laplacian on the graph and V (x) = − y∈G A(x, y). One can easily generalized such construction replacing the adjacency matrix by a weighted one and we can even add some energetic constrains as it is explained in [Dix15]. However, the mathematical framework of MERW has to be inquired further and a lot remains to be done. To begin with, there is stil few solvable models in which the spectral radius and the associated wave function are explicit. We allude to [OB12] dealing with Cayley trees for instance. Of course, when the graph size is small enough, one can compute them numerically in order to make computer simulations of the MERW but it is not always the case. Furthermore, and it is the main topic of this paper, in the case of infinite lattices, there is no consistent results as far as our knowledge. An infinite periodic lattice is mentioned in [BDLW10] and some diffusion coefficient is compute but many questions had not been raised yet. • What could be the right definition of the MERW in an infinite lattice ? • Is there existence and uniqueness of such MERW ? • What can be said about scaling limits of MERWs compared with those for GRWs ? One of our purpose is to show that many classical continuous-time processes can be reinterpreted as scaling limits of some MERW. The scaling limits archetype are the famous Donsker's results, initiated in [Don51], which gave rise to a fruitful literature helping us to understand how we go from some microscopic states to macroscopic ones. Originally, Donsker's Theorem states under suitable assumptions that scaling limits of GRWs lead to Brownian motions. Outline of the article In this paper we define MERWs on some general infinite lattice in section 2.1. Then we consider our model in section 2.2 : the natural number lattice N 0 = {0, 1, 2, · · · } perturbed at the origin with some weight γ ≥ 0. We refer to Figure 1. We highlight a phase transition phenomenon on the spectral radius and we show in 2.3 when γ = 0 that the MERW can be viewed as a the usual random walk conditioned to stay positive. In section 2.4 we state the functional scaling limits we obtain. When γ = 1 the limit stochastic process is the standard reflected Brownian motion whereas when 0 ≤ γ < 1 it is the celebrated threedimensional Bessel process (2.25). It can be viewed as a Brownian motion conditioned to stay positive and it can be obtain by a Doob h-transform of the Brownian motion. We can allude to [Wil74,Pit75] and to [Lam62,Ale11] regarding Bessel-like random walks. Those Markov processes are transient and thus we say that the origin is repulsive in the subcritical case. In the supercritical situation, when γ = 1 + λ/ √ n, we obtain the ergodic reflected diffusion (2.26). As an application, we study a pair of particle in subject to the Pauli exclusion principle and the maximal entropy principle. We can interpret the electrostatic repulsion as an entropic force as in [Wan10]. We point out that such questions are raised in [Dud12]. Furthermore, we explain in the much more tedious situation of continuous-time stochastic processes how it is possible to directly interpret the latter diffusions as those which maximize some pathwise entropy. Finally, we prove the functional scaling limits in section 3. General framework In order to introduced the MERWs we investigate, we begin with defined the spectral radius analogue to the finite situation. We refer to [VJ67] for more details about infinite positive matrix and especially to Theorem B, p. 364. Definition 2.1 (Combinatorial spectral radius). Let A be the weighted adjacency matrix of a (possibly infinite) irreducible weighted graph G such that V (x) = y∈G A(x, y) is uniformly bounded. Let x, y be any state in G. The combinatorial spectral radius ρ is the inverse of the positive radius of convergence of the resolvent ∞ n=0 A n (x, y) z n It does not depend on x, y. Roughly speaking, the combinatorial spectral radius ρ means that the leading asymptotic of the weighted-number of n-step trajectories from x to y is of order ρ n . Definition 2.2 (MERW). Let G be a graph satisfying the assumptions of Definition 2.1 with the same notations. A MERW on G is any random walk whose transition probabilities can be write as p(x, y) = A(x, y) ψ(y) ρ ψ(x) , (2.7) where ψ a positive eigenvector of A associated with ρ. There exists at least one MERW. We have to note that contrary to the finite situation, there exist a unique or an infinity of MERWs. A simple example can be explained when G = Z with the usual nearest neighbors relations A(x, x ± 1) = 1 for every x ∈ Z and the loops A(x, x) = 1 excepted when x = 0. In that case it can showned that ρ = 3 and ψ + (x) = 1 + x1 {x≥0} and ψ − (x) = ψ + (−x) are two positive eigenfunctions. As a matter of facts, any positive eigenfunction with ψ(0) = 1 can be written as a convex combination λψ + + (1 − λ)ψ − with 0 ≤ λ ≤ 1. The eigenfunctions ψ ± are extremal. This is close to the theory of positive harmonic functions and the Martin or Poisson boundary. See for instance [Doo01]. However, one a sufficient criterion for the uniqueness which can be found again in [VJ67]. Proposition 2.1 (R-recurrence). With the same assumptions on G as in the latter definitions, if A is R-recurrent, that is when n≥0 A n (x, y)ρ −n = ∞ for any or some x, y ∈ G, then there exists a unique MERW. In this case, it is positive-recurrent. The model Let γ ≥ 0 be and consider the nearest neighborhood weighted lattice on N 0 given by the weighted adjacency matrix A defined by A 0,0 = γ, A 0,1 = 1 and A n,n±1 = 1 for every n ≥ 1. When γ = 1 the graph is regular since every vertex is of degree 2. Otherwise, the degree of the origin is less or greater than 2. The latter can be called repulsive or attractive as we shall explain in the sequel. We first note there is a phase transition phenomenon regarding the combinatorial spectral radius. Proposition 2.2. The combinatorial spectral radius satisfies ρ = 2, if γ ≤ 1. γ + 1 γ , if γ ≥ 1. (2.8) Proof. First assume that 0 ≤ γ ≤ 1. One has C n ≤ A 2n 0,0 ≤ 4 n where C n is the n-th Catalan number. Note that A n 0,0 as soon as n is an odd number. Since C n ∼ π −1/2 n −3/2 4 n , we obtain that ρ = 2. Finally, assume that γ > 1. Set a n = A n 0,0 and thereafter G(z) = ∞ n=0 a n z n and S(z) = ∞ n=0 C n z 2n = 1 − √ 1 − 4z 2 2z 2 . (2.9) Let R = 1/ρ be the radius of convergence of G Following the powerful methods of algebraiccombinatorics, illustrated in particular in [Fla80,Ban01,BF02] for instance, one has G(z) = 1 1 − γz − z 2 S(z) , (2.10) for any \|z\| < R. One can see that these two terms are equal as formal power series to k∈N n −1 ,n 0 ,n 1 ··· ∞ i=−1 n i =k k n −1 , n 0 , n 1 , · · · γ n −1 ∞ i=0 a n i i z n −1 + ∞ i=0 n i (i+2) . (2.11) Indeed, it follows directly from the power series expansion of (1 − X) −1 and the multinomial theorem that the right-hand-side of (2.10) is equal to (2.11). Concerning the left-hand-side, we can describe any trajectory from 0 to 0 with some numbers n 0 , n 1 , · · · of trajectories into {1, 2, · · · }, whose respective length are 0, 1, · · · , starting and ending to 1, after the walker moves from 0 to 1 and followed by a return to 0. To complete the description, we also need to count the number n −1 of times the edge (0, 0) is taken. The length of the trajectory is nothing but than n −1 + i≥0 n i (i + 2) whereas k = n −1 + n 0 + n 1 + · · · is the number of sub-trajectories in the decomposition which can rearrange as many times as the multinomial coefficient in the later expression does. In particular, since all the coefficients involved are non-negative, the power series (2.11) is absolutely convergent on [0, R[ as G but also, as the right-hand side of (2.10), on [0, x * [ where x * is the maximal positive number for which 0 ≤ γx + x 2 S(x) < 1. One can easily check that x * = (γ + 1/γ) −1 . It is then straightforward to see that R = x * and thus ρ = γ + 1/γ. Proposition 2.3. Given γ ≥ 0 there exists a unique MERW on the non-negative integer. Let P be its probability transitions. 1. If γ < 1 then P n,n+1 = 1 2 2 − γ + (1 − γ)n 1 + (1 − γ)n , P n,n−1 = 1 2 γ + (1 − γ)n 1 + (1 − γ)n and P 0,0 = γ 2 . (2.12) Here n, n + 1, n − 1 belong to N 0 . Besides, the MERW is transient and π n = (1 + (1 − γ)n) 2 is an invariant measure for this Markov chain. 2. If γ = 1 then the transition probabilities are still given by (2.12) and thus P n,n+1 = 1/2 and P n,n−1 = 1/2 for every n ≥ 1 and P 0,0 = 1/2. It is null-recurrent and its invariant measure is up to some multiplicative constant the counting measure on N 0 . 3. If γ > 1 then P n,n+1 = 1 1 + γ 2 , P n,n−1 = γ 2 1 + γ 2 and P 0,0 = γ 2 1 + γ 2 . (2.13) Furthermore, the MERW is positive recurrent and its invariant probability measure is given by the geometric distribution on N 0 of parameter 1 − 1/γ 2 , that is π n = 1 γ 2n 1 − 1 γ 2 . (2.14) Remark 2.1. When 0 ≤ γ < 1 the MERW is a particular case Bessel-like random walk as they are defined in [Ale11]. Proof. We are looking for solutions ψ : N 0 −→ (0, ∞) of ψ n+1 + ψ n−1 = ρ ψ n , n > 0. ψ 1 + γψ 0 = ρ ψ 0 . (2.15) It follows from the boundary conditions that ψ 0 being fixed, say for instance ψ 0 = 1, ψ 1 is uniquely defined as all the ψ n , n ∈ N 0 . This proves the uniqueness of the associated MERW. If γ ≤ 1, it suffices to solved the characteristic polynomial X 2 − ρX + 1 = 0 and to used the boundary condition to get the latter expressions. When γ > 1, the same method applies but it is more tedious. Solving the characteristic polynomial X 2 − (γ + 1/γ)X + 1 = 0 we get that any solution takes the form ψ n = λ γ n + µ 1 γ n . (2.16) Since ψ 1 = (1/γ)ψ 0 it comes that λ = 0 and this leads to (2.13). Regarding the recurrence or transience properties, we begin with γ = 0. Recall that A 2n 0,0 = C n is the n-th Catalan number. It follows that the corresponding MERW is transient since n≥0 P n 0,0 = n≥0 C n 4 n < ∞. (2.17) More generally, when γ < 1, we use the generating function G in (2.9). As previously, one can check that G satisfies (2.10) on [0, 1/2]. Since A 2n 0,0 ≥ C n the radius of convergence R of G is equal to 1/2 and we obtain G(1/2) = n≥0 P n 0,0 = 2 1 − γ < ∞. (2.18) Similarly, when γ = 1, G(1/2) = ∞ and we deduce the recurrence of the MERW. The latter is null recurrent since the invariant measure is ψ 2 i = (1 + i) 2 . When γ > 1 one has ψ 2 i = γ −2i and this completes the proof. A random walk conditioned to stay positive Proposition 2.4. Let {S n } n≥0 be the simple symmetric random walk on Z. Then the distribution of S conditioned to stay in N 0 is the MERW when γ = 0. Remark 2.2. It is well-known that a Brownian conditioned to stay positve is nothing but a threedimensional Bessel process (see [Pit75]) for instance). Furthermore, functional scaling limits of random walks conditioned to stay positive have also been investigated since [Igl74,Bol76] and lead to the same limit under suitable assumptions. Even if Proposition (2.4) must be well-known, we do not find any reference and we present an elementary proof. Proof. Let us introduce τ = inf({n ≥ 0 : S n = −1}). One has P x (τ = n) = x + 1 n P 0 (S n = x + 1). (2.19) Indeed, we first write P x (S 1 > −1, · · · , S n−2 > −1, S n−1 = 0, S n = −1) = P 0 (S n−1 = x) − P 0 (S n−1 = x, ∃ 0 ≤ k ≤ n − 1 S k = x + 1) 2 . (2.20) Then applying the reflexion principle, we get P x (τ = n) = P 0 (S n−1 = x) − P 0 (S n−1 = x + 2) 2 . (2.21) Besides, one can check that P 0 (S n = y) = 1 2 n n n+y 2 ∼ n→∞ 1 √ πn , (2.22) when n + y ∈ 2Z and 0 ≤ n + y ≤ 2n. Furthermore, we deduce from the latter asymptotic that P x (τ > n) ∼ 2(x + 1) √ πn . (2.23) Finally, given any path x → x 1 → · · · → x k−1 → y into N 0 , one can write P x (S 1 = x 1 , · · · , S k−1 = x k−1 , S k = y\|τ > n) = P x (S 1 = x 1 , · · · , S k−1 = x k−1 , S k = y) P y (τ > n − k) P x (τ > n) ∼ n→∞ 1 2 k y + 1 x + 1 . (2.24) Functional scaling limits Before introduce our main result, we need to introduce some continuous-time stochastic processes involved in the scaling limits and we briefly recall some elementary settings about functional convergence in distribution. In the sequel, let x ∈ [0, ∞[ be fixed and denote by {B t } t≥0 a one dimensional standard Brownian motion. Also, we introduce a three-dimensional Bessel process {Y t } t≥0 starting at x. We recall that Y takes its values in [0, ∞[ and satisfies the stochastic differential equation dY t = dB t + 1 Y t dt, Y 0 = x. (2.25) For these two stochastic processes and much more we refer to [RY99]. Note that Y is transient and has for invariant measure m(dx) = x 2 dx on [0, ∞[. Also, we consider W t = \|x + B t \| the reflecting Brownian motion starting from x. Finally, let λ > 0 fixed and let {(Z t , L t )} t≥0 be the solution of the following reflected stochastic differential equation    dZ t = dB t − λdt + dL t , Z 0 = x Z t ≥ 0 (2.26) Here L t is a B t -measurable non-decreasing stochastic process such that L 0 = 0 and It is classical that such solution exists and is unique, one can refer to [Tan79] or [C98] for a more general setting. Besides, one can check that Z is a strong Markov process which is positive recurrent and whose invariant probability distribution is the exponential distribution of parameter 2λ. Finally, we endowed the space D of càdlàg real function on [0, ∞) with the topology of uniform convergence on compact sets. In the sequel, we denote by =⇒ the convergence in distribution of stochastic processes in D with the associated Borel σ-field. We refer to [Bil99] or [Whi02] for a more extensive survey on functional convergences in distribution. Theorem 2.1. Let {X k } k≥0 be the MERW given in (2.3). Assume X 0 = x n 0 is deterministic and depends on n in such way x n 0 √ n −−−→ n→∞ x. (2.28) Then, one has the following functional scaling limits. 1. If γ < 1 then X nt √ n t≥0 ===⇒ n→∞ {Y t } t≥0 . (2.29) 2. If γ = 1 then X nt √ n t≥0 ===⇒ n→∞ {W t } t≥0 . (2.30) 3. If γ := 1 + λ √ n then X nt √ n t≥0 ===⇒ n→∞ {Z t } t≥0 . (2.31) Remark 2.3. Since the increments of {X k } k≥0 are bounded, the same results hold replacing the stepwise constant stochastic processes by there continuous-times interpolations. Remark 2.4. When 0 ≤ γ < 1 this result can be obtained by using [Lam62]. However, we choose to present a more modern proof which relies on submartingale problems and which can be naturally extended to the critical situations when γ = 1 or γ = 1 + λ/ √ n. Application to some exclusion process Let us consider the following two-bodies problem. We consider two particles on the integers we can only jump to the right or the left but which can not occupy the same site. This is the lattice structure of the usual exclusion process on the integers with only two particles. We refer to [Der98,Sch01] for reviews on this subject. By symmetry, Ψ(x, y) = 1 + (y − x) is clearly a positive eigenfunction of the exclusion lattice associated with the combinatorial spectral radius ρ = 4. Here x denotes the position of the first particle and y the position of the second one and we suppose x < y. Such Ψ is not necessarily unique, as the example below Definition 2.2 shows, but if we assume only jumps in one direction (the totally asymmetric situation) the latter is unique and it is then associated to the spectral radius ρ = 2. Anyway, we get in the scaling limit d(Y t − X t ) = dB t + dt Y t − X t , (2.32) where X t < Y t denote the position of the particles at time t. Here the interesting point is we make appeared the usual electrostatic force assuming only the maximal entropy constrain. Continuous-time counterpart of MERW In the light of these scaling limits, we can ask whether the three-dimensional Bessel process, the reflected Brownian motion or the reflected drifted Brownian motion can be interpreted as maximal entropy stochastic processes without the intermediate of MERWs. To go further, the distribution γ n of the first n-steps of the MERW when γ = 1 is uniform since the latter is nothing but than a simple random walk on a regular graph. As a product, maximize the entropy in the right-hand side of (1.2) is equivalent to minimize the Kullback-Leibler divergence (also known as the relative entropy) D KL (µ n \|\|γ n ). We recall that given two probability measures µ, γ such that µ is absolutely continuous with respect to γ, D KL (µ γ) = ln dµ dγ dµ, (2.33) where dµ/dγ denotes the Radon-Nikodym derivative. We still denote by W the reflected Brownian motion on [0, ∞). It is well-known it can be written as W t = B t + L t where B is a standard brownian motion and L t is an increasing process which is σ(B t )-measurable and such that L 0 = 0 and the Stieltjes random mesure dL is carried by {t ≥ 0 : W t = 0}. In addition, we denote by P x the distribution of W when W 0 = x. This is a distribution on the canonical space C([0, ∞[, R) endowed with the standard σ-field F generated by cylinder sets. To go further, we denote by F t the standard filtration generated by the cylinder set up to the time t. Then, let ϕ be an absolutely continuous non-negative function on [0, ∞) such that U = {ϕ > 0} is an open set of [0, ∞). Introduce τ = inf{s ≥ 0 : W s ∈ U } and set for every t ≥ 0, M t = exp t 0 ϕ (W s ) ϕ(W s ) dB s − 1 2 t 0 ϕ (W s ) ϕ(W s ) 2 ds 1 {t<τ } . (2.34) Note that τ = ∞ as soon as ϕ is positive. By using the results in [FOT11, Chap. 6.3] one can see that {M t } t≥0 is an F t -martingale under P x for all x ∈ U . Let Q x be the distribution on C([0, ∞), R) defined by dQ (t) x = M t dP (t) x , (2.35) where Q (t) x and P (t) x stand for the restriction of Q x and P x to F t . The law of W under Q x is a ϕ 2 (x)dx-symmetric Markov process that never reaches ∂U when x ∈ U . Furthermore, by using the Girsanov theorem, which can be found in [RY99] for instance, it comes that B t = B t − t 0 ϕ (W s ) ϕ(W s ) ds, (2.36) defined a Brownian motion under Q x and the stochastic process W satisfies the refected stochastic differential equation d W t = d B t + ϕ ( W t ) ϕ( W t ) dt + d L t , (2.37) where W 0 = x and L is distributed as L under Q x . We deduce that D KL Q (t) x P (t) x = E x   t 0 ϕ ( W s ) ϕ( W s ) 2 ds   . (2.38) However, when ϕ 2 (x)dx is not a finite measure, it can happen that lim t→∞ 1 t D KL Q (t) x P (t) x = ∞. (2.39) Therefore, we are not able to retrieve the three-dimensional Bessel process only by minimizing the right-hand side of (2.38). On the contrary, we can state the following result whose proof is straighforward. Lemma 2.1. Assume that π(dx) = ϕ 2 (x)dx is a probability measure on U . Then for every x ∈ U and s > 0 the relative rate entropy h is equal to lim t→∞ 1 t D KL Q (t) x P (t) x = U (ϕ (x)) 2 dx = 1 s D KL Q (s) π P (s) π . (2.40) Hence, if we assume that ϕ(x) > 0 on ]0, L[ and ϕ(x) = 0 otherwise for some L > 0, and if we are looking for such function which minimizes (2.40), we obtain easily that h = π L 2 and ϕ(x) = 2 L sin π L x . (2.41) Thereafter, we retrieve the three-dimensional Bessel process by letting L goes to infinity since for all x > 0, ϕ (x) ϕ(x) ∼ L→∞ 1 x . (2.42) Regarding the case when γ = 1 + λ √ n , we need to add constraints on ϕ. We still assume that ϕ 2 (x)dx is a probability distribution on [0, ∞) but we require that ∞ 0 xϕ 2 (x)dx = 1 2λ . (2.43) Let ϕ be such minimizer and let δ be a compactly supported smooth function with δ(0) = 0 and ∞ 0 δ(x)dx = 0. Then by considering ϕ + εδ for sufficiently small ε > 0 and looking at the first and second order terms in front of ε and ε 2 , we obtain necessarily − ϕ (x) + xβϕ(x) = −αϕ(x) (2.44) and ∞ 0 (δ (x)) 2 dx + β ∞ 0 x(δ(x)) 2 dx + α ∞ 0 (δ(x)) 2 dx ≥ 0. (2.45) Here α, β are some Lagrange multiplier and the latter ordinary differential equation as to be understand in a weak sense when ϕ is not twice differentiable. Equation (2.44) is nothing but a Schrödinger equation in a linear (triangular if we symmetrise it) potential. When β = 0, solutions can be written as A Ai(z) + B Bi(z) with z = −β −1/3 (x + α) and A, B ∈ R where Ai and Bi are the Airy functions of first and second kinds. This can be proved by power series expansions or Fourier transform for instance. However, (2.45) implies that α, β ≥ 0 since δ is an arbitrary perturbation. It turns out that Ai(z) and Bi(z) are oscillating around zero when z goes to −∞ in such way that no non-negative solutions exist when β > 0. One has β = 0 and then the unique non-negative normalized square integrable solution satisfying (2.43) is ϕ(x) = √ 2λe −λx and we can check that ϕ (x) ϕ(x) = −λ. (2.46) We retrieve the reflected diffusion (2.26). Proof of Theorem 2.1 We prove this theorem following the usual scheme : to begin with, we prove the tightness, then we identify the limit by showing that it satisfies a martingale problem for which uniqueness holds. When γ < 1 we choose to work with square of X to remove the singularity in 1/x appearing in the drift of the three-dimensional Bessel process. Tightness 1. Case 0 ≤ γ ≤ 1. We shall apply [EK86, Theorem 8.6 & Remark 8.7, pp. 136-137 ]. We assume more generally that X 0 ∈ L 4 and X 2 0 n L 2 −−−→ n→∞ Y 0 , (3.47) for some random variable Y 0 . For the sake of readability, we omit the dependence on n. First, by using the discrete-time version of the Ito's formula which can be found in p. 180] or [Nor98,p. 132] for instance, we write X 2 nt n = X 2 0 n + 1 n nt −1 k=0 (P − I)g(X k ) + M nt , (3.48) Where I is the identity operator, g(x) = x 2 and {M n } n≥0 is some square integrable martingale with M 0 = 0. Besides, ones can check by using (2.12) that (P − I)g(x) = 2(1 − γ)x 1 + (1 − γ)x + 1 1 x>0 + 1 − γ 2 1 x=0 ∈ [0, 3]. (3.49) Set F n = σ(X 0 , · · · , X n ). We obtain for any t ≥ 0 and n ≥ 1, E X 2 nt n F 0 ≤ X 2 0 n + 3t. (3.50) This implies the tightness of X 2 nt /n n≥1 for any t ≥ 0. To go further, let 0 ≤ t ≤ T and 0 ≤ u ≤ δ with T, δ > 0. To go further, we shall bound E   X 2 n(t+u) n − X 2 nt n 2 F nt   . (3.51) To this end, by using (3.48) note that the square in the latter expression can be bounded by 2   1 n n(t+u) −1 k= nt (P − I)g(X k )   2 + 2 n 2 M n(t+u) − M nt 2 =: S 1 + S 2 . (3.52) Then, as previously, we easily obtain E S 1 \|F nt ≤ 18 nδ + 1 n 2 . (3.53) In order to bound the second term, we use again (3.48) and we remark that E[(M k+1 − M k ) 2 \|F k ] = V(X 2 k+1 − X 2 k \|F k ) ≤ (4X 2 k + 1). (3.54) Here we use that X k+1 − X k ∈ {±1}. It follows from the orthogonality of the increments of a square integrable martingale that E[S 2 \|F nt ] ≤ 4 n n(t+u) −1 k= nt E X 2 nk/n n F nt + n(t + u) − nt n 2 . (3.55) Therefore, by using (3.50) and the Markov property, we get that (3.51) is bounded by 2. Case γ = 1 + λ/ √ n. We keep the main notations of the case when 0 ≤ γ ≤ 1. We shall prove the tightness by a submarginal argument which can be found in [SV06, Chapter 1.4.]. As a matter of facts, this method applies to continuous stochastic processes in [SV06] but it can be directly extended to càdlàg ones. We first note that the drift of X satisfies by using Proposition 2.3 the asymptotic n − X 2 nt n 2 F nt   , (3.59) we deduce that E   X 2 n(t+u) n − X 2 nt n 2 F nt   ≤ E γ n (δ)\|F nt ,(3.d(x) = E[X k+1 − X k \|X k = x] ∼ n →∞ − λ √ n 1 x>0 + 1 2 1 x=0 . (3.62) Let f be a compactly supported non-negative smooth function on the real line. Set ∆f (x) = f x + 1 √ n − f x − 1 √ n and ∆ 2 f (x) = f x + 1 √ n − 2f x √ n + f x − 1 √ n . (3.63) Note that \|∆f (x)\| ≤ 2 f ∞ √ n and \|∆ 2 f (x)\| ≤ f ∞ n . (3.64) To go further, one can check by using by the discrete-time version of Ito's formula f X nt √ n = f X ns √ n + nt −1 k= ns 1 2 ∆ 2 f (X k ) + d(X k )∆f (X k ) 1 X k >0 + card({ ns ≤ k < nt : X k = 0})d(0) f 1 √ n − f (0) + M nt − M ns , (3.65) where M is some square integrable martingale. Thereafter, one can see that there exists C depending only on f ∞ \|, f ∞ and λ such that for all integers x, n ≥ 1, 1 2 ∆ 2 f (x) + d(x)∆f (x) 1 x>0 ≤ C n . (3.66) Fix ε > 0 and assume that f (x) ≡ 0 outside ] − ε, ε[ and f (x) ≡ 1 on [−ε/2, ε/2]. In the following, for any a ≥ 0 we set f a (x) = f (x − a). Then by using (3.66), (3.65) and the Markov property, we deduce that for all a ≥ 0 and all n ≥ 1 such that 1/ √ n ≤ ε/2, E f a X nt √ n − f a X ns √ n F ns ≥ −C(t − s) − C n . (3.67) Here we use d(0) ≥ 0 and f a (1/ √ n) − f a (0) ≥ 0. Then, by slightly adapting the proof of [SV06, Theorem 1.4.11, p. 41] -in particular Lemmas 1.4.4 and 1.4.10 -we deduce the tightness. As a matter of facts, it implies the tightness of the sequence of linear piecewise interpolation of t −→ X nt / √ n. However, it is well-known that in the space of càdlàg functions the Skorohod convergence is equivalent to the local-uniform one when the limit point is continuous, so both of the two-latter sequences of stochastic processes are tight or neither. Limit process We shall prove that any limit process is a solution to a submartingale problem in the spirit of [SV06,p. 144]. Submartingale arise naturally when we deal with reflected diffusions. As a matter of facts, we apply [KR17] to come down to a classical reflected stochastic differential equation for which existence and uniqueness is well known. The key point here is to show that the time spent at the boundary x = 0 is 0 for any limit process. For that we will use the following lemma whose proof is postponed to the end of this paper. n→∞ E 0 card({0 ≤ k < nu : X k ≤ η √ n}) n ≤ 2λ(u + v)(1 − e −2λη ) C v,λ . (3.68) Besides, one can choose C v,λ = P U + V 2λ √ v + 2λ √ v ≤ 0 , (3.69) with U ∼ N (0, 1) and V ∼ E(1) two independent random variables. Let us introduce, at least informally, the infinitesimal generators associated with the Markov processes Y , W and Z involved in Theorem 2.1 and respectively given by L Y f (x) = 1 2 f (x) + 1 x f (x), L W f (x) = 1 2 f (x) and L Z f (x) = 1 2 f (x) − λf (x). (3.70) In each of these three cases, we shall prove that any limit point is a solution of a martingale or submartingale problem associated with the corresponding generator. To be more precise, we deal with the generator of the square of Y given by where M is some square integrable martingale and L n f (x) = n(P − I)g n ( √ nx), with g n (x) = f x 2 n . L Y 2 f (x) = 1 2 4xf (x) + 3f (x). (3.73) First, we shall prove that L n f (x) −−−→ n→∞ L Y g( √ x), with g(x) = f (x 2 ), (3.74) uniformly on compact subset of (0, ∞) when f is continuously twice differentiable with bounded derivatives. Note that L Y g( √ x) = L Y 2 f (x). (3.75) As for (3.62) one can compute the drift of the corresponding MERW which is given by d(x) = (1 − γ) 1 + (1 − γ)x 1 x≥1 + 1 − γ 2 1 x=0 . (3.76) Then, one can check for every x ∈ {1/n, 2 2 /n, 3 2 /n, · · · } that L n f ( x) = A n f (x) + B n f (x) with A n f (x) = 1 2 f x + 1 n + 2 √ x √ n + f x + 1 n − 2 √ x √ n − 2f (x) 1/n and B n f (x) = d( √ nx) 2 f x + 1 n + 2 √ x √ n − f x + 1 n − 2 √ x √ n 1/n . (3.77) Besides, simple asymptotic expansion leads to Regarding the second term, we use A n f (x) = 1 2 4xf (x) + f (x) + O 1 √ n ,(3.d( √ nx) = 1 √ nx 1 − 1 (1 − γ) √ nx + O 1 nε , (3.79) as soon as x ≥ ε > 0 with a uniform constant in the O. It follows that B n f (x) = 2f (x) + O 1 √ n , (3.80) the O being uniformly bounded when x belongs to a compact subset of (0, ∞). As a consequence, we get (3.74) from (3.78) and (3.80). Thereafter, we shall prove that L n f (x) = L Y 2 f (x) + κ n (x)f (0) + O 1 √ n , (3.81) where κ n (x) ≥ 0, uniformly on any compact subset of R + . First note that L n f (0) = 1 − γ 2 f (1/n) − f (0) 1/n = 1 − γ 2 f (0) + O 1 n . (3.82) The asymptotic (3.78) still holds when 0 < x < ε whereas we need to be more sharp for B n f (x). Note that B n f (x) = 2f (x) + 2 1 1 + 1/ √ nx − 1 f (x) + O ε 1 √ n . (3.83) Since 0 < x < ε and nx ≥ 1 one can write f (x) = f (0) + xf (0) + O(x 2 ) and 2x 1 1 + 1/ √ nx − 1 = O ε n . (3.84) Finally, we deduce (3.81) with κ n (x) = 1 + 2 1 1 + 1/ √ nx − 1 1 x>0 + 1 − γ 2 1 x=0 . (3.85) We have to recall that since x ∈ {1/n, 2 2 /n, 3 2 /n, · · · } one has nx ≥ 1 and thus κ n (x) ≥ 0. To conclude, standard arguments as in [SV06,p. 144] apply and we obtain for arbitrary limit point {ω(t)} t≥0 that f (ω(t)) − t 0 L Y 2 f (ω(u)) 1 {ω(u)>0} du t≥0 , (3.86) is a submartingale as soon as f (0) ≥ 0. In order to apply [KR17] it remains to show that which implies the desired result by using the Fatou's lemma and (3.88). Therefore, we can omit the indicator function in (3.86). Then by using [KR17] and [Tan79] we can say that any limit point is a squared Bessel process. 2. Case γ = 1. Regarding the situation when γ = 0, the proof is more classical. It can be done with the same arguments, better, we do not have to consider the square of the process, as in the last situation. 3. Case γ = 1 + λ/ √ n. The proof is very similar. Let us focus on the few differences with the situation when γ ≤ 1. First, similarly to (3.72) we write f X nt √ n = f X 0 √ n + 1 n nt −1 k=0 L n f X n k/n √ n + M nt . (3.90) where M square integrable martingale and L n f (x) = n(P − I)g n (x √ n) with g n (x) = f x √ n . One can compute the drift of the corresponding MERW which is given by d(x) = 1 − γ 2 1 + γ 2 1 x≥1 + 1 1 + γ 2 1 x=0 . (3.91) Then for every x ∈ {1/ √ n, 2/ √ n, 3/ √ n, · · · } one has L n f (x) = A n f (x) + B n f (x) with A n f (x) = n 2 f x + 1 √ n + f x − 1 √ n − 2f (x) and B n f (x) = nd(x √ n) 2 f x + 1 √ n − f x − 1 √ n . (3.92) Simple asymptotic expansion leads to uniformly when x ∈ {1/ √ n, · · · } belong to a given compact subset of [0, ∞). One can see that A n f (x) = 1 2 f (x) + O 1 √ n ,(3.L n f (0) = 1 1 + γ 2 f (1/ √ n) − f (0) 1/ √ n = 1 2 f (0) + O 1 √ n . (3.96) We deduce that is a submartingale as soon as f (0) ≥ 0. We can omit the indicator function in (3.98) by using Lemma 3.1 when γ = 1 + λ/ √ n. Then thanks to [KR17] and [Tan79], we can say that any limit point is distributed as Z which completes the proof. L n f (x) = L Z f (x) + 1 2 1 {x=0} f (0) + O 1 √ n ,(3. Proof of Lemma 3.1. Let us start by proving the lemma for the case γ = 1 + λ/ √ n. Let N be a random variable distributed as a geometric random variable on N 0 with parameter 1 − 1/γ 2 ∼ 2λ/ √ n as n goes to infinity. Denote by π this distribution and note it is the invariant probability measure of the MERW. It comes This ends the proof. Finally, by a standard coupling argument, the result still holds when 0 ≤ γ ≤ 1. E π card({0 ≤ k < n(u + v) : X k ≤ η √ n}) ∼ n(u + v)(1 − e −2λη Figure 1: The model ∞ 0 1 0{Zt>0} dL t = 0 a.s.. (2.27) δ(1 + δ), (3.57) for some constant K. Applying the latter inequalities for t = 0 and theDoob' Case 0 ≤ γ < 1. Let f be some sufficiently smooth and bounded function. M nt , (3.72) 78) the constant in the O being chosen uniformly when x belong to a given compact subset of [0, ∞). end, by the Lemma 3.1 when 0 ≤ γ ≤ 1, 97)on any compact subset of [0, ∞). To conclude, same arguments as in the previous case apply, we obtain for arbitrary limit point {ω(t)} t≥0 that Lemma 3.1. Let {X k } k≥0 be the MERW given in (2.3). Let u, v, η > 0 be. There exists a positive constant C v,λ such thatlim sup 93 ) 93uniformly when x belong to a given compact subset of [0, ∞). Besides, one has uniformlyd(x √ n) = −λ √ n 1 {x∈{1/ √ n,··· }} + 1 2 1 {x=0} + O 1 √ n . (3.94) It follows that B n f (x) = −λf (x) + O 1 √ n , (3.95) Furthermore, let us introduce T = inf{k ≥ 0 : X k = 0}. One can write by using the strong Markov property that≥ P π (T < nv ) E 0Let (ξ k ) k≥1 be a sequence of i.i.d. Rademacher random variables with parameter 1/(1 + γ 2 ) independent on N . Note that E[ξ k ] ∼ −λ/ √ n. By comparison, one can easily see thatP π (T < nv ) ≥ P √ nvBesides, by using the characteristic function, one can check the Central Limit Theorem). (3.99) E π   n(u+v) −1 k=0 1 {X k ≤η √ n}   = i≥0 P π (T = i) E 0   n(u+v) −1−i k=0 1 {X k ≤η √ n}   (3.100)   nu −1 k=0 1 {X k ≤η √ n}   . (3.101) nv −1 k=0 ξ k nv + λ √ n ≤ − √ nv N nv + λ √ n (3.102) √ nv nv −1 k=0 ξ k nv + λ √ n ===⇒ n→∞ N (0, 1). (3.103) In addition, one has N √ n ===⇒ n→∞ E(2λ). (3.104) Evolutionary Formalism for Products of Positive Random Matrices. Ludwig Arnold, Matthias Volker, Lloyd Gundlach, Demetrius, The Annals of Applied Probability. 43Ludwig Arnold, Volker Matthias Gundlach, and Lloyd Demetrius. Evolutionary Formalism for Products of Positive Random Matrices. 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[ "Distributed Aggregative Optimization over Multi-Agent Networks", "Distributed Aggregative Optimization over Multi-Agent Networks" ]	[ "Xiuxian Li ", "Lihua Xie ", "Yiguang Hong " ]	[]	[]	This paper proposes a new framework for distributed optimization, called distributed aggregative optimization, which allows local objective functions to be dependent not only on their own decision variables, but also on the average of summable functions of decision variables of all other agents. To handle this problem, a distributed algorithm, called distributed gradient tracking (DGT), is proposed and analyzed, where the global objective function is strongly convex, and the communication graph is balanced and strongly connected. It is shown that the algorithm can converge to the optimal variable at a linear rate. A numerical example is provided to corroborate the theoretical result.	10.1109/tac.2021.3095456	null	218,900,751	2005.13436	5b0549efd428157848dba726720208812a8f8485	Distributed Aggregative Optimization over Multi-Agent Networks 27 May 2020 Xiuxian Li Lihua Xie Yiguang Hong Distributed Aggregative Optimization over Multi-Agent Networks 27 May 20201Index Terms-Distributed algorithmaggregative optimizationmulti- agent networksstrongly convex functionlinear convergence rate This paper proposes a new framework for distributed optimization, called distributed aggregative optimization, which allows local objective functions to be dependent not only on their own decision variables, but also on the average of summable functions of decision variables of all other agents. To handle this problem, a distributed algorithm, called distributed gradient tracking (DGT), is proposed and analyzed, where the global objective function is strongly convex, and the communication graph is balanced and strongly connected. It is shown that the algorithm can converge to the optimal variable at a linear rate. A numerical example is provided to corroborate the theoretical result. I. INTRODUCTION Distributed optimization has received immense attention in the past decade, mostly inspired by advanced and inexpensive sensors, big data, and large-scale networks, and so on. In distributed optimization, a network consisting of a family of agents is usually introduced to capture the communication pattern among all agents, where each agent is only accessible to partial (and maybe private) information on the global optimization problem. In this case, the agents in the network aim to cooperatively, by local information exchange, solve the global optimization problem. To date, a large volume of algorithms have been devised for distributed optimization problems. Generally speaking, the existing algorithms can be roughly summarized as two classes: consensusbased algorithms and dual-decomposition-based algorithms. Wherein, consensus-based algorithms employ the consensus idea to align the estimated variables of all agents, for which existing algorithms include distributed subgradient [1], diffusion adaptation strategy [2], fast distributed gradient [3], asynchronous distributed gradient [4], stochastic mirror descent [5], and distributed quasi-monotone subgradient algorithm [6], etc. With regard to dual-decomposedbased algorithms, dual variables are usually introduced by viewing the synchronization of all local variables as equality constraints, including alternating direction method of multipliers (ADMM) [7], EXTRA [8], augmented Lagrangian method [9], distributed dual proximal gradient [10], and distributed forward-backward Bregman splitting [11]. From another viewpoint, a variety of scenarios have so far been considered for distributed optimization. The simplest case is to minimize an objective/cost function without any constraints [1], [8], [12], including feasible set constraints, equality and inequality constraints, where the objective function is separable and composed of local objective functions. A little more complex case is to address distributed optimization with global/local feasible set constraints [13]- [15], that is, the decision variable must stay within some prespecified nonempty set that is often assumed to be closed and convex. X. Li Moreover, the scenario with local (affine) equality constraints are addressed, for example, in [16], while local inequality constraints are investigated such as in [17], and global inequality constraints that can be realized by all agents are taken into account in the literature, see [18] for an example. Furthermore, the case with globally coupled inequality constraints, where individual agent is only capable of accessing partial information on the global inequality constraints, is studied such as in [19]- [24], and meanwhile, time-varying objective functions and/or constraint functions are also considered in recent years [25]- [28]. With careful observation, it can be found that distributed optimization studied in the aforementioned works focus on the case where a global objective function is a sum of local objective functions, which are dependent only on their own decision variables. To be specific, the problem is in the form N i=1 fi(xi) such that xi = xj for all i = j, maybe subject to inequality constraints, from which it is easy to see that each fi is a function with respect to only xi, independent of any other variables xj, j = i. However, in a multitude of practical applications, local objective functions are also determined by other agents' variables. For example, in multi-agent formation control, each objective function often relies on variables (such as positions or velocities) of all its neighbors, and this scenario has been considered such as in [29] and [30] (cf. Remark 4). As another example, the average of all variables, i.e., N i=1 xi/N , is a vital parameter for all agents in a network, which can be discovered from a large number of applications, such as optimal placement problem, transportation network, and formation control, etc. For instance, in formation control, a group of networked agents desire to achieve a geometric pattern, and simultaneously, they may plan to encircle an important target, which can be cast as a target tracking problem for the center of all agents. Therefore, it is significant to deal with the scenario where the average of all variables is involved in local objective functions. From the theoretical perspective, when each local function fi also depends on variables of other agents (such as the average N i=1 xi/N ), the problem will be more challenging since other variables (such as the average N i=1 xi/N ) and related gradients are unavailable to agent i. Motivated by the above facts, this paper aims to formulate and study a new framework for distributed optimization, called distributed aggregative optimization, for which a distributed algorithm, called distributed gradient tracking (DGT), is developed and analyzed. It is shown that the proposed algorithm has a linear convergence speed under mild assumptions, such as strong convexity of the global objective function and a directed balanced communication graph. The contributions of this paper are as follows: (1) a new distributed aggregative optimization is formulated for the first time; (2) a linearly convergent distributed algorithm is proposed and analyzed rigorously; and (3) a numerical example is provided to support the theoretical result. The rest of this paper is structured as follows. Some preliminaries and the problem formulation are provided in Section II, followed by the main result in Section III. In Section IV, a numerical example is presented to corroborate the theoretical result, and the conclusion is drawn in Section V. Notations: Let R n and C be the set of vectors with dimension n > 0 and the set of complex numbers, respectively. Define [k] = {1, 2, . . . , k} for an integer k > 0. Denote by col(z1, . . . , z k ) the column vector by stacking up z1, . . . , z k . Let · , x ⊤ , and x, y be the standard Euclidean norm, the transpose of x ∈ R n , and standard inner product of x, y ∈ R n . Let 1 and 0 be column vectors of compatible dimension with all entries being 1 and 0, respectively, and I be the compatible identity matrix. Let ρ(M ) be the spectral radius of a square matrix M . ⊗ is the Kronecker product. Let J := 1 N 11 ⊤ and J := J ⊗ I with compatible dimension. II. PRELIMINARIES A. Graph Theory The communication pattern among all agents is captured by a simple graph in this paper, denoted by G = (V, E ) with the node set V = {1, . . . , N } and the edge set E ⊂ V × V. An edge (j, i) ∈ E means that node j can send information to node i, where j is called an in-neighbor of i. Denote by Ni = {j : (j, i) ∈ E } the in-neighbor set of node i. The graph G is called undirected if (i, j) ∈ E is equivalent to (j, i) ∈ E , and directed otherwise. The communication matrix A = (aij) ∈ R N×N is defined by: aij > 0 if (j, i) ∈ E , and aij = 0 otherwise. The following standard assumptions on the communication graph are postulated. Assumption 1. The following hold for the interaction graph: 1) The graph G is strongly connected; 2) The matrix A is doubly stochastic, i.e., N j=1 aij = 1 and N i=1 aij = 1 for all i, j ∈ [N ] . It should be noted that some approaches have been brought forward in the literature in order to hold the double stochasticity condition, for example, the uniform weights [31] and the least-mean-square consensus weight rules [32]. In addition, some distributed strategies have been proposed in [33] for strongly connected directed graphs to compute a doubly stochastic assignment in finite time. B. Convex Optimization For a convex function g : R n → R, the subdifferential, denoted as ∂g(x), of g at x is defined by ∂g(x) = {s ∈ R n : g(y) − g(x) ≥ s ⊤ (y − x), ∀y ∈ R n }, and each element in ∂g(x) is called a subgradient. When g is differentiable at x, the subdifferential ∂g(x) only contains one element, which is usually called gradient, denoted as ∇g(x). The differentiable function g is called µ-strongly convex if for all x, y ∈ R n , g(x) ≥ g(y) + ∇g(y) ⊤ (x − y) + µ 2 x − y 2 .(1) C. Problem Formulation This paper proposes a new framework for distributed optimization in a network composed of N agents, called distributed aggregative optimization, given as follows: min x∈R n f (x) := N i=1 fi(xi, σ(x)),(2)σ(x) := N i=1 φi(xi) N ,(3) where x = col(x1, . . . , xN ) is the global decision variable with xi ∈ R n i , n := N i=1 ni, and fi : R n → R is the local objective function. In problem (2), the global function f is not known to any agent, and each agent can only privately access the information on fi. Moreover, each agent i ∈ [N ] is only aware of the decision variable xi without any knowledge of xj's for all j = i. Moreover, the term σ(x) is an aggregative information of all agents' variables, and the function φi : R n i → R d is only accessible to agent i. The goal is to design distributed algorithms to seek an optimal decision variable for problem (2). Remark 1. It should be noted that distributed aggregative optimization is proposed here for the first time, to our best knowledge, which is different from aggregative games [34], [35]. The substantial difference lies in that all agents in problem (2) aim to cooperatively find an optimal variable for the sum of all local objective functions, while the objective of aggregative games is to find the Nash equilibrium in a noncooperative manner since each agent desires to minimize only its own objective function. This can be seen from the following simple example. Example 1. As a simple example, let us consider two agents in a network in the scalar space R without feasible set constraints. Let f1(x) = (x1 − 1) 2 + σ 2 (x) = (x1 − 1) 2 + (x1 + x2) 2 /4 and f2(x) = (x2 − 2) 2 + σ 2 (x) = (x2 − 2) 2 + (x1 + x2) 2 /4. As a result, for distributed aggregative optimization, the optimal variable of f (x) = f1(x) + f2(x) can be easily calculated, by ∇x 1 f (x) = 0 and ∇x 2 f (x) = 0, as x1 = 1/4, x2 = 5/4. On the other hand, as for aggregative games, the Nash equilibrium can be computed, by ∇x 1 f1(x) = 0 and ∇x 2 f2(x) = 0, as x1 = 1/2, x2 = 3/2. It is apparent to see that the Nash equilibrium x1 = 1/2, x2 = 3/2 is not the same as the global optimizer of f (x), i.e., x1 = 1/4, x2 = 5/4. In other words, the Nash equilibrium is generally not the optimal decision variable due to the noncooperative nature of all agents in aggregative games. To move forward, for brevity, let ∇1fi(xi, σ(x)) and ∇2fi(xi, σ(x)) denote ∇x i fi(xi, σ(x)) and ∇σfi(xi, σ(x)), respectively, for all i ∈ ∇1fN (xN , yN )) and ∇2f (x, y) := col (∇2f1(x1, y1), . . . , ∇2fN (xN , yN )). It is now necessary to list some assumptions. Assumption 2. The following hold for problem (2): 1) The global objective function f (x) is differentiable, µ-strongly convex, and L1-smooth on R n , that is, ∇f (x) − ∇f (x ′ ) ≤ L1 x − x ′ for all x, x ′ ∈ R n . Also, ∇1f (x, y) + ∇φ(x)1N ⊗ 1 N N i=1 ∇2fi(xi, yi) is L1-Lipschitz; 2) ∇2f (x, y) is L2-Lipschitz continuous, that is, ∇2f (x, y) − ∇2f (x ′ , y ′ ) ≤ L2( x − x ′ + y − y ′ ) for all x, x ′ ∈ R n and y, y ′ ∈ R Nd ; 3) All φi's are differentiable, and there exists a constant L3 > 0 such that ∇φi(xi) ≤ L3 for all xi ∈ R n i and i ∈ [N ]. It should be noted that the Lipschitz property of ∇f (x) and ∇1f (x, y) + ∇φ(x)1N ⊗ 1 N N i=1 ∇2fi(xi, yi) in Assumption 2.1 can be ensured by Assumptions 2.2 and 2.3 along with the boundedness of ∇2f (x, y) and the Lipschitz property of ∇1f (x, y) and ∇φi(xi), which are standard in distributed optimization and game theory (e.g., [1], [3], [15], [19], [24], [34], [35]). Please also note that it is only assumed the strong convexity of the global objective function f (x), without even the convexity of local objective functions fi's. To conclude this section, it is useful to display a few lemmas. Lemma 3. Let F : R n → R be µ-strongly convex and L-smooth. Then x − α∇F (x) − (y − α∇F (y)) ≤ (1 − µα) x − y for all x, y ∈ R n , where α ∈ (0, 1/L]. Proof. It is known that a convex function f : R n → R is l-smooth is equivalent to the convexity of l 2 x 2 −f (x). Thus, by L-smoothness of F , one has that L 2 x 2 − F (x) is convex, and then 1 2α x 2 − F (x) is ( 1 2α − L 2 )-strongly convex for α ∈ (0, 1/L], which further implies that H(x) := 1 2 x 2 − αF (x) is ( 1 2 − Lα 2 )-strongly convex. Meanwhile, it is easy to verify that 1−µα 2 x 2 − H(x) = α(F (x) − µ 2 x 2 ) is convex since F (x)− µ 2 x 2 is convex due to the µ-strong convexity of F . Therefore, H is (1 − µα)-smooth, i.e., ∇H(x) − ∇H(y) ≤ (1 − µα) x − y for all x, y ∈ R n , thus ending the proof. Proof. The assertion (i) is trivial to verify. For the assertion (ii), it is easy to see that Ax − J x = (A − J )x − (A − J )J x ≤ A − J x − J x = A − J x − J x . Invoking the doublestochasticity of A and the Perron-Frobenius theorem [36], one has A − J < 1. This ends the proof. Lemma 5. [37] Let X, E ∈ R n×n with λ being a simple eigenvalue of X. Let w and v be the left and right eigenvectors of X associated with the eigenvalue λ, respectively. Then, 1) for each ǫ > 0, there exists a δ > 0 such that, ∀t ∈ C with \|t\| < δ, there is a unique eigenvalue λ(t) of X + tE such that \|λ(t) − λ − t w ⊤ Ev w ⊤ v \| ≤ \|t\|ǫ, 2) λ(t) is continuous at t = 0, and limt→0 λ(t) = λ, 3) λ(t) is differentiable at t = 0, and dλ(t) dt t=0 = w ⊤ Ev w ⊤ v . III. MAIN RESULT This section presents the algorithm design and analysis. In doing so, a distributed algorithm, called distributed gradient tracking (DGT for short), for solving (2) is proposed for each agent i ∈ [N ] as in Algorithm 1. x i,k+1 = x i,k − α[∇1fi(x i,k , σ i,k ) + ∇φi(x i,k )y i,k ], (4a) σ i,k+1 = N j=1 aijσ j,k + φi(x i,k+1 ) − φi(x i,k ),(4b)y i,k+1 = N j=1 aijy j,k + ∇2fi(x i,k+1 , σ i,k+1 ) − ∇2fi(x i,k , σ i,k ),(4c) In algorithm (4), σ i,k is leveraged for agent i to track the average (3) since σ(x) is global information, which cannot be accessed directly for all agents, and meanwhile, y i,k is introduced for agent i to track the gradient sum 1 N N i=1 ∇2fi(xi, σ(x)), which is also unavailable to all agents. The initial variable xi,0 is arbitrary for all i ∈ [N ], and choosing σi,0 = φi(xi,0) and yi,0 = ∇2fi(xi,0, σi,0) for all i ∈ [N ]. The name "distributed gradient tracking" is attributed to the fact that algorithm (4) has combined the classical gradient descent algorithm with the variable tracking techniques. To proceed, for a vector x = col(x1, . . . , xN ) ∈ R n , it is helpful to define φ(x) := col (φ1(x1), . . . , φN (xN )). Also, for a differentiable function g(x) = col(g1(x), . . . , gm(x)), where gi's are real-valued functions, let us denote by ∇g(x) = (∇g1(x), . . . , ∇gm(x)). With the above notations and those after Example 1, DGT (4) can be written in a compact form x k+1 = x k − α[∇1f (x k , σ k ) + ∇φ(x k )y k ],(5)σ k+1 = Aσ k + φ(x k+1 ) − φ(x k ),(6)y k+1 = Ay k + ∇2f (x k+1 , σ k+1 ) − ∇2f (x k , σ k ),(7) with A = A⊗I d as defined in Lemma 4, x k := col(x i,k , . . . , x N,k ), and similar notations for σ k and y k . Before presenting the main result, it is necessary to first introduce a preliminary result. Lemma 6. Under Assumption 1, there hold: σ k := 1 N N i=1 σ i,k = 1 N N i=1 φi(x i,k ), y k := 1 N N i=1 y i,k = 1 N N i=1 ∇2fi(x i,k , σ i,k ). Proof. In view of (6) and double-stochasticity in Assumption 1, multiplying 1 ⊤ /N on both sides of (6) can lead to that σ k+1 =σ k + 1 N N i=1 φi(x i,k+1 ) − 1 N N i=1 φi(x i,k ), which further implies that σ k − 1 N N i=1 φi(x i,k ) =σ0 − 1 N N i=1 φi(xi,0). Combining the above equality and σi,0 = φi(xi,0) yields the first assertion of this lemma. Similar arguments can obtain the second one, which completes the proof. It is now ready to present the main result of this paper. Theorem 1. Under Assumptions 1 and 2, if 0 < α < min 1 L1 , αs ,(8) where αs := µ(1 − ρ) 2 L3Lµ[(1 − ρ)L0 + 2L2L3] ,(9) Lµ := µ + L1 + L2L3 and L0 := L1 + L2 + L2L3, then x k = col(x 1,k , . . . , x N,k ) generated by algorithm (4) can converge to the optimizer of problem (2) at a linear convergence rate. Proof. Let us bound x k+1 − x * , x k+1 − x k , σ k+1 − J σ k+1 , and y k+1 − J y k+1 in the sequel, where x * is the optimal variable of problem (2). Denote σ(x * ) as σ * for brevity in this proof. First, for x k+1 − x * , invoking (5) yields that x k+1 − x * = x k − x * − α[∇1f (x k , σ k ) + ∇φ(x k )y k ] ≤ x k − x * − α[∇1f (x k , 1N ⊗σ k ) + ∇φ(x k )1N ⊗ 1 N N i=1 ∇2fi(x i,k , 1N ⊗σ k )] + α∇f (x * ) + α ∇1f (x k , σ k ) + ∇φ(x k )1N ⊗ȳ k − ∇1f (x k , 1N ⊗σ k ) − ∇φ(x k )1N ⊗ 1 N N i=1 ∇2fi(x i,k , 1N ⊗σ k ) + α ∇φ(x k )y k − ∇φ(x k )1N ⊗ȳ k ≤ (1 − µα) x k − x * + αL1 σ k − 1N ⊗σ k + α ∇φ(x k ) y k − 1N ⊗ȳ k ≤ (1 − µα) x k − x * + αL1 σ k − J σ k + αL3 y k − J y k ,(10) where Assumption 2.1, Lemma 3, (5) and (8) have been utilized to obtain the second inequality, and the last inequality has applied the fact that ∇φ( x k ) ≤ max i∈[N] φi(x i,k ) ≤ L3 by Assumption 2.3, 1N ⊗σ k = J σ k , and 1N ⊗ȳ k = J y k . Second, for x k+1 − x k , by noting that ∇f (x * ) = ∇1f (x * , 1N ⊗ σ * ) + ∇φ(x * )[1N ⊗ 1 N N i=1 ∇2fi(x * , 1N ⊗ σ * )] = 0, invoking (5) yields that x k+1 − x k = α ∇1f (x k , σ k ) + ∇φ(x k )y k ≤ α ∇1f (x k , σ k ) + ∇φ(x k )J y k − ∇1f (x * , 1N ⊗ σ * ) − ∇φ(x * )[1N ⊗ 1 N N i=1 ∇2fi(x * , 1N ⊗ σ * )] + α ∇φ(x k )(y k − J y k ) ≤ αL1( x k − x * + σ k − 1N ⊗ σ * ) + αL3 y k − J y k ≤ αL1( x k − x * + σ k − J σ k ) + αL3 y k − J y k + αL1 J σ k − 1N ⊗ σ * ,(11) where Assumption 2.1 and ∇φ(x k ) ≤ L3 have been used in the second inequality. For the last term in (11), in view of Lemma 6, one has that J σ k − 1N ⊗ σ * 2 = 1N ⊗ (σ k − σ * ) 2 = N 1 N N i=1 (φi(x i,k ) − φi(x * i )) 2 ≤ 1 N ( N i=1 φi(x i,k ) − φi(x * i ) ) 2 ≤ 1 N ( N i=1 L3 x i,k − x * i ) 2 ≤ L 2 3 N i=1 x i,k − x * i 2 = L 2 3 x k − x * 2 , where Assumption 2.3 has been employed in the second inequality, and the last inequality has appealed to the fact that ( N i=1 ai) 2 ≤ N N i=1 a 2 i for any nonnegative scalars ai's. Therefore, combining the above inequality and (11) follows that x k+1 − x k ≤ αL1(1 + L3) x k − x * + αL1 σ k − J σ k + αL3 y k − J y k .(12) Third, regarding σ k+1 − J σ k+1 , by noting that J A = AJ = J , in light of (6), one can obtain that σ k+1 − J σ k+1 = Aσ k + φ(x k+1 ) − φ(x k ) − J Aσ k − J [φ(x k+1 ) − φ(x k )] ≤ ρ σ k − J σ k + I − J φ(x k+1 ) − φ(x k ) ≤ ρ σ k − J σ k + L3 I − J x k+1 − x k ,(13) where Lemma 4 has been leveraged in the first inequality, and Assumption 2.3 has been exploited in the last inequality. By noticing that I − J = 1 and inserting (12) into (13), it can be obtained that σ k+1 − J σ k+1 ≤ (ρ + αL1L3) σ k − J σ k + αL1L3(1 + L3) x k − x * + αL 2 3 y k − J y k .(14) Fourth, for y k+1 − J y k+1 , similar to (13), invoking (7) results in that y k+1 − J y k+1 ≤ ρ y k − J y k + ∇2f (x k+1 , σ k+1 ) − ∇2f (x k , σ k ) . (15) At this step, by (6) , one has that σ k+1 −σ k = (A−I⊗I d )(σ k − J σ k )+φ(x k+1 )−φ(x k ) ≤ A−I σ k −J σ k +L3 x k+1 −x k , where the fact Aσ k −σ k = (A−I ⊗I d )(σ k −J σ k ) has been used in the equality, and Assumption 2.3 has been leveraged in the inequality, which together with Assumption 2.2 yields that ∇2f (x k+1 , σ k+1 ) − ∇2f (x k , σ k ) ≤ L3( x k+1 − x k + σ k+1 − σ k ) ≤ L2(1 + L3) x k+1 − x k + L2 A − I σ k − J σ k .(16) Substituting (12) and (16) into (15) can give rise to y k+1 − J y k+1 ≤ (ρ + αL2L3(1 + L3)) y k − J y k + αL1L2(1 + L3) 2 x k − x * + (αL1L2(1 + L3) + L2 A − I ) σ k − J σ k .(17) Finally, define θ k := col( x k − x * , σ k − J σ k , y k − J y k ). By (10), (14), (17) and A − I ≤ 2, it can be concluded that θ k+1 ≤ M (α)θ k ,(18) where M (α) := X + αE, and X :=   1 0 0 0 ρ 0 0 2L2 ρ   ,(20)E :=   −µ L1 L3 L1L2(1 + L3) L1L3 L 2 3 L1L2(1 + L3) 2 L1L2(1 + L3) L2L3(1 + L3)   .(21) Denote by λ(α) the eigenvalues of M (α). It is easy to see that 1 is a simple eigenvalue of M (0), and its corresponding left and right eigenvectors are both w = col(1, 0, 0). As a result, invoking Lemma 6 leads to that dλ(α) dα α=0 = w ⊤ Ew w ⊤ w = −µ < 0,(22) which indicates that the spectral radius of M (α) will be less than 1 for sufficiently small positive α. One can also see that the graph corresponding to M (α) is strongly connected, which together with Theorem C.3 in [38] implies that M (α) is irreducible. By Lemma 1, M (α) is primitive, which in combination with Lemma 2 can ensure that 1 will be a simple eigenvalue of M (α) when α increases from 0 to some value. By calculating det(I − M (α)) = 0, one can obtain that α = αs, where αs is defined in (9). Therefore, all eigenvalues of M (α) have absolute values less than 1 when α ∈ (0, αs), which can guarantee the linear convergence rate of θ k , thus ending the proof. Remark 2. It is worth mentioning that, to our best knowledge, this paper is the first to investigate problem (2) in the presence of the aggregative term σ(x), for which a linearly convergent distributed algorithm has been developed here. IV. A NUMERICAL EXAMPLE This section aims at presenting an optimal placement problem for supporting the designed algorithm. In an optimal placement problem in R 2 , there are M entities that are located at fixed positions, and meanwhile, there are N free entities, each of which are only privately aware of some of the fixed M entities. The objective is to determine the optimal positions of N free entities in order to minimize the sum of all (square) distances from each free entity to its corresponding fixed entities and the (square) distances from each agent to the center of all free entities. For example, the entities can represent warehouses, the links between each free entity and its associated fixed entities as well as the center of all free entities stand for the transportation routes, and the center of all free entities means a goods factory or a central warehouse. In this example, free entities are called agents. For the above problem, let M = N = 5, and each agent i is only privately aware of the fixed entity i. In this case, the problem can be modeled as (2) by letting fi(xi, σ(x)) = γi xi − ri 2 + xi − σ(x) 2 , i ∈ [N ](23) where ri's are the fixed entities, and γi > 0 represents the weighting between the first and second terms. For the simulation, let φi be the identity mapping for all i ∈ [N ], α = 0.05, γi = i, r1 = col(3, 5), r2 = col (6,9), r3 = col (9,8), r4 = col(6, 2), and r5 = col(9, 2), and the communication graph is shown in Fig. 1, which is strongly connected. By randomly selecting the initial positions of agents, i.e., xi,0's, performing the developed DGT algorithm gives rise to evolutions of all x i,k 's and σ i,k 's, as shown in Figs. 2 and 3, respectively, showing that all agents can converge to their optimal positions very fast and the estimate σ i,k of each agent can converge to the optimal σ(x * ), where x * = col(x * 1 , . . . , x * N ) is the optimal position. Therefore, the simulation results support the theoretical result. The first coordinate The first coordinate V. CONCLUSION This paper has proposed and investigated a new framework for distributed optimization, i.e., distributed aggregative optimization, which allows local objective functions to be dependent not only on their own decision variables but also on an aggregative term σ(x), relying on decision variables of all other agents. To handle this problem, a distributed algorithm, i.e., DGT, has been developed and rigorously analyzed, where the global objective function is assumed to be strongly convex and smooth along with some Lipschitz property, and the communication graph is assumed to be fixed, balanced, and strongly connected. It has been shown that the algorithm can converge to the optimal variable at a linear rate. A numerical example has been provided to support the theoretical result. Basically, this paper opens up a new avenue to distributed optimization. Future works can be placed on various cases, such as unbalanced graphs, feasible constraint sets, and other interesting forms of objective functions, etc. [N ]. And for x ∈ R n and y = col(y1, . . . , yN ) ∈ R Nd , define f (x, y) := N i=1 fi(xi, yi), ∇1f (x, y) := col(∇1f1(x1, y1), . . . , Lemma 1 ( 1[36]).For an irreducible nonnegative matrix M ∈ R n×n , it is primitive if it has at least one non-zero diagonal entry. there hold (i) ρ(M ) > 0 is an eigenvalue of M , (ii) M x = ρ(M )x for some positive vector x, and (iii) ρ(M ) is an algebraically simple eigenvalue. Lemma 4 . 4Under Assumption 1, there hold (i) AJ = J A = J , and (ii) Ax − J x ≤ ρ x − J x for any x ∈ R Nd , where A := A ⊗ I d and ρ := A − J < 1. Initialization: Stepsize α > 0, and initial conditions xi,0 ∈ R n i , σi,0 = φi(xi,0), and yi,0 = ∇2fi(xi,0, σi,0) for all i ∈ [N ]. 2: Iterations: Step k ≥ 0: update for each i ∈ [N ]: Fig. 1 . 1The communication graph. Fig. 2 . 2Evolutions of x i,k 's, where squares and circles mean initial positions and final optimal positions of all agents, respectively. 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[ "Fault Diagnosis of Nonlinear Systems Using a Hybrid-Degree Dual Cubature-based Estimation Scheme", "Fault Diagnosis of Nonlinear Systems Using a Hybrid-Degree Dual Cubature-based Estimation Scheme" ]	[ "Yanyan Shen \nElectrical and Computer Engineering\nElectrical and Computer Engineering\nConcordia University\nH3G 1M8MontrealQuebecCanada\n\nConcordia University\nH3GMontrealQuebec\n", "Khashayar Khorasani \nElectrical and Computer Engineering\nElectrical and Computer Engineering\nConcordia University\nH3G 1M8MontrealQuebecCanada\n\nConcordia University\nH3GMontrealQuebec\n", "Khashayar Khorasani " ]	[ "Electrical and Computer Engineering\nElectrical and Computer Engineering\nConcordia University\nH3G 1M8MontrealQuebecCanada", "Concordia University\nH3GMontrealQuebec", "Electrical and Computer Engineering\nElectrical and Computer Engineering\nConcordia University\nH3G 1M8MontrealQuebecCanada", "Concordia University\nH3GMontrealQuebec" ]	[]	In this paper, a novel hybrid-degree dual estimation approach based on cubature rules and cubature-based nonlinear filters is proposed for fault diagnosis of nonlinear systems through simultaneous state and time-varying parameter estimation. Our proposed dual nonlinear filtering scheme is developed based on case-dependent cubature rules that are motivated by the following observations and facts, namely (i) dynamic characteristics of nonlinear system states and parameters generally are distinct and posses different degrees of complexities, and (ii) performance of cubature rules depend on the system dynamics and vary due to handling of high-dimensional integrations approximations. For improving the robustness capability of our proposed methodologies a modified cubature point propagation method is incorporated. The performance of our proposed dual estimation strategy is demonstrated and evaluated by application to a nonlinear gas turbine engine for addressing the component fault diagnosis problem within an integrated fault detection, isolation and identification framework. Robustness analysis is implemented to verify the capability of our proposed approaches to deal with parametric uncertainties and unmodeled dynamics. Extensive simulation case studies and discussions with respect to component fouling, erosion or abrupt faults are provided to substantiate and justify the superiority of our proposed fault diagnosis methodology when compared to other well-known alternative diagnostic techniques such as the Unscented Kalman Filters (UKF) and Particle Filters (PF) that are commonly available in the literature.Estimation as a quantitative evaluation process of unmeasured states and/or parameters from uncertain or imprecise observations is a fundamental problem for nonlinear systems in various disciplines such as control and fault diagnosis [1]. Model-based fault diagnosis (FD) techniques (that consist of fault detection, isolation, and identification (FDII)) relying on estimation approaches *	10.1109/cdc45484.2021.9683338	[ "https://arxiv.org/pdf/2111.07004v1.pdf" ]	244,117,522	2111.07004	ab5cfe43e73f8d608ba1edd20a911e9e8ea4531c	Fault Diagnosis of Nonlinear Systems Using a Hybrid-Degree Dual Cubature-based Estimation Scheme 13 Nov 2021 Yanyan Shen Electrical and Computer Engineering Electrical and Computer Engineering Concordia University H3G 1M8MontrealQuebecCanada Concordia University H3GMontrealQuebec Khashayar Khorasani Electrical and Computer Engineering Electrical and Computer Engineering Concordia University H3G 1M8MontrealQuebecCanada Concordia University H3GMontrealQuebec Khashayar Khorasani Fault Diagnosis of Nonlinear Systems Using a Hybrid-Degree Dual Cubature-based Estimation Scheme 13 Nov 2021Preprint submitted to Elsevier November 16, 2021Corresponding author:Hybrid-degreeCubature rulesCubature-based nonlinear filtersDual estimationFault diagnosisAircraft gas turbine engines In this paper, a novel hybrid-degree dual estimation approach based on cubature rules and cubature-based nonlinear filters is proposed for fault diagnosis of nonlinear systems through simultaneous state and time-varying parameter estimation. Our proposed dual nonlinear filtering scheme is developed based on case-dependent cubature rules that are motivated by the following observations and facts, namely (i) dynamic characteristics of nonlinear system states and parameters generally are distinct and posses different degrees of complexities, and (ii) performance of cubature rules depend on the system dynamics and vary due to handling of high-dimensional integrations approximations. For improving the robustness capability of our proposed methodologies a modified cubature point propagation method is incorporated. The performance of our proposed dual estimation strategy is demonstrated and evaluated by application to a nonlinear gas turbine engine for addressing the component fault diagnosis problem within an integrated fault detection, isolation and identification framework. Robustness analysis is implemented to verify the capability of our proposed approaches to deal with parametric uncertainties and unmodeled dynamics. Extensive simulation case studies and discussions with respect to component fouling, erosion or abrupt faults are provided to substantiate and justify the superiority of our proposed fault diagnosis methodology when compared to other well-known alternative diagnostic techniques such as the Unscented Kalman Filters (UKF) and Particle Filters (PF) that are commonly available in the literature.Estimation as a quantitative evaluation process of unmeasured states and/or parameters from uncertain or imprecise observations is a fundamental problem for nonlinear systems in various disciplines such as control and fault diagnosis [1]. Model-based fault diagnosis (FD) techniques (that consist of fault detection, isolation, and identification (FDII)) relying on estimation approaches * have been extensively investigated, albeit mostly for linear systems using e.g., Kalman filters (KF) [2] and observer-based methods [3]. Although linear approaches enable one to achieve acceptable estimation performance locally, their performance undergo deterioration as nonlinear dynamics dominate the system behavior and risk losing convergence given an accurate approximation requirement. As far as fault diagnosis of nonlinear systems are concerned linear approaches might be subject to high rates of false alarms and poor detection and diagnosis performance. A great deal of investigation on fault detection and isolation (FDI) problems that utilize nonlinear estimation approaches have been conducted in the literature. According to the employed nonlinear filters, these approaches can be broadly categorized into: 1) FDI methods developed using local nonlinear filters, e.g., Extended Kalman Filters (EKF) in [4], Unscented Kalman Filters (UKF) in [5], Cubature Kalman Filters (CKF) in [6], Gauss-Hermite Filters (GHF) in [7]; and 2) FDI methods developed using global nonlinear filters, e.g., Particle Filters (PF) in [8] and Ensemble Kalman Filters (EnKF) in [9]. These work have introduced basic frameworks for state estimation of nonlinear systems, and FDI schemes using the corresponding generated residuals. In this paper, component faults are represented as variations of health-related parameters, and their diagnosis will be achieved through simultaneous state and parameter estimation. The central idea of using simultaneous state and parameter estimation to accomplish fault diagnosis of nonlinear system is certainly not new. One significant set of publications focus on joint estimation schemes. The parameters and states are augmented into one vector for performing simultaneous estimation using e.g. nonlinear observer-based methods [10], EKF-based methods [11], and UKF-based methods [12]. Theoretically, joint estimation scheme casts the simultaneous estimation problem into a single filtering scheme; however, the consequent drawback is the large and high-dimensional matrix operation of the augmented model. Another set of publications focus on simultaneously estimating states and parameters using parallel filters, namely through dual estimation schemes. A higher accuracy can be expected as the estimation is performed in a closed-loop manner [13]. Additionally, Shi et al. in [14] have shown that the computational time of the dual EKF-based scheme is 25% less than the joint one. However, due to the linearization errors, the dual EKF-based scheme might fail to converge to accurate estimation in cases of highly nonlinear systems. Plett in [15] developed a dual UKFbased scheme for battery management systems with better performance as compared to EKF. Nevertheless, when the system order exceeds beyond three, perturbations are induced which cause the numerical inaccuracy; and moreover presence of negative weights might risk the filtering stability [16]. In [17], a dual PF-based fault diagnosis scheme was developed for a single-spool gas turbine engine. The superiorities on higher estimation accuracy and lower false alarms have been verified with guaranteed filtering stability. Other PF-based dual estimation works are found in [18,19] for different applications. Although their estimation performance have been justified to be highly accurate, their significant computational demands challenge utilization in wide range of real-time applications. One can observe from the above dual estimation works that the employed nonlinear filters are critical to the performance of simultaneous estimation and fault diagnosis problems. The CKF has recently been extensively studied for high-dimensional state estimation. Its derivative-free properties, reasonable estimation accuracy when subjected to Gaussian considerations, divergence avoidance, and dimensionality issues are the main advantages when compared to EKF, UKF, or PF estimation techniques [1,16]. It has also been successfully applied in various fields such as lithium-ion batteries state of charge estimation [20], missile attitude estimation [21], among others. Few studies can be found that take advantage of the CKF state estimation capability for FDI problems. For example, Kim et al. in [6] used a CKF to generate residuals to perform FDI of a multi-unmanned vehicle; Xu et al. in [22] has run various experiments to verify that CKF has the best diagnosis performance as compared to the EKF and strong tracking filter, and is more suitable for FDI of ECAS systems. However, CKF has limited estimation accuracy for certain nonlinearities due to its employed 3rd-degree cubature rules. Consequently, biased residual signals might be generated which can lead to misclassifying the health status. To remedy this drawback, and as a motivation to enhance the level of accuracy and to capture higher nonlinear dynamics, higher-degree cubature rules are proposed to be utilized. Jia et al. in [23] designed a high-degree CKF based on the spherical-radial rule. Its estimation accuracy has been shown to be superior to UKF, CKF, and PF. The downside with respect to the higher degree of cubature rules are the increase in both the computational cost and the risks in numerical instability. Consequently, investigating efficient and numerically stable cubature rules, and incorporating such cubature rules within the Bayesian filtering framework to propose our "cubature-based nonlinear filters (CNF)" and to solve the nonlinear FD estimation problem is one of the main objectives of this paper. Derivation of cubature rules have ascribed great importance in the field of numerical mathematics during the past few decades, e.g., [24,25]. Although a variety of cubature rules have been developed in numerical mathematics, not all of them are directly applicable to the general nonlinear filtering problems given that many were developed to only solve certain specific problems. Our focus in this paper is on those cubature rules that can either be generalized to fault diagnosis and health monitoring problems of nonlinear systems having arbitrary order as well as arbitrary degrees of polynomial accuracy and that can particularly perform well for the targeted nonlinear gas turbine engine (GTE) application. Early research on FDII of GTE have been one of the challenging application areas that have received much attention. Gas path analysis (GPA) is one of the most popular diagnostic procedures which relies on discernible changes in the observable parameters of the engine to detect presence of faults. Various fault diagnosis techniques have been developed for GTE based on GPA ranging from Kalman filters-based variants [26,9], neural networks [27], data-driven methods [28], and component adaptation method [29]. To the authors' best knowledge, the fault identification problem along with a unified integrated fault diagnosis scheme that employs CNF based on a dual estimation scheme has not been investigated in the literature. This represents as another objective of this paper where our goal is to develop an efficient and stable dual cubature-based estimation scheme to not only detect and isolate component faults but also to accurately identify severity of simultaneous multi-mode scenarios for the GTE system. However, several limitations and open areas are still outstanding for applying CNF to address component FD problems. Specifically, one can state the following challenges in and shortcomings of the literature: 1) The balancing selection and choice is between reasonable estimation accuracy and accept-able computational cost for a real-time implementation application. The cubature rules utilized in our corresponding CNF determine the estimation performance through the capacity of capturing the system nonlinear dynamics. Improved accuracy can be achieved for strong nonlinearities by using higher rule degrees, however, the computational demands would be growing quadratically or even of higher order. 2) Numerical stability of certain CNF decreases with the increasing degree of accuracy and the increasing system order due to the growing influence of negative weights. The accumulative impacts of numerical instabilities through iterative calculations do indeed impose high risks that lead to unstable filtering solutions [25]. Moreover, in dual estimation research that investigated Dual-EKF, Dual-UKF or Dual-CKF one actually utilizes the same order of accuracy for the state and parameter estimation modules. However, due to the fact that the dynamics of states and parameters are completely different, higher false alarms or low accuracy rates can occur given their incapability of capturing the corresponding different degrees of nonlinearities. Motivated by the above discussion on FD of nonlinear systems, in this paper a novel hybriddegree dual estimation scheme is implemented for the first time in the literature for real-time health monitoring of GTE. The term "hybrid" indicates and refers to the notion that different degrees or cubature rules are considered for developing the nonlinear state and parameter estimation schemes. The hybrid-degree dual estimation scheme is motivated by the fact that the nonlinear dynamics of the system states and parameters in general, and the GTE in particular, are practically completely different. For example, the GTE state dynamics contain higher degrees of nonlinearities where a conventional 3rd-degree cubature rule cannot completely capture and represent, thereby necessitating one to utilize a higher-degree cubature rule, whereas the GTE health-related parameter dynamics are in effect less complex so that a high-degree cubature rule is not necessary given the practical implementation and limited computational resources available in real-time. To summarize the main contributions of this paper can be stated as follows: 1) A novel stable and efficient hybrid-degree dual cubature-based filtering approach for FD of nonlinear systems. In contrast to the same degree-based Dual-EKF, Dual-UKF, and Dual-CKF, our proposed case-dependent hybrid degree solution will improve the estimation accuracy and FD performance. Furthermore, our proposed hybrid-degree strategy has the flexibility to simultaneously achieve improved accuracy and computational efficiency by considering the prior knowledge of system dynamics and user's specifications and requirements. We have also performed quantitative comparative evaluation and analysis of various approaches available in the literature that are utilized as reference benchmark. 2) Compared to Dual-UKF, Dual-CKF, and Dual-PF algorithms, advantages and superiorities of our proposed FD scheme are justified and validated in terms of fault detection promptness, isolation, and identification accuracy, false alarm rates, precision, and computational cost. 3) Robustness capabilities of the proposed methodology with respect to modeling uncertainties have been formally and quantitatively analyzed. For improving the robustness and reliability of the proposed methodology against parametric, unmodelled dynamic uncertainties, a modified cubature points propagation methodology is incorporated into the proposed framework. Comparative evaluations in terms of false alarm rates, fault detection time, and accuracy performance metrics are provided. 4) Performance of the proposed unified component FD framework is verified and validated by application to multi-mode simultaneous/concurrent fault diagnosis problem of GTE system that is subject to both abrupt and incipient fault types. 5) The boundedness properties of the estimated health-related parameter errors are formally investigated and analyzed. The remainder of this paper is organized as follows. In Section 1, the statement of dual estimation problem for system states and parameters based on numerical cubature rules is presented. The cubature-based nonlinear filters (CNF) are developed in Section 2, and a detailed design procedure of our proposed hybrid-degree dual estimation scheme is provided in Section 3. The state and parameter estimation problems as well as FD strategy formulations are provided. The effectiveness of the proposed framework is verified by its application to a component FD problem of a GTE system. Comparative studies are conducted in Section 4 where performance of our proposed CNF and FD strategies are evaluated in terms of metrics of accuracy, stability factor, and computational cost. Conclusions are provided in Section 5. Problem Statement Consider the general discrete-time nonlinear system x k+1 = f (x k , θ k , u k ) + w k (1) z k = g(x k , θ k , u k ) + v k(2) where x k ∈ R n x , z k ∈ R n z , θ k ∈ R n ϑ denote the system states, measurements, and health-related parameters, respectively. Also, u k ∈ R n u denotes the control input, f : R n x × R n θ × R n u → R n x denotes the nominal nonlinear system dynamics, g : R n x × R n θ × R n u → R n z denotes a known nonlinear function, w k and v k represent the zero-mean uncorrelated Gaussian white noise sequences for states and measurements, with E[w k w T l ] = Σ w,k δ k,l , E[v k v T l ] = Σ v,k δ k,l and E[w k v T l ] = 0, respectively. It is assumed that the dynamic characteristics of the multiplicative health-related parameters are represented by θ k = h(θ k−1 ) + τ k(3) where h denotes the degradation dynamics, and τ k represents a zero-mean Gaussian white noise with E[τ k τ T l ] = Σ τ,k δ k,l . Problem Statement 1: This paper aims to develop a unified multi-mode FD methodology that simultaneously handles fault detection, isolation and identification problems. Our goals for the proposed FD methodology are to provide a fast detection, low false alarm rates and missed detections, and low estimation errors, while being computationally feasible for real-time implementa-tion. This is accomplished by monitoring the status of health-related parameters θ k by developing an efficient dual estimation scheme. The objective of dual estimation scheme here can be formulated as that of approximating the conditional expectations with respect to states and parameters as governed by E(φ 1 (x k )\|z 1:k , θ k−1 ) = R nx φ 1 (x k )p(x k \|z 1:k , θ k−1 )dx k (4) E(φ 2 (θ k )\|z 1:k , x k ) = R n θ φ 2 (θ k )p(θ k \|z 1:k , x k )dθ k(5) where φ 1 (x k ) and φ 2 (θ k ) are functions that are to be simultaneously estimated, and z 1:k = {z 1 , z 2 , · · · , z k } denotes the available observations up to the time instant k, p(x k \|z 1:k , θ k−1 ) and p(θ k \|z 1:k , x k ) denote the conditional probability density functions (pdfs) that are expected to be approximated by the developed nonlinear filters. The key difficulty for obtaining the pdfs by using Bayesian filtering scheme is the involved intractable multivariate integrals in both prediction and update stages. Proceeding with the assumption that the pdfs of states and parameters are Gaussian, only the moments of their means and covariances are needed to be computed. The computation of the mean and covariance requires multivariate integrals that are of the tractable Gaussian weighted form. The underlying problem now is to accurately approximate the Gaussian weighted integrals in each stage of the Bayesian filtering. Let us take φ 1 (x k ) = x k and φ 2 (θ k ) = θ k , and denote the prior Gaussian distributions of states and parameters at time k − 1 as N x k−1 (x k−1 ;x k−1 , P xx k−1 ) and N θ k (θ k ;θ k−1 , P θθ k−1 ), respectively, wherex k−1 andθ k−1 denote the estimated mean, P xx k−1 and P θθ k−1 represent the corresponding covariance matrices. Let us consider the Gaussian integrals for the mean of predictive density of states and parameters in the prediction stage where they can be approximated by cubature rules as: R nx φ 1 (x k )N x k−1 dx k−1 ≈ N dx i=1 φ 1 (ξ d x i,k\|k−1 )w d x i (6) R n θ φ 2 (θ k )N θ k−1 dθ k−1 ≈ N d θ j=1 φ 2 (ξ d θ j,k\|k−1 )w d θ j(7) where the variables w, N andξ k\|k−1 denote the weights, the total number of points, and the propagated cubature points at time k\|k − 1 based on the sampled points ξ. The subscripts or superscripts d x and d θ represent the degree of cubature rules for state and parameter filters, respectively. Let us define two sets {ξ d x i , w d x i }, i = 1, · · · , N d x and {ξ d θ j , w d θ i }, j = 1, · · · , N d θ . Obviously, design of these variables affect the approximation performance of (6) and (7). They are determined according to the degree of cubature rules, d x and d θ according to the employed cubature rules. This paper designs the degree of cubature rules according to the prior knowledge on φ 1 (·) and φ 2 (·) functions. Specifically, the function in (6) can be obtained by the d x th-degree cubature rules if it is exact for the nonzero function φ 1 (x k ) whose components are linear combinations of 6 monomials having coefficients a α and monomials x α = n x i=1 x α i i , with the total degree up to d x . In other words, the monomials integers in state dynamics satisfy the following d x = max{ n x i=1 \|α i \| : a α 0}(8) In case of α 1 + · · · + α n x > d x in φ 1 (·), the d x th-degree cubature rules cannot be exactly approximated by the monomials with the d x th-degree accuracy, since the d x th-degree cubature rule can reach a d x th-degree of accuracy [23]. Similarly, the parameter estimation is accomplished by the d θ th-degree cubature rules if the parameter dynamics fulfill d θ = max{ n θ j=1 \|β j \| : b β 0}, where θ β = n θ i=1 θ β j j denotes the monomials. From approximation accuracy point of view, the choice of degrees are essentially case-dependent given the priori knowledge on the underlying system dynamics. Problem Reformulation: Construct appropriate sets {ξ d x i , w d x i } and {ξ d θ j , w d θ j }, i = 1, · · · , N d x , j = 1, · · · , N d θ to approximate the involved Gaussian weighted integrals (6) and (7) in the dual estimation scheme by proposing proper d x th-degree and d θ th-degree cubature rules for state and parameter estimation problems, respectively. Importantly, the designed case-dependent degree of cubature rules and the employed cubature formulas will eventually lead to different CNFs. For achieving reliable fault diagnosis performance though designing efficient, accurate and stable dual-CNF scheme, this paper will not limit to one solution, but provide comparisons and evluations on various choices. In this case, we can provide a generalized solution to arbitrary nonlinear system order with arbitrary degree of accuracy, or that can particularly perform quite satisfactorily on specific cases of nonlinear systems such as the GTE for addressing the fault diagnosis problem. With respect to the application to GTE system, the unmodelled dynamics, model mismatches, parametric uncertainties, and noise discrepancy between the actual GTE and the on-board engine model (OBEM) can increase the fault detection time and lead to occurrence of false alarms and incorrect fault severity estimation results. The details are provided in Section IV.D. In order to verify the robustness of our proposed methodology in presence of measurement uncertainties, robustness analysis of the proposed CNF and FD methodologies on fault estimation performance against them is conducted, based on the Assumption 1 below. Assumption 1: The measurement uncertainty is bounded by ζ(x k , u k ) ≤ζ, forζ > 0, which is present in the measurement equation as below: z k = g(x k , θ k , u k ) + ζ(x k , u k ) + v k(9) Acceptable ranges on modeling uncertainties that do not lead to false alarms will be specified under various healthy and faulty scenarios in Section V. Cubature-Based Nonlinear Filters (CNFs) The goal of this section is twofold. First, a class of cubature rules are presented in an accessible manner to aid in implementation of these methods. Second, a class of CNFs is constructed based 7 on cubature rules for the nonlinear estimation problem. 2.1. The Construction of Sets {ξ d x i , w d x i } and {ξ d θ j , w d θ j } As stated in Section 1, the objective of the dual estimation problem can be transformed into designing the sets {ξ d x i , w d x i } and {ξ d θ j , w d θ j }. This section presents a class of cubature rules where for sake of generality an n-dimensional nonlinear estimation problem is considered and the general set {ξ d i , w d i } will be constructed. The integral with respect to a general Gaussian distribution N(x;x, P) is approximated by I(φ) = R n φ(x)N(x;x, P)dx ≈ N i=1 w i φ(S ξ i +x)(10) where φ denotes an arbitrary nonlinear function, and P = S S T . The weight w i , the sampled cubature points ξ i and the total number of points N are determined and specified in the following subsections by analyzing various cubature rules. Generally, the choice of cubature points ξ d i depends on the domain of integration. This paper investigates the cubature rules over both spherical surfaces and the entire n-D space surfaces. Of interest is the spherical surface that is concerned with both Genz's theorem [30] and the Mysovskikh's theorem [31] for obtaining the corresponding cubature rules. Cubature Rules Over Spherical Surface The integral in (10) can be transformed into the following spherical-radial coordinate system I(φ) = ∞ 0 U n φ(rs)r n−1 exp(−r 2 )dσdr(11) where the spherical surface U n = {s ∈ R n : s T s = 1}, with x = rs and r = √ x T x, s = [s 1 , · · · , s n ] T , and σ(·) denotes the spherical surface measure. The integrals in (11) can be addressed by separately approximating two sub-integrals, namely (a) the spherical integral U n φ s (s)dσ(s), and (b) the radial integral ∞ 0 φ r (r)r n−1 exp(−r 2 )dr [16]. • The Spherical Rule: It is utilized to solve the spherical integral as N s p=1 w s,p φ(s p ), where s p and w s,p denote spherical points and weights. We concentrate on two rules to compute the spherical integral, namely: (i) Genz spherical rule [30] that allows a system with an arbitrary order achieves an arbitrary degree of accuracy, and (ii) Mysovskikh spherical rule [32] that enables one to realize a more efficient approximation when compared to the Genz spherical rule for systems having order n ≥ 4. Specifically, the Genz method constructs (2m + 1)th-degree spherical rule over the surface of the sphere U n , where m = ρ 1 + ρ 2 + · · · + ρ n , with ρ p denoting nonnegative integers. For designing the dth-degree spherical rule, one sets d = 2m + 1 and analyze each non-negative integer. Mysovskikh in [31] derived a rule based on the transformation group of regular simplex with vertices a (p) = [a (p) 1 , a (p) 2 , · · · , a (p) n ] T , p = 1, 2, · · · , n+1. For designing the dth-degree spherical rule, various topologies are considered, as provided in Algorithm 1. • The Radial Rule: It aims to solve the radial integral in (11) as N r q=1 w r,q φ(r q ), where r q and w r,q denote radial points and weights. Moment matching method is employed for computing the radial 8 rule. The key idea is to determine radial points and weights that satisfy the moment equations based on the rule degree and the system order. For more details refer to [16]. Consequently, (11) can further be formulated as I(φ) ≈ N r q=1 N s p=1 w r,q w s,p φ(r q s p )(12) where the sets (s p , w s,p ) and (r q , w r,q ) are obtained by using the spherical and radial rules, with N s and N r denotes the corresponding required number of points. respectively, as provided in Algorithm 1. Definition 1. The set {ξ d i , w d i } based on the spherical surface by using the spherical-radial cubature rules is constructed as ξ d i ∈ { 2r q s p , p = 1,· · ·, N s , q = 1,· · ·, N r } w d i ∈ {w r,q w s,p /π n/2 , p = 1,· · ·, N s , q = 1,· · ·, N r }(13) where i = 1, · · · N, with N = N r × N s if r q 0, and N = (N r − 1) × N s + 1 if one of r q in (12) is zero. Theorem 1 ([33] ). The number of nodes N of a cubature of degree d = 2s − 1 satisfies N ≥ N min with N min =            n+ s − 1 n + n−1 k=1 2 k−n k+ s − 1 k , s even n+ s − 1 n + n−1 k=1 (1−2 k−n ) k+ s − 2 k , s odd It is easy to check that for d = 3, N min = 2n, while d = 5 yields N min = n 2 + n + 1. Given that one of our main goals is to investigate the FD problem of GTE system, efficiency concerns in determining whether our approach is applicable to a practical problem is of significant importance. In this case, a special but efficiency 5th-degree cubature rule that is integrated over the entire n-D space is considered as follows. Efficient Cubature Rule Over the Entire n-D Space Given that one of our main goals is to investigate the FD problem of GTE system, efficiency concerns in determining whether our approach is applicable to a practical problem is of significant importance. In this case, a special but efficiency 5th-degree cubature rule is considered. ξ d i ∈ { ν 1 , −ν 1 2 , ν 2 , −ν 2 2C 1 n =n , ν 3 , −ν 3 2C 2 n =n(n−1)/2 }(14) where ν 1 , ν 2 , and ν 3 denote the points that are related to the system order n that have been given below in Algorithm 1, for i = 1, · · · N with N = n 2 + n + 2. For further detail on the weights w d i and all other coefficients refer to [34]. One downside of this rule is that it is only valid for a limited range of system orders of 2 ≤ n ≤ 7, it is perhaps the most efficient one possible among the 5th-degree rules since it requires only one point more than the lower bound that is given by [33]. A variety of dynamical systems fulfill such condition, including the state dynamics of our GTE system. Therefore, the efficient cubature rule that is derived over the entire n-D space as proposed in Stroud [34] will also be discussed in this paper. Finally, given the logic of determining the degrees d x and d θ in Section 1, and the known dimensions of n x and n θ , the objective sets Determine all possible nonnegative ρ = [ρ 1 , · · · , ρ n ] 5: {ξ d x i , w d x i } and {ξ d θ j , w d θ j } could be consequently constructed by using the set {ξ d i , w d i }. with \|ρ\| = m and d = 2m + 1 6: For each possible [ρ 1 , · · · , ρ n ], derive [30] 7: s d p = [v 1 u ρ 1 , · · ·, v n u ρ n ] T , 8: with u ρ p = ρ p /m, v p = ±1 9: Calculate the weight w d s,p = 2 −c(u p ) w ρ 10: End 11: [s d p , w d s,p ] = Mysovskikh(n, d): 12: s d p : Transformed topologies of regular simplex with: 13: [a p,1 , · · · , a p,n ], p = 1, · · ·, n+1, i = 1,· · ·, n 15: a p,i          − √ (n+1)/ [n(n−i+2)(n−i + 1)], i < p − (n+1)(n− p+1)/ [n(n−i+2)], i = p 0, i >Solve N r q=1 w r,q φ(r q ) = 1 2 Γ( n+l 2 ), l = 0, 2,· · ·, 2m [16] 20: ⇒ I(φ)=SphericalSurface (n, d, s d q , w d s,q , s d p , w d r,p ) using 21: N r q=1 N s p=1 w r,qq w s,p φ(r q s p ) 22: J Efficient 5th-degree cubature rules over n-D space 23: I(φ)=EntireSurface (n, ν 1 , ν 2 , ν 3 , w 1 , w 2 , w 3 ) 24: Re-defined: ν 1 = [η, η, · · · , η], ν 2 = [λ, ξ, · · · , ξ], 25: ν 3 = [υ, υ, γ, · · · , γ], 26: υ = (−3 ± √ 16 − 2n)γ, γ 2 = 3± √ 7−n 2(16−n±4 √ 16−2n) , 27: η 2 = n(n−7)∓(n 2 −3n−16) √ 7−n 2(2n 3 −7n 2 −16n+128) 28: ⇒ I(φ) = w 1 φ(ν 1 ) + φ(−ν 1 ) + w 2 φ(ν 2 ) + φ(−ν 2 ) +w 3 φ(ν 3 ) + φ(−ν 3 ) Remark 1. Each class of cubature rules exhibits different advantages and disadvantages in dealing with various nonlinear systems. This paper compares their performance of approximation accuracy, numerical stability and computational cost when applied to a complex GTE system. This should provide guidelines to be used as benchmark and reference for handling other systems and applications. Remark 2. For tackling FD of GTE system that is considered in this paper the degree of the cubature rule selected is up to the 5th-degree. Using higher degrees of cubature rules do not yield improved or better solutions for the GTE case study. A significant observation is that a cubature rule over spherical surface that is constituted by the dth-degree spherical rule and the dth-degree radial rule can achieve a dth-degree estimation accuracy. Nevertheless, the degree of the spherical rule as shown in Algorithm 1 is not necessarily equal to the radial rule degree which facilitates and motivates the mixture-degree of cubature rules over the spherical surface [23,35]. Cubature-based Nonlinear Filters (CNF) The proposed CNF in this paper represent as Bayesian filtering approaches that are developed on the basis of the class of dth-degree cubature rules. We have utilized the Genz and Mysovskikh theorem-based cubature rules, and a specially designated but efficient 5th-degree cubature rule for systems with order 2 ≤ n ≤ 7. The pseudo-code of procedures for designing the dth-degree cubature rules are provided in Algorithm 1 that are derived from references [30,23,36,31,32,34,16]. The appropriate choice of cubature rules for developing a nonlinear estimation filter essentially relies on the prior knowledge of the system and the user's requirements. First, the degree d that is obtained from the a priori knowledge of the system dynamics order (based on Eq. (8)) enables one to achieve approximations that are empowered with high accuracy and second, the prior knowledge of the order n enables one to seek a more specific theorem for developing the cubature rules that provide the most proper achievable approximation given the order range and third, the trade-offs to be made between user's computational efficiency requirements, degree of accuracy, and finally estimation error boundedness guarantees (since higher d and n result in higher computational complexity and increase in likelihood of negative weights). CNF-I 3rd-degree Genz-spherical rule & 3rd-degree radial rule CNF-II 5th-degree Genz-spherical rule & 5th-degree radial rule CNF-III 3rd-degree Mysovskikh-spherical rule & 3rd-degree radial rule CNF-IV 5th-degree Mysovskikh-spherical rule & 5th-degree radial rule CNF-V 3rd-degree Mysovskikh-spherical rule & 5th-degree radial rule CNF-VI 5th-degree Stroud-based cubature rule The procedure for developing CNF consists of prediction and update steps that are identified in conventional CKF using 3rd-degree cubature rules, whereas for our schemes multiple cubature degree rules are employed with associated different projected cubature points and weights. The six CNF schemes constructed for application to GTE are given in Table 1. Hybrid-Degree Dual Estimation-based Fault Diagnosis Methodology A novel hybrid-degree dual nonlinear filtering strategy is proposed in this section. To improve the robustness to unmodelled dynamics and uncertainties, a modified cubature point propagation is further incorporated into the hybrid solution. Finally, the fault diagnosis framework is formulated. Hybrid-Degree Dual Estimation Strategy Our proposed dual estimation scheme is developed by running two filters concurrently. At every time step, the first CNF-based state filter estimates the states by using the current available estimate of the parameters,θ k−1\|k−1 , wheares the second CNF-based parameter filter estimates the health-related parameters by using the current estimate of the states,x k\|k . Remark 3. The key feature and novelty of our proposed hybrid strategy is that the degree of accuracy for state and parameter estimations are case-dependent or determined based on certain performance metrics by using case-dependent cubature rules. Given that the system state process model is highly nonlinear, in general, higher-degree cubature rules are necessary for designing the state estimation filter, whereas for performing parameter estimation lower-degree cubature rules can be acceptable and sufficient. In the next subsections, design of concurrent state/parameter estimation filters are provided and the fault diagnosis methodology is introduced and formally specified. d x th-degree Cubature-based State Estimation The goal pursued in this subsection is to approximate the objective function in E(φ 1 (x k )\|z 1:k , θ k−1 ) that is specified in Eq. (4). The parameter vector θ k−1 is assumed to be given and fixed atθ k−1 during the state estimation process. Assume that the d x th-degree cubature rules are implemented for designing the state estimation filter given the cubature points and weights that are specified and set as {ξ d x i , w d x i }, i = 1, · · · , N d x . The class of CNF schemes that are considered for implementing the state estimation filter is introduced next. Given φ 1 (x k ) = x k and the distribution at time k−1 asx k−1\|k−1 ∼ N(x k−1 ;x k−1\|k−1 , P xx k−1\|k−1 ), an approximation to predictive E(x k \|θ k−1 , z 1:k−1 ) can be first obtained before z k arrives. For simplicity, let us denoteŪ k−1 = [θ T k−1 , u T k−1 ] T . CNF-I: For the 3rd-degree nonlinear filter the predictive expectation E(x k \|z 1:k−1 , θ k−1 ) is represented by: E(·) = w(n x ) m(n x ) i=1 f ( κ x I P xx k−1 [e] i +x k−1 ),Ū k−1 + f (− κ x I P xx k−1 [e] i +x k−1 ),Ū k−1(15) where w(n x ) = 1/2n x , m(n x ) = n x , [e] i denotes the unit vector in R n x with the ith element being 1, and κ x I = n x . CNF-II: For the 5th-degree nonlinear filter the predictive expectation E(x k \|z 1:k−1 , θ k−1 ) is ap-proximated by: E(·) = w 0 (n x ) f (x k−1 ,Ū k−1 )+w 1 (n x ) m 1 (n x ) i=1 f ( κ x II P xx k−1 [s] + i +x k−1 ),Ū k−1 + f (− κ x II P xx k−1 [s] + i +x k−1 ),Ū k−1 +w 1 (n x ) m 2 (n x ) i=1 f ( κ x II P xx k−1 [s] − i +x k−1 ),Ū k−1 + f (− κ x II P xx k−1 [s] − i +x k−1 ),Ū k−1 +w 2 (n x ) m 3 (n x ) i=1 f ( κ x II P xx k−1 [e] i +x k−1 ),Ū k−1 + f (− κ x II P xx k−1 [e] i +x k−1 ),Ū k−1(16) where m 1 (n x ) = m 2 (n x ) = n x (n x − 1)/2, and m 3 (n x ) = n x . The weights are provided in √ 1/2([e] k + [e] l ) : k < l, k, l = 1, · · · n x and √ 1/2([e] k − [e] l ) : k < l, k, l = 1, · · · n x , respectively, and κ x II = n x + 2 is the scaling factor. CNF-III: For the 3rd-degree nonlinear filter the predictive expectation E(x k \|z 1:k−1 , θ k−1 ) is represented by: E(·) = w(n x ) m(n x ) i=1 f ( κ x III P xx k−1 [a] i +x k−1 ),Ū k−1 + f (− κ x III P xx k−1 [a] i +x k−1 ),Ū k−1 (17) with w(n x ) = 1/2(n x + 1), m(n x ) = n x + 1, and κ x III = n x . CNF-IV: For the 5th-degree nonlinear filter the predictive expectation E(x k \|z 1:k−1 , θ k−1 ) is ap- proximated by: E(·) = w 0 (n x ) f x k−1 ,Ū k−1 +w 1 (n x ) m 1 (n x ) i=1 f ( κ x IV P xx k−1 ×[a] i +x k−1 ),Ū k−1 + f (− κ x IV P xx k−1 [a] i +x k−1 ),Ū k−1 +w 2 (n x ) m 2 (n x ) i=1 f ( κ x IV P xx k−1 [b] i +x k−1 ),Ū k−1 + f (− κ x IV P xx k−1 [b] i +x k−1 ),Ū k−1(18) where m 1 (n x ) = n x + 1 and m 2 (n x ) = n x (n x + 1)/2. The weights are defined in Table 2, and κ x IV = n x + 2. CNF-V: For the mixture-degree nonlinear filter the predictive expectation E(x k \|z 1:k−1 , θ k−1 ) is approximated by: E(·) = w 0 (n x ) f (x k−1 ,Ū k−1 )+w 1 (n x ) m(n x ) i=1 f ( κ x V P xx k−1 [a] i +x k−1 ),Ū k−1 + f (− κ x V P xx k−1 [a] i +x k−1 ),Ū k−1(19) where κ x V = n x + 2, and the weights are defined in Table 2. CNF-VI: For the re-defined 5th-degree nonlinear filter based on the Stroud's theorem [34] the predictive expectation E(x k \|z 1:k−1 , θ k−1 ) is represented by: E(·) = w 1 (n x ) m 1 (n x ) i=1 f ( P xx k−1 [ν 1 ] i +x k−1 ),Ū k−1 + f (− P xx k−1 [ν 1 ] i +x k−1 ),Ū k−1 + w 2 (n x ) m 2 (n x ) i=1 f ( P xx k−1 [ν 2 ] i +x k−1 ),Ū k−1 f (− P xx k−1 [ν 2 ] i +x k−1 ), +Ū k−1 + w 3 (n x ) m 3 (n x ) i=1 f ( P xx k−1 [ν 3 ] i +x k−1 ), U k−1 + f (− P xx k−1 [ν 3 ] i +x k−1 ),Ū k−1(20) 13 where the weights w 1 (n x ), w 2 (n x ) and w 3 (n x ) are deterministic values with respect to the specific system order. For completeness, the cubature points and weights for all orders can be obtained from [34], and for n > 7, some of the cubature points take on complex values. Filter d x ξ d x i w d x i CR CNF-I 3rd {[e] i } 1/(2n x ) [30, 16] CNF-II 0 2/(n x + 2) {[s] + i } 1/((n x + 2) 2 ) 5th {[s] − i } 1/((n x + 2) 2 ) [23] {[e] i } (4−n x )/(2(n x + 2) 2 ) CNF-III 3rd {[a] i } 1/(2(n x + 1)) [32, 24] CNF-IV 0 2/(n x + 2) 5th {[a] i } n 2 x (7−n x )/(2(n x +1) 2 (n x +2) 2 ) [36] {[b] i } 2(n x −1) 2 /((n x +1) 2 (n x +2) 2 ) CNF-V 0 2/(n x + 2) mixture {[a] i } n x /(2(n x + 1)(n x + 2)) [16, 35] CNF-VI {[ν 1 ]} i w 0 (n x ) 5th {[ν 2 ] i } w 1 (n x ) [34] {[ν 3 ] i } w 2 (n x ) Note: "CR" denotes the employed cubature rules in the corresponding filters. Based onx k\|k−1 = E(x k \|z 1:k−1 , θ k−1 ) and Table 2, one can further implement the procedure that are identified in conventional CKF using 3rd-degree cubature rules, whereas for our schemes multiple cubature degree rules are employed with associated different projected cubature points and weights. The differences among the class of CNFs can be identified in terms of the cubature points and weights as determined by the cubature rules that are projected onto the integration domain, leading to different performance on computing the integration of the conditional expectation x k\|k = E(x k \|z 1:k , θ k−1 ). Remark 4. This subsection explicitly derives a class of CNFs using the compiled cubature rules in Section 1, which enables one in an accessible manner implementation of these methods. The class of CNFs exhibits different advantages and disadvantages in dealing with various nonlinear systems. The performance of the proposed schemes with respect to the approximation accuracy, estimation error boundedness, robustness to unmodelled dynamics and uncertainties, and computational cost when applied to the complex GTE system are quantitatively evaluated and compared in Section 4. d θ th-degree Cubature-based Parameter Estimation 3.3.1. Modeling of Parameter Evolution In terms of long-term degradation, many works consider an exponential growth with respect to the operating time. For our proposed model-based fault parameter estimation module in this work, we consider simple linear model (Model I) and exponential model (Model II) for the short-term and long-term degradations, respectively. Model I: For a linear fault or degradation evolution model with uniform time-step the parameter evolution is considered to be governed by θ k = θ k−1 + α∆t + τ k(21) where α denotes the growth coefficient and ∆t denotes a known time-step length. Model II: For an exponential evolution of the parameter, the model takes the following form θ k = e α∆t θ k−1 + β(1 − e α∆t ) + τ k(22) where α and β denote model coefficients corresponding to the parameter evolution. The coefficient β is a scaling factor that can in practice be tuned to better fit the measurement records. For sake of generality, the state-space model corresponding to parameters are represented to be governed by Eq. (3), where h(·) can be linear as in Model I or exponential as in Model II. The states are assumed to be fixed atx k\|k that is determined ad specified from the state estimation filter module. CNF for Parameter Estimation The main goal here is to approximate the high-dimensional expectation integral E(φ 2 (θ k )\|z 1:k , x k ) given by Eq. (5) in the region x ∈ R n θ on the premise of a Gaussian assumption. In this subsection, it is assumed that state variablesx k\|k are available in order to design our proposed parameter estimation filter. Let us assume that the d θ th-degree cubature rules are implemented for designing the parameter estimation filter, where the cubature points and weights set are given by {ξ d θ j , w d θ j }, j = 1, · · · , N d θ . The process is referred to as state estimation for obtaining the conditional expectation of E(θ k \|z 1:k , x k ) and CNF for parameter estimation as summarized in Table 3. Table 3: Summary of CNF proposed for parameter estimation. [23,24] Unlike the state estimation problem, the parameter estimation problem introduced in this subsection is addressed by using the 3rd-degree or the mixture-degree cubature rules. This is justified based on observation that the health parameter dynamics in our engine application are modeled by linear Model I and exponential Model II that are as in general of lower complexity than that of the state dynamics [37]. The 5th-degree or higher-degree cubature rules theoretically can be used, however, from the computational efficiency perspective, the higher-degree cubature rules are not Table 4: The proposed 'Hybrid{i − j}' strategy for dual estimation with i, j ∈ {I,II,III,IV,V,VI} denoting the CNF as given in Table 1. Filter d θ ξ d θ i w d θ i CR CNF-I 3rd {[e] j } 1/2n θ [30, 16] 0 2/n θ + 2 CNF-III 3rd {[a] j } w : 1/(2(n θ + 1)) [32] 0 2/(n θ + 2) CNF-V mixture {[a] j } n θ /(2(n θ + 1)(n θ + 2))Group Methodology State Filter CNF − i Parameter Filter CNF − j i Degree d j Degree d G-I Hybrid{i − I} i ∈ {I, III} 3rd-degree j ∈ {I} 3rd-degree i ∈ {II, IV, VI} 5th-degree i ∈ {V} mixture-degree G-II Hybrid{i − III} i ∈ {I, III} 3rd-degree j ∈ {III} 3rd-degree i ∈ {II, IV, VI} 5th-degree i ∈ {V} mixture-degree G-III Hybrid{i − V} i ∈ {I, III} 3rd-degree j ∈ {V} mixture-degree i ∈ {II, IV, VI} 5th-degree i ∈ {V} mixture-degree G-IV Dual-PF PF - PF - Dual-UKF UKF - UKF - recommended as they could lead to substantial computational burden without yielding proportionally improved accuracy and performance. Let us denote the following error matrices as follows Ξ 1 k\|k−1 = h(ξ d θ k−1\|k−1, j )−θ k\|k−1 ,· · ·h(ξ d θ k−1\|k−1,N d θ )−θ k\|k−1 T Ξ 2 k\|k−1 = ξ d θ k\|k−1,1 −θ k\|k−1 ,· · ·ξ d θ k\|k−1,N d θ −θ k\|k−1 T Ξ 3 k\|k−1 = g(ξ d θ k\|k−1,1 )−ẑ k\|k−1 ,· · ·g(ξ d θ k\|k−1,N d θ )−ẑ k\|k−1 T(23) We are now in a position to present the following algorithm. Algorithm 2: The procedure for our proposed square-root d θ th-degree CNF for parameter estimation is now introduced. 1) Draw N d θ cubature points ξ d θ j and weights w d θ j based on the d θ th-degree cubature rule and previous distribution N(θ k−1\|k−1 , P θθ k−1\|k−1 ), with j = 1, · · · N d θ and P θθ k−1\|k−1 = S θθ k−1\|k−1 (S θθ k−1\|k−1 ) T . 2) Propagate the sampled cubature points ξ d θ j as follows ξ d θ k−1\|k−1 = S θθ k−1\|k−1 ξ d θ +θ k−1\|k−1(24) 3) Evaluate and predict the states bŷ θ k\|k−1 = N d θ j=1 w d θ j h ξ d θ k−1\|k−1, j and obtain the square-root version of the prediction error covariance by S θθ k\|k−1 = qr([Ξ 1 k\|k−1 / N d θ S Σ τ,k−1 ] ). 4) Draw and re-propagate the cubature points with the predicted value bỹ ξ d θ k\|k−1, j = S θθ k\|k−1 ξ d θ j +θ k\|k−1 . 16 5) Estimate the predicted measurement by evaluatinĝ z θ k\|k−1 = N d θ j=1 w d θ j g ξ d θ k\|k−1, j ,x k\|k , u k and obtain the square-root version of innovation covariance matrix as S zz,k\|k−1 = qr([Ξ 3 k\|k−1 / N d θ S Σ v,k ]). 6) Compute the cross-covariance matrix by firstly consider the square-root version of C k\|k−1 = Ξ 2 k\|k−1 // N d θ , and secondly obtain the cross-covariance matrix as P θz,k\|k−1 = Ξ 2 k\|k−1 (Ξ 3 k\|k−1 ) T /N d θ . 7) Update the parameters by invokingθ k\|k =θ k\|k−1 +K θ k (z k −ẑ k\|k−1 ), with K k = P θz,k\|k−1 (P zz,k\|k−1 ) −1 . The square-root error covariance matrix is now given by S θθ k\|k = qr([Ξ 2 k\|k−1 / N d θ − K θ k Ξ 3 k\|k−1 / N d θ K θ k S Σ v,k ]). Modified Cubature Points Propagation This subsection presents a modified cubature point propagation update strategy to enhance the robustness capability of our proposed dual cubature-based scheme to deal with modeling uncertainties. For the parameter estimation module, the d θ th-degree CNF can indeed capture the process dynamics by analyzing the known function φ 2 (·). However, in case of non-negligible uncertainties and unmodelled dynamics in the measurement model, the actual estimation can become compromised by only using the 3rd-degree cubature rules. In order to enable cubature points to at least account for both the mean and covariance of the process functions approximate errors, the following conditions are now proposed to be employed in our methodology as modified cubature point propagation update strategy [38,39], namely consider Ξ 1 k\|k−1 w d θ = 0 Ξ 1 k\|k−1 W d θ (Ξ 1 k\|k−1 ) T = P θθ k\|k−1 − Σ τ,k(25)Ξ 1 k\|k w d θ = 0 Ξ 1 k\|k W d θ (Ξ 1 k\|k ) T = P θθ k\|k − ∆E(26) where Ξ 1 k\|k = h(ξ d θ k\|k,1 ) −θ k\|k , · · · , h(ξ d θ k\|k,N d θ ) −θ k\|k T , and W d θ = diag([w d θ 1 , · · · , w d θ N d θ ] ). Let us as- sume Ξ 1 k\|k = Υ k Ξ 1 k\|k−1 , and substitute it into Eq. (26), to obtain Ξ 1 k\|k W d θ (Ξ 1 k\|k ) T = Υ k (P θθ k\|k−1 − Σ τ,k )Υ T k . Given the following equations L − k (L − k ) T = P θθ k\|k−1 − Σ τ,k−1(27)L + k (L + k ) T = P θθ k\|k − ∆E k(28) one can then obtain Υ k = L + k (L − k ) −1 . The estimation error covariance is now expressed as P θθ k\|k = (I n θ − K θ k B k )P k\|k−1 (I n θ − K θ k B k ) T + K θ k (E((ζ k + ψ z )(ζ k + ψ z ) T ) + Σ v,k )(K θ k ) T(29) where ζ k and ψ z denote the modeling uncertainty and high order terms resulting from the Taylor series expansion, and B k = ∂g(x k\|k , θ k , u k )/∂θ. Therefore, the term ∆E k in Eq. (28) is defined as Λ k K k Σ v,k K T k , where Λ k is selected as the largest eigenvalue of P θθ k\|k at the time instant k. We are now in a position to present our Algorithm 3. Algorithm 3: The procedure for the modified propagation of cubature points is constructed as follows: 1) Generate Ξ k−1\|k−1 by using N(θ k−1\|k−1 , P θθ k−1\|k−1 ) 2) Modified cubature points are then generated according tõ ξ d θ k−1\|k−1 = Ξ k−1\|k−1 +θ k−1\|k−1(30) 3) Run the Step 3) of Algorithm 1 4) Obtain Ξ 1 k\|k−1 from Eq. (23) and obtain L − k by Eq. (27) 5) Run the Steps 5) to 7) of Algorithm 1 6) Compute L + k as in Eq. (28) and compute Ξ k\|k . Then setξ k\|k = Ξ k\|k +θ k\|k . Remark 5. By comparing Eq. (24) with Eq. (30), it follows that the normal cubature point propagation method depends on the Gaussian assumption of the posterior probability density function (pdf), whereas the modified method relaxes the limitation on the Gaussian assumption of the posterior pdf. Remark 6. The 5th-degree cubature rules are significantly more robust to non-Gaussian noise and uncertainties when compared to the 3rd-degree CKF, UKF and PF [23]. In presence of measurement uncertainties, the modified cubature points propagation method can be utilized for the parameter estimation scheme given that the 3rd-degree cubature rules are employed. Due to the fact that this paper concentrates on our proposed dual estimation-based FD methodologies, further comparisons will be implemented on the hybrid-degree solutions. Nevertheless, the robustness analysis with respect to measurement uncertainties and unmodelled dynamics are conducted in Section 4.5 to demonstrate and illustrate the capabilities and benefits of our accomplished solutions. The Proposed Fault Diagnosis (FD) Formulation Diagnosis of drifts in unmeasurable health-related component parameters requires prior knowledge of parameters under healthy condition. Our FD logic and decision making protocol is developed based on analysis of residuals by comparing estimated parameters obtained by CNF schemes with parameters that are estimated under the initial fault free operation of the system. The FD problem under consideration deals with the nonlinear system whose dynamics is now governed by x k+1 = f (x k , θ T k λ θ (x k ), u k ) + w k z k = g(x k , θ T k λ θ (x k ), u k ) + ζ(x k , u k ) + v k(31) where λ θ (x k ) denotes the system health parameters representing a known differentiable function that determines the relationship between the system states and the fault parameters. The component FD problem is tackled and solved by considering that each health parameter is affected by an unknown and time-varying multiplicative fault parameter vector θ k . Since true values of parameters are assumed unknown, the required residuals for determining the FD criteria are obtained through the so-called residual signals. These signals are constructed as the difference between the estimated parameters under the fault-free operational mode (during the very start of the system operation) that is denoted byθ h , and the estimated parameters subsequent to the initial start of the system operation under the possibly faulty mode that is denoted byθ k\|k , that is r k =θ h −θ k\|k(32) where r k ∈ R n θ . For implementation of our proposed FD strategy developed based on the hybriddegree dual estimation scheme, the parameter estimates error will be considered as the main indicator for diagnosing faults in the system components. The decision-making logic for detecting, isolating, and identifying the faults are given as follows. Fault Detection Decision Logic: The decision on occurrence of a fault is made when at least one element of the residual signal in Eq. (32) exceeds its corresponding threshold, i.e., if ∀m, E( r m k ) ≤ r m max , the system is classified as healthy; otherwise if ∃m, E( r m k ) > r m max , the system is classified as being in the faulty condition, where m ∈ {M1, · · · , M8} denotes the fault mode as explicitly defined in Section V.C. Fault Isolation Decision Logic: The mth fault mode is isolated if E( r m k ) > r m max . In case of multi-mode fault scenarios, multiple residuals will exceed their corresponding thresholds. The variable r m max denotes the upper bound threshold for the mth residual signal as given by Eq. (32). The threshold for each residual signal is selected by conducting Monte Carlo simulation runs using the healthy system I/O data such that missed alarms and false alarms are minimized corresponding to the healthy mode of the system operation. Fault Identification Logic: Once the fault modes are detected and isolated, their severity levels through the parameter estimation module are identified based on the magnitude of the residual signals r m k . Our proposed hybrid-degree dual estimation strategy features case-dependent cubature rules and corresponding CNF for both state estimation and parameter estimation. The details are depicted in Table 4, where i denotes the specific CNF for the state estimation module and j refers to the specific parameter estimation module. Four groups G-I to G-IV of hybrid-degree methodologies are compared and investigated for different purposes. G-I aims to evaluate and compare estimation and fault diagnosis performance under the fixed 3rd-degree CNF-I for parameter estimation but varying degree of cubature rules for state estimation. G-II and G-III replace the parameter estimation filter in G-I, attempting to evaluate and compare whether different cubature theorems affect the performance of parameter estimation schemes. Dual estimation performance are compared with the well-known PF and UKF that are included in the group G-IV in order to evaluate the accuracy and computational cost of our methodology with these state-of-the-art nonlinear estimation techniques. Fault Diagnosis of a Gas Turbine Engine (GTE) Modeling Overview of Gas Turbine Engines The capabilities, advantages, and benefits of our proposed hybrid-degree CNF-based dual estimation strategy are now investigated and demonstrated by applying it to the FD problem of a twin-spool GTE. The FD performance is verified when the GTE is subjected to degradations in its component health parameters by injecting various concurrent/simultaneous abrupt or slowlyvarying faults. For a high fidelity representation of the GTE dynamical characteristics the volume dynamics and rotor dynamics are considered, as well as the heat transfer dynamics since they contribute to the nonlinear behavior of the twin-spool GTE [40]. The mathematical model as constructed in [41] is a set of nonlinear equations of motion that are expressed by Eq. (33). For the physical significance of the model parameters and details refer to [40,41]. The state variable for the GTE is given by x = [T CC , N 1 , N 2 , P LT , P CC , P LC , P HT ] T and the measurement is designated by z = [N 1 , N 2 , P HC , T HC , T LC , P LC , T LT , T HT ] T , where T CC , T HC , T LC , T LT and T HT represent the temperature variables in combustion chamber (CC), high pressure compressor (HPC), low pressure compressor (LPC), low pressure turbine (LPT) and high pressure turbine (HPT), respectively. N 1 and N 2 denote the rotational speeds of the spool connecting the HPC to HPT, and the spool connecting the LPC to LPT, respectively. P LT , P CC , P LC , P HT , P HC , and P LC denote the pressure variables in the subscripted components. The input or the control signal of the twin-spool GTE is the power level angle (PLA) which is related to the fuel mass flow rate (ṁ f ) through a variable gain. We now have, J 1 N 1 ( π 30 ) 2 N 2 = η 2 mech θ m LTṁ LT c p (T HT −T LT )−θ m LCṁ LC c p (T LC − T d ) J 2 N 2 ( π 30 ) 2 P LT = RT M V M (θ m LTṁ LT + β 1 + β θ m LCṁ LC −ṁ n ) P CC = P CC T CCṪ CC + γRT CC V CC (θ m HCṁ HC +ṁ f − θ m HTṁ HT ) P LC = RT LC V LC ( 1 1 + β θ m LCṁ LC − θ m HCṁ HC ) P HT = RT HT V HT (θ m HTṁ HT − θ m LTṁ LT )(33) During the engine lifetime the compressor and turbine undergo degradations that can originate from various sources, such as fouling, erosion and corrosion that are aerodynamic or performancerelated challenges and derivations. These performance-related anomalies can affect the component behavior and eventually the overall behavior of the GTE system. Component faults that are of 20 concern in this paper are caused by fouling and erosion degradations since they contribute to significant deterioration in the engine life cycle [26]. Fouling always occur in compressors (both at the HPC and LPC segments), which cause changes in compressor mass flow rate and efficiency. Erosion phenomena exert effects on reduction of efficiency and increase of mass flow rate in HPT or LPT segments. Consequently, the health parameters that are considered in this paper relate to efficiency and mass flow rates in compressor caused by fouling, as well as in turbine caused by erosion. A fault vector [θ η LC , θ η HC , θ η LT , θ η HT , θṁ LC , θṁ HC , θṁ LT , θṁ HT ] T is incorporated into the mathematical model (33) to manifest impacts of health parameters in corresponding components. The subscript η implies the change of efficiency and the subscriptṁ implies the change of mass flow rate. Verification and Validation of the Model Subject to Uncertainties To verify and validate the effectiveness of our proposed strategy the design of our nonlinear filters is based on a simplified mathematical model as provided in Eq. (33), however all the simulations shown subsequently have been applied to a more detailed, complex, and accurate model of the GTE that is obtained from GSP10 [40,41,29]. The differences between the simplified model (33) and the high fidelity representation of the GTE that is obtained from GSP10 [40,41,29] capture uncertainties and unmodeled dynamics. These are attributed to the manner performance maps are constructed that can express relationships between the health parameters and the system states as denoted by ζ(x k , u k ) in Eq. (31). Specifically, the performance maps for efficiencies and mass flow rates of the compressors (including both the HPC and LPC segments) that correspond toṁ HC , η HC ,ṁ LC and η LC in the model (33), as well as the performance maps for efficiencies and mass flow rates of the turbines (including both the HPT and LPT segments) that correspond toṁ HT , η HT ,ṁ LT and η LT in the model (33) need to be estimated and identified. Performance maps used in the GTE thermodynamic model are generated through various methodologies in the literature, such as [29]. In this paper, the methodology that is used for generating performance maps for the compressors and turbines is through twelve multi-layer feedforward neural networks. The networks are used for identifying the relationships between the concerned health parameters and the pressure ratio, as well as the states. An extensive set of simulation studies are conducted to ensure that the simplified model used in Eq. (33) is sufficiently reliable with respect to the more detailed, accurate, and high fidelity model of the GTE for further conducting our case studies robustness to uncertainties and unmodelled dynamics. These details are provided in Section 4.5. Hybrid-Degree Fault Diagnosis Performance Analysis The goal of this subsection is to justify and verify the rationalization and effectiveness of our proposed hybrid-degree dual CNF schemes through simulations under various fault scenarios. All the simulation scenarios correspond to the cruise flight mode of the GTE, and the process and measurement noise levels correspond to the same values as provided in [41], where standard deviations are given as percentage of the nominal values at typical cruise operating conditions. The PLA is assumed to be at 0.9, the Mach number is 0.74, and the ambient conditions are set to standard conditions. Importantly, since our goal is to compare capabilities of our proposed nonlinear filters it is justifiable that all comparative studies associated with the considered methodologies are implemented on the basis of the same process and measurement noise distributions for both state and parameter estimation problems. Our main objective is focused on FD performance of the GTE system. The hybrid-degree combinations are provided in Table 4, where the Dual-UKF and Dual-PF, and Hybrid I-I are effectively three concurrently running UKF, two PF and two CKF. For implementing the Dual-PF, the number of particles is selected through a quantitative analysis that is derived based on the mean absolute error (MAE%) accuracy criterion with respect to the estimation process steady state values. The number of particles is chosen as 500 corresponding to both the state and parameter estimation filters for the GTE. The number of cubature points for the CNF and the unscented points for the UKF are deterministic values and are provided in Table 9. Below we provide details on our considered three (3) distinct case studies where the fault modes are explicitly defined in Table 5: Case I: Abrupt Faults in the HPC In this scenario, effects of abrupt faults are studied by injecting a 3% mass flow rate loss (representing the fault severity) affecting the HPC component at the instant t = 3s. The residual signals with respect to the mass flow rate in the HPC are shown in Fig. 1 corresponding to the groups G-I to G-IV. The blue dotted lines depict the confidence bounds for residuals that are determined based on 50 independent Monte Carlo simulation runs under various healthy scenarios. By analyzing the residuals, the fault can be clearly detected and diagnosed. Fig. 1 (a) and Fig. 2 (b) depict the comparative results with respect to group G-I where they share the same filter for the parameter estimation (CNF-I), and group G-IV that involves Dual-PF and Dual-UKF. It follows from these results that residuals corresponding to our proposed combinations of the "5th-degree cubature rules for state estimation and 3rd-degree for parameter estimation", including Hybrid {II-I}, Hybrid {IV-I} and Hybrid {VI-I}, as well as the Dual-PF schemes can detect changes after the fault occurrence and converge to the injected fault severity. However, the Dual-CKF (i.e., Hybrid {I-I}) fails to detect the fault occurrence and the Hybrid {III-I} shows both false positive and false negative alarms during the indicated time window. What is in common for Hybrid {I-I} and Hybrid {III-I} methodologies is that both are using the 3rddegree cubature rule for designing the state estimation nonlinear filter. In this case, the 3rd-degree cubature rules are not appropriate/suitable for designing the state estimator under the given noise levels. Moreover, the Dual-UKF scheme is not capable of detecting the fault, whereas the Dual-PF scheme performs well in terms of detection and residual change tracking. The differences among the 5th-degree filters and the Dual-PF are not visibly distinguished, therefore the quantified MAE% is provided in Table 6. Observations from this table indicate that the approximation accuracy of the 5th-degree filters are quite close to that of the Dual-PF, where the combination of Hybrid {II-I} and Hybrid {VI-I} are slightly more accurate than others. Fig. 1 (c) compares the residuals corresponding to the 3rd-degree Mysovskikh-based CNF (CNF-III) for the parameter estimation purpose. The goal is to analyze whether the theorems affect the performance of dual estimation schemes and generated residuals. Effectively, modification of the 3rd-degree cubature rule for the parameter estimation does not provide obvious influence on the resulting residuals. The quantified estimation accuracy through MAE% is provided in Table 6 by comparing the Hybrid {VI-I} with the Hybrid {VI-III}, and the Hybrid {II-I} with the Hybrid {II-III}. The observation from this table is that the 3rd-degree cubature rule for parameter estimation based on the Genz's theorem yields relatively a higher precision than that is based on the Mysovskikh's theorem for our GTE application. Fig. 1 (d) aims to analyze the performance of the mixture-degree filter for parameter estimation. It follows that the hybrid combination constituted by a mixture-degree filter (CNF-V) for parameter estimation cannot detect the fault and converge to the fault severity within the selected time window. Remark 7. The 3rd-degree cubature rules (i.e., CR-I and CR-III) obtain poor engine state approximation of the statistical moments. Therefore, the corresponding CNF significantly deteriorates the dual estimation and fault diagnosis results (refer to e.g., Hybrid I-I or Hybrid III-I). Consequently, the importance of a a vlaid method for approximating the statistical moments is demonstrated and emphasized. Case II: Simultaneous Abrupt Faults in the HPT The effects of simultaneous abrupt faults are investigated by injecting a 2% mass flow rate increase and a 2% efficiency decrease affecting the HPT segment at the instant t = 6s. The residual signals resulting from the class of hybrid-degree combinations for the HPT mass flow rate and efficiency are shown in Fig. 2. Results for G-II, G-III and G-IV are not provided since these methodologies cannot detect faults, and methodologies in G-II are not listed as well since their performance have not been improved. The observations from Fig. 2 can be summarized as follows: (i) The 3rd-degree cubature rules used for both state and parameter estimation cannot achieve the FD objectives, where the Hybrid {I-I} yields false alarms and cannot converge to the correct fault severity. The Hybrid {III-I} is not capable of detecting the fault occurrence in the mass flow rate (Fig. 2 (a)) and yields both false positive and negative alarms in efficiency (Fig. 2 (b)). (ii) The proposed hybrid combinations, i.e. Hybrid {II-I} and Hybrid {IV-I} can detect the fault immediately after its occurrence and do ultimately converge to the correct fault severity. Besides, the Dual-UKF scheme cannot react to the fault occurrence, while the Dual-PF scheme achieves accurate estimation that are close to our proposed hybrid combinations. Quantitative estimation accuracy results for the Case II using the MAE% metric is shown in Table 7. Discussions on FD Performance for Abrupt Fault Cases: The purpose of this subsection is to provide comparison on the FD performance of all the methodologies provided in Table 1 before proceeding to more case studies. The metrics for evaluating the reliability of FD schemes consist of the estimation accuracy, computational cost and numerical stability factor (SF). The estimation accuracy is quantitatively measured through MAE% corresponding to the last two seconds of simulations after convergence of the filters. The computational cost is evaluated as the number of points or particles, and the numerical stability factor is quantified by SF = N d i=1 \|w d i \|/ R n w d (x)dx = N d i=1 \|w d i \|/ N d i=1 w d i . The metric SF manifests the numerical stability capability of the cubature rules, where SF = 1 denotes an optimal value, since it implies that the cubature rule holds the weights all-positive. The estimation accuracy based on MAE% is provided in Table 6 and Table 7 for the two fault cases. The computational cost as judged by the number of points/particles and SF values for various methods are shown in Table 9. Comparisons lead to the following observations and conclusions: • In view of estimation accuracy, our proposed hybrid schemes with combination of 5thdegree CNF for the state estimation and the 3rd-degree CNF for the parameter estimation can reach a high accuracy level with respect to MAE% using the Dual-PF method. The downside of the other hybrid choices are that some parameter estimates cannot converge to the actual fault severity and they provide a large number of false alarms (combinations based on the 3rd-degree for both the state and parameter estimations), and some cannot even detect the fault after its occurrence (combinations based on the 5th-degree or mixture-degree for parameter estimation). The specific theorems affect the performance slightly among the 5th-degree cubature rules, but generally the accuracy improves significantly by using the 3rd-degree cubature rules for the state estimation module. • From the perspective of computational cost, to achieve the expected estimation accuracy using the Dual-PF one should employ 500 particles to perform either state estimation or parameter estimation. This is by far higher than our proposed hybrid-degree schemes. Particularly, the Hybrid {VI-I} is the most computationally efficient combination within the approaches that can simultaneously detect, isolate and identify the faults (Table 9). • In view of numerical stability, the 5th-degree cubature rules (either the Genz's or Mysovskikh's theorems) risk of having higher probabilities of instabilities, although the Mysovskikh's theorem is more robust since the negative weights occur when the system order is greater than 7, while Genz's theorem experiences negative weights when n ≥ 4. The UKF that is utilized in our GTE application suffer from higher risk of numerical instability for both state and parameter estimation scheme (as shown in Table 9). Importantly, our proposed efficient 5thdegree rule based on the Stroud's theorem and the CNF-VI filter maintain positive weights for our GTE system which enables them to guarantee their numerical stability. In the following discussions on FD capabilities for the GTE system, we concentrate on comparisons and evaluations of Hybrid {I-I} (two concurrently running CKF), Hybrid {VI-I} and Dual-PF schemes. The simulation scenarios consist of multi-mode concurrent fault cases and simultaneous fouling and erosion degradation scenarios. For the fault parameter estimation module dealing with the compressors fouling degradation, the linear model in Eq. (21) is selected, whereas, the exponential model in Eq. (22) is utilized for the long-term turbine degradation prediction. Case III: Multi-Mode Concurrent Faults Effects of concurrent faults are investigated by injecting sequential fault patterns into the GTE system first at time t = 30s where the mass flow rate and efficiency in the LPC segment simultaneously decrease by 3%; second at the time instance t = 80s the mass flow rate in the LPT is increased by 2% and the efficiency is decreased by 2%; third the HPC segment experiences a 1% mass flow rate loss and a 4% efficiency loss at the time instant t = 120s; and finally at t = 160s, the mass flow rate in HPT is increased by 2% and the efficiency is decreased by 2%. The resulting residual signals are shown in Fig. 3, where the Dual-UKF scheme is not shown since it cannot detect fault occurrences in this case. It can be observed that our proposed Hybrid {VI-I} demonstrates the best performance as compared to the Dual-PF and Hybrid {I-I}, since it can detect and isolate multi-mode faults at instances of fault occurrences. Moreover, estimated fault severities converge to their corresponding true injected fault values. Although the Dual-PF can also achieve the FD objectives for majority of generated residuals, however for the mass flow rate change in HPT at t = 160s it fails to converge to the expected 2% mass flow rate increase. The Hybrid {I-I} generates false negative in LPC mass flow rate fault, and the convergence rate of estimated parameters is much slower than the other two methodologies. Therefore, residuals cannot converge to actual fault severities in the selected time windows. Moreover, a given parameter change can cause other parameter changes and slightly affect estimation of other fault severities. For instance, in Fig. 3 (h), residuals within the expected faultfree time window (before t = 160s) change slightly when the other faults occur, but residuals do not exceed their thresholds. This behavior can be explained as a result of uncertainties and discrepancies between the actual engine model and the simplified mathematical model that was used for the filter design, as well as unavoidable coupling effects among components that generate the actual engine data. Nevertheless, our proposed methodologies are still able to detect, isolate and identify fault scenarios and their severities. B-F C-II B-F C-II B-F C-II B-F C-II B-F C-II B-F C-II Fault Diagnosis Comparative Results In this subsection, a quantitative study is conducted by utilizing the confusion matrix analysis to evaluate the reliability, accuracy, precision, false alarm and/or misclassification rates corresponding to methodologies that are proposed in this work. For each algorithm (i.e., Hybrid {I-I}, Hybrid {VI-I} and Dual-PF), the confusion matrices are obtained by performing 100 independent Monte Carlo simulation runs. Fault scenarios are generated by considering severities that range from 1% to 10% of loss of effectiveness. The rows in confusion matrices show the actual number of fault scenarios applied to the GTE system and the columns represent the number of estimated fault categories. The diagonal elements represent the true positive rate (T P) for each fault occurrence. The evaluation metrics of the accuracy (ACC = 9 j=1 c j j /( 9 i=1 9 j=1 c i j )), precision (P j = c j j / 9 i=1 c i j ) and false positive (FP) (FP = 8 j=1 c 9 j / 9 j=1 c 9 j ) are also provided in Table 8, where c i j with i, j = 1, · · · 9 denote the value of rows and columns of the confusion matrix. The results are summarized in Table 8 which demonstrate that the FD accuracy of our Hybrid {VI-I} estimation (86.42%) outperforms that of the Hybrid {I-I} approach (59.67%), and the false positive alarm rate of our proposed method (8.61%) is much lower than that of the Hybrid {I-I} method (33.40%). The precision of our scheme for all the eight fault parameters is higher than that of Hybrid {I-I} approach. The performance of Dual-PF scheme is quite close to that of our designed Hybrid {VI-I} approach in terms of ACC, TP and precision, however the former approach needs a much higher computational cost to achieve the same performance estimation levels. Robustness Analysis in Presence of Uncertainties The purpose of this section is to evaluate robustness of the designed hybrid-degree dual cubaturebased nonlinear filtering schemes with respect to parametric uncertainties and unmodelled dynamics that arise from modeling. To verify the robustness of our proposed FD framework, the following uncertainties are first considered: ζ 1 (x k , u k ) = (k 1 (T CC − T HT ) − k 2 (T HC − T LC ))/(N 2 (π/30) 2 ∆J 1 ) ζ 2 (x k , u k ) = (k 3 (T HT − T LT ) − k 4 (T LC − T d ))/(N 1 (π/30) 2 ∆J 2 ) ζ 3 (x k , u k ) = ∆γRT CC (ṁ HC +ṁ f −ṁ HT )/V CC where k 1 = η 1 mechṁ HT c p , k 2 =ṁ HC c p , k 3 = η 2 mechṁ LT c p , and k 4 =ṁ LC c p . The parameters ∆γ, ∆J 1 and ∆J 2 correspond to inaccuracies in the ratio, the inertia of the high and low spool shafts, respectively. Therefore, the modeling uncertainty is represented by ζ(x k , u k ) = ζ 1 (x k , u k ), ζ 2 (x k , u k ), ζ 3 (x k , u k ), 0, 0, 0, 0, 0 T At time t = 6s , the fault to the LPT component is injected with a 2% increase in the mass flow rate. The parametric uncertainties of ∆γ, ∆J 1 and ∆J 2 in ζ(x k , u k ) are first assumed to be present at time t = 7s with an error of 3%. The FD performance in terms of residuals in presence of the above modeling uncertainties are shown in Fig. 4. It can be observed that our proposed Hybrid {VI-I}, Dual-PF and Hybrid {VI-I} with modified cubature points propagation can still detect, isolate and identify the faults having different levels of fluctuations, where the Hybrid {VI-I} with modified cubature points propagation method can be more robust to uncertainties. Several false alarms have occurred by using the Dual-PF. In contrast, higher false alarms are generated by the considered Dual-CKF and Dual-UKF. Table 10 shows the robustness analysis when a 6% increase of the LPT mass flow rate is injected in presence of different levels of uncertainties. It follows from this table that the Hybrid {VI-I} with modified cubature points propagation exhibits the lowest false alarm rates and the best accuracy in terms of MAE(%), whereas the fault detection time is longer than the Hybrid {VI-I} and Dual-PF. As compared to Dual-PF, Dual-CKF, and Dual-UKF schemes, the Hybrid {VI-I} scheme can detect occurrence of a fault most quickly and shows a more robust capabilities with respect to false alarm rates. The Dual-CKF and Dual-UKF are more sensitive to parametric uncertainties. One can observe that if level of uncertainties is increased e.g. to 7% inaccuracy, then all methodologies produce false alarms. Therefore, this testing case study scenario can be regarded as a reference benchmark on limits of our proposed strategy when handling significant levels of simultaneous severe faults and modelling uncertainties. In terms of the robustness capability against unmodelled dynamics, additive nonlinearities are added to the measurement model with respect to the spool speed N 1 . It was shown that (figures not shown due to space limitations) if the magnitude of uncertainty exceeds beyond a certain range our proposed FD framework could produce erroneous decisions. Nevertheless, it generates significantly improved FD performance as compared to available methodologies in the literature that we have considered in this work, and it enables one to deal with unknown dynamics within a given bounded range. Boundedness Analysis of Parameter Estimation Error The following lemmas are essential in establishing our main technical analysis and results. Lemma 1 ([42]). For 0 ≤ k ≤ N, suppose that X = X T ≥ 0, S k (X) = S T k (X) ∈ R n×n and H k (X) = H T k (X) ∈ R n×n . If S k (Y) ≥ S k (X), ∀X ≤ Y = Y T(34) and H k (Y) ≥ S k (X)(35) Then the solutions M k and N k to the following difference equations M k+1 = S k (M k ), N k+1 = H k (N k ), M 0 = N 0 > 0 (36) satisfy M k ≤ N k(37) Lemma 2 ( [43]). Given matrices A, H, E and F with appropriate dimensions such that FF T ≤ I. Let X be a symmetric positive definite matrix and γ be an arbitrary positive constant such that γ −1 I − EXE T > 0. Then the following inequality holds 44]). If both A and B are symmetric positive definite matrices, then (A + HFE)X(A + HFE) T ≤ A(X −1 − γE T E) −1 A T + γ −1 HH T (38) Lemma 3 ([(A + B) −1 > A −1 − A −1 BA −1(39) This paper concentrates on the boundedness analysis of parameter estimation error which is of utmost importance to accomplish the fault diagnosis objective, rather than the joint convergence analysis. The conventional strategy for dual state/parameter estimation scheme is to first optimize one with the other one fixed, and then alternate. Different from the other direct decoupling approaches, the error-coupling effects between states and parameters will be considered for the boundedness analysis of parameter estimation error in the next subsection. Before proceeding to the boundedness analysis, the following assumption is made regarding the dynamical system (1) and (2). Assumption 2: The variable {x k , θ k } satisfies the range over a compact set, for which the functions f (x k , θ k , u k ) in (1) and g(x k , θ k , u k ) in (2) are continuously differentiable with respect to the state x k and the parameter θ k , respectively. The parameter estimation methodology that is developed in this work is based on 3rd-degree cubature rules. Our goal is to investigate boundedness properties of the estimated parameters in presence of both approximation errors of the cubature rules, modeling uncertainties and error from the state estimation. Let us consider the following reformulated system Ω θ for the parameter estimation problem Ω θ : θ k = h(θ k−1 ) + τ k−1 z k = g(x k\|k , θ k , u k ) +g x,k + ζ(x k\|k , u k ) + v k(40) whereg x,k g(x k , θ k , u k ) − g(x k\|k , θ k , u k ) denotes the nonlinear interactive error which is introduced to account for the bias of state estimatex k\|k . Bounded Parameter Estimation Error Covariance The goal here is to verify the boundedness of the parameter estimation error covariance. Theorem 2. Consider the nonlinear system (40), and let the following conditions hold: (1) There exist positive constants b min , b max , δ v,max , such that the following bounds are satisfied for k ≥ 0: b 2 min I ≤ B k B T k ≤ b 2 max I, Σ v,k ≤ δ v,max I(41) (2) Taking the high-order terms of the Taylor series expansion into consideration, there exist positive constants, γ min , γ max , d i,min , d i,max , l min , l max , such that the following bounds can be fulfilled: γ 2 min I ≤ Γ k Γ T k ≤ γ 2 max I d 2 i,min I ≤ D i,k D T i,k ≤ d 2 i,max I, i = 1, 2 l 2 min I ≤ L k L T k ≤ l 2 max I,(42) Then, the parameter estimation error covariance matrix can be bounded by P θθ k\|k ≤ λ θ max I(43) Proof. Based on the Taylor series expansion, the parameter and measurement prediction errors by using the proposed CNFs for parameter estimation can be obtained as k\|k−1 = A k−1 k−1\|k−1 +ψ θ (θ k−1\|k−1 , θ k−1 , x k )+τ k (44) k\|k = (I − K θ k B k ) k\|k−1 + K θ k ψ θ z,k (θ k\|k−1 , θ k , x k ) − K θ k ζ k − K θ kg x,k − K θ k v k(45) where ψ θ (θ k−1\|k−1 , θ k−1 , x k ) and ψ z (θ k\|k−1 , θ k , x k ) represent the higher order terms which involve truncation errors associated with the approximation, and A k−1 = ∂h(·)/∂θ k−1 , B k = ∂g(·)/∂θ k . For simplicity, the high-order terms ψ θ (θ k−1\|k−1 , θ k−1 , x k ), ψ z (θ k\|k−1 , θ k , x k ) and ζ(x k , u k ) in the following deviations are simplified as ψ θ,k−1 , ψ θ z,k and ζ k , respectively. In order to facilitate the expression in the process of boundedness analysis, the conditions on the interactive error termg x,k , the high-order terms ψ θ,k−1 and ψ θ z,k are first analyzed. The condition on uncertainty ζ k has been provided in Assumption 1. • The interactive error termg x,k can be bounded given the following assumption. Assumption 3: The state estimation error and its corresponding error covariance matrix at the time instant k are bounded by ε x and ς max I, respectively, with ε x > 0 and ς max > 0. The rationality of the Assumption 3 will be discussed in Section 5.3. Based on the Taylor series expansion ofg x,k atx k\|k , one obtainsg x,k = α g x G x,kxk\|k , where G x,k = ∂g(x k , θ k , u k )/∂x k and α g x = diag(α g x ,1,k , α g x ,2,k , · · · , α g x ,n x ,k ) denotes an unknown instrumental diagonal matrix to compensate the high-order terms of expansion. Given Assumption 2, the two terms G x,k and α g x are assumed to hold conditions of G x,k ≤ḡ x and α g x ≤ᾱ g x I, respectively. In this case, further considering Assumption 3, one can obtain inequalities g x,k ≤ḡ x ε x and g x,kg T x,k ≤ḡ 2 x ς max I, withḡ x =ᾱ g xḡ x . • The high-order terms ψ θ,k−1 and ψ θ z,k can be transformed into the following formulations [45] ψ θ,k−1 = Θ k−1 D 1,k−1 L k−1 k−1\|k−1 ψ θ z,k = Γ k D 2,k L k k\|k−1 where Θ k−1 and Γ k denote problem-dependent scaling matrices, L k is introduced to provide an extra degree of freedom to tune the filter, and D i,k , i = 1, 2 denotes an unknown timevarying matrix accounting for the linearization errors of dynamical model which satisfies D i,k D T i,k ≤ I. The conditions on these matrices are given in (42). Considering the parameter error covariance matrices P θθ k\|k−1 = E{ k\|k−1 T k\|k−1 } and P θθ k\|k = E{ k\|k T k\|k }, with the Gaussian-assumed update procedures, the error covariance can be approximated by P θθ k\|k = (A k−1 + Θ k−1 D 1,k−1 L k−1 )P θθ k−1\|k−1 (A k−1 + Θ k−1 D 2,k−1 L k−1 ) T − (A k−1 + Θ k−1 D 1,k−1 L k−1 )P θθ k−1\|k−1 ×(A k−1 + Θ k−1 D 2,k−1 L k−1 ) T + Σ τ,k−1 (B k + Γ k D 2,k L k ) T (B k + Γ k D 2,k L k ) (A k−1 + Θ k−1 D 1,k−1 L k−1 ) ×P θθ k−1\|k−1 (A k−1 + Θ k−1 D 1,k−1 L k−1 ) T + Σ τ,k (B k + Γ k D 2,k L k ) T + E{ζ k ζ T k } + E{g x,kg T x,k } + Σ v k −1 (B k + Γ k D 2,k L k ) × (A k−1 + Θ k−1 D 1,k−1 L k−1 )P θθ k−1\|k−1 (A k−1 + Θ k−1 D 1,k−1 × L k−1 ) T + Σ τ,k(46) where the termg x,k is uncorrelated with the modeling uncertainty ζ k and the predictive error k\|k−1 . According to Lemma 3, one can approximate P θθ k\|k ≤ (B k + Γ k D 2,k L k ) −1 E{ζ k ζ T k } + E{g x,kg T x,k } + Σ v k (B k + Γ k D 2,k L k ) −T(47) Given the conditions in (41) and (42), we can have P θθ k\|k ≤ζ 2 I +ḡ 2 x ς max I + δ v,max I (b min + γ min d 2,min l min ) 2 (48) Computing the Euclidean norm on both sides of the above inequality leads to: P θθ k\|k ≤ (ζ 2 +ḡ 2 x ς max + δ v,max )/(b min + γ min d 2,min l min ) 2 (49) Therefore, the parameter estimation error covariance P θθ k\|k is bounded in the case that the modeling uncertainty ζ k , the interactive error term E{g x,kg T x,k } and the noise term Σ v,k are bounded. Let the right hand side of the inequality (48) be denoted by λ θ max I, then the proof of Theorem 2 is completed. Bounded Parameter Estimation Error The second task for parameter estimation error boundedness analysis is to provide a sufficient condition to verify the exponential boundedness of the parameter estimation error in the mean square sense. The following Assumption 4 states some standard results on boundedness of stochastic processes that are utilized as presented in our main result in Theorem 3. Assumption 4: It is assumed that (a) the matrix A satisfies A k A T k ≤ a 2 max I, (b) the prior error covariance matrix satisfies λ θ min I ≤ P θθ k−1\|k−1 ≤ λ θ max I, (c) the high-order terms-related matrix is bounded by Θ k−1 Θ T k−1 ≤ ϑ 2 max I, (d) the inequalities Σ τ,k ≤ δ τ,max I hold, and where all the bounds are positive constants. Theorem 3. Consider the parameter estimation filter as proposed in the dual methodology consisting of a 3rd-degree (d θ = 3) CNF (CNF-I or CNF-III), and let Theorem 2 and Assumption 4 hold. The parameter estimation error k\|k satisfies the following conditions, E[J( k\|k )] − J( k−1\|k−1 ) ≤ µ q − µ p J( k−1\|k−1 ) (50) 1 λ θ max k−1\|k−1 2 ≤ J( k−1\|k−1 ) ≤ 1 λ θ min k−1\|k−1 2 (51) Therefore, k\|k is boundeded in mean square sense where E{ k\|k 2 } ≤ λ θ max λ θ min E{ 0\|0 2 }(1 − µ p ) k + µ q λ θ min k−1 i=1 (1 − µ p ) i (52) if the initial conditions of the system satisfy 0\|0 ≤ f , and µ q > 0, 0 < µ p ≤ 1. Proof. Let us define a performance index for parameter estimation as J( k\|k ) = T k\|k (P θθ k\|k ) −1 k\|k . Following the Assumption 3, it gives 1 λ θ max k−1\|k−1 2 ≤ J( k−1\|k−1 ) ≤ 1 λ θ min k−1\|k−1 2 Substituting Eq. (44) intoθ k\|k and Eq. (45), one obtains k\|k = (A k−1 + Θ k−1 D 1,k−1 L k−1 )(I − K θ k (B k + Γ k D 2,k L k )) k−1\|k−1 + o u,k + o n,k(53) where o n,k = (I − K θ k (B k + Γ k D 2,k L k ))τ k − K k v k denotes the noise term and o u,k = −K k ζ k − K kgx,k denotes uncertainties from the approximation error and modeling. Consequently, the parameter error covariance matrix becomes P θθ k\|k = Π k P θθ k−1\|k−1 Π T k + E{Π k k−1\|k−1 o T u,k + o u,k T k−1\|k−1 Π T k } + E{o n,k o T n,k } + E{o u,k o T u,k } + ∆P k\|k (54) where Π k = (A k−1 + Θ k−1 D 1,k−1 L k−1 )(I − K θ k (B k + Γ k D 2,k L k )), which satisfy Π k ≤ (a max + ϑ max d 1,max l max )(1 +k θ (b max + γ max d 2,max l max )) π wherek θ denotes the upper bound of the gain withk θ ≤ λ θ max (b max + γ max d 2,max l max )/δ v,max . By substituting k\|k into J( k\|k ), one obtains J( k\|k ) = T k−1\|k−1 Π T k (P θθ k\|k ) −1 Π k k−1\|k−1 + T k−1\|k−1 Π T k (P θθ k\|k ) −1 o u,k + o T u,k (P θθ k\|k ) −1 Π k k−1\|k−1 + o T u,k (P θθ k\|k ) −1 o u,k + o T n,k (P θθ k\|k ) −1 o n,k(55) Each term in (55) can be shown to be bounded by utilizing certain conditions of the proposed assumptions. The simplified derivation process is as follows. Simplify (54) as P θθ k\|k = Π k P θθ k−1\|k−1 Π T k + Γ * k\|k , then P θθ k\|k = Π k P θθ k−1\|k−1 + Π −1 k Γ * k\|k Π −T k Π T k(56) where Π −1 k Γ * k\|k Π −T k ≥ δ τ,max /π 2 . It follows that the next inequality can be obtained: T k−1\|k−1 Π T k (P θθ k\|k ) −1 Π k k−1\|k−1 ≤ (1 − µ p ) T k−1\|k−1 (P θθ k−1\|k−1 ) −1 k−1\|k−1 (57) where 1 − µ p = 1 + δ τ,max /π 2 −1 . Given that J( k−1\|k−1 ) = T k−1\|k−1 (P θθ k−1\|k−1 ) −1 k−1\|k−1(58) the first term in (55) can be shown to be bounded by (1 − µ p )J( k−1\|k−1 ). Regarding the uncertainty term o u,k , we have o u,k ≤k θ (ζ +ḡ x ε x ) ō u(59) where the uncertainty term satisfies o T u,k (P θθ k\|k ) −1 o u,k ≤ō 2 u /λ θ max(60) where λ θ max I denotes the upper bound for P θθ k\|k , which has been shown earlier. Then T k−1\|k−1 Π T k (P θθ k\|k ) −1 o u,k + o T u,k (P θθ k\|k ) −1 Π k k−1\|k−1 is consequently bounded by 2πō u k−1\|k−1 /λ θ max . Considering k−1\|k−1 ≤ f , it follows that the second and third terms of Eq. (55) can satisfy T k−1\|k−1 Π T k (P θθ k\|k ) −1 o u,k + o T u,k (P θθ k\|k ) −1 Π k k−1\|k−1 ≤ 2πō u f /λ θ max Regarding the noise term o n,k , we have o n,k o T n,k ≤ (1 +k θ (b max + γ max d max l max )) 2 δ τ,max +k 2 θ δ v,max ō n (61) Therefore, the following inequality holds o T n,k (P θθ k\|k ) −1 o n,k ≤ō n /λ θ max(62) Consequently, one can obtain J( k\|k ) ≤ (1 − µ p )J( k−1\|k−1 ) + µ q(63) where µ q = 2πō u f /λ θ max +ō 2 u /λ θ max +ō n /λ θ max . Therefore, the following inequality can be observed: E[J( k\|k )] − J( k−1\|k−1 ) ≤ −µ p J( k−1\|k−1 ) + µ q(64) where µ q > 0 and 0 < µ p < 1. The parameter estimation error k\|k in presence of bounded sensor modeling uncertainties by using the 3rd-degree CNF satisfies the root mean square boundedness, i.e., E{ k\|k 2 } ≤ λ θ max λ θ min E{ 0\|0 2 }(1 − µ p ) k + µ q λ θ min k−1 i=1 (1 − µ p ) i ≤ λ θ max λ θ min E{ 0\|0 2 }(1 − µ p ) k + µ q λ θ min ∞ i=1 (1 − µ p ) i = λ θ max λ θ min E{ 0\|0 2 }(1 − µ p ) k + µ q λ θ min µ p(65) when the initial error 0\|0 is bounded by f . Considering the Jensen's inequality, we have E{( k\|k ) 2 } ≤ E{ k\|k 2 }(66) where the upper bound of the parameter estimation error can be given as E{ k\|k } ≤ E{ k\|k 2 } ≤ λ θ max λ θ min E{ 0\|0 2 }(1 − µ p ) k + µ q λ θ min µ p(67) This completes the proof of the theorem. Remark 7: Compared to the existing estimation error boundedness analysis of CKF, our analysis has the following unique features. First, distinct from [46], which is based on a nonlinear system but with linear measurements, our boundedness analysis is conducted for nonlinear stochastic systems with nonlinear measurement equations. Analyzing the error boundedness of the 3rd-degree CNF with nonlinear measurement expressions is more challenging than analyzing that of linear measurement equations due to the resulting higher approximation errors associated with the cubature. Second, distinct from the work presented in [47], we further added the term (ζ(x k , u k )) representing uncertainties into the boundedness analysis. Consequently, uncertainty from both the approximation error of the cubature rules as well as measurement uncertainties are taken into account and considered. Importantly, we have analyzed the interactive error effects g x,k from the state estimation, which has not been considered in the boundedness analysis of the relevant literature. Since the above boundedness analysis on parameter estimation error can be guaranteed with one important premise that the state estimation error and its covariance are bounded (i.e., Assumption 3), the following section will verify such rationality. Let us define a performance index as J( x k\|k ) = ( x k\|k ) T (P xx k\|k ) −1 x k\|k , which can be expressed as J( x k\|k ) = ( x k−1\|k−1 ) T (Π 1,k Π 2,k ) T (P xx k\|k ) −1 Π 1,k Π 2,k x k−1\|k−1 + ( x k−1\|k−1 ) T (Π 1,k Π 2,k ) T (P xx k\|k ) −1 o u,k + o T u,k (P xx k\|k ) −1 × Π k x k−1\|k−1 + o T u,k (P xx k\|k ) −1 o u,k + o T n,k (P xx k\|k ) −1 o n,k (72) where Π x 1,k = I − K x k (D k + Z k S 2,k M k ) Π x 2,k = C k−1 + X k−1 S k−1 M k−1 o x n,k = Π x 1,k w k − K x k v k o x u,k = Π x 1,kf θ,k−1 − K x kg θ,k−1 − K x k ζ k It is easy to observe the boundedness of the above variables, which are defined as Π x 1,k ≤π 1 , Π x 1,k ≤π 2 , o x n,k ≤ō x n and o x u,k ≤ō x u . Through tedious algebraic manipulations and assuming that x k−1\|k−1 ≤ x , each term in Eq. (72) can be shown to be bounded by utilizing certain conditions of the Assumption 5, which can be expressed as follows: J( x k\|k ) ≤ (1 − p )J( k−1\|k−1 ) + q(73) where, p = 1 − 1 + δ τ,max /π 2 1π 2 2 −1 q = 2π 1π2ō x u x /σ max + {(ō x u ) 2 +ō x n }/σ max withō x u =π 1fθ +k xḡθ +k xζ andō x n =π 2 1 δ w,max +k 2 x δ v,max . Therefore, the following inequality can be observed: E[J( x k\|k )] − J( x k−1\|k−1 ) ≤ − p J( x k−1\|k−1 ) + q(74) where q > 0 and 0 < p < 1. Based on the Assumption 5 and Eq. (58), one obtains 1 σ max x k−1\|k−1 2 ≤ J( x k−1\|k−1 ) ≤ 1 σ min x k−1\|k−1 2(75) The state estimation error x k\|k in presence of error coupling effects from the parameter estimation satisfies the root mean square boundedness, i.e., E{ x k\|k 2 } ≤ σ max σ min E{ x 0\|0 2 }(1 − p ) k + q σ min p(76) when the initial error x 0\|0 is bounded by ε x 0 . Considering the Jensen's inequality, the upper bound of the state estimation bias can be given as E{( x k\|k )} ≤ E{ x k\|k 2 } ≤ σ max σ min E{ ε x 0 2 }(1 − p ) k + q σ min p(77) Towards this end, our concerned parameter estimation error can be ultimately to be justified as bounded. Specifically, by assuming that both the initial errors and the initial error covariance matrices for states and parameters are bounded, the state estimation at next time instant can be bounded by the condition in (77). This condition will be subsequently utilized into the parameter estimation error boundedness analysis, leading to the bounded error that is expressed in (67). Such analysis approach is motivated by the proposed dual estimation scheme in this paper. That is, the developed state filter and parameter filter are concurrently running, which indicates that one estimate is obtained and optimized at one time and then alternate to estimate the other. Remark 8: It should be noted that the nonlinearities, faults and modeling uncertainties lead to the deviation of the possible equilibrium points. Therefore, we aim to consider the exponential boundedness in mean square (rather than the convergence) of the estimation error for both states and parameters. As shown in the proposed theorems, sufficient conditions under certain assumptions are given to achieve the desired performance requirements. Importantly, upper bounds on the estimation bias for the developed dual estimation scheme are provided, even taking into account the interactive error coupling effects between states and parameters. Further research directions include development of global convergence criterion for the joint state and fault estimation algorithm. Discussion and Conclusions In this paper, a novel hybrid-degree dual estimation framework is proposed by using casedependent cubature rules and our proposed cubature-based nonlinear filters for performing simultaneously state and parameter estimation objectives. The performance of our proposed hybriddegree dual estimation strategy is demonstrated and evaluated by its application to a nonlinear gas turbine engine system for solving component fault diagnosis problem. From the perspective of dual estimation performance, our proposed hybrid-degree scheme with the 5th-degree for state estimation and the 3rd-degree for parameter estimation demonstrates its superiorities in terms of estimation accuracy and robustness to unmodelled dynamics and parametric uncertainties as compared to cubature Kalman filters and unscented Kalman filters, and computational efficiency as compared to the well-known particle filters. The superiority, especially of the hybrid combination of Hybrid {VI-I}, is reflected by the promptness in fault detection time, lower false alarm rates, reasonable fault identification accuracy, guarantee of computational efficiency and estimation error boundedness. By incorporating a modified cubature point propagation method into our proposed hybrid solution, the robustness capabilities against modeling uncertainties can be improved in terms of lower false alarms. The above characteristics justify and substantiate the observation that our proposed strategy is more suitable for the purpose of fault diagnosis of safety critical nonlinear systems that require lower fault detection times, lower false alarm rates, and accurate identification of the current health status. The limitations of using deterministic sampling and weighting in cubature-based nonlinear filters suggest that considering more effective and adaptive tuning of free parameters may lead to a promising solution for improving the overall diagnostics and estimation performance, especially robustness with respect to modeling uncertainties, model mismatches, and disturbances. In addition, another one of our future work will be concentrated on efficiently estimating the noise statistics to improve the adaptivity and robustness of the developed cubature-based nonlinear filters. This is motivated by the fact that it has been shown by some research work in the literature that incorporating the noise statistic estimator into the filtering process can actually in an adaptive manner adjust the noise tuning parameters. The verification and validation of our proposed results to a real gas turbine engine is another topic for our future research. Definition 2 . 2The set {ξ d i , w d i } based on n-D surface by using the fifth-degree modified Stroud's Theorem is constructed as [(c p T HC θ m HCṁ HC +η CC H uṁ f −c p T CC θ m HTṁ HT ) −c v T CC (θ m HCṁ HC +ṁ f −θ m HTṁ HT )] N 1 = η 1 mech θ m HTṁ HT c p (T CC −T HT )−θ m HCṁ HC c p (T HC −T LC ) Figure 1 : 1Residuals r k (y-axis) corresponding to the abrupt 3% mass flow rate fault scenario in the HPC injected at the time t = 3s. Hybrid {I-I} Hybrid {II-I} Hybrid {IV-I} Hybrid {VI-I} Dual-PF Hybrid {II-III} Hybrid {VI-III} B-F and C-I denote 'Before-Fault' and 'after Case-I', respectively. Figure 2 : 2Residuals r k (y-axis) (%) for (a) mass flow rate; (b) mass flow rate; (c) efficiency; and (d) efficiency, simultaneous fault scenario: Case II. B-F and C-II denote 'Before-Fault' and 'after Case-II', respectively. Figure 3 : 3Residuals r k (the y-axis) (%) corresponding to concurrent and simultaneous fault scenarios in the component subsystems LPC, LPT, HPC and HPT. Figure 4 : 4Residuals r k (the y-axis) (%) corresponding to degradation in the LPT mass flow rate scenarios in presence of modeling uncertainties. FDT denotes 'Fault Detection Time', and FAR denote 'False Alarm Rate'. Table 1 : 1The CNF used for GTE System based on class of cubature rules.CNF Description Table 2 . 2The Table 2 : 2Summary of CNF proposed for the state estimation. Table 5 : 5Degradation Modes Considered in the Gas Turbine Engine.Component HP Description Mode HPC θ η HC Changes in efficiency of HPC M1 θṁ HC Changes in mass flow rate of HPC M2 HPT θ η HT Changes in efficiency of HPT M3 θṁ HT Changes in mass flow rate of HPT M4 LPC θ η LC Changes in efficiency of LPC M5 θṁ LC Changes in mass flow rate of LPC M6 LPT θ η LT Changes in efficiency of LPT M7 θṁ LT Changes in mass flow rate of LPT M8 Table 6 : 6State/parameter estimation accuracy (MAE%) for Case I corresponding to abrupt faults in the HPC. Table 7 : 7State/parameter estimation accuracy (MAE%) for Case II corresponding to simultaneous abrupt faults in the HPT.Var. Hybrid {I-I} Hybrid {II-I} Hybrid {IV-I} Hybrid {VI-I} Dual-PF Hybrid {VI-III} Table 8 : 8Confusion matrix analysis (%).ACC FP P θ mHC P θ η HC P θ mLC P θ η LC P θ mHT P θ η HT P θ mLT P θ η LT Hybrid {I-I} 59.67 33.40 65.22 69.57 62,96 76.47 60.00 47,62 47.83 58.33 Hybrid {VI-I} 86.42 8.61 92.86 90.00 93.33 88.46 92.59 80.77 77.77 67.74 Dual-PF 87.01 7.687 91.25 88.37 94.56 89.87 91.74 88.93 77.85 67.83 Table 9 : 9Computational cost with respect to number of points and stability factor (SF) for dual estimation methodologies. Hy {I-I} Hy {II-I} Hy {III-I} Hy {IV-I} Hy {VI-I} Hy {II-III} Hy {VI-III} Dual-PF Dual-UKFState Estimation Ref 2n x 2n 2 x + 1 2n x + 2 n 2 x + 3n x + 3 n 2 x + n x + 2 2n 2 x + 1 n 2 x + n x + 2 - 2n x + 1 GTE 14 99 16 73 58 99 73 500 15 SF 1 1.23 1 1 1 1 1 - 3.67 Parameter Estimation Ref 2n x 2n x 2n x 2n x 2n x 2n x + 2 2n x + 2 - 2n θ + 1 GTE 16 16 16 16 16 18 18 500 17 SF 1 1 1 1 1 1 1 - 4.33 Note: "Hy" indicates Hybrid, and "Ref" indicates the general reference number of points; 'GTE' indicates the specific number of points for the gas turbine engine. Table 10 : 10Robustness analysis for 6% increase in the LPT mass flow rate corresponding to various uncertainty levels.Method FDT 2% 3% 4% 5% 6% 7% MAE% FAR% MAE% FAR% MAE% FAR MAE% FAR% MAE% FAR% MAE% FAR% Boundedness Analysis of State Estimation ErrorThe goal of this section is to verify the boundedness of the state estimation error for achieving the boundedness of the estimated parameters. Considering the following reformulated nonlinear system for state estimation problem at time k given the estimatedθ k−1 Ω x :xThe error due to the estimateθ k−1 is accounted for by introducingf θ,k−1 andg θ,k−1 .Based on the Taylor series expansion off θ,k−1 atθ k−1\|k−1 , one obtains f θ,α f θ ,n θ ,k ) denotes an unknown instrumental diagonal matrix to compensate the high-order terms of expansion, which is assumed to satisfy α f θ ≤ᾱ f θ . Given that the prior parameter error and its covariance matrix are assumed to be bounded at the time instantSimilarly, g θ,k−1 ≤ḡ θ θ can be also devised by using the Taylor series expansion ofg θ,k−1 at θ k−1\|k−1 and each of the matrices can be bounded.In this paper, we have developed 5th-degree cubature rule-based nonlinear filters for state estimation. Due to the fact that different 5th-degree cubature-based nonlinear filters can have different weights and cubature points, therefore, the boundedness analysis differs from each other. Here the boundedness analysis will focus on the employed CNF-IV.Based on the Taylor series expansion of the nonlinear functions f (·) and g(·), we havewhere C k−1 = ∂ f (·)/∂x k−1 and D k = ∂g(·)/∂x k . ψ x (x k−1\|k−1 , x k−1 ) and ψ x z (x k\|k−1 , x k ) represent the higher order terms which involve truncation errors associated with the approximation. This can be transformed into easy-to-handle formulations, i.e., ψ x,k−1 = X k−1 S 1,k−1 M k−1 x k−1\|k−1 and ψ x z,k = Z k S 2,k M k x k\|k−1 , where X k−1 and Z k denote problem-dependent scaling matrices, M k is introduced to provide an extra degree of freedom to tune the filter, S 1,k−1 and S 2,k denote unknown timevarying matrices accounting for the linearization errors of the dynamical model which satisfy S 1,k−1 S T 1,k−1 ≤ I and S 2,k S T 2,k ≤ I, respectively. 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[ "Stringy corrections to the entropy of electrically charged supersymmetric black holes with AdS 5 × S 5 asymptotics", "Stringy corrections to the entropy of electrically charged supersymmetric black holes with AdS 5 × S 5 asymptotics" ]	[ "João F Melo \nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUK\n", "Jorge E Santos \nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUK\n\nInstitute for Advanced Study\n08540PrincetonNJUSA\n" ]	[ "DAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUK", "DAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUK", "Institute for Advanced Study\n08540PrincetonNJUSA" ]	[]	We study the leading α corrections to the entropy of certain black holes with AdS5 × S 5 asymptotics. We find that, in the supersymmetric limit, the entropy does not receive α corrections. This result strengthens recent calculations that match the index of N = 4 Super-Yang-Mills with the corresponding partition function in the supersymmetric limit. In the small temperature regime, we find that the entropy corrections are concordant with the weak gravity conjecture.	10.1103/physrevd.103.066008	[ "https://arxiv.org/pdf/2007.06582v3.pdf" ]	234,356,575	2007.06582	26945f6cf7a733d1f395d6e7a94a09c474deafcf	Stringy corrections to the entropy of electrically charged supersymmetric black holes with AdS 5 × S 5 asymptotics João F Melo DAMTP Centre for Mathematical Sciences University of Cambridge Wilberforce RoadCB3 0WACambridgeUK Jorge E Santos DAMTP Centre for Mathematical Sciences University of Cambridge Wilberforce RoadCB3 0WACambridgeUK Institute for Advanced Study 08540PrincetonNJUSA Stringy corrections to the entropy of electrically charged supersymmetric black holes with AdS 5 × S 5 asymptotics We study the leading α corrections to the entropy of certain black holes with AdS5 × S 5 asymptotics. We find that, in the supersymmetric limit, the entropy does not receive α corrections. This result strengthens recent calculations that match the index of N = 4 Super-Yang-Mills with the corresponding partition function in the supersymmetric limit. In the small temperature regime, we find that the entropy corrections are concordant with the weak gravity conjecture. Introduction. Quantum gravity remains a largely unexplored frontier. However, due to the seminal work in black hole thermodynamics in the seventies [1][2][3][4][5][6][7][8][9] we know that, whatever the ultimate unifying theory is, it should reproduce the Hawking effect and give a microscopic derivation of the Bekenstein-Hawking black hole entropy in the appropriate semi-classical limit. To date, string theory appears to be the only candidate for a quantum theory of gravity that explains both of these effects in an ambiguity free manner at a microscopic level [10][11][12][13]. In particular, the seminal work of [10] provided a beautiful matching between the Bekenstein-Hawking entropy of certain five-dimensional supersymmetric black holes with asymptotically flat boundary conditions and the counting of specific supersymmetric states. Since then, a number of generalisations of this work have been accomplished for black holes with more complex topologies (see e.g. [14]). However, this matching has only been accomplished for black holes with asymptotically flat boundary conditions. One might wonder how to extend these results to asymptotically anti-de Sitter (AdS) spacetimes, for which we have the so-called AdS/CFT correspondence [15][16][17][18]. In its original form, the AdS/CFT correspondence relates four-dimensional N = 4 Super-Yang-Mills (SYM) with gauge group SU (N ) and 't Hooft coupling λ, to type IIB superstring theory with string coupling g s , string length s ≡ √ α on AdS 5 ×S 5 with radius L and N units of F (5) flux through the S 5 . The field theory is thought to live at the conformal boundary of AdS 5 , and for this reason the correspondence is said to be holographic in nature. The string theory side is often referred to as the 'bulk' and the field theory side as the 'boundary'. The parameters on each side of the AdS/CFT correspondence are related via λ N = 2πg s and 2λ = L 4 4 s .(1) However, it remains a challenge to understand string theory for generic values of g s , so one usually takes N → +∞, at fixed λ, so that g s → 0. Under these assumptions, the bulk theory reduces to a classical theory of strings. To simplify matters further, we can also take λ to be large, but not necessarily infinite. On the field theory side, we are thus looking at strong coupling effects, and on the gravity side we have a supergravity theory. Corrections to the strict λ → +∞ limit appear in the bulk as higher derivative terms which account for finite size string corrections. The problem of reproducing the entropy of certain black hole solutions in global AdS 5 on the string theory side is now mapped into a counting problem of certain states on the field theory. Because we are interested in global AdS 5 , the field theory is thought to live on R t ×S 3 . The holographic description of electrically-charged supersymmetric black holes with AdS × S 5 asymptotics is in terms of states of the dual N = 4 SYM that preserve only one of the available sixteen supercharges. Such states should be counted (with sign) by the superconformal index. However, early attempts to compute this index gave an order one result [19], whereas the entropy of AdS 5 black holes scales with N 2 . It was not until recently that this long-standing problem was partially solved. In particular, [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] have argued that, upon using complex chemical potentials, the cancellations between fermionic and bosonic degrees of freedom observed in [19] can be avoided. This leads to an index of order N 2 , whose associated entropy matches those of known supersymmetric black holes [37][38][39]. This body of work thus provides overwhelming evidence that whether we compute the entropy via the index or via a more standard calculation using the partition function of N = 4 SYM, the results should agree with each other. It should be noted that this latter quantity can only be computed via an indirect bulk calculation using the Bekenstein-Hawking entropy. The matching between the partition function calculation and index, leads to a number of fascinating predictions. In particular, since the index cannot exhibit a dependence on continuous parameters 1 , we expect the counting on the field theory side to not depend on the 't Hooft coupling λ. On the bulk side of the story, because we are computing directly a partition function, this is not an obvious fact given we know that the classical equations of motion of type IIB supergravity do admit corrections in α , due to finite size stringy effects. These, in the small α limit, appear as higher-derivative corrections to the equations of motion of type IIB supergravity. The first non-trivial corrections for supergravity configurations that only involve the metric g and five-form F (5) were worked out in [42] 2 , following the seminal results of [43]. The black holes. We focus on black hole solutions of five-dimensional minimal gauged supergravity, whose action comprises a five-dimensional metric g and a field strength F = dA and reads S 5D = 1 16πG 5 M d 5 x √ −g R + 12 L 2 − 1 4 F ab F ab + 1 12 √ 3 ε abcde F ab F cd A e . (2) Known black hole solutions in this theory carry one electric charge Q, and two angular momenta J 1 , J 2 . For simplicity, we focus on the case where J 1 = J 2 = J. The equations of motion derived from Eq. (2) read R ab − g ab 2 R − 6 L 2 g ab = 1 2 F c a F bc − g ab 4 F cd F cd ,(3a)∇ a F ab = 1 4 √ 3 ε bcdef F cd F ef .(3b) We are interested in the α corrections to the entropy of the black holes constructed in [44], which read ds 2 5D = − f h dt 2 + dr 2 f + r 2 4 (σ 2 1 + σ 2 2 ) + r 2 4 h (σ 3 − W dt) 2 , (4a) A = √ 3Q r 2 dt −J 2 σ 3 ,(4b) where σ 1 , σ 2 , σ 3 are the usual left-invariant 1-forms of S 3 2 We would like to note, however, that [42] has a number of typos in their section 4, which summarises their results. σ 1 = − sin ψ dθ + cos ψ sin θ dφ ,(5a)σ 2 = cos ψ dθ + sin ψ sin θ dφ ,(5b)σ 3 = dψ + cos θ dφ ,(5c)and f = r 2 L 2 + 1 − 2M r 2 (1 − χ) +Q 2 r 4 1 −J 2 L 2 + 2M L 2 χ Q 2 , (6a) W = 2J r 2 h 2M +Q r 2 −Q 2 r 4 , (6b) h = 1 −J 2Q2 r 6 + 2J 2 (M +Q) r 4 ,(6c) where L 2 χ ≡J 2 (1 +Q/M ). The constantsM ,Q andJ parametrise the energy M , electric charge Q and angular momentum J as M = 3M π 4G 5 1 + χ 3 , (7a) J =J π 4G 5 (2M +Q) , (7b) Q = √ 3LπQ 4G 5 . (7c) The black hole event horizon is the null hypersurface r = r + , with r + being the largest real positive root of f (r). The associated Hawking temperature T , entropy S, chemical potential µ and angular velocity Ω can be found in [44]. It is then a simple exercise to check that all thermodynamic quantities satisfy the first law of black hole mechanics dE = T dS + µ dQ + Ω dJ .(8) The Gibbs free energy is then constructed in the usual manner via G = E − T S − µ Q − Ω J. One can show that G/T agrees with the Euclidean on-shell action (2) up to the usual Gibbons-Hawking-York [45,46] term and boundary counterterms [47,48]. Finally, with our normalizations for F , the BPS condition is given by 3 ∆ ≡ M − 2 L J − √ 3 L Q ≥ 0 .(9) The saturation of the BPS condition occurs only for supersymmetric solutions. Similar BPS bound have been shown not to receive α corrections even for asymptotically flat black holes [49]. The AdS BPS condition (9), together with the first law, implies T = 0, Ω = 2/L and µ = √ 3/L, which in turn yield Q =Q BPS ≡ r 2 + 1 + r 2 + 2L 2 ,(10a)J =J BPS ≡ Lr 2 + r 2 + + 2L 2 .(10b) Note that even though the solutions (4) appear to depend on three parameters (M ,Q,J), the BPS condition reduces this family to a one-parameter family, despite the fact that extremal black holes form a two-parameter family of solutions. We remark that [50,51] provided strong numerical evidence for the existence of a new twoparameter family of supersymmetric black holes, whose role in this story remains to be understood. One can also show that demanding the absence of naked singularities in (4a) implies that L >J 4 . Since the α corrections are only know in type IIB supergravity, we uplift the solutions (4) to ten dimensions. Using the results of [52][53][54], one can show that Eq. (4) oxidises to the following solution of type IIB supergravity: ds 2 = ds 2 5D + L 2 dΨ + A − A √ 3L 2 + dCP 2 (11a) G (5) = r 3 2L dt ∧ dr ∧ σ 1 ∧ σ 2 ∧ σ 3 + L 3 2 √ 3 J ∧ 5 F ,(11b)F (5) = G (5) + 10 G (5) (11c) where 5 is the five-dimensional Hodge dual obtained using the line element (4a), 10 is the Hodge dual obtained using the ten-dimensional line element (11a), dCP 2 is the standard Fubini-Study metric on CP 2 and J = dA is its associated Kähler form. Evaluating the corrections. The action 5 with the leading order α correction is [42]: S IIB = 1 16πG 10 M10 d 10 x √ −g R − 1 4 × 5! F 2 (5) + γ W (12) where W is given by W ≡ 1 86016 20 i=1 n i M i(13) with all twenty monomials given in table I and 6 T abcdef = i∇ a F bcdef + 1 16 F abcmn F mn def − 3F abf mn F mn dec .(14) 4 These is not the only restrictions on the three-dimensional moduli space of black hole solutions {J,Q, r + } that bulk regularity demands, but it is the only one we will need to show that δS > 0. 5 As usual, we use this term with a certain abuse of notation, because the five-form F (5) is only made self-dual at the level of the equations of motion. After the inclusion of the correction term proportional to γ, the self-duality condition is accordingly changed. 6 Note that after computing T with this expression, one still needs to antisymmetrise over the first three indices and the last three indices and then symmetrise for their exchange, before plugging into the monomials. ni Mi -43008 corrections of any solution in type IIB supergravity with nontrivial metric g and fiveform F (5) . Following [42], all tensor monomials are written with all indices lower. C abcd C abef C cegh C dgf h 86016 C abcd C aecf C bgeh C dgf h 129024 C abcd C aef g C bf hi T cdeghi 30240 C abcd C abce T df ghij T ef hgij 7392 C abcd C abef T cdghij T ef ghij -4032 C abcd C aecf T beghij T df ghij -4032 C abcd C aecf T bghdij T eghf ij -118272 C abcd C aef g T bcehij T df hgij -26880 C abcd C aef g T bcehij T dhif gj 112896 C abcd C aef g T bcf hij T dehgij -96768 C abcd C aef g T bcheij T df hgij 1344 C abcd T abef gh T cdeijk T f ghijk -12096 C abcd T abef gh T cdf ijk T eghijk -48384 C abcd T abef gh T cdf ijk T egihjk 24192 C abcd T abef gh T cef ijk T dghijk 2386 T abcdef T abcdgh T egijkl T f ijhkl -3669 T abcdef T abcdgh T eijgkl T f ikhjl -1296 T abcdef T abcghi T dejgkl T f hkijl 10368 T abcdef T abcghi T dgjekl T f hkijl 2688 T abcdef T abdegh T cgijkl T f jkhil Finally, we also have γ = α 3 16 π 3 8 ζ(3) .(15) We notice that table I corrects some typos in the final table of [42]. Our objective is to use these results to compute the leading correction to the entropy of the black hole solution detailed in (4). Naively, one might think that we would need to solve the equations of motion from the action (12) and only then evaluate the correction to the entropy. However, due to the work in [55] (whose results straightforwardly generalise to the case at hand), one in fact only needs to know the 0 th order solution, and evaluate that on the corrected action to get the leading corrections to the entropy. This is a major simplification and is one of the main reasons this work is possible. However, it is still not a trivial task to evaluate all the monomials from table I without accidentally inserting typos. Therefore, one of the key steps we had to take was validating our calculations. We wrote two pieces of code independently from one another, only comparing them at the end to make sure they agreed. We started by confirming the results of [42] to make sure there were no mistakes when copying the monomials from table I. Only after we had two matching codes that confirmed the results in [42] did we insert the solution (4). And even then, to be completely certain we had no typos or no convention compatibility issues, not only did we include many consistency checks throughout the code, e.g. confirming we indeed solved the correct equations of motion, but we used two different parametrisations. One of them using a CP 2 fibration and another using a more direct method using the coordinates as originally written in [53]. The CP 2 fibration is the more efficient method and therefore is the one included in the supplemental material. However, the direct method is more amenable to generalisation for the case of different angular momenta [53,56], which we leave for future work. After the colossal amount of dust settles, all twenty terms in table I are non-vanishing on our solutions, and yet the final result appears simple, which gives further confidence in our answer. Using the relation between the Gibbs free energy G and the Euclidean action obtained from (12), we find that the stringy correction to the Gibbs free energy at fixed chemical potential µ, angular velocity Ω and temperature T reads (δG) µ,Ω,T = − 12π 3 α 3 M +Q 2 ζ(3) N 2 L 12 r 15 + 9L 2 −J 2 × × L 2 −J 2 3 ∆ ∆ + 4 L J ≤ 0 . (16) It is a simple matter to compute the variation in entropy, (δS) Q,J,M , at fixed asymptotic charges Q, J and M from (δG) µ,Ω,T . In particular, we can follow the same steps as in [55] to show that (δS) Q,J,M = −T −1 (δG) µ,Ω,T .(17) Equations (16) and (17) are the main result of this manuscript, whose physical significance we discuss next. Interpretation of results. The first thing we note is the fact that (δS) Q,J,M = 0 on the supersymmetric black hole solutions found in [37]. One might wonder why that is the case, given that (17) has a factor of T in the denominator, and for supersymmetric solutions T = 0. However, we note that if we takeQ =Q BPS + δQ andJ =J BPS + δJ, with δQ, δJ 1, we get T = O(δQ, δJ), whereas ∆ = O(δQ 2 , δQδJ, δJ 2 ). This means (δS) Q,J,M = O(δQ, δJ) in Eq. (17), i.e. it vanishes in the supersymmetric limit. Another way to see this result is to note that one can read off the change in entropy due to stringy corrections at constant chemical potential µ, temperature T and angular velocity Ω using the standard thermodynamic relation S = −(∂G/∂T ) Ω,µ . In this limit, we get that the correction to the entropy is finite at extremality, being zero in the supersymmetric limit. To our knowledge there is no a priori reason, based on bulk physics, for why the entropy in the supersymmetic limit is not corrected via stringy effects. This lends support in favour of the index picture advocated in [21][22][23][25][26][27][28][29][30][31][32][33][34][35][36]. Second, the sign of (δS) Q,J,M appears consistent with the weak gravity conjecture [57], similarly to the analogous calculations in flat space [58][59][60][61][62] and with AdS asymptotics [63]. In particular, one can show using the generalisation of the Goon-Penco relation to AdS [62,63] that the leading correction to the extremality bound at fixed energy M , charge Q and angular momentum J necessarily decreases with respect to the uncorrected solution. This relation is in perfect agreement with the weak gravity conjecture [58,60,62]. Thirdly, we point out that our final expression (16) only assumes equal angular momenta and equal charges. Notably, it is non-vanishing for a generic nonsupersymmetric extremal black hole, and is even valid away from extremality. It would be interesting to understand whether the methods used in [64] could be extended to capture the leading α corrections presented in this letter. Further, this then offers a prediction for the quantum field theoretic calculation. Even though the counting of the supersymmetric states is not corrected at finite λ, the counting including non-supersymmetric states should be, and its form should be given by (16). However, as of yet, there are no techniques capable of computing a partition function at strong coupling without the aid of supersymmetry. Though we should mention that in [65] some progress has been reported in going slightly beyond the supersymmetric limit. Our results rely heavily on [55], since we solely use the uncorrected solution to determine the thermodynamic properties of the corrected solution. In principle, we could use the equations of motion that follow from (12) together with the modified self-duality condition of [42] to determine directly the stringy corrected black holes. Under such circumstances, we could determine all thermodynamic properties from the solutions per se instead of using the arguments presented in [55]. Perhaps our current results suggest that the uncorrected supersymmetric solution might be a solution of the corrected equations of motion. This phenomenon has been recently observed in [66] for a number of corrections and black hole solutions. We leave this avenue of research for the future. Finally, an interesting avenue for future work is to generalise this calculation to the case when all the angular momenta and charges are distinct, using the results from [56]. The complexity of this solution is quite daunting, and computing these corrections would necessarily require more computing power and a more efficient algorithm 7 . TABLE I . ITable detailing the α 3 Except perhaps when wall-crossing is observed, see e.g. [40, 41]. arXiv:2007.06582v3 [hep-th] 3 Mar 2022 To avoid cluttering in the notation, from here onward we take Q ≥ 0 and J ≥ 0. For the interested reader, even just checking that the solution[56] indeed solves the equations of motion as claimed takes a few hours with a rather optimised Mathematica code. AcknowledgmentsWe thank Joonho Kim Reversible and irreversible transforations in black hole physics. D Christodoulou, Phys. Rev. Lett. 25D. Christodoulou, Reversible and irreversible transforations in black hole physics, Phys. Rev. Lett. 25 (1970) 1596-1597. Gravitational radiation from colliding black holes. S Hawking, Phys. Rev. Lett. 26S. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett. 26 (1971) 1344-1346. Rigidity of a black hole. B Carter, Nature Physical Science. 23883B. Carter, Rigidity of a black hole, Nature Physical Science 238 (1972), no. 83 71-72. 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[ "Adrian et al. Application of Information System for COVID Reporting APPLICATION OF EXECUTIVE INFORMATION SYSTEM FOR COVID-19 REPORTING SYSTEM AND MANAGEMENT: AN EXAMPLE FROM DKI JAKARTA, INDONESIA", "Adrian et al. Application of Information System for COVID Reporting APPLICATION OF EXECUTIVE INFORMATION SYSTEM FOR COVID-19 REPORTING SYSTEM AND MANAGEMENT: AN EXAMPLE FROM DKI JAKARTA, INDONESIA" ]	[ "Verry Adrian \nDepartment of Health\nDepartment of Health\nDepartment of Health\nJakarta, Jakarta, Jakarta\n", "Intan Rachmita Sari \nDepartment of Health\nDepartment of Health\nDepartment of Health\nJakarta, Jakarta, Jakarta\n", "Hardya Gustada Hikmahrachim \nDepartment of Health\nDepartment of Health\nDepartment of Health\nJakarta, Jakarta, Jakarta\n" ]	[ "Department of Health\nDepartment of Health\nDepartment of Health\nJakarta, Jakarta, Jakarta", "Department of Health\nDepartment of Health\nDepartment of Health\nJakarta, Jakarta, Jakarta", "Department of Health\nDepartment of Health\nDepartment of Health\nJakarta, Jakarta, Jakarta" ]	[]	SARS CoV-2 infection and transmission are problematic in developing countries such as Indonesia. Due to the lack of an information system, Provinces must be able to innovate in developing information systems related to surveillance of SARS CoV-2 infection. Jakarta Department of Health built a data management system called Executive Information System (EIS) of COVID-19 Reporting. EIS aimed to provide actual data so that current epidemiological analysis is accurate. The main idea of EIS is to provide valid and actual information to stakeholders, which can then be presented in the form of a dashboard. EIS is utilized to push data flow and management for rapid surveillance purposes. This could be the first time in Indonesia that a system reports near-actual data of nearly half a million people daily using an integrated system through a transparent system. The main data presented is important to monitor and evaluate COVID-19 transmission is the cumulative case dan daily case number. Data in EIS also can offer data geographically so that a more detailed analysis could be done. EIS's data and the dashboard help the government in pandemic control by presenting actual data on bed occupancy and availability across hospitals, especially isolation wards. Stakeholders, academic institutions should utilize EIS data and other elements to help Indonesia fight COVID-19.	null	[ "https://arxiv.org/pdf/2108.09738v1.pdf" ]	237,267,045	2108.09738	f0558515d552b3ed81e5e6a788df6dd72dd98840	Adrian et al. Application of Information System for COVID Reporting APPLICATION OF EXECUTIVE INFORMATION SYSTEM FOR COVID-19 REPORTING SYSTEM AND MANAGEMENT: AN EXAMPLE FROM DKI JAKARTA, INDONESIA Verry Adrian Department of Health Department of Health Department of Health Jakarta, Jakarta, Jakarta Intan Rachmita Sari Department of Health Department of Health Department of Health Jakarta, Jakarta, Jakarta Hardya Gustada Hikmahrachim Department of Health Department of Health Department of Health Jakarta, Jakarta, Jakarta Adrian et al. Application of Information System for COVID Reporting APPLICATION OF EXECUTIVE INFORMATION SYSTEM FOR COVID-19 REPORTING SYSTEM AND MANAGEMENT: AN EXAMPLE FROM DKI JAKARTA, INDONESIA COVID-19Information SystemData ReportingPublic HealthJakarta 1 SARS CoV-2 infection and transmission are problematic in developing countries such as Indonesia. Due to the lack of an information system, Provinces must be able to innovate in developing information systems related to surveillance of SARS CoV-2 infection. Jakarta Department of Health built a data management system called Executive Information System (EIS) of COVID-19 Reporting. EIS aimed to provide actual data so that current epidemiological analysis is accurate. The main idea of EIS is to provide valid and actual information to stakeholders, which can then be presented in the form of a dashboard. EIS is utilized to push data flow and management for rapid surveillance purposes. This could be the first time in Indonesia that a system reports near-actual data of nearly half a million people daily using an integrated system through a transparent system. The main data presented is important to monitor and evaluate COVID-19 transmission is the cumulative case dan daily case number. Data in EIS also can offer data geographically so that a more detailed analysis could be done. EIS's data and the dashboard help the government in pandemic control by presenting actual data on bed occupancy and availability across hospitals, especially isolation wards. Stakeholders, academic institutions should utilize EIS data and other elements to help Indonesia fight COVID-19. INTRODUCTION The first coronavirus disease (COVID) 19 case in Indonesia was reported in March 2020, specifically in Depok, a city near Jakarta. Since that, massive infection and local transmission are unstoppable in Indonesia. In the middle of March 2021, around 1.5 million people were infected by SARS CoV-2 virus in Indonesia, with currently 122.000 active cases and a 2.7% mortality rate. The highest cases were found in Jakarta. Yet, Indonesia had not overcome COVID-19. Due to the lack of an information system, provinces are pushed to do any innovation related to SARS CoV-2 infection surveillance. Daily data reported are mandatory for every province, consist of new cases, recovered cases, and death cases. Those data were then collected to be summarized as a daily national report. To provide valid and reliable data, Jakarta Department of Health had been developing a data management system called Executive Information System (EIS) of COVID-19 Reporting. EIS aimed to provide actual data so that current epidemiological analysis is accurate. Any decision for pandemic control should be based on high-quality data. This paper aimed to present the design and outcomes of EIS utilization in Jakarta. This system had been build at the beginning of 2018. The main idea of EIS is to provide valid and actual information to stakeholders in the form of a dashboard. This idea is growing as demand increases during the pandemic, especially when COVID-19 case reporting emerged. Figure 1. Information flow in EIF build by Jakarta During the COVID-19 pandemic in Jakarta, EIS is utilized to push data flow and management for rapid surveillance purposes. As seen in Figure 1, Jakarta's data sources are public health services, hospitals, and laboratories. Both patient data and the result of PCR data are reported by those healthcare facilities by using an online database, excel, csv, and other tools. Data were then processed into EIS so that further steps can be integrated. Document management primarily uses line unit and line unit folder and is brought to automated data feed on sharing. Epidemiologic analysis and presentation (information products) of EIS in Jakarta are daily COVID updates (will be explained later), GIS dashboard and maps, weekly external SitMap and Epi Analysis, and also pandemic dashboard. This data would then be utilized by epidemiologists and public health analysts for pandemic-related policy, writing of scientific reports, and knowledge sharing via social media. Using this system, actual data were presented and accessible for all elements in Jakarta. Due to a large amount of data incoming across the province, the EIS system optimizes data verification by collaboration with the civil registration department to prevent data duplication, mostly using a single civil registration number (Nomor Induk Kependudukan or NIK) (Fig 2). Although it still needs improvement, this scheme prevents potential data duplication during COVID-19 case reporting. The significant difference of data collection and analysis before and after the establishment of EIS is presented in Table 1. Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Table 1. Difference in data collection to data analysis before and after the establishment of EIS Aspects Before After Data collection Written in paper-based form, then recapitulated manually. Direct input into system and automatic data summary by system. Reporting Manually recapitulated data sent into stakeholders in higher level for data input into system. Data connected via API, directly visualized in dashboard. Time consumption Needs longer time due to manual processing from data input to data analysis. Less time needed for data analysis due to interconnected system. Data quality Moderate quality data due to incomplete data, duplication, and human error during data collection process. Better data quality due to mandatory variable input in EIS system. Not possible for data duplication due to similar data will be automatically merged. Accumulative calculation is done by system to prevent human error during data calculation. Data analysis More difficult step for data analysis and data visualization due to manual process. Automatic data analysis and visualization in the form of table, graph, or any presentation that been set before. Data sharing More difficult due to manual sending data to every stakeholder. Data is accessible both for stakeholders and public as presented in the dashboard. 2. DATA PRESENTATION AND VISUALIZATION This EIS COVID-19 Reporting system's main objective is to provide data for Jakarta Provincial Government and then assisted by the Department of Communication, Informatics, and Statistics; data will be reprocessed and visualized in the form of a dashboard for use by stakeholders, academic institutions, and public consumption. Data should be regularly updated and give a lot of useful information. This is probably the first time in Indonesia that a system could report near-actual data of nearly half a million people daily using an integrated system through a transparent system. EIS is the back end of this system information, while the front end is presented on Jakarta COVID-19 website. Current dashboard that presented at corona.jakarta.go.id: Figure 2. Daily Summary of Cummulative COVID-19 Cases in Indonesia (left) and Jakarta (right). Kasus terkonfirmasi = Confirmed cases; kasus positif = positive cases; kasus aktif = active cases; sembuh = recovered; meninggal = death; dirawat = hospitalized; isolasi mandiri = self-isolation; tanpa gejala = asymptomatic; bergejala = symptomatic; belum diketahui = unknown. The main data presented that is important to monitor and evaluate COVID-19 transmission is cumulative case dan daily case number. Using EIS, Jakarta could present a more specific data compared to national data by reporting self-isolation case, asymptomatic cases, and symptomatic cases. Those proportion is important to estimate how severe this disease are among people of Jakarta. Figure 3. Detailed information about suspect (suspek), probable, travelers (pelaku perjalanan), close contact (kontak erat), and discarded cases (cumulative). Selesai isolasi = finished self-isolation; isolasi di rumah = self-isolation at home; isolasi di RS = isolation at hospital; meninggal = death. Data above present a stratification analysis about COVID-19 in Jakarta, especially about suspected cases, probable cases, travellers cases, close-contact cases, and discarded cases. This information is crucial as Indonesia yet do not have an optimal testing number and strategies and near half of them are conducted in Jakarta. Cumulative number of self-isolation compared to hospital admission can be an information to be used for policy making about Jakarta capacity to fight for this pandemic. Another important role of EIS is that it can visualize an actual report of daily new cases and its trend (Fig 5). This graphic has been adapted based on other international health organization report such as US CDC and UK NHS. Accumulative cases could also be accesses freely as it also presented at COVID-19 website after being processed in EID (Fig 6). As seen in Fig 7, EIS also can report the daily positivity rate of COVID-19 diagnostic test in Jakarta. The trend of specimen tested and positivity rate helps epidemiologist in deciding for the further strategies. Any strategies related to pandemic control would not be explained in detail in this paper. Data in EIS also can present data geographically so that a more detailed analysis could be done. Data reported as cumulative (Fig 8) or daily new cases (Fig 9). New cases reported in detailed information such as age, gender, and hospitalization status. Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Information at EIS dashboard that can be accessed by public are: 1. National COVID confirm cases-cumulative (active, recovered, and death cases) 2. Jakarta COVID confirm cases -cumulative: i. Hospitalized ii. Recovered iii. Death iv. Self-isolation v. Asymptomatic vi. Symptomatic vii. Unknown symptom status 3. Mapping of suspected, close-contact, and positive cases in Jakarta 4. Crosstabulation data on gender and age group 5. Daily new suspected, close-contact, and positive cases 6. Comparison between national and Jakarta new cases trend data 7. Daily mortuary rate with COVID-19 protocol or without protocol 8. Positivitiy rate of daily COVID-19 diagnostic workup 9. Data on law violation by companies 10. Data on Jakarta air quality and traffic information 11. Network graph of COVID-19 Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Data on bed availability and referral system An innovation delivered by EIS is that this system help government in pandemic control by presenting actual data on bed occupancy and availability across hospitals, especially isolation wards. EIS integrate both government-owned and private-owned hospitals in Jakarta. This transpiration on data had been proven to prevent delay in referral and further improve survival of critical COVID-19 patients that needs ICU, PICU, or NICU (Fig 10). Data presented on EIS including isolation ward, ICU with and without negative pressure, pediatric ICU, neonatal ICU, operating theater with negative pressure, and hemodialysis facilities for COVID-19. Data were directly inputted by hospitals into the system and being updated hourly. More detail data also presented (Fig 11) specifically for each hospital. This might be one of the first system to present almost actual data on bed occupancy in Indonesia. DISCUSSION EIS brings Jakarta into a city with more developed data system on public health. This breakthrough should become an example for other cities of provinces in Indonesia and a further collaboration is always welcome. Some developing countries also have an information system to support policy making during COVID-19 pandemic. The key to fight pandemic is how to present data in transparency. 1,2 Data reporting are a major issue in developing countries. India reported that only 10 out of 29 states that provide visual representation of COVID-19 cases trends, while most of states did not report data stratification in age, gender, district nor comorbidities. 3 Previous studies had reported that gender and age played an important role in COVID-19 data reporting due to its impact on severity and mortality. 4,5 An objective indicator on how data reporting quality can be assessed using COVID-19 Data Reporting Score (CDRS) as developed by Stanford or other scoring systems, in which related to health development index. 3 This disparity would be common among developing countries, especially with geographical challenges like Jakarta, so that an intersectoral collaboration is necessary. IT infrastructure and system also the major source of problem during system reporting. EIS might be easily implemented in Jakarta due to prior IT readiness, but a contrast finding could be met in other Indonesian cities. It needs a policy commitment to build an efficient IT system in a short time during this pandemic era. 6 Some other concern that might impair data collection are about privacy. Some patients, both in developing and developed countries, are not giving permission of his civil ID number to be registered in national system as a COVID-19 cases. This phenomenon is strongly related to local social and spiritual belief, or a feeling of ashamed to be infected by SARS CoV-2. Thus, during EIS development, data privacy should be handled carefully to prevent further ethical issues. 7 Future direction The large amount of data, known as big data, should be optimized by academicians to publish scientific paper in both national and international level. Data utilization is not limited to health sectors, but social and economic institution can also bring this to a scientific evidence in order to help government for policy making. EIS should also be integrated to central government system, such as New All Record Kemenkes, Peduli Lindungi application for contact tracing, self-isolation monitoring apps, and other new mobile-based apps. This collaboration can boost the quality of information extracted from the present data and open for any innovation in the future. Figure 2 . 2Concept of Data Management during COVID Pandemic Figure 4 . 4Daily New (Penambahan Kasus Harian) COVID-19 Case and Mortality Report. Meninggal harian = death case daily; positif harian = new positive case daily; sembuh harian = recovered case daily Figure 5 . 5Cumulative (Akumulasi) Data on COVID-19 Active (Masih perawatan), Death (meninggal), Self-Isolation (isolasi mandiri), and Recoved Cases (Sembuh) Figure 6 . 6Positivity Rate and Other Related Data on COVID-19 Testing in Jakarta. Tanggal = date; jumlah orang di test = people tested; orang positif harian = daily new positive cases; orang negative harian = daily negative cases; kasus baru harian = new active cases daily; total specimen di test = number of specimen tested; positivity rate specimen harian = daily positivity rate. Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Figure 7 . 7Positive COVID-19 Cases based on Administrative Regions. Positif = positive; Kecamatan = district; Wilayah Kota = administrative city; jumlah positif = positive cases. Figure 8 . 8Mapping of New Positive Case Weight by Case Number (upper) and by Basic Demographic Data (lower) Figure 9 . 9Bed Availability and Occupancy Data Presented in Executive Information System Jakarta. ICU tekanan negatif = negative pressure ICU; ICU tanpa tekanan negatif = without negative pressure ICU; Kamar isolasi = isolation room; OK Khusus COVID-19 = operating theater for COVID-19 cases; HD Khusus COVID-19 = hemodialysis facilities for COVID-19 cases; tersedia = availability in number; sisa kapasitas = capacity in percentage; dengan ventilator = with ventilator; tanpa ventilator = without ventilator Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Figure 10 . 10Bed Availability and Occupancy Status (per Hospital). Ketersediaan tempat tidur = bed availability; wilayah = region; nama RS = hospital name; ICU tekanan negatif = negative pressure ICU; ICU tanpa tekanan negatif = without negative pressure ICU; Kamar isolasi = isolation room; OK = operating theater; dengan ventilator = with ventilator; tanpa ventilator = without ventilator 3. Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Proceedings of the 1st Virtual Conference on Implications of Information and Digital Technologies for Development, 2021 Transparency during public health emergencies: from rhetoric to reality. P O'malley, J Rainford, A Thompson, Bull World Health Organ. 878O'Malley P, Rainford J, Thompson A. Transparency during public health emergencies: from rhetoric to reality. Bull World Health Organ. 2009;87(8):614-8. Early epidemiological analysis of the coronavirus disease 2019 outbreak based on crowdsourced data: a population-level observational study. The Lancet Digital Health. K Sun, J Chen, C Viboud, 2Sun K, Chen J, Viboud C. Early epidemiological analysis of the coronavirus disease 2019 outbreak based on crowdsourced data: a population-level observational study. The Lancet Digital Health. 2020;2(4):e201-e8. Disparity in the quality of COVID-19 data reporting across India. V Vasudevan, A Gnanasekaran, V Sankar, S A Vasudevan, J Zou, medRxiv. 2020:2020.07.19Vasudevan V, Gnanasekaran A, Sankar V, Vasudevan SA, Zou J. Disparity in the quality of COVID-19 data reporting across India. medRxiv. 2020:2020.07.19.20157248. Gender Differences in Patients With COVID-19: Focus on Severity and Mortality. Jin J-M Bai, P He, W Wu, F Liu, X-F Han, D-M , Frontiers in Public Health. 8152Jin J-M, Bai P, He W, Wu F, Liu X-F, Han D-M, et al. Gender Differences in Patients With COVID-19: Focus on Severity and Mortality. Frontiers in Public Health. 2020;8(152). Estimates of the severity of coronavirus disease 2019: a model-based analysis. The Lancet Infectious Diseases. R Verity, L C Okell, I Dorigatti, P Winskill, C Whittaker, N Imai, 20Verity R, Okell LC, Dorigatti I, Winskill P, Whittaker C, Imai N, et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. The Lancet Infectious Diseases. 2020;20(6):669-77. Barriers to hospital electronic public health reporting and implications for the COVID-19 pandemic. A J Holmgren, N C Apathy, J Adler-Milstein, J Am Med Inform Assoc. 278Holmgren AJ, Apathy NC, Adler-Milstein J. Barriers to hospital electronic public health reporting and implications for the COVID-19 pandemic. J Am Med Inform Assoc. 2020;27(8):1306- 9. Ethical Collection, Storage, and Use of Public Health Data: A Proposal for a National Privacy Protection. L M Lee, L O Gostin, JAMA. 3021Lee LM, Gostin LO. Ethical Collection, Storage, and Use of Public Health Data: A Proposal for a National Privacy Protection. JAMA. 2009;302(1):82-4.	[]
[ "Leakage and the Reproducibility Crisis in ML-based Science", "Leakage and the Reproducibility Crisis in ML-based Science" ]	[ "Sayash Kapoor \nDepartment of Computer Science and Center for Informa-tion Technology Policy\nPrinceton University\n\n", "Arvind Narayanan \nDepartment of Computer Science and Center for Informa-tion Technology Policy\nPrinceton University\n\n" ]	[ "Department of Computer Science and Center for Informa-tion Technology Policy\nPrinceton University\n", "Department of Computer Science and Center for Informa-tion Technology Policy\nPrinceton University\n" ]	[]	The use of machine learning (ML) methods for prediction and forecasting has become widespread across the quantitative sciences. However, there are many known methodological pitfalls, including data leakage, in ML-based science. In this paper, we systematically investigate reproducibility issues in ML-based science. We show that data leakage is indeed a widespread problem and has led to severe reproducibility failures. Specifically, through a survey of literature in research communities that adopted ML methods, we find 17 fields where errors have been found, collectively affecting 329 papers and in some cases leading to wildly overoptimistic conclusions. Based on our survey, we present a finegrained taxonomy of 8 types of leakage that range from textbook errors to open research problems.We argue for fundamental methodological changes to ML-based science so that cases of leakage can be caught before publication. To that end, we propose model info sheets for reporting scientific claims based on ML models that would address all types of leakage identified in our survey. To investigate the impact of reproducibility errors and the efficacy of model info sheets, we undertake a reproducibility study in a field where complex ML models are believed to vastly outperform older statistical models such as Logistic Regression (LR): civil war prediction. We find that all papers claiming the superior performance of complex ML models compared to LR models fail to reproduce due to data leakage, and complex ML models don't perform substantively better than decades-old LR models. While none of these errors could have been caught by reading the papers, model info sheets would enable the detection of leakage in each case.	10.48550/arxiv.2207.07048	[ "https://arxiv.org/pdf/2207.07048v1.pdf" ]	250,526,411	2207.07048	8ceb0fc9197e4b3225f13eeda45b37f51cdfea3b	Leakage and the Reproducibility Crisis in ML-based Science Sayash Kapoor Department of Computer Science and Center for Informa-tion Technology Policy Princeton University Arvind Narayanan Department of Computer Science and Center for Informa-tion Technology Policy Princeton University Leakage and the Reproducibility Crisis in ML-based Science Correspondence to: Sayash Kapoor <[email protected]>. The use of machine learning (ML) methods for prediction and forecasting has become widespread across the quantitative sciences. However, there are many known methodological pitfalls, including data leakage, in ML-based science. In this paper, we systematically investigate reproducibility issues in ML-based science. We show that data leakage is indeed a widespread problem and has led to severe reproducibility failures. Specifically, through a survey of literature in research communities that adopted ML methods, we find 17 fields where errors have been found, collectively affecting 329 papers and in some cases leading to wildly overoptimistic conclusions. Based on our survey, we present a finegrained taxonomy of 8 types of leakage that range from textbook errors to open research problems.We argue for fundamental methodological changes to ML-based science so that cases of leakage can be caught before publication. To that end, we propose model info sheets for reporting scientific claims based on ML models that would address all types of leakage identified in our survey. To investigate the impact of reproducibility errors and the efficacy of model info sheets, we undertake a reproducibility study in a field where complex ML models are believed to vastly outperform older statistical models such as Logistic Regression (LR): civil war prediction. We find that all papers claiming the superior performance of complex ML models compared to LR models fail to reproduce due to data leakage, and complex ML models don't perform substantively better than decades-old LR models. While none of these errors could have been caught by reading the papers, model info sheets would enable the detection of leakage in each case. The use of machine learning (ML) methods for prediction and forecasting has become widespread across the quantitative sciences. However, there are many known methodological pitfalls, including data leakage, in ML-based science. In this paper, we systematically investigate reproducibility issues in ML-based science. We show that data leakage is indeed a widespread problem and has led to severe reproducibility failures. Specifically, through a survey of literature in research communities that adopted ML methods, we find 17 fields where errors have been found, collectively affecting 329 papers and in some cases leading to wildly overoptimistic conclusions. Based on our survey, we present a finegrained taxonomy of 8 types of leakage that range from textbook errors to open research problems. We argue for fundamental methodological changes to ML-based science so that cases of leakage can be caught before publication. To that end, we propose model info sheets for reporting scientific claims based on ML models that would address all types of leakage identified in our survey. To investigate the impact of reproducibility errors and the efficacy of model info sheets, we undertake a reproducibility study in a field where complex ML models are believed to vastly outperform older statistical models such as Logistic Regression (LR): civil war prediction. We find that all papers claiming the superior performance of complex ML models compared to LR models fail to reproduce due to data leakage, and complex ML models don't perform substantively better than decades-old LR models. While none of these errors could have been caught by reading the papers, model info sheets would enable the detection of leakage in each case. Overview There has been a marked shift towards the paradigm of predictive modeling across quantitative science fields. This shift has been facilitated by the widespread use of machine learning (ML) methods. However, pitfalls in using ML methods have led to exaggerated claims about their performance. Such errors can lead to a feedback loop of overoptimism about the paradigm of prediction-especially as non-replicable publications tend to be cited more often than replicable ones (Serra-Garcia & Gneezy, 2021). It is therefore important to examine the reproducibility of findings in communities adopting ML methods. Scope. We focus on reproducibility issues in ML-based science, which involves making a scientific claim using the performance of the ML model as evidence. There is a much better known reproducibility crisis in research that uses traditional statistical methods (Open Science Collaboration, 2015). We also situate our work in contrast to other ML domains, such as methods research (creating and improving widely-applicable ML methods), ethics research (studying the ethical implications of ML methods), engineering applications (building or improving a product or service), and modeling contests (improving predictive performance on a fixed dataset created by an independent third party). Investigating the validity of claims in all of these areas is important, and there is ongoing work to address reproducibility issues in these domains (Hullman et al., 2022;Pineau et al., 2020;Erik Gundersen, 2021;Bell & Kampman, 2021). We define a research finding as reproducible if the code and data used to obtain the finding are available and the data is correctly analyzed (Hofman et al., 2021a;Leek & Peng, 2015;Pineau et al., 2020). This is a broader definition than computational reproducibility -when the results in a paper can be replicated using the exact code and dataset provided by the authors (see Appendix A). Leakage. Data leakage has long been recognized as a leading cause of errors in ML applications (Nisbet et al., 2009). In formative work on leakage, Kaufman et al. (2012) provide an overview of different types of errors and give several recommendations for mitigating these errors. Since this paper was published, the ML community has investigated leakage in several engineering applications and modeling competitions (Fraser, 2016;Ghani et al., 2020;Becker, 2018;Brownlee, 2016;Collins-Thompson). However, leakage occurring in ML-based science has not been comprehensively arXiv:2207.07048v1 [cs.LG] 14 Jul 2022 - Table 1. Survey of 20 papers that identify pitfalls in the adoption of ML methods across 17 fields, collectively affecting 329 papers. In each field, papers adopting ML methods suffer from data leakage. The column headings for types of data leakage, shown in bold, are based on our taxonomy of data leakage. We also highlight other issues that are reported in the papers, including issues with computational reproducibility (the availability of code, data, and computing environment to reproduce the exact results reported in the paper), data quality (for example, small size or large amounts of missing data), metric choice (using incorrect metrics for the task at hand, for example, using accuracy for measuring model performance in the presence of heavy class imbalance), and standard dataset use, where issues are found despite the use of standard datasets in a field. investigated. As a result, mitigations for data leakage in scientific applications of ML remain understudied. In this paper, we systematically investigate reproducibility issues in ML-based science due to data leakage. We make three main contributions: 1) A survey and taxonomy of reproducibility issues due to leakage. We provide evidence for a growing reproducibility crisis in ML-based science. Through a survey of literature in research communities that adopted ML methods, we find 20 papers across 17 fields where errors have been found, collectively affecting 329 papers (Table 1). Each of these fields suffers from leakage. We highlight that data leakage mitigation strategies developed for other ML applications such as modeling contests and engineering applications often do not translate to ML-based science. Based on our survey, we present a fine-grained taxonomy of 8 types of leakage that range from textbook errors to open research problems (Section 2.4). 2) Model info sheets to detect and prevent leakage. Current standards for reporting model performance in ML-based science often fall short in addressing issues due to leakage. Specifically, checklists and model cards are one way to provide standard best practices for reporting details about ML models (Mongan et al., 2020;Collins et al., 2015;Mitchell et al., 2019). However, current efforts do not address issues arising due to leakage. Further, most checklists currently in use are not developed for ML-based science in general, but rather for specific scientific or research communities (Pineau et al., 2020;Mongan et al., 2020). As a result, best practices for model reporting in ML-based science are underspecified. In this paper, we introduce model info sheets to detect and prevent leakage in ML-based science (Section 3). They are inspired by the model cards in Mitchell et al. (2019). Filling out a model info sheet requires the researcher to provide precise arguments to justify that models used towards making scientific claims do not suffer from leakage. Model info sheets address all types of leakage identified in our survey. We advocate for model info sheets to be included with every paper making a scientific claim using an ML model. 3) Empirical case study of leakage in civil war prediction. For an in-depth look at the impact of reproducibility errors and the efficacy of model info sheets, we undertake a reproducibility study in civil war prediction, a subfield of political science where ML models are believed to vastly outperform older statistical models such as Logistic Regression. We perform a systematic review to find papers on civil war prediction and find that all papers in our review claiming the superior performance of ML models compared to Logistic Regression models fail to reproduce due to data leakage (Figure 1) 1 . Each of these papers was published in 1 A note on terminology. We use "ML models" as a shorthand for models other than Logistic Regression, specifically, Random Forests, Gradient-Boosted Trees, and Adaboost. To be clear, all of these models including Logistic Regression involve learning from data in the predictive modeling approach. However, the top political science journals. Further, when the errors are corrected, ML models don't perform substantively better than decades-old Logistic Regression models, calling into question the shift from explanatory modeling to predictive modeling in this field. While none of these errors could have been caught by reading the papers, model info sheets enable the detection of leakage in each case. Evidence of a reproducibility crisis Many scientific fields have adopted ML methods and the paradigm of predictive modeling (Athey & Imbens, 2019; Schrider & Kern, 2018;Valletta et al., 2017;Iniesta et al., 2016;Tonidandel et al., 2018;Yarkoni & Westfall, 2017). We find at least three main uses of ML models in scientific literature. First, models which are better at prediction are thought to enable an improved understanding of scientific phenomena (Hofman et al., 2021b). Second, especially when used in medical fields, models with higher predictive terminology we use is common in fields that distinguish ML from statistical methods that invoke an assumption about the true data generating process, such as Logistic Regression (Christodoulou et al., 2019). accuracy can aid in research and development of better diagnostic tools (McDermott et al., 2021). Finally, ML-based methods have also been used to investigate the inherent predictability of phenomena, especially for predicting social outcomes (Salganik et al., 2020). The increased adoption of ML methods in science motivates our investigation of reproducibility issues in ML-based science. Data leakage causes irreproducible results Data leakage is a spurious relationship between the independent variables and the target variable that arises as an artifact of the data collection, sampling, or pre-processing strategy. Since the spurious relationship won't be present in the distribution about which scientific claims are made, leakage usually leads to inflated estimates of model performance. Researchers in many communities have already documented reproducibility failures in ML-based science within their fields. Here we conduct a cross-disciplinary analysis by building on these individual reviews. This enables us to highlight the scale and scope of the crisis, identify common patterns, and make progress toward a solution. When searching for past literature that documents reproducibility failures in ML-based science, we found that different fields often use different terms to describe pitfalls and errors. This makes it difficult to conduct a systematic search to find papers with errors. Therefore, we do not present our results as a systematic meta-review of leakage from a coherent sample of papers, but rather as a lower bound of reproducibility issues in ML-based science. Additionally, most reviews only look at the content of the papers, and not the code and data provided with the papers to check for errors. This leads to under-counting the number of affected papers, since the code might have errors that are not apparent from reading the papers. Our findings present a worrying trend for the reproducibility of ML-based science. We find 20 papers from 17 fields that outline errors in ML-based science in their field, collectively affecting 329 papers. A prominent finding that emerges is that data leakage is a pitfall in every single case. The results from our survey are presented in Table 1. Columns in bold represent different types of leakage (Section 2.4). The last four columns represent other common trends in the papers we study (Section 2.5). For systematic reviews, we report the number of papers reviewed. Each paper in our survey highlights issues with leakage, with 6 papers highlighting the presence of multiple types of leakage in their field. Data leakage mitigations for other ML applications do not apply to scientific research Most previous research and writing on data leakage has focused on mitigating data leakage primarily for engineer-ing settings or predictive modeling competitions (Kaufman et al., 2012). However, the taxonomy of data leakage outlined in this body of work does not address all kinds of leakage that we identify in our survey. In particular, we find that leakage can result from a difference between the distribution of the test set and the distribution of scientific interest (Section 2.4). Robustness to distribution shift is an area of ongoing research in ML methods, and is as such an open problem (Geirhos et al., 2020). Additionally, prior work primarily focuses on mitigating leakage in modeling competitions and engineering applications. Both of these settings are very different from scientific research, and mitigations for data leakage in modeling competitions as well as engineering applications of ML often do not translate into strategies for mitigating data leakage in ML-based science. Leakage in modeling competitions. In predictive modeling competitions, dataset creation and model evaluation is left to impartial third parties who have the expertise and incentives to avoid errors. Within this framework, none of the participants have access to the held-out evaluation set before the competition ends. In contrast, in most ML-based science the researcher has access to the entire dataset while creating the ML models. Leakage often occurs due to the researcher having access to the entire dataset during the modeling process. Leakage in engineering applications. One of the most common recommendations for detecting and mitigating leakage is to deploy the ML model at a limited scale in production. This advice is only applicable to engineering applications of ML, where the end goal is not to gain insights about a particular process, but rather to serve as a component in a product. Often, a rough idea of model performance is enough to decide whether a model is good enough to be deployed in a product. Contrarily, ML-based science involves making a scientific claim using the performance of the ML model as evidence. In addition, engineering applications of ML often operate in a rapidly changing context and have access to large datasets, so small differences in performances are often not as important, whereas scientific claims are sensitive to small performance differences between ML models. Why do we call it a reproducibility crisis? We say that ML-based science is suffering from a reproducibility crisis for two related reasons: First, our results show that reproducibility failures in ML-based science are systemic. In nearly every scientific field that has carried out a systematic study of reproducibility issues, papers are plagued by common pitfalls. In many systematic reviews, a majority of the papers reviewed suffer from these pitfalls. Thus, we find that similar problems are likely to arise in many fields that are adopting ML methods. Second, despite the urgency of addressing reproducibility failures, there are no systemic solutions that have been deployed for these failures. Scientific communities are discovering the same failure modes across disciplines, but have yet to converge on best practices for avoiding reproducibility failures. Calling attention to and addressing these widespread failures is vital to maintaining public confidence in ML-based science. At the same time, the use of ML methods is still in its infancy in many scientific fields. Addressing reproducibility failures pre-emptively in such fields can correct a lot of scientific research that would otherwise be flawed. Towards a solution: A taxonomy of data leakage We now provide our taxonomy of data leakage errors in ML-based science. Such a taxonomy can enable a better understanding of why leakage occurs and inform potential solutions. Our taxonomy is comprehensive and addresses data leakage arising during the data collection, pre-processing, modeling and evaluation steps. In particular, our taxonomy addresses all cases of data leakage that we found in our survey (Table 1). [L1] Lack of clean separation of training and test dataset. If the training dataset is not separated from the test dataset during all pre-processing, modeling and evaluation steps, the model has access to information in the test set before its performance is evaluated. Since the model has access to information from the test set at training time, the model learns relationships between the predictors and the outcome that would not be available in additional data drawn from the distribution of interest. The performance of the model on this data therefore does not reflect how well the model would perform on a new test set drawn from the same distribution of data. [L1.1] No test set. Using the same dataset for training as well as testing the model is a text-book example of overfitting, which leads to overoptimisic performance estimates (Kuhn & Johnson, 2013). [L1.2] Pre-processing on training and test set. Using the entire dataset for any pre-processing steps such as imputation or over/under sampling. For instance, using oversampling before splitting the data into training and test sets leads to an imperfect separation between the training and test sets since data generated using oversampling from the training set will also be present in the test set. The judgement of whether the use of a given feature is legitimate for a modeling task requires domain knowledge and can be highly problem specific. As a result, we do not provide sub-categories for this sort of leakage. Instead, we suggest that researchers decide which features are suitable for a modeling task and justify their choice using domain expertise. [L3] Test set is not drawn from the distribution of scientific interest. The distribution of data on which the performance of an ML model is evaluated differs from the distribution of data about which the scientific claims are made. The performance of the model on the test set does not correspond to its performance on data drawn from the distribution of scientific interest. [L3.1] Temporal leakage. When an ML model is used to make predictions about a future outcome of interest, the test set should not contain any data from a date before the training set. If the test set contains data from before the training set, the model is built using data "from the future" that it should not have access to during training, and can cause leakage. [L3.2] Nonindependence between train and test samples. Nonindependence between train and test samples constitutes leakage, unless the scientific claim is about a distribution that has the same dependence structure. In the extreme (but unfortunately common) case, train and test samples come from the same people or units. For example, Oner et al. (2020) find that a recent study on histopathology uses different observations of the same patient in the training and test sets. In this case, the scientific claim is being made about the ability to predict gene mutations in new patients; however, it is evaluated on data from old patients (i.e., data from patients in the training set), leading to a mismatch between the test set distribution and the scientific claim. The traintest split should account for the dependencies in the data to ensure correct performance evaluation. Methods such as 'block cross validation' can partition the dataset strategically so that the performance evaluation does not suffer from data leakage and overoptimism (Roberts et al., 2017;Valavi et al., 2021). Handling nonindependence between the training and test sets in general-i.e., without any assumptions about independence in the data-is a hard problem, since we might not know the underlying dependency structure of the task in many cases (Malik, 2020). [ L3.3] Sampling bias in test distribution. Sampling bias in the choice of test dataset can lead to data leakage. One example of sampling bias is spatial bias, which refers to choosing the test data from a geographic location but making claims about model performance in other geographic locations as well. Another example is selection bias, which entails choosing a non-representative subset of the dataset for evaluation. For example, Bone et al. (2015) highlight that in a study on predicting autism using ML models, excluding the data corresponding to borderline cases of autism leads to leakage since the test set is no longer representative of the general population about which claims are made. In addition, borderline cases of autism are often the most tricky to diagnose, so excluding them the evaluation set is likely to lead to overoptimistic results. Cases of leakage due to sampling bias can often be subtle. For example, Zech et al. (2018) find that models for pneumonia prediction trained on images from one hospital do not generalize to images from another hospital due to subtle differences in how images are generated in each hospital. A model may have leakage when the distribution about which the scientific claim is made does not match the distribution from which the evaluation set is drawn. ML models may also suffer from a related, but distinct limitation: the lack of generalization when we try to apply a result about one population to another similar but distinct population. (Geirhos et al., 2020). In ML-based science, where the aim is to create generalizable knowledge, we should take results that claim to generalize to a different population from the one models were evaluated on with caution. (Table 1) Computational reproducibility issues. Computational reproducibility of a finding refers to sharing the complete code and data needed to reproduce the findings reported in a paper exactly. This is important to enable external researchers to reproduce results and verify their correctness. Five papers in our survey outlined the lack of computational reproducibility in their field. Other issues identified in our survey Data quality issues. Access to good quality data is essential for creating ML models (Paullada et al., 2020;Scheuerman et al., 2021). Issues with the quality of the dataset could affect the results of ML-based science. 10 papers in our survey highlighted data quality issues such as not addressing missing values in the data, the small size of datasets compared to the number of predictors, and the outcome variable being a poor proxy for the phenomenon being studied. Metric choice issues. A mismatch between the metric used to evaluate performance and the scientific problem of interest leads to issues with performance claims. For example, using accuracy as the evaluation metric with a heavily imbalanced dataset leads to overoptimistic results, since the model can get a high accuracy score by always predicting the majority class. Four papers in our survey highlighted metric choice issues. Use of standard datasets. Reproducibility issues arose despite the use of standard, widely-used datasets, often because of the lack of standard modeling and evaluation procedures such as fixing the train-test split and evaluation metric for the dataset. Seven papers in our survey highlighted that issues arose despite the use of standard datasets. Model info sheets for detecting and preventing leakage Our taxonomy of data leakage highlights several failure modes which are prevalent in ML-based science. To detect cases of leakage, we provide a template for a model info sheet to accompany scientific claims using predictive modeling 2 . The template consists of precise arguments needed to justify the absence of leakage. Model info sheets would address every type of leakage identified in our survey. Prior work on model cards and reporting standards Our proposal is inspired by prior work on model cards and checklists, which we now review. Mitchell et al. (2019) introduced model cards for reporting details about ML models, with a focus on precisely reporting the intended use cases of ML models. They also addressed fairness and transparency concerns: they require that the performance of ML models on different groups of users (e.g., on the basis of race, gender, age) is reported and documented transparently. These model cards complement the datasheets introduced by Gebru et al. (2021) to document details about datasets in a standard format. The use of checklists has also been impactful in improving reporting practices in the few fields that have adopted them (Han et al., 2017). While checklists and model cards provide concrete best practices for reporting standards (Mongan et al., 2020;Collins et al., 2015;Mitchell et al., 2019;Garbin & Marques, 2022), current efforts do not address pitfalls arising due to leakage. Further, even though several scientific fields-especially those related to medicine-have adopted checklists to improve reporting standards, most checklists are developed for specific scientific or research communities instead of ML-based science in general. Scientific arguments to surface and prevent leakage When ML models are used to make scientific claims, it is not enough to simply separate the training and test sets and report performance metrics on the test set. Unlike research in ML methods, where a model's performance on a hypothetical task (i.e., one that is not linked to a specific scientific claim) is still of interest to the researcher in some cases (Raji et al., 2021), in ML-based science, claims about a model's performance need to be connected to scientific claims using explicit arguments. The burden of proof for ensuring the correctness of these arguments is on the researcher making the scientific claims (Lundberg et al., 2021). In our models, we ask researchers to present three arguments that are essential for determining that scientific results which use ML methods do not suffer from data leakage. Note that most ML-based science papers do not present any of the three arguments, although they sometimes partially address the first argument (clean train-test separation) by reporting out-of-sample prediction performance. The arguments below are based on our taxonomy of data leakage issues presented in Section 2.4, and inform the main sections of the model info sheet. [L1] Clean train-test separation. The researcher needs to argue why the test set does not interact with training data during any of the preprocessing, modeling or evaluation steps to ensure a clean train-test separation. [L2] Each feature in the model is legitimate. The researcher needs to argue why each feature used in their model is legitimate, i.e., a claim made using each feature is of scientific interest. Note that some models might use hundreds of features. In such cases, it is even more important to reason about the correctness of the features used, since the incorrect use of a single feature in the model can cause leakage. That said, the same argument for why a feature is legitimate can often apply to a whole set of features. For example, for a study using individuals' location history as a feature vector, the use of the entire vector can be justified together. Note that we do not ask for the researcher to list each feature used in their model. Rather, we ask that the justification provided for the legitimacy of the features used in their model should cover every feature used in their model. [L3] Test set is drawn from the distribution of scientific interest. If the distribution about which the scientific claims are made is different from the one on which the model is tested, then any claims about the performance of an ML model on the evaluation step fall short. The researcher needs to justify that the test set is drawn from the distribution of scientific interest and there is no selection or sampling bias in the data collection process. This step can help clarify the distribution about which scientific claims are being made and detect temporal leakage. Model info sheets and our theory of change Model info sheets can influence research practices in two ways: first, researchers who introduce a scientific model alongside a paper can use model info sheets to detect and prevent leakage in their models. These info sheets can be included as supplementary materials with their paper for transparently reporting details about their models. In scientific fields where the use of ML methods is not yet widespread, using transparent reporting practices at an early stage could enable easier adoption and more trust in ML methods. This would also help assuage reviewer concerns about reproducibility. Second, journal submission guidelines could encourage or require authors to fill out model info sheets if a paper does not transparently report how the model was created. In this case, model info sheets can be used to start a conversation between authors and reviewers about the details of the models introduced in a paper. Current peer-review practices often do not require the authors to disclose any code or data during the review process (Liu & Salganik, 2019). Even if the code and data are available to reviewers, reproducing results and spotting errors in code is a time consuming process that often cannot be carried out under current peer-review practices. Model info sheets offer a middle ground: they could enable a closer scrutiny of methods without making the process onerous for reviewers. Limitations of model info sheets While model info sheets can enable the detection of all types of leakage we identify in our survey, they suffer from limitations owing to the lack of computational reproducibility of results in scientific research, incorrect claims made in model info sheets, and the lack of expertise of authors and reviewers. First, the claims made in model info sheets cannot be verified in the absence of computational reproducibility. That is, unless the code, data and computing environment required to reproduce the results in a paper are made available, there is no way to ascertain if model info sheets are filled out correctly. Ensuring the computational reproducibility of results therefore remains an important goal for improving scientific research standards. Second, incorrect claims made in model info sheets might provide false assurances to reviewers about the correctness of the claims made in a paper. However, by requiring authors to precisely state details about their modeling process, model info sheets enable incorrect claims to be challenged more directly than in status quo, where details about the modeling process are often left undisclosed. Filling out and evaluating model info sheets requires some expertise in ML. In fields where both authors and reviewers lack any ML expertise, subtle cases of leakage might slip under the radar despite the use of model info sheets. In such cases, we hope that model info sheets released publicly along with papers will enable discourse within scientific communities on the shortcomings of scientific models. Finally, we acknowledge that our understanding of leakage may evolve, and model info sheets may need to evolve with it. To that end, we have versioned model info sheets, and plan to update them as we continue to better understand leakage in ML-based science. A case study of civil war prediction To understand the impact of data leakage and the efficacy of model info sheets in addressing it, we undertake a reproducibility study in a field where ML models are believed to vastly outperform older statistical models such as Logistic Regression (LR) for predictive modeling: civil war prediction (Bara, 2020). Over the last few years, this field has switched to predictive modeling using complex ML models such as Random Forests and Adaboost instead of LR (see Figure 2), with several papers claiming near-perfect performance of these models for civil war prediction (Muchlinski et al., 2016;Colaresi & Mahmood, 2017;Kaufman et al., 2019). While the literature we reviewed in our survey highlighted the pitfalls in adopting ML methods (Table 1), we go further than most previous research to investigate whether the claims made in the reviewed studies survive once the errors are corrected. Systematic search of predictive modeling literature in civil war research. We conducted a systematic search to find relevant literature (detailed in Appendix B.1). This yielded 124 papers. We narrowed this list to the 12 papers that focused on predicting civil war, evaluated performance using a train-test split, and shared the complete code and data. For these 12, we attempted to identify errors and reproducibility issues from the text and through reviewing the code provided with the papers. When we identified errors, we re-analyzed the data with the errors corrected. Finding 1: Data leakage causes irreproducible results. We present our results in Figure 1. We found errors in 4 of the 12 papers-exactly the 4 papers that claimed superior performance of complex ML models over baseline LR models for predicting civil war. All papers suffer from different forms of leakage. All 4 papers were published in top-10 journals in the field of "Political Science and International Relations" (sci, 2020). When the errors are corrected, complex ML models perform no better than baseline LR models in each case except Wang (2019), where the difference between the AUC of the complex ML models and LR models drops from 0.14 to 0.01. Further, while none of these errors could have been caught by reading the paper, model info sheets enable the detection of leakage in each case (Appendix C). Beyond reproducibility, our results show that complex ML models are not substantively better at civil war prediction than decades old LR models. This is consistent with similar sobering findings in other tasks involving predicting social outcomes such as children's life outcomes (Salganik et al., 2020) and recidivism (Dressel & Farid, 2018). Our findings strongly suggest the need for tempering the optimism about predictive modeling in the field of civil war prediction and question the use of ML models in this field. We provide a detailed overview of our methodology for correcting the errors and show that our results hold under several robustness checks in Appendix B. Finding 2: No significance testing or uncertainty quantification. We found that 9 of the 12 papers for which complete code and data were available included no significance tests or uncertainty quantification for classifier performance comparison (Table A6). Especially when sample sizes are small, significance testing and uncertainty quantification are important steps towards reproducibility (McDermott et al., 2021;Gorman & Bedrick, 2019). As an illustration, we examine this issue in detail in the case of Blair & Sambanis (2020) since their test dataset has a particularly small number of instances of civil war onset (only 11). They propose a model of civil war onset that uses theoretically informed features and report that it outperforms other baseline models of civil war onset using the AUC metric on an out-of-sample dataset. We find that the performance of their model is not significantly better than other baseline models for civil war prediction. 3 Further, all models have large confidence intervals for their out-of-sample performance. For instance, while the smoothed AUC performance reported by the authors is 0.85, the 95% confidence interval calculated using bootstrapped test set re-sampling is [0.66-0.95]. Beyond leakage: enhancing the reproducibility of ML-based science We have outlined how the use of model info sheets can address data leakage in ML-based science. In addition to leakage, we found a number of other reproducibility issues in our survey. Here, we present five diagnoses for reproducibility failures in fields adopting ML methods. Each of our diagnoses is paired with a recommendation to address it. [D1] Lack of understanding of the limits to prediction. Recent research for predicting social outcomes has shown that even with complex models and large datasets, there are strong limits to predictive performance (Salganik et al., 2020;Dressel & Farid, 2018). However, results like the better-than-human performance of ML models in perception tasks such as image classification (He et al., 2015;Szeliski, 2021) give the impression of ML models surpassing human performance across tasks, which can confuse researchers about the performance they should realistically expect from ML models. [R1] Understand and communicate limits to prediction. A research agenda which investigates the efficacy of ML models in tasks across scientific fields would increase our understanding of the limits to prediction. This can alleviate the overoptimism that arises from confusing progress in one task (e.g., image classification) with another (e.g., predicting social outcomes). If we can identify upper bounds on the predictive accuracy of tasks (i.e., lower bound the Bayes Error Rate for a task), then once the achievable accuracy has been reached, we can avoid a futile effort to increase it further and can apply increased skepticism towards results that claim to violate known bounds. [D2] Hype, overoptimism and publication biases. The hype about commercial AI applications can spill over into ML-based science, leading to overoptimism about their performance. Non-replicable findings are cited more than replicable ones ( [R2] Treat results from ML-based science as tentative. When overoptimism is prevalent in a field, it is important to engage with results emerging from the field critically. Until reproducibility issues in ML-based science are widely addressed and resolved, results from this body of work should be treated with caution. [D3] Inadequate expertise. The rapid adoption of ML methods in a scientific field can lead to errors. These can be caused due to the lack of expertise of domain experts in using ML methods and vice-versa. [R3] Inter-disciplinary collaborations and communication of best-practices. Literature in the ML community should address the different failure modes that arise during the modeling process. Researchers with expertise in ML methods should clearly communicate best practices in deploying ML for scientific research (Lones, 2021). Having an interdisciplinary team consisting of researchers with domain expertise as well as ML expertise can avoid errors. [D4] Lack of standardization. Several applied ML fields, such as engineering applications and modeling contests, have adopted practices such as standardized train-test splits, evaluation metrics, and modeling tasks to ensure the validity of the modeling and evaluation process (Russakovsky et al., 2015;Koh et al., 2021). However, many of these have not yet been adopted widely in ML-based science. This leads to subtle errors in the modeling process that can be hard to detect. [R4] Adopt the common task framework when possible. The common task framework allows us to compare the performance of competing ML models using an agreed-upon training dataset and evaluation metrics, a secret holdout dataset, and a public leaderboard (Rocca & Yarkoni, 2021;Donoho, 2017). Dataset creation and model evaluation is left to impartial third parties who have the expertise and incentives to avoid errors. However, one undesirable outcome that has been observed in communities that have adopted the common task framework is a singular focus on optimizing a particular accuracy metric to the exclusion of other scientific and normatively desirable properties of models (Paullada et al., 2020;Marie et al., 2021;Gorman & Bedrick, 2019). [D5] Lack of computational reproducibility. The lack of computational reproducibility hinders verification of results by independent researchers (Section 2.5). While computational reproducibility does not mean that the code is errorfree, it can make the process of finding errors easier, since researchers attempting to reproduce results do not have to spend time getting the code to run. [R5] Ensure computational reproducibility. Platforms such as CodeOcean (Clyburne-Sherin et al., 2019), a cloud computing platform which replicates the exact computational environment used to create the original results, can be used to ensure the long term reproducibility of results. We follow several academic journals and researchers in recommending that future research in fields using ML methods should use similar methods to ensure computational reproducibility (noa, 2018; Liu & Salganik, 2019). Conclusion The attractiveness of adopting ML methods in scientific research is in part due to the widespread availability of offthe-shelf tools to create models without expertise in ML methods (Hutson, 2019). However, this laissez faire approach leads to common pitfalls spreading to all scientific fields that use ML. So far, each research community has independently rediscovered these pitfalls. Without fundamental changes to research and reporting practices, we risk losing public trust owing to the severity and prevalence of the reproducibility crisis across disciplines. Our paper is a call for interdisciplinary efforts to address the crisis by developing and driving the adoption of best practices for ML-based science. Model info sheets for detecting and preventing leakage are a first step in that direction. Overview of the Appendix. In Appendix A, we justify our choice of the word reproducibility. In Appendix B, we provide a detailed description of the methods we used to select papers for our review of civil war prediction and fix reproducibility issues in the papers with errors. In Appendix C, we show that each type of leakage identified in our survey (Section 2.4) is addressed by model info sheets. We provide a template of model info sheets and a list of all 124 papers that we considered for our literature review in civil war prediction on our website (https://reproducible.cs.princeton.edu). A. Why do we call these reproducibility issues? We acknowledge that there isn't consensus about the term reproducibility, and there have been a number of recent attempts to define the term and create consensus (National Academies of Sciences, 2019). One possible definition is computational reproducibility-when the results in a paper can be replicated using the exact code and dataset provided by the authors (Liu & Salganik, 2019). We argue that this definition is too narrow because even cases of outright bugs in the code would not be considered irreproducible under this definition. Therefore we advocate for a standard where bugs and other errors in data analysis that change or challenge a paper's findings constitute irreproducibility. The goal of predictive modeling is to estimate (and improve) the accuracy of predictions that one might make in a real-world scenario. This is true regardless of the specific research question one wishes to study by building a predictive model. In practice one sets up the data analysis to mimic this real-world scenario as closely as possible. There are limits to how well we can do this and consequently there is always methodological debate on some issues, but there are also some clear rules. If an analysis choice can be shown to lead to incorrect estimates of predictive accuracy, there is usually consensus in the ML community that it is an error. For example, violating the train-test split (or the learn-predict separation) is an error because the test set is intended to provide an accurate estimate of 'out-of-sample' performance-model performance on a dataset that was not used for training (Kuhn & Johnson, 2013). Thus, to define what is an error, we look to this consensus in the ML community (e.g. in textbooks) and offer our own arguments when necessary. B. Materials and Methods: Reproducibility issues in civil war prediction Different researchers might have different aims when comparing the performance on civil war prediction -determining the absolute performance, or comparing the relative performance of different models of civil war prediction. Whether the aim is to determine the relative or absolute performance of models of civil war prediction, data leakage causes a deeper issue in the findings of each of the 4 papers with errors that leads to inaccurate estimates of both relative and absolute out-of-sample performance. In correcting the papers with errors (Muchlinski et al., 2016;Colaresi & Mahmood, 2017;Kaufman et al., 2019), our aim is to report out-of-sample performance of the various models of civil war prediction after correcting the data leakage, while keeping all other factors as close to the original implementation as possible. Fixing the errors allows a more accurate estimate of out-of-sample performance. At the same time, we caution that just because our corrected results offer a more accurate estimate of out-of-sample performance doesn't mean that we endorse all other methodological choices made in the papers. For example, to correct the results reported by Muchlinski et al. (2016), we use imputation on an out-of-sample dataset that has 95% missing values. While an imputation model created only using the training data avoids data leakage, it does not mean that using a dataset with 95% missing values to measure out-of-sample performance is desirable. B.1. Paper selection for review To find relevant papers on civil war prediction for our review, we used the search results from a dataset of academic literature (Hook et al., 2018) for papers with the terms 'civil' AND 'war' AND ('prediction' OR 'predicting' OR 'forecast') in their title or abstract, as well as papers that were cited in a recent review of the field (Bara, 2020). To keep the number of papers tractable, we limited ourselves to those that were published in the last 5 years, specifically, papers published between 1st January 2016 and 14th May 2021. This yielded 124 papers. We narrowed this list to the 15 papers that were focused on predicting civil war and evaluated performance using a train-test split. Of the 15 papers that meet our inclusion criteria, 12 share the complete code and data. For these 12, we attempted to identify errors and reproducibility issues from the text and through reviewing the code provided with the papers. When we identified errors, we re-analyzed the data with the errors corrected. We now address the reproducibility issues we found in each paper in detail. B.2. Muchlinski et al. (2016) Imputation is commonly used to fill in missing values in datasets (Donders et al., 2006). Imputing the training and test datasets together refers to using data from the training as well as the test datasets to create an imputation model that fills in all missing values in the dataset. This is an erroneous imputation method for the predictive modeling paradigm, since it can lead to data leakage, which results in incorrect, over-optimistic performance claims. This pitfall is well known in the predictive modeling community -discussed in ML textbooks (Kuhn & Johnson, 2013), blogs (Ghani et al., 2020 and popular online forums (noa). Muchlinski et al. (2016) claim that a Random Forests model vastly outperforms Logistic Regression models in terms of out-of-sample performance using the AUC metric (Fawcett, 2006). However, since they impute the training and test datasets together, their results suffer from data leakage. The impact of leakage is especially severe because of the level of missingness in their out-of-sample test dataset: over 95% of the values are missing (which is not reported in the paper), and 70 of the 90 variables used in their model are missing for all instances in the out-of-sample test set. 4 When their imputation method is corrected, their Random Forests model performs no better than the Logistic Regression models that they compared against. We focus on reproducing the out-of-sample results reported by Muchlinski et al. (2016). Table A1 (2006) when training their models, and provide a separate out-of-sample test set for evaluation. To address missing values, they use a Random Forests based imputation method in R called rfImpute. However, the training and test sets are imputed together, which leads to a data leakage. This results in overoptimistic performance claims. Below, we detail the steps we take to correct their results, provide a visualization of the data leakage, and provide a simulation showcasing how the data leakage can result in overoptimistic claims of performance. Correcting the data imputation. To correct this error, we use the mice package in R which uses multiple imputation for imputing missing data. This is because the mice package allows us to specify which rows in the dataset are a part of the test set and it does not use those rows for creating the imputation model, whereas rfImpute -the original method used to impute the missing data in the original results by Muchlinski et al. (2016) -does not have this feature. The authors imputed the training set together with the out-of-sample test set using rfImpute, which led to data leakage. Table A1 provides the comparisons between the results reported in Muchlinski et al. (2016), our reproductions of their reported (incorrect) results, as well as the corrected version of their results. Using multiple imputation fills in missing values without regarding the underlying variable's original distribution. For example, using multiple imputation fills in different missing values for the variable representing the percentage of rough terrain in a country in different years (Beger, 2021), whereas this particular variable (percentage of rough terrain) is constant over time. However, when multiple imputation is used with a train-test split, there is still no leakage between the training and test sets, since the imputation model only uses data from the training set to fill in missing values in the test set. Why can't we use rfImpute in the corrected results? Instead of using the mice package, another way to impute the data correctly, i.e., without data leakage, would be to run the imputation using rfImpute on the training and test data separately -creating two separate imputation models -one for the training data and one for the test data. We could not use this imputation method because 70 of the 90 variables used in Muchlinski et al. (2016)'s model as features do not have any values in the out-of-sample test data provided -i.e. they are missing for all observations in the out-of-sample datasetand rfImpute requires at least some values for each variable to not be missing. In other words, the mice package allows us to train an imputation model on the training set and use it to fill in missing values in the test set. Subtle differences between explanatory and predictive modeling. In the explanatory modeling paradigm, the aim is to draw inferences from data, as opposed to optimizing and evaluating out-of-sample predictive performance. In this case, data imputation would be considered a part of the data pre-processing step, even though it is still important to keep in mind the various assumptions being made in this process Schafer (1999). Contrarily, in the predictive modeling paradigm, the imputation is a part of the modeling step (Kuhn & Johnson, 2013) because the aim of the modeling exercise is to validate performance on an out-of-sample test set, which the model does not have access to during the training. In this case, imputing the training and test datasets together leads to leaking information from the test set to the training set and thus the performance evaluation on the purportedly "out-of-sample" test set would be an over-estimate. What is the precise mechanism by which the leakage occurs in Muchlinski et al. (2016)? When Muchlinski et al. (2016 impute the missing values in the out-of-sample test set, the imputation model has access to the entire training data as well as the labels of the target variables in the test data -they also include the target variable in the list of variables which the imputation model treats as independent variables when carrying out the imputation. The model therefore uses correlations between the target variable and independent variables in the training dataset and uses them to fill in the missing values in the test dataset -i.e. the model uses the labels of the target variables in the test data and correlations from the training data to fill in missing values. This leads to the test dataset having similar correlations between the target and independent variables as the ones present in the training data. Further, the missing data is filled in in such a way that it favors ML models such as Random Forests over Logistic Regression models, as we show in the visualization below. Visualizing the leakage. We can visually observe an instance of data leakage in Figure A1. We focus on the distribution of the feature agexp, which represents the proportion of agricultural exports in the GDP of a country. We choose this feature because in the Muchlinski et al. paper, this feature had the highest gini index for the random forests model -which means that it was an important feature for the model. While we only visualize one feature here, similar results hold across multiple features used in the model. Below, we reconstruct the process by which the data leakage was generated -following the exact steps Muchlinski et al. (2016) used to create and evaluate the dataset: • Figure A1a represents the distribution of the agexp variable for war and peace data points in the original dataset by Hegre & Sambanis (2006), ignoring missing values. • Figure A1b shows the same distribution after including the imputed values of agexp. In particular, we see two peaks in the dataset for war and peace data points alike, one due to war instances and one due to peace instances. • If we look only at the data points that were imputed using the rfImpute method ( Figure A1c), we see that the distribution of the imputed data points for war and peace are completely separated, in contrast to the original distribution where there was a significant overlap between the distributions. • Finally, Figure A1d shows the effect of imputing this already-imputed dataset with the out-of-sample test set -we see that the out-of-sample dataset only has the peak for peace datapoints, whereas the distribution for war is almost uniform. Further, the random forests model can learn the peak for the agexp variable in the peace instances from the training dataset after imputation, since the peak for the training and test sets is similar. It can distinguish between war and peace datapoints much more easily compared to a logistic regression model that only uses one parameter per feature -logistic regression models are monotonic functions of the independent variables and therefore cannot learn that a variable only lies within a small range for a given label. This highlights the reason behind Random Forests outperforming Logistic Regression in this setting -imputing the training and test datasets together leads to variable values being artifically concentrated within a very small range for both the training and test datasets -and further, being neatly separated across war and peace instances. The impact of the imputation becomes even clearer when we consider that the out-of-sample test dataset provided by Muchlinski et al. (2016) has over 95% of the data missing, and 70 out of 90 variables are missing for all instances in the out-of-sample dataset. A simulation showcasing the impact of missingness on performance estimates in the presence of leakage. We can observe a visual example of how data leakage affects performance evaluation in Figure A2. We describe the simulation below: (d) Distribution of the agexp variable for peace and war data points for the out-of-sample test set Figure A1. Distribution of the agexp variable for peace and war data points for different imputation steps in Muchlinski et al. (2016). Note that the distribution of peace instances in the test set (D) has a peak that is close to the distribution in the imputed training set (B, C)which allows the random forests model to learn the small range of values where peace data points are concentrated. While we report results for the agexp variable, similar trends appear across independent variables in the dataset. • there are two variables -the target variable onset and the independent variable gdp. • onset is a binary variable. gdp is drawn from a normal distribution and depends on onset as follows: gdp = N (0, 1) + onset. • We generate 1000 samples with onset=0 and 1000 samples with onset=1 to create the dataset. • We randomly split the data into training (50%) and test (50%) sets, and create a random forests model that is trained on the training set and evaluated on the test set. • To observe the impact of imputing the training and test sets together, we randomly delete a certain percentage of values of gdp, and impute it using the imputation method used in Muchlinski et al. (2016). • We vary the proportion of missing values from 0% to 95% in increments of 5% and plot the accuracy of the random forests classifier on the test set. • We run the entire process 100 times and report the mean and 95% CI of the accuracy in Figure A2; the 95% CI is too small to be seen in the Figure. We find that imputing the training and test sets together leads to an increasing improvement in the purportedly "out-of-sample" accuracy of the model. Estimates of model performance in this case are artificially high. This example also highlights the impact of the high percentage of missing values -since the out-of-sample test set used by Muchlinski et al. (2016) contains over 95% missing values, the impact of imputing the training and test sets together is very high. Figure A2. Results of a simulation that showcase how imputing the training and test sets together leads to overoptimistic estimates of model performance. The 95% Confidence Intervals are too small to be seen. B.3. Colaresi & Mahmood (2017) Colaresi & Mahmood (2017) We focus on reproducing the final round of results reported in the paper Colaresi & Mahmood (2017), which consists of a comparison of 3 models of civil war onset -the Random Forests model proposed in Muchlinski et al. (2016), the Random Forests model proposed in Colaresi & Mahmood (2017) as well as the Logistic Regression model proposed in Fearon & Laitin (2003). Their dataset has 17.4% values missing, and the test set has 19% values missing. The proportion of missing values in individual variables can be even higher -for example, the agexp, which represents the proportion of agricultural exports in the GDP of a country, is missing for 54.3% of the rows in the test set. In our corrected results, we use the original dataset from Hegre & Sambanis (2006) and impute the training and test data separately using the rfImpute function. The test set consists of data from the years after 1988. One of the independent variables, milper, is missing for all instances in the test set of Colaresi & Mahmood (2017) so we exclude this variable from our models. Table A2 provides the comparisons between the results reported in Colaresi & Mahmood (2017), our reproductions of their reported (incorrect) results, as well as the corrected version of their results. Colaresi & Mahmood (2017) and Table A3. Original and corrected results in the Wang (2019). We find that using an out-of-sample test set further favors Logistic Regression models over ML models. The metric for all results is AUC. These results were not reported using nested cross-validation in . In our reproduction of these reported results, we use nested cross-validation, which ensures that we do not get over-estimates of performance. use the original dataset from Hegre & Sambanis (2006) and impute the training and test data separately using the rfImpute function within each cross validation fold. This ensures that there is no data leakage between the training and test sets in each fold. Table A3 provides the comparisons between the results reported in Wang (2019), our reproductions of their reported (incorrect) results, as well as the corrected version of their results. We also conduct an additional robustness analysis in which we use a separate out-of-sample test set instead of k−fold cross validation, since using k−fold cross validation with temporal data can also lead to leakage across the train-test split. To maintain comparability between the original and corrected results by testing on the same instances of civil war, we continue to use k−fold cross validation in the corrected results in Figure 1. We report the results after making this change in Table A3. We use the same train-test split as Colaresi & Mahmood (2017) year < 1988 as training data and the rest as test data -for the out-of-sample test set. The test set consists of data from the years after 1988. One of the independent variables, milper, is missing for all instances in the test set of Colaresi & Mahmood (2017) so we exclude this variable from our models. Note that the imputation method that should be used depends on the exact model deployment scenario, and should mimic it as closely as possible for accurate performance estimates. For example, in some model deployment settings samples for prediction come in one at a time and in some cases they come in batches. In the former setting, imputing the entire test set together may result in overoptimistic performance evaluations as well, since the deployed model doesn't have access to a batch of samples. Our results may thus offer an upper bound on the performance of civil war prediction models in the case of Colaresi & Mahmood (2017) and . B.5. Kaufman et al. (2019) We focus on reproducing the results on civil war prediction in Kaufman et al. (2019). There are several issues in the paper's results. We outline each issue below and provide a comparison of various scenarios in Table A4 that highlight the precise cause of the performance difference between the original and corrected results, and visualize the robustness of our corrected results. We find that even though there are several issues in Kaufman et al. (2019), the main difference in performance between the original results they report and our corrected results is due to data leakage. Data leakage due to proxy variables. The dataset used by Kaufman et al. (2019) has several variables that, if used as independent variables in models of civil war prediction, could cause data leakage, since they are proxies of the outcome variable. Using k−fold cross validation with temporal data. k−fold cross validation shuffles the dataset before it is divided into training and test datasets. When the dataset contains temporal data, the training dataset could contain data from a later date than the test dataset because of being shuffled. To maintain comparability between the original and corrected results by testing on the same instances of civil war, we continue to use k−fold cross validation in the corrected results in Figure 1. To evaluate out-of-sample performance without using cross-validation, we use a separate train-test split instead of k−fold cross validation and report the difference in results for this scenario in the row Corrected (out-of-sample) in Table A4. We find that there is no substantial difference between the results when using the out-of-sample test set and k−fold cross validation -in each case, none of the models outperforms a baseline that predicts the outcome of the previous year. We use the same train-test split as Colaresi & Mahmood (2017) year < 1988 as training data and the rest as test data. Replacing missing values with zeros. Kaufman et al. (2019) replace missing values in their dataset with zeros, instead of imputing the missing data or removing the rows with missing values. This is a methodologically unsound way of dealing with missing data: for example, the models would not be able to discern whether a variable has a value of zero because of missing data or because it was the true value of the variable for that instance. This risks getting underestimates of performance, as opposed to overoptimistic performance claims. As a robustness check, we impute the training and test data separately in each cross-validation fold using the rfImpute function in R and report the results in the Corrected (imputation) row of Table A4. We find that the choice of imputation method does not cause a difference in performance, perhaps because only 0.6% of the values of variables are missing in the dataset. Choice of cut-offs for calculating accuracy. Instead of calculating model cutoffs based on the best cutoff in the training set, Kaufman et al. use the distribution of model scores to decide the cutoffs for calculating accuracy. We include robustness results when we change the cutoff selection procedure to choosing the best cutoffs for the training set in the Corrected (cutoff choice) row of Table A5. We find that the choice of cutoff does not impact the main claim -the performance of the best model is still worse than a baseline that predicts the outcome of the previous year. Weak Baseline. Kaufman et al. (2019) compare their results against a baseline model that always predicts peace. We find that a baseline that predicts war if the outcome of the target variable was civil war in the previous year and predicts peace otherwise is a stronger baseline (Accuracy: 97.5% vs. 86.1%; χ 2 =633.7, p = 7.836e-140 using McNemar's test as detailed in Dietterich (1998)), and report results against this stronger baseline in Table A4. Confusion about the target variable. Kaufman et al. (2019) use ongoing civil war instead of civil war onset as the target variable in their models. While their abstract mentions that the prediction task they attempt is civil war onset prediction, they switch to using the term civil war incidence in later sections, without formally defining this term. To attempt to determine what they mean by this term, we looked at the papers they cite; one of them has the term civil war incidence in the title Collier & Hoeffler (2002), and defines civil war incidence as 'observations [that] experienced a start of a civil war'. At the same time, in the introduction, they state that they are 'predicting whether civil war occurs in a country in a given year'which refers to ongoing civil war instead of civil war onset. This might confuse a reader about the specific prediction task they undertake. (2003) include lagged versions of the first 4 variables in the list as independent variables in their model to avoid leakage. Following their use of lagged versions of these variables, we do the same in our correction to avoid leakage. The other variables are proxies for the outcomes of interest and hence we remove them from the models to avoid data leakage. B.6. Blair & Sambanis (2020) Blair & Sambanis (2020) state that their escalation model outperforms other models across a variety of settings. However, they do not test the performance evaluations to see if the difference is statistically significant. We find that there is no significant difference between the smoothed AUC values of the escalation model's performance and other models they compare it to when we use a test for significance. Further, we provide a visualization of the 95% confidence intervals of specificities and sensitivities in the smoothed ROC curve they report for their model (escalation) as well as for a baseline model (cameo) -and find that the 95% confidence intervals are large (see Figure A3). Uncertainty quantification, p-values and Z-values for tests of statistical significance. • We report p-values and Z values for a one-tailed significance test comparing the smoothed AUC performance of the escalation model with other baseline models reported in their paper -quad, goldstein, cameo and average respectively. Note that we do not correct for multiple comparisons; such a correction would further reduce the significance of the results. We implement the comparison test for smoothed ROC curves detailed in Robin et al. (2011). - While a small p-value is used to reject the null hypothesis (in this case -that the out-of-sample performance does not differ between the models being compared), a singular focus on a test for statistical significance at a pre-defined threshold can be harmful (see, for example Imbens (2021)). Blair and Sambanis do report performance evaluations for a variety of different model specifications. However, the purpose of such robustness checks is to determine whether model performance sensitive to the parameter choices; it is unclear whether it helps deal with issues arising from sampling variance. At any rate, Blair and Sambanis's results turn out to be highly sensitive to another modeling choice: the fact that they compute the AUC metric on the smoothed ROC curve instead of the empirical curve that their model produces. Smoothing refers to a transformation of the ROC curve to make the predicted probabilities for the war and peace instances normally distributed instead of using the empirical ROC curve (see Robin et al. (2011)). This issue was pointed out by Beger et al. (2021) and completely changes their original results; Blair & Sambanis (2021) discuss it in their rebuttal. B.7. Overview of papers in Table A6 Table A6 provides the list of 12 papers included in our review, showing information about whether they report confidence intervals, conduct tests of statistical significance when comparing classifier performance, which metrics they report, the number of rows and the number of positive instances (i.e. instances of war/conflict) in the test set, and whether their main claim relies on out-of-sample evaluation of classifier performance. We detail information about the numbers we report in Table A6 below. • Hegre et al. (2016): We report the number of rows and number of positive instances of civil war incidence for the dates between 2001 and 2013 in the UCDP dataset, i.e. all years for which out-of-sample estimates are provided. We report the out-of-sample AUC performance difference for the Major conflict setting. Out-of-sample evaluation results are not included in the main text of the paper, hence we report that the paper's main claim does not rely on out-of-sample evaluations. • Muchlinski et al. (2016): We report the number of rows and number of positive instances of civil war onset for the dates after 2000 in the out-of-sample dataset provided by Muchlinski et al. We report the out-of-sample AUC performance difference between the Random Forests and the best Logistic Regression setting. Out-of-sample evaluation results are used to justify the performance improvement of using Random Forests models, hence we report that the paper's main claim relies on out-of-sample evaluations. • Chiba & Gleditsch (2017): We report the total number of instances and the number of positive instances of governmental onsets in the years 2013-14 (the test set dates). We report the difference between the territorial onset AUC's reported in the paper. Note that while Chiba & Gleditsch (2017) do report small number of data points that are used in one of their settings, they do not address how to estimate variance or perform tests of statistical significance. Out-of-sample evaluation results are not used as the main evidence of better performance in the main text of the paper, hence we report that the paper's main claim does not rely on out-of-sample evaluations. • Colaresi & Mahmood (2017): We report the number of rows and onsets of civil war after the year 1988 (the test set dates). We report the out-of-sample AUC difference between the two random forests models compared in the paper. Out-of-sample evaluation results are used to justify the performance improvement of using an iterative method for model improvement, hence we report that the paper's main claim relies on out-of-sample evaluations. • Hirose et al. (2017): We report the number of locations included in the out-of-sample results. Since the paper does not attempt binary classification, we do not report the number of positive instances in this case. We report the out-ofsample performance gain of adding relative ISAF support to the baseline model in the IED attack setting of the paper. Out-of-sample evaluation results are used as important evidence of better model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. • Schutte (2017): We report the number of rows in the entire dataset, since the paper uses k-fold cross validation and therefore all instances are used for testing. Since the paper does not attempt binary classification, we do not report the number of positive instances in this case. We report the out-of-sample normalized MAE difference between the population model and the best performing model compared in the paper. Out-of-sample evaluation results are used as (a) Visualizing the 95% confidence intervals of the specificities for the 1 month forecast in the smoothed ROC curve reported in Blair & Sambanis (2020). (b) Visualizing the 95% confidence intervals of the sensitivities for the 1 month forecast in the smoothed ROC curve reported in Blair & Sambanis (2020). (c) Visualizing the 95% confidence intervals of the specificities for the 6 month forecast in the smoothed ROC curve reported in Blair & Sambanis (2020). (d) Visualizing the 95% confidence intervals of the sensitivities for the 6 month forecast in the smoothed ROC curve reported in Blair & Sambanis (2020). Figure A3. The wide confidence intervals for sensitivities and specificities reported in Blair and Sambanis. Here, we visualize the escalation and cameo models for the 1 month and 6 month forecast in the base specification (reported in Figure 1 of their paper). important evidence of better model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. • Hegre et al. (2019b): We report the number of rows and number of positive instances of civil war incidence for the dates between 2001 and 2013 in the UCDP dataset, i.e. all years for which out-of-sample estimates are provided. We report the out-of-sample AUC performance difference for the Major conflict setting. Out-of-sample evaluation results are not used as the primary evidence of better model performance in the main text of the paper, hence we report that the Table A6. A list of papers for which code and dataset were available, showing information about whether they report confidence intervals, conduct tests of statistical significance when comparing classifier performance, which metrics they report, the number of rows and the number of positive instances (i.e. instances of war or conflict or onset thereof) in the test set, and whether their main claim relies on out-of-sample evaluation of classifier performance. AUC = Area Under ROC, MAE = Mean Absolute Error, RMSE = Root Mean Squared Error, AUPR = Area Under Precision-Recall Curve, TPR = True Positive Rate, FPR = False Positive Rate, OOS performance delta = the performance difference for the most salient performance comparison reported in the paper (details in Section B.7). Hirose et al. state that the out-of-sample performance is significantly better in the Supplement of their paper, but we could not find the figure they cite as evidence of this claim in their Supplement. paper's main claim does not rely on out-of-sample evaluations. • Hegre et al. (2019a): We report the number instances with state based conflict in the ViEWS Monthly Outcomes at PRIO-Grid Level data between 2015 and 2017 -the years for which the out-of-sample results are reported in the paper. We report the out-of-sample AUC performance difference for the state-based conflict setting. Out-of-sample evaluation results are used as the primary evidence of better model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. • Kaufman et al. (2019): We report the total number of rows and all instances of civil war incidence in the dataset used by Kaufman et al., since they use k-fold cross validation and therefore all instances are used for testing. We report the out-of-sample accuracy difference between the Adaboost and Logistic Regression settings. Out-of-sample evaluation results are used as the primary evidence of better model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. • Wang (2019): We report the total number of rows and onsets of civil war used in the dataset used by Wang since they use k-fold cross validation and therefore all instances are used for testing. We report the out-of-sample AUC performance difference between the Adaboost and Logistic Regression models. Out-of-sample evaluation results are used as the primary evidence of better model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. • Blair & Sambanis (2020): We report the number of rows and onsets of civil war after the year 2007 (the test set dates). We report the out-of-sample AUC performance difference between the escalation and cameo models for the one-month base setting. Out-of-sample evaluation results are used as the primary evidence of better model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. • Hegre et al. (2021): We report the number of rows and number of positive instances for civil war onset the dates between 2001 and 2018, i.e. all years for which out-of-sample estimates are provided. We don't report the out-of-sample performance difference because the paper does not perform comparisons between models. Out-of-sample evaluation results are used as the primary evidence of model performance in the main text of the paper, hence we report that the paper's main claim relies on out-of-sample evaluations. [ Figure 2 . 2Number of political science papers containing the terms "civil war" AND "machine learning" in the dimensions database of academic research(Hook et al., 2018). Note the sharp increase in papers using ML methods in the last few years. Serra-Garcia & Gneezy, 2021), which can result in feedback loops of overoptimism in ML-based science. Besides, publication biases that have been documented in several scientific fields(Shi & Lin, 2019; Gurevitch et al., 2018) can also affect ML-based science(Hofman et al., 2017; Islam et al., 2017). Hegre, H., Allansson, M., Basedau, M., Colaresi, M., Croicu, M., Fjelde, H., Hoyles, F., Hultman, L., Högbladh, S., Jansen, R., Mouhleb, N., Muhammad, S. A., Nilsson, D., Nygård, H. M., Olafsdottir, G., Petrova, K., Randahl, D., Rød, E. 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Successes and struggles with computational reproducibility: Lessons from the fragile families challenge. Socius, 5:2378023119849803, 2019. provides the comparisons between the results reported in Muchlinski et al. (2016), our reproductions of their reported (incorrect) results, as well as the corrected version of their results. Muchlinski et al. (2016) received two critiques of the methods used in their paper (Wang, 2019; Neunhoeffer & Sternberg, 2019). 5 . In response, they published a reply with clarifications and revised code addressing both critiques (Muchlinski et al., 2019). We use the revised version of their code. We find that the error in their imputation methods exists in the revised code as well as the original code, and was not identified by the previous critiques. Muchlinski et al. (2016) re-use the dataset from Hegre & Sambanis Distribution of the agexp variable for peace and war data points for the original Hegre et al. Distribution of the agexp variable for peace and war data points for the imputed Hegre et al. dataset used by Muchlinski et al. Distribution of the agexp variable for peace and war data points only for the data points that were added during imputation (i.e. the data points that were missing in the original dataset) Figure 1. A comparison of reported and corrected results in civil war prediction papers published in top political science journals. The main findings of each of these papers are invalid due to various forms of data leakage: Muchlinski et al. (2016) impute the training and test data together, Colaresi & Mahmood (2017) and Wang (2019) incorrectly reuse an imputed dataset, and Kaufman et al. (2019) use proxies for the target variable which causes data leakage. The use of model info sheets (Section 3) would detect leakage in every paper.When we correct these errors, complex ML models (such as Adaboost and Random Forests) do not perform substantively better than decades-old Logistic Regression models for civil war prediction in each case. Each column in the table outlines the impact of leakage on the results of a paper. The figure above each column shows the difference in performance that results from fixing leakage issues.Paper Muchlinski et al. Colaresi and Mahmood Wang Kaufman et al. Claim Random Forests model drastically outperforms Logistic regression models Random Forests models drastically outperform Logistic regression model Adaboost and Gradient Boosted Trees (GBT) drastically outperform other models Adaboost outperforms other models Error [L1.2] Pre-proc. on train-test (Incorrect imputation) [L1.2] Pre-proc. on train-test (Incorrect reuse of an imputed dataset) [L1.2] Pre-proc. on train-test. (Incorrect reuse of an imputed dataset) [L3.1] Temporal leakage (k-fold cross validation with temporal data) [L2] Illegitimate features (Data leakage due to proxy variables) [L3.1] Temporal leakage (k-fold cross validation with temporal data) Impact Random Forests perform no better than Logistic Regression Random Forests perform no better than Logistic Regression Difference in AUC between Adaboost and Logistic Regression drops from 0.14 to 0.01 Adaboost no longer outperforms Logistic Regression. None of the models outperform a baseline model that predicts the outcome of the previous year Discussion Impact of the incorrect imputation is severe since 95% of the out-of-sample dataset is missing and is filled in using the incorrect imputation method Re-use the dataset provided by Muchlinski et al., which uses an incorrect imputation method Re-use the dataset provided by Muchlinski et al., which uses an incorrect imputation method Use several proxy variables for the outcome as predictors (e.g., colwars, cowwars, sdwars, all proxies for civil war), leading to near perfect accuracy 0.5 0.6 0.7 0.8 0.9 1.0 Reported results (AUC) Corrected results (AUC) Reported results (AUC) Corrected results (AUC) Reported results (AUC) Corrected results (AUC) Reported results (Accuracy) Corrected results (Accuracy) Logistic Regression 1 Logistic Regression 2 Logistic Regression 3 Random Forests Logistic Regression Random Forests 1 Random Forests 2 AdaBoost GBT Logistic Regression 1 Logistic Regression 2 Logistic Regression 3 Random Forests AdaBoost Extratrees Lasso Logistic Regression Random Forest SVM Materials and methods. 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Kaufman et al.(2019)use an incorrect parameter selection technique when creating their Lasso model that leads to the model always predicting peace (i.e. all coefficients of the variables in the model are always zero). We correct this using a standard technique for parameter selection. Instead of choosing model parameters such that the model always predicts peace, we use the cv.glmnet function in R to choose a suitable value for model parameters based on the training data. Table A4. Results for the various scenarios inKaufman et al. (2019). We report results up to 3 significant figures in this table because the small difference in performance between AdaBoost and Logistic Regression that is ascribed signifance in Kaufman et al.(2019)can only be observed in the third decimal point. The first 2 values of 'Stronger Baseline' are reported as 0 because this baseline was not included in the results of Kaufman et al. (2019).Table A5. This table highlights the variables included as independent variables in Kaufman et al. (2019) which cause a data leakage. In the original use of the dataset, Fearon & LaitinScenario ADT RF SVM ERF Lasso LR Baseline Stronger Baseline Reported 0.990 0.989 0.983 0.990 0.862 0.987 0.861 0.000 Reported (reproduction) 0.990 0.990 0.983 0.989 0.861 0.987 0.861 0.000 Corrected 0.974 0.959 0.974 0.957 0.975 0.972 0.861 0.975 Corrected (out-of-sample) 0.966 0.936 0.962 0.927 0.966 0.963 0.796 0.966 Corrected (imputation) 0.974 0.959 0.974 0.957 0.975 0.975 0.861 0.975 Corrected (cutoff choice) 0.974 0.972 0.966 0.967 0.975 0.971 0.861 0.975 Variable name Reason for leakage Variable definition in data documentation pop affected by target variable population; in 1000s lpop affected by target variable log of population polity2 affected by target variable revised polity score gdpen affected by target variable gdp/pop based on pwt5.6; wdi2001;cow energy data onset codes civil war onset 1 for civil war onset ethonset codes civil war onset 1 if onset = 1 and ethwar ∼= 0 durest NA if onset = 0 estimated war duration aim NA if onset = 0 1 = rebels aim at center; 3 = aim at exit or autonomy; 2 = mixed or ambig. ended NA if onset = 0 war ends = 1; 0 = ongoing ethwar NA if onset = 0 0 = not ethnic; 1 = ambig/mixed; 2 = ethnic emponset codes civil war onset onset coded for data with empires sdwars codes ongoing civil war Number of Sambanis/Doyle civ wars in progress sdonset codes civil war onset onset of Sambanis/Doyle war colwars codes ongoing civil war Number of Collier/Hoeffler wars in progress colonset codes civil war onset onset of Collier/Hoeffler war cowwars codes ongoing civil war Number of COW civ wars in progress cowonset codes civil war onset onset of COW civ war Department of Computer Science and Center for Information Technology Policy, Princeton University. Correspondence to: Sayash Kapoor <[email protected]>. The model info sheet template is available on our website: https://reproducible.cs.princeton.edu Z = 0.64, 1.09, 0.42, 0.67; p = 0.26, 0.14, 0.34, 0.25 for a onetailed significance test comparing the smoothed AUC performance of the model proposed in the paper-the escalation model-with other baseline models reported in their paper-quad, goldstein, cameo and average respectively. We implement the comparison test for smoothed ROC curves detailed byRobin et al. (Robin et al., 2011). Note that we do not correct for multiple comparisons; such a correction would further reduce the significance of the results. While leakage is particularly serious in predictive modeling, a dataset with 95% of values missing is problematic even for explanatory modeling. 5 Hofman et al. (2021a) also outline the shortcomings in the initial code released by Muchlinski et al. (2016). Acknowledgements. We are grateful to Jessica Hullman, Matthew J. Salganik and Brandon Stewart for their valuable feedback on drafts of this paper. We thank Robert Blair, Aaron Kaufman, David Muchlinski and Yu Wang for quick and helpful responses to drafts of this paper. We are especially thankful to Matthew Sun, who reviewed our code and provided helpful suggestions and corrections for ensuring the computational reproducibility of our own results, and to Angelina Wang, Orestis Papakyriakopolous, and Anne Kohlbrenner for their feedback on model info sheets. Common pitfalls and recommendations for using machine learning to detect and prognosticate for COVID-19 using chest radiographs and CT scans. 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M J Salganik, I Lundberg, A T Kindel, C E Ahearn, K Al-Ghoneim, A Almaatouq, D M Altschul, J E Brand, N B Carnegie, R J Compton, D Datta, T Davidson, A Filippova, C Gilroy, B J Goode, E Jahani, R Kashyap, A Kirchner, S Mckay, A C Morgan, A Pentland, K Polimis, L Raes, D E Rigobon, C V Roberts, D M Stanescu, Y Suhara, A Usmani, E H Wang, M Adem, A Alhajri, B Alshebli, R Amin, R B Amos, L P Argyle, L Baer-Bositis, M Büchi, B.-R Chung, W Eggert, G Faletto, Z Fan, J Freese, T Gadgil, J Gagné, Y Gao, A Halpern-Manners, S P Hashim, S Hausen, G He, K Higuera, B Hogan, I M Horwitz, L M Hummel, N Jain, K Jin, D Jurgens, P Kaminski, A Karapetyan, E H Kim, B Leizman, N Liu, M Möser, A E Mack, M Mahajan, N Mandell, H Marahrens, D Mercado-Garcia, V Mocz, K Mueller-Gastell, A Musse, Q Niu, W Nowak, H Omidvar, A Or, K Ouyang, K M Pinto, E Porter, K E Porter, C Qian, T Rauf, A Sargsyan, T Schaffner, L Schnabel, B Schonfeld, B Sender, J D Tang, E Tsurkov, A V Loon, O Varol, X Wang, Z Wang, Salganik, M. J., Lundberg, I., Kindel, A. T., Ahearn, C. E., Al-Ghoneim, K., Almaatouq, A., Altschul, D. M., Brand, J. E., Carnegie, N. B., Compton, R. J., Datta, D., David- son, T., Filippova, A., Gilroy, C., Goode, B. J., Jahani, E., Kashyap, R., Kirchner, A., McKay, S., Morgan, A. C., Pentland, A., Polimis, K., Raes, L., Rigobon, D. E., Roberts, C. V., Stanescu, D. M., Suhara, Y., Usmani, A., Wang, E. H., Adem, M., Alhajri, A., AlShebli, B., Amin, R., Amos, R. B., Argyle, L. P., Baer-Bositis, L., Büchi, M., Chung, B.-R., Eggert, W., Faletto, G., Fan, Z., Freese, J., Gadgil, T., Gagné, J., Gao, Y., Halpern-Manners, A., Hashim, S. P., Hausen, S., He, G., Higuera, K., Hogan, B., Horwitz, I. M., Hummel, L. M., Jain, N., Jin, K., Jurgens, D., Kaminski, P., Karapetyan, A., Kim, E. H., Leizman, B., Liu, N., Möser, M., Mack, A. E., Mahajan, M., Man- dell, N., Marahrens, H., Mercado-Garcia, D., Mocz, V., Mueller-Gastell, K., Musse, A., Niu, Q., Nowak, W., Omidvar, H., Or, A., Ouyang, K., Pinto, K. M., Porter, E., Porter, K. E., Qian, C., Rauf, T., Sargsyan, A., Schaffner, T., Schnabel, L., Schonfeld, B., Sender, B., Tang, J. D., Tsurkov, E., Loon, A. v., Varol, O., Wang, X., Wang, Z., Measuring the predictability of life outcomes with a scientific mass collaboration. J Wang, F Wang, S Weissman, K Whitaker, M K Wolters, W L Woon, J Wu, C Wu, K Yang, J Yin, B Zhao, C Zhu, J Brooks-Gunn, B E Engelhardt, M Hardt, D Knox, K Levy, A Narayanan, B M Stewart, D J Watts, S Mclanahan, 1091-6490Publisher: National Academy of Sciences Section: Social Sciences. 117Wang, J., Wang, F., Weissman, S., Whitaker, K., Wolters, M. K., Woon, W. L., Wu, J., Wu, C., Yang, K., Yin, J., Zhao, B., Zhu, C., Brooks-Gunn, J., Engelhardt, B. E., Hardt, M., Knox, D., Levy, K., Narayanan, A., Stewart, B. M., Watts, D. J., and McLanahan, S. Measuring the predictability of life outcomes with a scientific mass col- laboration. Proceedings of the National Academy of Sci- ences, 117(15):8398-8403, April 2020. ISSN 0027-8424, 1091-6490. Publisher: National Academy of Sciences Section: Social Sciences. Multiple imputation: a primer. J L Schafer, 0962-2802Statistical Methods in Medical Research. 81SAGE Publications Ltd STMSchafer, J. L. Multiple imputation: a primer. Statistical Methods in Medical Research, 8(1):3-15, February 1999. ISSN 0962-2802. Publisher: SAGE Publications Ltd STM. Do Datasets Have Politics? Disciplinary Values in Computer Vision Dataset Development. M K Scheuerman, A Hanna, E Denton, 317:1-317:37Proceedings of the ACM on Human-Computer Interaction. 5CSCW2Scheuerman, M. K., Hanna, A., and Denton, E. Do Datasets Have Politics? Disciplinary Values in Computer Vi- sion Dataset Development. Proceedings of the ACM on Human-Computer Interaction, 5(CSCW2):317:1-317:37, October 2021. Supervised Machine Learning for Population Genetics: A New Paradigm. D R Schrider, A D Kern, 0168-9525Trends in Genetics. 344Schrider, D. R. and Kern, A. D. Supervised Machine Learn- ing for Population Genetics: A New Paradigm. Trends in Genetics, 34(4):301-312, April 2018. ISSN 0168-9525. Regions at Risk: Predicting Conflict Zones in African Insurgencies. S Schutte, 2049-8470Political Science Research and Methods. 53Cambridge University PressSchutte, S. Regions at Risk: Predicting Conflict Zones in African Insurgencies. Political Science Research and Methods, 5(3):447-465, July 2017. ISSN 2049-8470, 2049-8489. Publisher: Cambridge University Press. Nonreplicable publications are cited more than replicable ones. M Serra-Garcia, U Gneezy, 2375-2548Science Advances. 7211705Publisher: American Association for the Advancement of Science Section: Research ArticleSerra-Garcia, M. and Gneezy, U. Nonreplicable publications are cited more than replicable ones. Science Advances, 7 (21):eabd1705, May 2021. ISSN 2375-2548. Publisher: American Association for the Advancement of Science Section: Research Article. Block cross-validation for species distribution modelling. R Valavi, J Elith, J Lahoz-Monfort, G Guillera-Arroita, Valavi, R., Elith, J., Lahoz-Monfort, J., and Guillera-Arroita, G. Block cross-validation for species distribution mod- elling, 2021. Applications of machine learning in animal behaviour studies. J J Valletta, C Torney, M Kings, A Thornton, J Madden, 0003-3472Animal Behaviour. 124Valletta, J. J., Torney, C., Kings, M., Thornton, A., and Madden, J. Applications of machine learning in ani- mal behaviour studies. Animal Behaviour, 124:203-220, February 2017. ISSN 0003-3472. Overly optimistic prediction results on imbalanced data: a case study of flaws and benefits when applying over-sampling. G Vandewiele, I Dehaene, G Kovács, L Sterckx, O Janssens, F Ongenae, F De Backere, F De Turck, K Roelens, J Decruyenaere, S Van Hoecke, T Demeester, 0933-3657Artificial Intelligence in Medicine. 111101987Vandewiele, G., Dehaene, I., Kovács, G., Sterckx, L., Janssens, O., Ongenae, F., De Backere, F., De Turck, F., Roelens, K., Decruyenaere, J., Van Hoecke, S., and Demeester, T. Overly optimistic prediction results on im- balanced data: a case study of flaws and benefits when ap- plying over-sampling. Artificial Intelligence in Medicine, 111:101987, January 2021. ISSN 0933-3657. Comparing Random Forest with Logistic Regression for Predicting Class-Imbalanced Civil War Onset Data: A Comment. Y Wang, 1047-1987Political Analysis. 271Cambridge University PressWang, Y. Comparing Random Forest with Logistic Regres- sion for Predicting Class-Imbalanced Civil War Onset Data: A Comment. Political Analysis, 27(1):107-110, January 2019. ISSN 1047-1987, 1476-4989. Publisher: Cambridge University Press. When Optimism Hurts: Inflated Predictions in Psychiatric Neuroimaging. R Whelan, H Garavan, 0006-3223Biological Psychiatry. 759Whelan, R. and Garavan, H. When Optimism Hurts: Inflated Predictions in Psychiatric Neuroimaging. Biological Psy- chiatry, 75(9):746-748, May 2014. ISSN 0006-3223. Choosing Prediction Over Explanation in Psychology: Lessons From Machine Learning. T Yarkoni, J Westfall, 1745-6916Perspectives on Psychological Science. 126SAGE Publications IncYarkoni, T. and Westfall, J. Choosing Prediction Over Expla- nation in Psychology: Lessons From Machine Learning. Perspectives on Psychological Science, 12(6):1100-1122, November 2017. ISSN 1745-6916. Publisher: SAGE Publications Inc. Variable generalization performance of a deep learning model to detect pneumonia in chest radiographs: A cross-sectional study. J R Zech, M A Badgeley, M Liu, A B Costa, J J Titano, E K Oermann, 1549- 1676PLOS Medicine. 15111002683Publisher: Public Library of ScienceZech, J. R., Badgeley, M. A., Liu, M., Costa, A. B., Ti- tano, J. J., and Oermann, E. K. Variable generalization performance of a deep learning model to detect pneumo- nia in chest radiographs: A cross-sectional study. PLOS Medicine, 15(11):e1002683, November 2018. ISSN 1549- 1676. Publisher: Public Library of Science. Here, we detail how model info sheets would address each type of leakage that we found in our survey, as well as the types of leakage we found in our case study of civil war prediction. C , Model info sheets for detecting and preventing leakage in ML-based science We include the model info sheet template as a Microsoft Word document on our websiteC. Model info sheets for detecting and preventing leakage in ML-based science We include the model info sheet template as a Microsoft Word document on our website (https://reproducible. cs.princeton.edu). Here, we detail how model info sheets would address each type of leakage that we found in our survey, as well as the types of leakage we found in our case study of civil war prediction. Model info sheets require an explanation of how the train and test set is split during all steps in the modeling process (Q9-17 of model info sheets). • L1, 1 No test set• L1.1 No test set. Model info sheets require an explanation of how the train and test set is split during all steps in the modeling process (Q9-17 of model info sheets). Details of how the train and test set are separated during the preprocessing selection step need to be included in the model info sheet (Q12-13). This would address leakage due to incorrect imputation Muchlinski. • L1, 2 Pre-processing on training and test set• L1.2 Pre-processing on training and test set. Details of how the train and test set are separated during the preprocess- ing selection step need to be included in the model info sheet (Q12-13). This would address leakage due to incorrect imputation Muchlinski et al. (2016); . Wang, Wang (2019); . & Colaresi, Mahmood, Colaresi & Mahmood (2017). 3 Feature selection on training and test set. Details of how the train and test set are separated during the feature selection step need to be included in the model info sheet. • L1, • L1.3 Feature selection on training and test set. Details of how the train and test set are separated during the feature selection step need to be included in the model info sheet (Q14-15). Model info sheets require details of whether there are duplicates in the dataset, and if so. • L1, 4 Duplicates in datasets. how they are handled (Q10)• L1.4 Duplicates in datasets. Model info sheets require details of whether there are duplicates in the dataset, and if so, how they are handled (Q10). • L2 Model uses features that are not legitimate. For each feature used in the model, researchers need to argue why the feature is legitimate to be used for the modeling task at hand (Q21). Kaufman et al.This addresses the leakage due to the use of proxy variables in• L2 Model uses features that are not legitimate. For each feature used in the model, researchers need to argue why the feature is legitimate to be used for the modeling task at hand (Q21). This addresses the leakage due to the use of proxy variables in Kaufman et al. (2019). In case the claim is about predicting future outcomes of interest based on ML methods, researchers need to provide an explanation for why the time windows used in the training and test set are separate, and why data in the test set is always a later timestamp compared to the data in the training set (Q20). • L3, Kaufman et al.This addresses the temporal leakage in• L3.1 Temporal leakage. In case the claim is about predicting future outcomes of interest based on ML methods, researchers need to provide an explanation for why the time windows used in the training and test set are separate, and why data in the test set is always a later timestamp compared to the data in the training set (Q20). This addresses the temporal leakage in Kaufman et al. (2019); . Wang, Wang (2019). 2 Dependencies in training and test data. Researchers need to reason about the dependencies that may exist in their dataset and outline how dependencies across training and test sets are addressed. • L3, • L3.2 Dependencies in training and test data. Researchers need to reason about the dependencies that may exist in their dataset and outline how dependencies across training and test sets are addressed (Q11). 3 Sampling bias in test distribution. Researchers need to reason about the presence of selection bias in their dataset and outline how the rows included for data analysis were selected, and how the test set matches the distribution about which the scientific claims are made. • L3, • L3.3 Sampling bias in test distribution. Researchers need to reason about the presence of selection bias in their dataset and outline how the rows included for data analysis were selected, and how the test set matches the distribution about which the scientific claims are made (Q18-19).	[]
[ "Early stages of magnetization relaxation in superconductors", "Early stages of magnetization relaxation in superconductors" ]	[ "Mihajlo Vanević \nDepartment of Physics\nUniversity of Belgrade\nStudentski trg 1211158BelgradeSerbia\n", "Zoran Radović \nDepartment of Physics\nUniversity of Belgrade\nStudentski trg 1211158BelgradeSerbia\n", "Vladimir G Kogan \nAmes Laboratory DOE\n50011AmesIowaUSA\n" ]	[ "Department of Physics\nUniversity of Belgrade\nStudentski trg 1211158BelgradeSerbia", "Department of Physics\nUniversity of Belgrade\nStudentski trg 1211158BelgradeSerbia", "Ames Laboratory DOE\n50011AmesIowaUSA" ]	[]	Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by m(t) ∝ 1 − t/τ in the narrow initial time interval and by m(t) ∝ (1 + t/τ ) −1 at later times before the flux creep sets in. The characteristic timesτ and τ are proportional to the viscous drag coefficient η. Quantitative agreement with available experimental data is obtained for both conventional and hightemperature superconductors with η exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicates a strong influence of pinning centers on the flux diffusion. Notwithstanding the complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient. PACS numbers: 74.25.-q, 74.25.WxMagnetic flux penetrates a type-II superconductor in the form of discrete quantized vortices. Vortex structures in conventional and high-temperature superconductors display remarkable complexity both in equilibrium 1,2 and dynamic regimes. 3-12 Relaxation of the magnetic moment of superconductors is achieved through initial viscous flux flow 13-18 and slow, logarithmic in time, thermally activated creep. 19-23 Thermally-assisted hopping of vortices and vortex bundles between local minima in the random pinning potential is characteristic of both the creep and the flux flow under a driving force. In the latter, the hopping gives rise to the viscous drag coefficient η ∝ e U/kT , where U is the effective activation energy and T is the temperature. 21 A free flux flow regime can be realized at microwave frequencies (10 − 100GHz) when the effect of the pinning is negligible. Measurements of surface impedance give viscous drag coefficients η 0 ∼ 10 −6 −10 −7 Ns/m 2 at low temperatures for all superconductors, e.g., conventional NbSe 2 , 24 cuprates YBCO, BSCO, 25,26 and pnictide LiFeAs. 10 The order of magnitude is in accordance with the Bardeen-Stephen result for the viscous drag, η 0 = Φ 0 H c2 /ρ n c 2 , caused by dissipation in the vortex core (Φ 0 = hc/2e is the flux quantum, ρ n is the normal-state resistivity, and H c2 is the upper critical field). 4In this paper we study early stages of the flux dynamics after switching off the external magnetic field. We use a simple hydrodynamic approach: The local force the vortex experiences due to interaction with other vortices, the surface, and the local quenched disorder (pinning centers) is described by an effective viscosity η ≫ η 0 .The same approach successfully describes the vortex creep, if supplemented by a phenomenological model of current-dependent or time-dependent activation energy, U = U c ln(j c /j) or U = kT ln(t/t 0 ), where j c is the critical current and t 0 is the characteristic time scale for flux creep.[19][20][21][22][23]We consider a model of massless vortex motion where the driving Lorentz force equals the viscous drag (1/c)J× Φ 0 − ηv = 0. Here, J is the current density, v is the vortex velocity, and η is a viscous drag coefficient. For magnetic induction B = nΦ 0 related to the vortex density n, the force balance equation reads (1/c)J×B−η\|B\|v/Φ 0 = 0, with J = (c/4π)∇ × B. Taking into account the continuity equation ∂B/∂t + ∇ · (Bv) = 0, the dynamics of the magnetic flux in a superconductor is described by the well-known nonlinear diffusion equation 20-22We have solved Eq. (1) for three sample geometries: a slab, a square-shaped plate, and a disk (seeFig. 1). We assume the sample thickness along the field is sufficiently large and neglect stray fields on the top and bottom of the sample. Magnetic induction B(r, t) is directed along the sample symmetry axis z and satisfies the following initial and boundary conditions: (i) B is uniform within the sample at t = 0, B(x, y; t = 0) = B 0 , and (ii) B vanishes at the sample edges for t > 0.For a long superconducting slab of width L, Eq.(1)FIG. 1. Vortex dynamics is studied for three sample geometries: (a) a slab, (b) a square-shaped plate, and (c) a disk.	10.1103/physrevb.87.144501	[ "https://arxiv.org/pdf/1302.4312v2.pdf" ]	118,347,027	1302.4312	3d7f26c1f5efd3e89498cb89c11889f6b3d9f77d	Early stages of magnetization relaxation in superconductors 29 Mar 2013 Mihajlo Vanević Department of Physics University of Belgrade Studentski trg 1211158BelgradeSerbia Zoran Radović Department of Physics University of Belgrade Studentski trg 1211158BelgradeSerbia Vladimir G Kogan Ames Laboratory DOE 50011AmesIowaUSA Early stages of magnetization relaxation in superconductors 29 Mar 2013(Dated: April 1, 2013)arXiv:1302.4312v2 [cond-mat.supr-con] Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by m(t) ∝ 1 − t/τ in the narrow initial time interval and by m(t) ∝ (1 + t/τ ) −1 at later times before the flux creep sets in. The characteristic timesτ and τ are proportional to the viscous drag coefficient η. Quantitative agreement with available experimental data is obtained for both conventional and hightemperature superconductors with η exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicates a strong influence of pinning centers on the flux diffusion. Notwithstanding the complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient. PACS numbers: 74.25.-q, 74.25.WxMagnetic flux penetrates a type-II superconductor in the form of discrete quantized vortices. Vortex structures in conventional and high-temperature superconductors display remarkable complexity both in equilibrium 1,2 and dynamic regimes. 3-12 Relaxation of the magnetic moment of superconductors is achieved through initial viscous flux flow 13-18 and slow, logarithmic in time, thermally activated creep. 19-23 Thermally-assisted hopping of vortices and vortex bundles between local minima in the random pinning potential is characteristic of both the creep and the flux flow under a driving force. In the latter, the hopping gives rise to the viscous drag coefficient η ∝ e U/kT , where U is the effective activation energy and T is the temperature. 21 A free flux flow regime can be realized at microwave frequencies (10 − 100GHz) when the effect of the pinning is negligible. Measurements of surface impedance give viscous drag coefficients η 0 ∼ 10 −6 −10 −7 Ns/m 2 at low temperatures for all superconductors, e.g., conventional NbSe 2 , 24 cuprates YBCO, BSCO, 25,26 and pnictide LiFeAs. 10 The order of magnitude is in accordance with the Bardeen-Stephen result for the viscous drag, η 0 = Φ 0 H c2 /ρ n c 2 , caused by dissipation in the vortex core (Φ 0 = hc/2e is the flux quantum, ρ n is the normal-state resistivity, and H c2 is the upper critical field). 4In this paper we study early stages of the flux dynamics after switching off the external magnetic field. We use a simple hydrodynamic approach: The local force the vortex experiences due to interaction with other vortices, the surface, and the local quenched disorder (pinning centers) is described by an effective viscosity η ≫ η 0 .The same approach successfully describes the vortex creep, if supplemented by a phenomenological model of current-dependent or time-dependent activation energy, U = U c ln(j c /j) or U = kT ln(t/t 0 ), where j c is the critical current and t 0 is the characteristic time scale for flux creep.[19][20][21][22][23]We consider a model of massless vortex motion where the driving Lorentz force equals the viscous drag (1/c)J× Φ 0 − ηv = 0. Here, J is the current density, v is the vortex velocity, and η is a viscous drag coefficient. For magnetic induction B = nΦ 0 related to the vortex density n, the force balance equation reads (1/c)J×B−η\|B\|v/Φ 0 = 0, with J = (c/4π)∇ × B. Taking into account the continuity equation ∂B/∂t + ∇ · (Bv) = 0, the dynamics of the magnetic flux in a superconductor is described by the well-known nonlinear diffusion equation 20-22We have solved Eq. (1) for three sample geometries: a slab, a square-shaped plate, and a disk (seeFig. 1). We assume the sample thickness along the field is sufficiently large and neglect stray fields on the top and bottom of the sample. Magnetic induction B(r, t) is directed along the sample symmetry axis z and satisfies the following initial and boundary conditions: (i) B is uniform within the sample at t = 0, B(x, y; t = 0) = B 0 , and (ii) B vanishes at the sample edges for t > 0.For a long superconducting slab of width L, Eq.(1)FIG. 1. Vortex dynamics is studied for three sample geometries: (a) a slab, (b) a square-shaped plate, and (c) a disk. Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by m(t) ∝ 1 − t/τ in the narrow initial time interval and by m(t) ∝ (1 + t/τ ) −1 at later times before the flux creep sets in. The characteristic timesτ and τ are proportional to the viscous drag coefficient η. Quantitative agreement with available experimental data is obtained for both conventional and hightemperature superconductors with η exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicates a strong influence of pinning centers on the flux diffusion. Notwithstanding the complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient. Magnetic flux penetrates a type-II superconductor in the form of discrete quantized vortices. Vortex structures in conventional and high-temperature superconductors display remarkable complexity both in equilibrium 1,2 and dynamic regimes. [3][4][5][6][7][8][9][10][11][12] Relaxation of the magnetic moment of superconductors is achieved through initial viscous flux flow [13][14][15][16][17][18] and slow, logarithmic in time, thermally activated creep. [19][20][21][22][23] Thermally-assisted hopping of vortices and vortex bundles between local minima in the random pinning potential is characteristic of both the creep and the flux flow under a driving force. In the latter, the hopping gives rise to the viscous drag coefficient η ∝ e U/kT , where U is the effective activation energy and T is the temperature. 21 A free flux flow regime can be realized at microwave frequencies (10 − 100GHz) when the effect of the pinning is negligible. Measurements of surface impedance give viscous drag coefficients η 0 ∼ 10 −6 −10 −7 Ns/m 2 at low temperatures for all superconductors, e.g., conventional NbSe 2 , 24 cuprates YBCO, BSCO, 25,26 and pnictide LiFeAs. 10 The order of magnitude is in accordance with the Bardeen-Stephen result for the viscous drag, η 0 = Φ 0 H c2 /ρ n c 2 , caused by dissipation in the vortex core (Φ 0 = hc/2e is the flux quantum, ρ n is the normal-state resistivity, and H c2 is the upper critical field). 4 In this paper we study early stages of the flux dynamics after switching off the external magnetic field. We use a simple hydrodynamic approach: The local force the vortex experiences due to interaction with other vortices, the surface, and the local quenched disorder (pinning centers) is described by an effective viscosity η ≫ η 0 .The same approach successfully describes the vortex creep, if supplemented by a phenomenological model of current-dependent or time-dependent activation energy, U = U c ln(j c /j) or U = kT ln(t/t 0 ), where j c is the critical current and t 0 is the characteristic time scale for flux creep. [19][20][21][22][23] We consider a model of massless vortex motion where the driving Lorentz force equals the viscous drag (1/c)J× Φ 0 − ηv = 0. Here, J is the current density, v is the vortex velocity, and η is a viscous drag coefficient. For magnetic induction B = nΦ 0 related to the vortex density n, the force balance equation reads (1/c)J×B−η\|B\|v/Φ 0 = 0, with J = (c/4π)∇ × B. Taking into account the continuity equation ∂B/∂t + ∇ · (Bv) = 0, the dynamics of the magnetic flux in a superconductor is described by the well-known nonlinear diffusion equation [20][21][22] ∂B ∂t = Φ 0 4πη ∇ · (\|B\| ∇B).(1) We have solved Eq. (1) for three sample geometries: a slab, a square-shaped plate, and a disk (see Fig. 1). We assume the sample thickness along the field is sufficiently large and neglect stray fields on the top and bottom of the sample. Magnetic induction B(r, t) is directed along the sample symmetry axis z and satisfies the following initial and boundary conditions: (i) B is uniform within the sample at t = 0, B(x, y; t = 0) = B 0 , and (ii) B vanishes at the sample edges for t > 0. reads ∂B ∂t = Φ 0 4πη ∂ ∂x \|B\| ∂B ∂x ,(2) where B(x, t) = 0 at x = ±L/2 for t > 0. We can seek the solution in the form B(x, t) = ∞ k=1 B k (t) sin[kπ(x/L + 1/2)],(3) with functions B k (t) to be determined from Eq. (2) and B(x, t = 0) = B 0 . This gives the following set of differential equations: dB k (t) dt = 1 B 0 τ 0 ∞ i,j=1 B i (t) F k (i, j) B j (t),(4) with the initial conditions B k (0 ) = (2B 0 /πk) [1 − (−1) k ] (k = 1, 2, . . .). Here, the coefficients F k (i, j) are given by F k (i, j) = kπ 4 (i − j) 2 (i − j) 2 − k 2 − (i + j) 2 (i + j) 2 − k 2(5) for \|i±j\| = k and i+j+k odd, and F k (i, j) = 0 otherwise. The characteristic time constant is τ 0 = πL 2 η Φ 0 B 0 .(6) Equations (4) are solved by truncating the system at sufficiently large k (k ∼ 40). The induction B(x, t) for the slab is shown in Fig. 2 at various times t/τ 0 = 0.01, 0.1, 0.2, ..., 3.2. We observe that the flux flow near the sample edges in the initial time interval is very fast, reaching the center of the slab (x = 0) at time t ∼ 0.1 τ 0 after switching off the field. This regime is followed by a slower flux flow taking place in the bulk of the sample. The spatial dependence of the magnetic induction is in accordance with the previous results for the flux flow regime with constant activation energy. 22 In the presence of flux creep, which may take place in the center of the slab for t ≪ 0.1τ 0 , or at large times t ≫ τ 0 when remanent magnetization is small, a phenomenological model of current and field-dependent activation energy should be used. [19][20][21][22][23] Note that the obtained B(x, t) shown in Fig. 2 is qualitatively different from the solution of Eq. (2) when the field is switched on at t = 0. In that case the magnetic field enters the sample in the form of a flux front propagating from the edges. 20,22,27 Magnetic induction in the vicinity of the front is a linear function of the coordinate, B(x, t) = (4πη/Φ 0 ) v f \|x−x f \|, with x f (t) and v f being the position and the velocity of the front. In our case, the field is switched off at t = 0 and the flux escapes the sample with no front in B(x, t) formed even at t ≪ τ 0 . Indeed, at a sufficiently large distance u from the edge, Eq. (2) can be linearized with respect to δB = B 0 − B, which gives the exponential de- cay δB(u, t) ∝ (u/2κ √ t) −1 e −u 2 /4κ 2 t (κ = Φ 0 B 0 /4πη) characteristic of the linear diffusion. In the following we study the dynamics of the average magnetic inductionB(t) = A −1 dxdy B(x, y, t) (A is the sample area) which is proportional to the magnetic moment m(t) that can be measured. There are two regimes of the flux dynamics in the system. At very short times t ≪ τ 0 after switching off the field, the flux flow is localized near the edges and is unaffected by the sample size. In this case, the solution for a half-infinite superconductor is a good approximation, B(u, t) = B 0 f (u/κ √ t) . 28 Here, f is a dimensionless function to be determined from Eq. (2) for the half-infinite superconductor with the boundary conditions f (0) = 0 and f (∞) = 1. Using the above expression for B(u, t) and taking into account that it deviates significantly from B 0 in the vicinity of the edges, we find for the average induc- of the sample. This gives the magnetic moment relaxation m(t) = m 0 1 − t/τ , t ≪τ ,(7) with the time constant τ = 9π(A/P ) 2 η/Φ 0 B 0 ,(8) where the numerical prefactor characterizes the spatial spread of B away from the edges. Comparison with the numerical solution for m(t) is shown in the inset of Fig. 3 for different sample geometries. We find that Eq. (7) is a good approximation of the exact m(t) in the short initial time interval t/τ 0.1 before the flux flow reaches the center of the sample. The flux flow in this time interval is very fast, leading to a 30% reduction of the overall magnetic moment. At times t τ 0 the flux flow extends through the whole sample, giving rise to the magnetization relaxation which depends on geometry. For the superconducting slab, the first-order approximation of Eqs. (4) for k = 1 readsB (1) (t) = (8B 0 /π 2 )[1 + t/(0.75τ 0 )] −1 . Truncating Eqs. (4) at k ∼ 40, a practically exact solution is obtained. This solution can be approximated by a simple formula,B(t) = B 0 [1 + t/(0.62τ 0 )] −1 , which is very close to the exact one for t τ 0 . This suggests that the exact solution for the magnetic moment m(t) can be accurately approximated by m(t) = m 0 1 + t/τ , τ = ατ 0 ,(9) where α is a number which depends on geometry. Fitting the exact numerical solution for m(t) to Eq. (9) we find α = 0.620, 0.244, and 0.226 for the slab, square, and disk geometries, respectively (Fig. 3). The fitting ensures the smallest absolute error between exact and fitted m(t) for 1 < t/τ < 3. As expected, the decay of m(t) is slower (that is, geometric factor α is larger) for the slab than for the disk, other parameters being equal. In what follows, we analyze available experimental data on m(t) and extract the characteristic time constant as well as the effective drag η. Relaxation of the magnetic moment in BSCO single crystals is studied in Ref. 14. Experimental data are shown in Fig. 4 (open circles) fitted to Eq. (9) (solid curve) with m 0 = 1.1 × 10 −5 Am 2 and τ = 0.43 min. The fitting is performed for the initial time interval before logarithmic in time, thermally activated flux creep sets in. The linear time dependence of the inverse magnetic moment is shown in the inset of Fig. 4; the crossover between flux flow and flux creep regimes is seen as a dramatic change of the slope at t/τ ≈ 16. Let us now extract η. The dimensions of the sample used in the experiment are 0.14 × 1.37 × 2.06 mm, which gives B 0 = 4πm 0 /V = 35 mT, where V is the volume. Taking α = 0.620 for the slab of the width L = 0.14 mm, we obtain η = 0.5 Ns/m 2 . This value for the effective vortex viscosity exceeds by six orders of magnitude the Bardeen-Stephen drag coefficient η 0 ∼ 10 −7 Ns/m 2 measured in BSCO. 26 Huge enhancement of the drag indicates a strong influence of the pinning on the vortex diffusion. Despite the complexity of the vortex motion in the presence of pinning and thermal agitation, the magnetization follows a simple algebraic time dependence, Eq. (9). Vortex dynamics has been studied in NbSe 2 using the decoration technique for visualization of flowing vortex lattices. 15 Magnetization measurements have been per- formed using the SQUID (superconducting quantum interference device) magnetometry. A crossover has been observed as a function of increasing flux density from a layered (smectic) flowing flux lattice in the disorderdominated low-field limit to a more ordered (Bragg glass) lattice structure in the interaction-dominated high-field case. The observed time dependence of magnetization relaxation in the high-field limit (B 0 = 3.3 mT) is shown in Fig. 5. The regimes indicated in Fig. 5 correspond to the flux flow and to the quasistatic vortex motion. The solid curve in Fig. 5 is the fit of m(t) to Eq. (7) for the NbSe 2 sample 0.5 × 0.5 × 0.2 mm in size, which gives the relaxation timeτ = 1.13 × 10 3 min and the viscous drag coefficient η = 11 Ns/m 2 . We observe that the simple hydrodynamic model with an effective viscous drag force fits the data in the initial stages of magnetization relaxation where the vortex density is large and the flux flow takes place. The flux flow is localized near the edges, as corroborated experimentally by a small reduction of the magnetic moment of the sample over the measurement time and, more directly, by observing the static vortex structure in the center of the sample. 15 Large effective η is clearly due to hopping caused by successive pinning and thermally-assisted depinning of vortices, as evidenced by studying single-vortex dynamics in pristine NbSe 2 monocrystals by scanning tunneling microscopy. 16 Magnetic moment relaxations in YBCO polycrystal 17 and monocrystal 18 ation can be fitted by Eq. (7) describing the flux flow in the vicinity of the edges (Fig. 6, solid curves). This is in agreement with the observed small reduction of the overall magnetic moment during the measurement. The obtained effective viscosity strongly depends on temperature, ranging between η ∼ 100 and 0.1 Ns/m 2 as the temperature is increased from 30K to 77K, see inset of Fig. 6. The extracted value η(77K) = 0.19 Ns/m 2 is consistent with the value η = 0.12 Ns/m 2 measured independently at the same temperature by studying the spatiotemporal change of the magnetization profile in a bulk YBCO sample in the flux-flow regime. 29 Taking η ∝ e U/kT and neglecting the temperature dependence of the effective activation energy U as well as of the prefactor, we find U ≈ 360 K in accordance with the previous results. 21 Magnetic moment relaxation in small YBCO monocrystal (1 × 1 × 0.02 mm) is shown in Fig. 7. 18 The data can be fitted with the effective viscous drag coefficients η = 0.27 and 0.04 Ns/m 2 at temperatures of 85K and 87K, respectively. The decrease of η in such a narrow temperature range may be due to the proximity of the critical temperature (T c ≈ 88K) where fluctuations are pronounced. In addition, the sharp change in the relaxation rate observed at 87K and t ≈ 30 min suggests that the flux flow is inhomogeneous and made of large domains which, upon depinning, abruptly increase the magnetic moment relaxation rate. In conclusion, we have studied vortex dynamics in type-II superconductors in the initial time interval before the flux creep sets in. We have used a simple phenomenological (hydrodynamic) model of nonlinear diffusion of massless vortices where pinning of the flux lines by material inhomogeneities, interaction with other vortices and the surface, and the Bardeen-Stephen dissipation in the vortex core are described by an effective viscous drag coefficient, η. After switching off the external magnetic field, the vortex dynamics exhibits two distinct regimes before the creep sets in with logarithmic in time decay of remanent magnetization. In the beginning, the flux flow is localized near the edges and is independent of the sample size. At later times, this regime is followed by a slower flux flow involving the bulk of the sample. We find that magnetic moment relaxation in these regimes can be accurately approximated by m(t) = m 0 (1− t/τ ) for t ≪τ and m(t) = m 0 (1 + t/τ ) −1 for t τ , where geometry-dependentτ and τ are proportional to η. We have analyzed available experimental data on early stages of magnetization relaxation after the magnetic field is instantaneously removed. We obtained quantitative agreement for both conventional and hightemperature superconductors, albeit with η exceeding the Bardeen-Stephen value η 0 by many orders of magni-tude. Huge enhancement of η with respect to η 0 , as well as its exponential temperature dependence, indicates a strong influence of pinning and thermally assisted depinning of vortices on flux diffusion. We argue that early stages of magnetization relaxation can be modeled as a flux flow with an effective drag coefficient. This allows for a simple experimental determination of the bulk vortex viscosity, which cannot be accessed by the surface impedance measurements. This research was supported by the Serbian Ministry of Science, Project No. 171027. Work by V.K. at the Ames Laboratory is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-07CH11358. M.V. acknowledges support by the DFG through SFB 767 and the hospitality of the Quantum Transport Group, Universität Konstanz, Germany, where part of this work was done. PACS numbers: 74.25.-q, 74.25.Wx FIG. 2 . 2For a long superconducting slab of width L, Eq. (1) FIG. 1. Vortex dynamics is studied for three sample geometries: (a) a slab, (b) a square-shaped plate, and (c) a disk. Magnetic induction B(x) across one half of a superconducting slab of width L is shown for times t/τ0 = 0.01, 0.1, 0.2,. . . , 3.2 (top to bottom). FIG. 3 . 3Numerical solutions for magnetic moment relaxation for the slab (×), disk (•), and square ( ) geometries, compared to the analytic approximation, Eq. (9) (solid line). Inset: A better fit for the initial time interval t ≪ τ to the analytic expression, Eq.(7). In this case, only magnetization at the edges is affected by the flux flow. FIG. 4 . 4tion 1 −B(t)/B 0 ∝ (P/A)κ √ t,where P is the perimeter Experimental data 14 (•) for the magnetic moment m(t) in BSCO single crystal fitted to Eq. (9) with m0 = 1.1 × 10 −5 Am 2 and τ = 0.43 min (solid curve). This corresponds to η = 0.5 Ns/m 2 . Inset: The inverse magnetic moment as a function of time. The crossover between flux flow and flux creep regimes is seen as a dramatic change of the slope at t ≈ 7 min (dashed line). The sample is a slab 0.14 × 1.37 × 2.06 mm in size, the initial magnetic induction B0 = 35 mT, and T = 77K. FIG. 5 . 5Experimental data 15 (•) for magnetic moment relaxation in NbSe2 monocrystal fitted to Eq. (7) (solid curve) withτ = 1.13 × 10 3 min, corresponding to η = 11 Ns/m 2 . The dashed line indicates a crossover between the flux flow regime, Eq. (1), and the slow quasistatic motion before the flux creep. The sample has a square geometry 0.5 × 0.5 × 0.2 mm, the initial magnetic induction B0 = 3.3 mT, and T = 4.2K. FIG. 6 . 6Experimental data 17 (•) for magnetic moment relaxation in YBCO polycrystal at temperatures T = 30, 41, 50, 61, and 77 K (top to bottom), fitted to Eq. (7) (solid curves) withτ = 2.1×10 5 , 5.0×10 4 , 1.4×10 4 , 3.0×10 3 , and 7.4×10 2 min, respectively. This corresponds to the drag coefficients η = 284, 66, 17, 2.93, and 0.19 Ns/m 2 . Inset: Logarithm of the drag coefficient, normalized to η(77K) = 0.19 Ns/m 2 , as a function of the temperature. The sample has a rectangular geometry of 66 × 34 × 15 mm. The initial magnetic induction is B0 = 3.95, 3.77, 3.48, 2.80, 0.735 T, respectively. FIG. 7 . 7are shown in Figs. 6 and 7. 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[ "ARPD: Anchor-free Rotation-aware People Detection using Topview Fisheye Camera", "ARPD: Anchor-free Rotation-aware People Detection using Topview Fisheye Camera" ]	[ "Quan Nguyen \nHanoi University of Science and Technology\n\n\nViettel High Technology Industries Corporation\n\n", "Minh ", "Bang Le Van \nViettel High Technology Industries Corporation\n\n", "Can Nguyen [email protected] ", "Anh Le \nFPT University\n\n", "Viet Dung Nguyen [email protected] \nHanoi University of Science and Technology\n\n\nViettel High Technology Industries Corporation\n\n" ]	[ "Hanoi University of Science and Technology\n", "Viettel High Technology Industries Corporation\n", "Viettel High Technology Industries Corporation\n", "FPT University\n", "Hanoi University of Science and Technology\n", "Viettel High Technology Industries Corporation\n" ]	[]	People detection in top-view, fish-eye images is challenging as people in fish-eye images often appear in arbitrary directions and are distorted differently. Due to this unique radial geometry, axis-aligned people detectors often work poorly on fish-eye frames. Recent works account for this variability by modifying existing anchorbased detectors or relying on complex pre/post-processing. Anchor-based methods spread a set of pre-defined bounding boxes on the input image, most of which are invalid. In addition to being inefficient, this approach could lead to a significant imbalance between the positive and negative anchor boxes. In this work, we propose ARPD, a singlestage anchor-free fully convolutional network to detect arbitrarily rotated people in fish-eye images. Our network uses keypoint estimation to find the center point of each object and regress the object's other properties directly. To capture the various orientation of people in fish-eye cameras, in addition to the center and size, ARPD also predicts the angle of each bounding box. We also propose a periodic loss function that accounts for angle periodicity and relieves the difficulty of learning small-angle oscillations. Experimental results show that our method competes favorably with state-of-the-art algorithms while running significantly faster.	10.1109/avss52988.2021.9663768	[ "https://arxiv.org/pdf/2201.10107v1.pdf" ]	245,708,559	2201.10107	4681fc9a8808c1ae7af47df89aa37ab0aa4e7d55	ARPD: Anchor-free Rotation-aware People Detection using Topview Fisheye Camera Quan Nguyen Hanoi University of Science and Technology Viettel High Technology Industries Corporation Minh Bang Le Van Viettel High Technology Industries Corporation Can Nguyen [email protected] Anh Le FPT University Viet Dung Nguyen [email protected] Hanoi University of Science and Technology Viettel High Technology Industries Corporation ARPD: Anchor-free Rotation-aware People Detection using Topview Fisheye Camera People detection in top-view, fish-eye images is challenging as people in fish-eye images often appear in arbitrary directions and are distorted differently. Due to this unique radial geometry, axis-aligned people detectors often work poorly on fish-eye frames. Recent works account for this variability by modifying existing anchorbased detectors or relying on complex pre/post-processing. Anchor-based methods spread a set of pre-defined bounding boxes on the input image, most of which are invalid. In addition to being inefficient, this approach could lead to a significant imbalance between the positive and negative anchor boxes. In this work, we propose ARPD, a singlestage anchor-free fully convolutional network to detect arbitrarily rotated people in fish-eye images. Our network uses keypoint estimation to find the center point of each object and regress the object's other properties directly. To capture the various orientation of people in fish-eye cameras, in addition to the center and size, ARPD also predicts the angle of each bounding box. We also propose a periodic loss function that accounts for angle periodicity and relieves the difficulty of learning small-angle oscillations. Experimental results show that our method competes favorably with state-of-the-art algorithms while running significantly faster. I. . Introduction Omni-directional cameras, most notably fish-eye cameras, are widely used for surveillance applications. One top-view fish-eye camera covers the same area as many conventional perspective cameras due to their 360 • field of view. Another advantage of a ceiling-mounted fish-eye camera is minimal occlusion between objects in a frame. However, images from overhead-mounted fish-eye cameras may have human bodies at various orientations and poses. Another difficulty is severe geometric distortions at the peripheral areas of a fish-eye image, which can cause an object's appearance to varying. Pedestrian detectors trained with perspective images [1], [2], [3] cannot accommodate these variations and deformations, hence do not work well on top-view fish-eye images, often missing non-up-right bodies. (Fig. 1a) Various methods have been proposed to tackle these problems. In early works, features are extracted using rudimentary methods such as background subtraction or edge detection [4], [5]. These features have poor discrimination power and are sensitive to environmental changes Fig. 2. Difference between anchor-based detectors (a) and our center point detector (b). Best viewed in color. As horizontal proposals are located along the image edge, the extracted feature of an object may contain features of background and nearby objects. and noise. Another approach utilizes traditional person detectors such as HOG and LBP, making slight adjustments to accommodate fish-eye geometry [6], [7], [8]. Recently, CNN-based people detection methods have been proposed for over-head, fish-eye images [9], [10], [11]. To deal with the radial geometry, these works often require complex pre-/post processing. For example, Li et al. [9] applied YOLOv3 [1] to 24 rotated, overlapping windows, and the results are re-mapped to the original image. However, this requires YOLOv3 to run 24 times for each frame. Tamura et al. [10] tries to train a rotation-invariant model by introducing rotation augmentation during training. RAPiD [11] modifies YOLOv3 to detect people in fish-eye images using oriented bounding boxes. However, applying horizontal-anchor-based detectors such as YOLOv3 or Faster-RCNN [3] to the oriented object detection task would lead to misalignment between the extracted features and object's features (Fig. 2a). A straightforward way to deal with this problem is to use oriented anchor boxes [12], [13], [14]. However, rotated anchors need to take the predefined angle (or orientation of object) into account. Hence, the mumber of anchors can be dramatically increased, increasing computational cost. Furthermore, the vast imbalance between the positive and negative anchors would lead to slow training and inferior performance. [15] In this paper, we propose ARPD, a novel single-stage anchor-free convolutional neural network that detects arbitrarily rotated bounding boxes of people in top-view fish-eye images. Our work extends the model proposed in CenterNet [16], one of the novel keypoint-based object detection algorithms [17], [18], [16] for standard images. We model each object as the center point of its bounding box. Other properties are directly inferred from the center keypoint feature. To capture the various alignment of people in ceiling-mounted fish-eye images, we predict the angle of each bounding box in addition to center and size. We also introduce a new rotation-aware periodic loss function that takes into account angle periodicity. The addition of an orientation head and a novel angle loss function allows ARPD to directly infer the oriented bounding boxes in fish-eye images without the need for prior boxes or non-maximal suppression. We evaluate the performance of ARPD on three publicly available person detection datasets captured by ceiling-mounted fish-eye cameras: HABBOF [9], Mirror World [19] and CEPDOF [11]. The main contributions of this paper can be summarized as follows: • We propose ARPD, a single-stage anchor-free detector for rotation-aware people detection in overhead fish-eye images. Our method eliminates the need for multiple anchors and complex pre/post-processing. In experiments on multiple fish-eye datasets, ARPD achieved competitive performance compared to stateof-the-art methods and keeps a real-time inference speed. • We introduce a periodic loss function that allows our network to learn the symmetry of rotated people in top-view fish-eye images. Our loss function does not suffer from the training instability and performance degeneration caused by the loss discontinuity when using standard regression functions. II. . Related Works Horizontal object detection: Horizontal object detectors can be roughly classified into two categories: anchor-based and keypoint-based. The former consists of two-stage and one-stage methods. Two-stage detectors, most notably models from the R-CNN family [21], [3], [22] are regionbased. First, the model proposes a set of regions of interest. Then a classifier only processes the region candidates. One-stage detectors such as YOLOv3 [1] or SSD [2] skips the region proposal stage and runs detection directly over a dense sampling of possible locations, known as anchor boxes. Recently, keypoint-based detectors have been proposed to overcome the disadvantages of anchor-based solutions. CornerNet [17] predicts the upper-left and lower-right corners of bounding boxes for every pixel along with an embedding, which is then used to determine the objects. ExtremeNet [18] predicts the center of objects as well as farthest left, right, top, and bottom points. These points are then matched based on their geometry. However, the post-grouping process is time-consuming. Zhou's CenterNet [16] considers the center of a box as an object as well as a key point and then uses this predicted center to find the coordinates/offsets of the bounding box [20] as the encoder-decoder network as it gives the best speed and accuracy tradeoff [16]. We replace the upsampling layers with 3x3 deformable convolution layers and add more skip connections from the lower layers to help increase the feature map resolution symmetrically. The output feature map is transformed into four branches: heatmap, off-set, size, and orientation. In addition to center point and size, the OBB is also represented by its angle of rotation. without post-grouping, thus making prediction faster. The keypoint-based object detectors show advantages over the anchor-based ones in terms of speed and accuracy. [16] Oriented Object Detection: Different from horizontal object detectors, these algorithms use rotated bounding boxes to represent oriented objects. R-DFPN [12] designs a rotation anchor strategy to predict the minimum circumscribed rectangle of the object and build dense connections to create high-level semantic feature maps for all scales. RoI Transformer [13] designed a Rotated RoI learner to transform a Horizontal Region of Interest into a Rotated Region of Interest. To solve the misalignment problems between the region of interest and the object's feature, R3Det [14] proposes an end-to-end refined single-stage rotated object detector. All of the methods above use five parameter coordinates to describe oriented bounding box: center, width, height and rotation angle, with the angle defined in − π 2 , 0 and use common regression loss. However, due to the inherent symmetry of rotated bounding boxes, this approach will lead to loss discontinuity (i.e., loss value will jump when the angle reaches its range boundary) and instability during training. People detection using overhead, fish-eye cameras: Person detection methods using ceiling-mounted fish-eye cameras have been much less studied than conventional algorithms using standard perspective cameras, with most research appearing in recent years. Early works use techniques such as background subtraction or optical flow to obtain the location of moving people [5], [4]. Methods based on Histogram of Gradients (HOG) or Local Binary Patterns (LBP) [6], [8], [7] have also been put forward. Chiang and Wang [7] applied HOG descriptor with SVM classifier on sections sliding windows extracted from fish-eye frames. In a recent work, instead of directly de-warping the fish-eye images, Krams et al. [8] de-warp features extracted from the fish-eye image using an ACF classifier. Recently, deep learning-based algorithms have been applied to person detection in fish-eye images [9], [11], [10]. Li et al. [9] proposes a method in which YOLOv3 is applied to 24 rotated, overlapping windows, and the results are post-processed to produce detection results. One of the top-performing algorithms, RAPiD [11] propose a fully convolutional network also based on YOLOv3 that detects people in fish-eye images using rotated bounding boxes. All of the methods mentioned above either require excessive computation or make modifications to horizontal anchor-based detectors, which are prone to feature misalignment problems as well as severe imbalance issues. III. . Proposed Method We propose ARPD, a novel single-stage anchorfree CNN that extends Zhou's CenterNet [16]. Unlike Centernet which only predicts the location and size of each object, ARPD also predicts the angle of bounding boxes of people in a top-down, fish-eye image. We also introduce a periodic loss function based on an extension of common smooth L1 loss to deal with the loss discontinuity caused by angle periodicity. In this section, we first describe the overall architecture of the proposed method and the output maps in detail. The results of the output maps are then gathered and decoded to determine the center-point, size, and orientation of each person in an overhead fish-eye image. A. . Network Architecture The proposed network (see Fig. 3) consists of a feature extraction network and a bounding box regression network , also known as detection head. The feature extraction network takes an image I ∈ R W ×H×3 as input and returns an output feature map. The output feature map X ∈ R B. . Center point estimation Given an input image I ∈ R W ×H×3 , we first create a keypoint heatmapK ∈ [0, 1] W s × H s ×C . HereK is the function of x, y, c. A predictionK x,y,c = 1 corresponds to detected center for class c.K x,y,c = 0 is considered as background. To generate ground-truth heatmaps for training, each key-point ground-truth p ∈ R 2 are converted to lowresolution equivalentp = p s . These centers are then splat using a 2D Gaussian Kernel exp − (px−px) 2 +(py−py) 2 2σ 2 p , where σ is an object-size adaptive standard deviation [17]. The final result is the ground truth heatmap K ∈ [0, 1] W s × H s ×C . In case two or more Gaussians of the same class overlap, we get the element-wise maximum [23]. When training the keypoint estimator, directly learning the positive center points would be difficult due to the imbalance between the positive and negative samples. To handle this problem, we decrease the penalty for the points inside the Gaussian bumps and use focal loss [24] to train the heatmap: L K = −1 N xyc          1 −K xyc α log K xyc if K xyc = 1 (1 − K xyc ) β K xyc α log 1 −K xyc otherwise (1) where N is the number of objects in image I. α and β are hyper-parameters of the focal loss. We use α = 2 and β = 4 empirically as in [17] in all our experiments. After the extraction of peak points from heatmaps, we have to map these coordinates to a higher dimensional input image. This will cause a discretization error as the original image pixel indices are integers and we will be predicting the float values. The offset can be calculated asô = px s − px s ,p y s − py s . To predict this value, we use an offset predictorÔ ∈ R W s × H s ×2 . This offset predictor is optimized with L1 loss: L of f = 1 N N k=1 \|o k −ô k \|(2) In the next section, we will show how to extend this keypoint estimator to the arbitrarily oriented person detection task. C. . Oriented bounding box regression Let x k 1 , y k 1 , x k 2 , y k 2 be the bounding box of object k. In addition to predicting the center point, we directly regress sizeŝ k = x k 2 − x k 1 , y k 2 − y k 1 ofL size = 1 N N k=1 \|ŝ k − s k \|(3) To capture the oriented bounding boxes, we also need to predict angle of an bounding box from it's center point. We define the orientation map asθ ∈ R H s × W s ×1 . To train this orientation map , we first define the ground-truth of the orientation class. Previous works on oriented object detection often use a 5-component vector (c x , c y , w, h, θ) to represent the ground truth of rotated bounding boxes, where θ ∈ − π 2 , 0 . Due to rotational symmetry, bounding box b 1 = (c x , c y , w, h, θ) with center points c x , c y , width w, height h and angle θ is indistinguishable from bounding box b 2 = (c x , c y , h, w, θ − π 2 ) with width h, height w and angle θ − π 2 . However, common regression loss do not account for this symmetry, hence can lead to large cost even when prediction is close to ground truth (Fig. 4). We solve this by enforcing the ground truth angle θ to be in range [− π 2 , π θ = π * Tanh t θ (4) We limit the predicted angleθ to be in range [−π, π). This parametrization will later be explained in Section III-D. D. . Rotation-Aware Loss Function Our loss function is inspired by that used in CenterNet [16], with an additional periodic loss for angle Fig. 6. In certain cases, gradient descent will cause the predicted angleθ (green arrow) to move further away from the ground truth angle θ (blue arrow). In this case, we want the network to learn to predict θ + π instead of θ. To facilitate this behavior, we need to extend the angle range to include θ + π (dashed blue arrow) otherwiseθ will stop at π 2 . prediction: L det = L K + λ size L size + λ off L off + λ angle L angle (5) where L angle is the angle loss function and λ size , λ off , λ angle are constants. Traditionally, common regression functions based on L1 or L2 [12], [13], [14] are used for angle prediction. However, these loss functions are not take into account the fact that since a bounding box remains identical after a rotation of k * π, the angle loss function must also satisfy that L angle ∆θ = L angle (∆θ + kπ), where ∆θ is the difference between the ground truth and the predicted angle. To this end, we propose a new, periodic angle loss function: L angle = 1 N N k=1 \|∆θ periodic \| − 0.5 if \|∆θ periodic \| ≥ 1 0.5 * ∆θ periodic 2 otherwise (6) where ∆θ periodic = arctan sin ∆θ cos ∆θ Our angle loss function is π-periodic with respect to θ. The function is also defined and differentiable except for angles such as ∆θ = kπ + π 2 ( Fig. 5). We ignore these angles during backpropagation. Since ground truth angle θ ∈ − π 2 , π TABLE I. Performance comparision of ARPD with other state of the art methods on 3 fish-eye datasetsII. P, R and F1 denote Precision, Recall and F1-score, respectively. Results and inference speed of ARPD and RAPiD is measured without any test time augmentation at confidence threshold t conf = 0.3 using a single GTX 1070 Ti GPU. Results and FPS * of Tamura et al. [10] and Li et al. [9] are as tested in RAPiD [11]. when ∆θ ∈ − π 2 , 0 . When ∆θ ∈ − π 2 , 0 , the derivative of the angle loss function L angle will be negative, which will in turn cause gradient descent to moveθ away from θ (Fig. 6). Since a bounding box is identical after rotation of π, in this situation, it is preferable that the network learn to predictθ +π instead. To allow this behavior, we extend the range of predicted angleθ to [−π, π). To provide steady gradients for large values of ∆θ and less oscillations during updates when ∆θ is small, we use smooth L1 norm for our angle loss (Fig. 5). MW IV. . Experimental results A. . Datasets and Implemetation details We evaluate our method on three publicly available fish-eye datasets: Mirror World (MW) [19], HABBOF [9] and CEPDOF [11]. All three datasets contain videos of people captured by ceiling-mounted fish-eye cameras in various scenarios. Further information about the datasets are given in Table II. To measure the performance of our algorithm, we adopt the Average Precision metric used by COCO [25] in addition to F1 score, precision and recall. Since a single person can be represented by multiple ground-truth rotated bounding boxes with different angles, we only consider the AP at IoU = 0.5 (AP 50 ). We use a confidence threshold of 0.3 for all our tests. For the training and testing stage, we resize the input image to 512 × 512. In addition to random flip, random cropping, random scaling, and color jittering, we also use random rotation augmentation. We first train our network on MS COCO 2017 [25] for 90 epochs. In the training phase, perspective images (containing people) from the COCO dataset are randomly rotated before being input into the network. We split the three datasetsII into train/test split, i.e., two datasets are used for training, and the remaining are used for testing. For example, we fine-tune ARPD on HABBOF and CEPDOF for 10 epochs, and test it on MW. This process is repeated for all permutations. We use Adam optimizer with an initial learning rate of 1.25 × 10 −4 , with learning rate drop at B. . Main results Results from Table. I shows that ARPD's performance come close to the top-performing algorithm , while running many times faster than all other methods tested. This makes ARPD superior for the real-time person detection task. Our method outperforms Tamura et al.'s method [10] by a considerable margin on all three fish-eye datasets, and is slightly better in terms of AP compared to the method of Li et al. [9], while running tens of times faster. We achieve an execution speed of 21 frames per second, which is nearly two times faster than RAPiD while not having to sacrifice considerably in terms of accuracy. ARPD achieves an AP 50 score of more than 95 on MW and HABBOF, both of which mostly consist of people walking and standing appearing radially-oriented (Fig. 7 ae). ARPD is also capable of detecting various challenging scenarios in CEPDOF such as extreme body poses or occlusion (Fig. 7 f-g). However, scenarios such as lowlight or small objects remain difficult (Fig. 7 h). C. . Ablation Experiments In this section, we conduct various experiments to analyze how individual parts of ARPD contributes to the overall performance, and the effectiveness of novel Rotation aware Angle loss: We compare the performance of our novel periodic loss function against commonly used regression loss functions. Results from Table III proves that our loss function is better for oriented person detection . Smooth L1 norm also performs slightly better than L1. This can be explained due to the fact that Smooth L1 allows for better optimization when ∆θ is small. Parameterization of oriented bounding box: As shown in Table III, there is a notable performance improvement when we extend the prediction range from − π 2 , π 2 to [−π, π), however increasing it further does not significantly impact accuracy. Impact of different orientation weight: We test the approach's sensitivity to the orientation weight λ angle . λ angle = 0.1 yields the best result. Performance noticeably deteriorate when λ angle > 0.1. V. . Conclusions In this paper, we propose ARPD, a new oriented person detection method in fish-eye images based on center point detection. ARPD is single-stage, free of anchor or NMS post-processing. In addition to the location and dimension of the bounding boxes, ARPD also predicts its angle of rotation. We also introduce an angle-aware periodic loss function based on smooth L1 norm, which considers angle periodicity and is more sensitive towards small-angle variation between ground truth and prediction. Experimental results show that our method achieves the best speed-to-accuracy trade-off on multiple fish-eye datasets. Our method would benefit real-world applications and serve as a baseline algorithm for real-time person detection on fish-eye cameras. Fig. 1 . 1Illustration of typical people-detection results on ceilingmounted, fish-eye images of conventional person detector (a) and the proposed method (b). Human-aligned bounding boxes fit bodies more accurately compared to axis-aligned bounding boxes. Fig. 3 . 3The overall architecture and the oriented bounding box (OBB) descriptions of the proposed method. We use a modified version of DLA-34 s ×1 ), where C is the number of classes and s = 4 refers to the stride. For the person detection task, C = 1. The output maps are gathered and decoded to generate the oriented bounding boxes of people in overhead fish-eye images. Fig. 4 . 4Demonstration of the loss discontinuity. Given the ground truth bounding box bgt = (cx, cy, h, w, −1 • ) and the predicted bounding box b pred = (cx, cy, w, h, 89 • ). Due to the way rotated bounding box is represented traditionally, cost value is large even though prediction is close to ground truth. Fig. 5 . 5Periodic loss function with smooth L1 norm and its derivative. Fig. 7 . 7Qualitative results of ARPD on MW (a-d), HABBOF (e) and CEPDOF (f-h). ARPD works well on both easy and challenging cases, such as heavy occlusion, various poses and background. For cases of small people detection (c), in some cases the bounding boxes does not fully enclose the person. Unsurprisingly, low-light scenarios (h) remains challenging. each object. For this, we train a dimension headŜ ∈ RW s × H s ×2 using standard L1 distance norm: TABLE II . IIStatistics of three publicly available overhead fisheye image datasets. All images have a 360 • field of view and 1:1 aspect ratio. epochs 60 and 80. Unless otherwise specified, we set λ off = 1 and λ size = λ angle = 0.1 in all of our experiments. We train the network with the batch size of 32 on two NVIDIA 2080Ti. The inference speed is measured on a single NVIDIA 1070Ti.Dataset # of videos # of frames # of GT boxes MW 19 8752 22758 CEPDOF 4 25504 173428 HABBOF 8 5873 20430 TABLE III . IIIAblation experiments conducted on Mirror World dataset. 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[ "First principle derivation of semiclassical force for electroweak baryogenesis", "First principle derivation of semiclassical force for electroweak baryogenesis" ]	[ "Kimmo Kainulainen \nNORDITA\nBlegdamsvej 17DK-2100Copenhagen ØDenmark\n", "Tomislav Prokopec \nInstitute for Theoretical Physics\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany\n", "Michael G Schmidt \nInstitute for Theoretical Physics\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany\n", "Steffen Weinstock \nInstitute for Theoretical Physics\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany\n" ]	[ "NORDITA\nBlegdamsvej 17DK-2100Copenhagen ØDenmark", "Institute for Theoretical Physics\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany", "Institute for Theoretical Physics\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany", "Institute for Theoretical Physics\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany" ]	[]	We perform a systematic gradient expansion on kinetic equations and derive the CPviolating semiclassical force for fermions propagating in presence of a CP-violating wall at a first order electroweak phase transition. The force appears at orderh in the flow term of the kinetic equation and agrees with the semiclassical force used for baryogenesis computations. In particular we consider the force for charginos in both the MSSM and NMSSM. We then study the continuity equations for the vector and axial vector currents and stress the role of the latter as the one containing baryogenesis sources. We also show that there is no CP-violating force for bosons to orderh in gradient expansion.• [email protected] ⋄ T.Prokopec@, M.G.Schmidt@, [email protected]	10.1088/1126-6708/2001/06/031	[ "https://arxiv.org/pdf/hep-ph/0105295v2.pdf" ]	16,423,229	hep-ph/0105295	021eca137f778dab1bb6584ba5fbcf1020f7ef2c	First principle derivation of semiclassical force for electroweak baryogenesis Jul 2001 May 2001 Kimmo Kainulainen NORDITA Blegdamsvej 17DK-2100Copenhagen ØDenmark Tomislav Prokopec Institute for Theoretical Physics Heidelberg University Philosophenweg 16D-69120HeidelbergGermany Michael G Schmidt Institute for Theoretical Physics Heidelberg University Philosophenweg 16D-69120HeidelbergGermany Steffen Weinstock Institute for Theoretical Physics Heidelberg University Philosophenweg 16D-69120HeidelbergGermany First principle derivation of semiclassical force for electroweak baryogenesis Jul 2001 May 2001arXiv:hep-ph/0105295v2 11 We perform a systematic gradient expansion on kinetic equations and derive the CPviolating semiclassical force for fermions propagating in presence of a CP-violating wall at a first order electroweak phase transition. The force appears at orderh in the flow term of the kinetic equation and agrees with the semiclassical force used for baryogenesis computations. In particular we consider the force for charginos in both the MSSM and NMSSM. We then study the continuity equations for the vector and axial vector currents and stress the role of the latter as the one containing baryogenesis sources. We also show that there is no CP-violating force for bosons to orderh in gradient expansion.• [email protected] ⋄ T.Prokopec@, M.G.Schmidt@, [email protected] Introduction The creation of a baryon asymmetry at a first-order electroweak phase transition in the early universe is an attractive proposal [1] because the elementary particles and interactions involved in this process can be tested soon in accelerator experiments. For a successful baryogenesis a coalition between CP violation, nonequilibrium thermodynamics and baryon number violation is needed [2]. Model calculations require a study of generation and transport of CP-violating flows arising from interactions of fermions with the expanding phase transition fronts. As the problem involves the dynamics of quantum fields in a spatially varying background it cannot be treated within the classical transport theory. While fully general quantum Boltzmann equations can quite easily be formulated by making use of suitably truncated Dyson-Schwinger equations for the out-of-equilibrium two-point functions, some approximation scheme is needed to derive a set of practically solvable, yet sufficiently general transport equations for electroweak baryogenesis (EWBG). A fast baryon number violating rate in the unbroken phase is a necessary ingredient of any EWBG model. However, to avoid a wash-out of the newly created asymmetry, the baryon number violation must turn off in the Higgs phase. As is well known, this is the case provided the transition is strong enough [3]. Since for the present experimental bounds on the Higgs mass the electroweak phase transition in the Standard Model (SM) is not firstorder [4], one is lead to consider extensions of the Standard Model. The natural candidates are supersymmetric models, which include the Minimal Supersymmetric Standard Model (MSSM) [5] and the nonminimal extension (NMSSM) [6] with an additional Higgs-singlet field. These models contain additional scalars which can strengthen the phase transition as required for baryogenesis. In supersymmetric models the bubble walls are found to be quite slow and thick [7,8,9,10,11], in the sense that v wall ≪ c and ℓ wall ≫ ℓ dB , where ℓ dB ∼ 1/T denotes a typical de Broglie wave length of thermal particles. The latter condition is of particular importance, because it implies that a gradient expansion in terms of ℓ dB /ℓ wall represents a controlled, rapidly converging approximation scheme for most of the excitations in the electroweak plasma. In the past many heuristic attempts have been made to derive approximate transport equations for EWBG [12,13,14,15,16,17]. Common to all these methods is the strategy to somehow isolate the essential quantum features of the transport in the form of "sources" to be inserted into classical transport equations. Baryon production has in this way been computed in two doublet models [13,14,15,18], MSSM [16,19,20,21,22,23] and NMSSM [24,25]. Different approaches, when applied to the same physical problem, have been found to disagree however. In particular the sources from chargino and squark sectors in the MSSM, found using WKB-methods [21,22,24], are parametrically different from those derived by the use of the continuity equations and the relaxation time approximation [19,20], and by other earlier attempts [16,26]. In this paper we present a rigorous first-principle derivation of quantum transport equations appropriate for baryogenesis in the limit of thick phase boundaries ignoring collisions. We start our analysis by writing the exact Dirac equation of motion for the dynamical Green function (Wightman function) G < αβ (u, v) ≡ i ψ β (v)ψ α (u) with a CP-violating spatially varying pseudoscalar mass term. For simplicity here we consider particles moving perpendicular to the phase boundary, which effectively reduces our problem to 1+1 dimensions. The results discussed here are not affected in any important way when the general 3+1 dimensional case is considered [27]. By performing a Wigner transform we obtain a controlled expansion in gradients, or more appropriately, in powers ofh. We show that, to first order inh, G < admits a spectral decomposition in terms of on-shell quasiparticle excitations. The on-shell momenta are set by a dispersion relation derived from the equations of motion and agree with the results of [28], where the spectral function A was considered in gradient expansion. The on-shell distribution functions f s+ and f s− for particles and antiparticles of spin s, respectively, are then shown to obey the following kinetic Liouville equations: ∂ t f s± + v s± ∂ z f s± + F s± ∂ kz f s± = 0.(1) The quantum information in (1) is entirely contained in the expression for the quasiparticle energy ω s± , which shows up in the expressions for the group velocity v s± ≡ k z /ω s± and the semiclassical force F s± = ω s± dv s± /dt, where k z denotes the kinetic momentum. For example, we show that in the case of a single chiral fermion moving in a CP-violating background with planar symmetry, represented by a z-dependent complex mass m(z) = \|m(z)\|e iθ(z) , a quasiparticle moving in z-direction with momentum k z has the energy ω s± = ω 0 ∓h s\|m\| 2 θ ′ 2ω 2 0 ,(2) where ω 0 ≡ k 2 z + \|m\| 2 , and experiences the force F s± = − \|m\| 2 ′ 2ω s± ±h s(\|m\| 2 θ ′ ) ′ 2ω 2 0 .(3) We also derive explicit expressions for the semiclassical force for a general case of N mixing fermions and in particular for charginos both in the MSSM and NMSSM. We then show that in the case of N mixing bosonic fields, such as the squarks in the (N)MSSM, there is no CP-violating force to first order inh. Our results agree with recent results obtained by the use of the WKB-approach [22,24]. The WKB-method was originally introduced by Joyce, Prokopec and Turok in [29] and [15] and then applied to the MSSM in [21]. The CP-violating velocities and accelerations for fermions interacting with a phase transition wall were correctly computed from the WKBdispersion relations in Refs. [29,15,21]. The velocity and force in kinetic equations were obtained from the Hamilton equations based on canonical momentum. However, when the dispersion relation is derived by considering the spectral function in gradient approximation [28], the momentum appearing in the Wigner representation is the kinetic momentum. The relevance of the kinetic momentum as the true physical variable in the WKB-picture was first realized by Cline, Joyce and Kainulainen [22], who also showed how it can be consistently incorporated into kinetic theory leading to equations identical to (1)(2)(3). The outstanding contribution of the present work is a controlled first principle derivation of the kinetic equation (1). This is important because of a considerable controversy in literature concerning the transport equations to be used for EWBG calculations. Moreover, our treatment in principle allows a study of the plasma dynamics beyond first order inh, which cannot be achieved by WKB-methods. As an example, at second order inh the full equations do not admit the spectral decomposition solution for G < ; for a scalar field this problem is considered in [30]. Let us mention that in a related work [31] the Liouville equations for fermions in presence of a classical gauge field have been considered in gradient approximation. The crucial role of the constraint equations in the derivation of the kinetic equations was then stressed in [32,33]. The problem of a pseudo-scalar mass term in kinetic equations has been considered in Refs. [34,32], but these authors discussed the flow term only to zeroth (classical) order in h, whereas the spin dependent force essential for EWBG arises only at quantum level, as we show here. Inclusion of interaction terms gives rise to yet another source which is of first order in h, namely the spontaneous baryogenesis (SBG) source of Ref. [35]. The SBG source appears because the CP-violating split in the dispersion relation causes the CP-conjugate states to relax towards different local equilibria in the bubble wall. Thus a first principle derivation of the SBG source requires not only a consistent expansion inh, which we have done, but also a consistent expansion in relevant coupling constants. The latter is necessarily a model dependent problem and shall be considered elsewhere. However, to facilitate comparison with literature we derive the vector current to orderh in gradient expansion which displays the SBG source in the relaxation time approximation. When applied to the MSSM, our results differ from Refs. [36], [17] and [20]. The paper is organized as follows. In section 2 we derive the Liouville equations for a single Dirac fermion with a spatially varying complex mass term. In section 3 we generalize these results to the case of N mixing fermionic fields and study the case of mixing charginos in both the MSSM and NMSSM. We then in section 4 consider the case of N mixing scalar fields and show that there is no CP-violating source to first order inh. In section 5 we study the continuity equations for both vector and axial vector current, and spontaneous baryogenesis in the relaxation time approximation, and make a comparison with literature. For example, in contrast to what is claimed in [37] and [17], we find that the continuity equation for the vector current contains no CP-violating source in the absence of collisions. On the other hand, in the continuity equation for the axial current there are CP-violating sources that can be related to higher moments expansion of the semiclassical Boltzmann equation. Finally, section 6 contains a discussion and summary. Fermionic field with a complex mass We first consider the dynamics of a fermionic field with a complex spatially varying mass term. More precisely, we take our system to be described by the effective lagrangian L = iψ ∂ / ψ −ψ L mψ R −ψ R m * ψ L + L int ,(4) where L int contains interactions and m(x) = m R (x) + im I (x) = \|m(x)\|e iθ(x)(5) is a space-time dependent mass term arising from an interaction with some CP-violating scalar field condensate. We are primarily interested in the case where m arises from the Higgs field condensate at a first order electroweak phase transition. As the bubbles of broken phase grow several orders of magnitude larger than the wall width before coalescence, the wall can be approximated by a planar interface to good accuracy. We therefore consider a mass term in the bubble wall frame which is only a function of the spatial coordinate orthogonal to the wall, m = m(z). Our focus in this paper is on the semiclassical treatment of fermions in presence of background fields. In particular we derive the flow term for a fermionic kinetic equation with a nontrivial force induced by the CP-violating mass parameter (5). To keep the discussion simple, we relegate the explicit treatment of collision terms and self-energy corrections which are induced by the specific interactions included in L int , to later publications [27]. Moreover, we are interested in the case of wide walls, so that the de Broglie wave length ℓ dB of a typical excitation is small in comparison with the wall width, ℓ dB ≪ ℓ wall . This condition is amply satisfied at the electroweak phase transition, where typically ℓ dB ∼ 1/T and ℓ wall ∼ 10/T [8]. With the above assumptions we now develop the equations of motion for the Wightman function G < αβ (u, v) ≡ i ψ β (v)ψ α (u)(6) in a consistent expansion in gradients of the background fields. Here · denotes the expectation value with respect to the initial state. The function G < describes the statistical properties of an out-of-equilibrium system. It corresponds to the off-diagonal part of the fermionic two-point function in the Schwinger-Keldysh formalism [38], which indeed is the method of choice to derive the equations of motion for G < including interactions. However, in their absence all one needs is the familiar Dirac equation. Dropping interactions we have from Eq. (4): i ∂ / u − m R (u) − iγ 5 m I (u) ψ(u) = 0 .(7) Multiplying (7) from the left by the spinor iψ(v) and taking the expectation value one finds: i ∂ / u − m R (u) − iγ 5 m I (u) G < (u, v) = 0 .(8) This equation and the hermiticity property iγ 0 G < (u, v) † = iγ 0 G < (v, u),(9) which can be immediately inferred from the definition (6), completely specifies G < . In order to study equation (8) in gradient expansion we perform a Wigner transform of G < to the mixed representation, i.e. a Fourier transform with respect to the relative coordinate r ≡ u − v: G < (x, k) ≡ d 4 r e ik·r G < (x + r/2, x − r/2),(10) where x = (u + v)/2 denotes the center-of-mass coordinate. The crucial advantage of the representation (10) is that it separates the internal fluctuation scales, described by momenta k, from the external ones which show up as a dependence of G < on x, and thus gives us the chance of exploiting possible hierarchies between these scales. In the Wigner representation equation (8) becomes k / −m 0 (x) − im 5 (x)γ 5 G < = 0,(11) where we use the following convenient shorthand notation k µ ≡ k µ + i 2 ∂ µ (12) m 0(5) ≡ m R(I) e − i 2 ← ∂x· ∂ k .(13) The original local equation (8) for G < is thus transformed into an equation involving an infinite series in gradients. This in fact can be viewed as an expansion in powers of the Planck constanth. We have seth → 1, but a dimensionfulh can at any stage be easily restored by the simple replacements ∂ x →h ∂ x and G < →h −1 G < . Because of the planar symmetry (here we do not consider initial states of the plasma that break this symmetry) G < can depend only on the spatial coordinate orthogonal to the wall, z ≡ x 3 . We also consider equation (11) only in a frame where the momentum parallel to the wall vanishes, k = 0. This last assumption effectively reduces the problem to 1+1 dimensions, and we can cast equation (11) into the form k 0 +k z γ 0 γ 3 −m 0 γ 0 + im 5 γ 0 γ 5 iγ 0 G < = 0,(14) wherek 0 = k 0 + i 2 ∂ t andk z = k z − i 2 ∂ z . The differential operator in (14) is entirely spanned by a closed 1+1-dimensional subalgebra of the full 3+1-dimensional Clifford algebra. Moreover, it commutes with the operator S 3 = γ 0 γ 3 γ 5 , which measures spin s in z-direction. s is thus a good quantum number in the frame k = 0, which motivates to seek solutions for iγ 0 G < which satisfy S 3 iγ 0 G < s = iγ 0 G < s S 3 = siγ 0 G < s . Working in a convenient chiral representation this condition leads immediately to the following spinor structure: − iγ 0 G < s = 1 2 (1 + sσ 3 ) ⊗ g < s ,(15) where σ 3 is the Pauli matrix referring to spin in z-direction. g < s has indices in the remaining two dimensional chiral space and can also be written in terms of the Pauli matrices ρ i as follows: g < s ≡ 1 2 g s 0 + g s i ρ i .(16) The decomposition (15) contains the implicit assumption that iγ 0 G < does not mix spins. We expect that this is a good approximation in the electroweak plasma. The signs and normalizations in (15)(16) are chosen such that g s 0 measures the number density of particles with spin s in phase space. With these simplifications we can now reduce our original 4 × 4 problem to a two-dimensional one by effecting the replacements γ 0 → ρ 1 , −iγ 0 γ 5 → ρ 2 , −γ 5 → ρ 3(17) and we find k 0 − sk z ρ 3 − ρ 1m 0 − ρ 2m 5 g < s = 0.(18) A similar procedure is used in [28] for a treatment of the fermionic propagator. Equation (18) may look simple, but it still constitutes a set of four coupled complex (or eight coupled real) differential equations, which we shall now analyze. We first find the four independent complex equations for g s a (a = 0, i) by multiplying (18) successively by 1 and ρ i , and taking the trace:k 0 g s 0 − sk z g s 3 −m 0 g s 1 −m 5 g s 2 = 0 (19) k 0 g s 3 − sk z g s 0 − im 0 g s 2 + im 5 g s 1 = 0 (20) k 0 g s 1 + isk z g s 2 −m 0 g s 0 − im 5 g s 3 = 0 (21) k 0 g s 2 − isk z g s 1 + im 0 g s 3 −m 5 g s 0 = 0.(22) As a consequence of Eq. (9) the matrices g < s are hermitean so that g s a are real functions. We then have twice as many equations as independent functions corresponding to real and imaginary parts of Eqs. (19)(20)(21)(22), and hence one half of the equations must correspond to the constraints on the solutions of the other half; those equations are kinetic equations. This was first pointed out in the context of kinetics of fermions by Zhuang and Heinz [32]. As one sees from (12), no time derivatives appear in the real parts of (19-22) so that they indeed provide four constraint equations (CE) on the solutions of four kinetic equations (KE). These contain time derivatives and correspond to the imaginary parts of (19-22). Because we have put no restrictions to the form of them-operators, Eqs. (19)(20)(21)(22) are still valid to any order in gradients in the frame where k = 0. In what follows we assume that the mass is a slowly varying function of x and truncate gradient expansion at second order, which is the lowest order at which CP-violating effects can be discussed consistently. This method is not adequate for problems involving quantum mechanical reflection, which require nonperturbative treatment inh. With the truncation,m 0(5) in (19)(20)(21)(22) simplifies tô m 0(5) ≃ m R(I) + i 2 m ′ R(I) ∂ kz − 1 8 m ′′ R(I) ∂ 2 kz .(23) Even with this truncation, we are facing a problem involving eight coupled second order partial differential equations. Our next task is to reduce these to a single equation governing the dynamics of the fermionic two-point function. Constraint equations Let us first consider the constraint equations. They consist of four homogeneous equations for four functions, which implies that there is one constraint which gives rise to the dispersion relation. While this property remains true to any order in gradients (or equivalently inh), we only need to work to first order to find the nontrivial result we are looking for. To this order we have k 0 g s 0 − sk z g s 3 − m R g s 1 − m I g s 2 = 0 (24) k 0 g s 3 − sk z g s 0 + 1 2 m ′ R ∂ kz g s 2 − 1 2 m ′ I ∂ kz g s 1 = 0 (25) k 0 g s 1 + s 2 ∂ z g s 2 − m R g s 0 + 1 2 m ′ I ∂ kz g s 3 = 0 (26) k 0 g s 2 − s 2 ∂ z g s 1 − m I g s 0 − 1 2 m ′ R ∂ kz g s 3 = 0.(27) We first use the constraint equations (25)(26)(27) iteratively up to first order to express g s 1 , g s 2 and g s 3 in terms of g s 0 and ∂ kz g s 0 , and then insert the results into (24). Remarkably all terms proportional to ∂ kz g s 0 cancel and we find that to first order in gradients g s 0 satisfies the algebraic equation k 2 0 − k 2 z − \|m\| 2 + s k 0 \|m\| 2 θ ′ g s 0 = 0 .(28) This admits the spectral solution g s 0 = πn s \|k 0 \|δ(Ω 2 s ), which can also be written as g s 0 = ± π 2Z s± n s δ(k 0 ∓ ω s± ) .(29) Here n s (k 0 , k z , z) are nonsingular functions which are, as we show below, related to the onshell distribution functions. The indices ± refer to the sign of k 0 , to be eventually related to particles and antiparticles. The energy ω s± is specified by the roots of the equation Ω 2 s ≡ k 2 0 − k 2 z − \|m\| 2 + s k 0 \|m\| 2 θ ′ = 0,(30) and the normalization factor Z s± is defined as Z s± ≡ 1 2ω s± \|∂ k 0 Ω 2 s \| k 0 =±ω s± .(31) To first order in gradients these can be solved iteratively: ω s± = ω 0 ∓ s \|m\| 2 θ ′ 2ω 2 0 , ω 0 = k 2 z + \|m\| 2 (32) Z s± = 1 ∓ s \|m\| 2 θ ′ 2ω 3 0 = ω s± ω 0 .(33) Equation (32) Solution (29) nicely illustrates how the constraint equations operate. Solving (24)(25)(26)(27) consistently to first order accuracy constrains the solutions of the kinetic equation to sharp, locally varying energy shells given by (32). One way of understanding this is as follows. The CP-violating phase θ ′ in (28) can be related to an axial gauge field [29,15] which, to leading order in gradients, lifts the degeneracy in the dispersion relation, but does not spoil the quasiparticle picture, just as it is the case with a vector gauge field. We should note however, that the confinement to sharp energy shells does not persist beyond first order in gradients. While for noninteracting fermions one can always express g s 1,2,3 in terms of g s 0 , at higher orders more complicated derivative structures arise, as the constraint cannot be written as a simple algebraic equation with a spectral solution. For a treatment of such a situation in the case of a scalar field see [30]. Eq. (32) is identical with the results derived earlier by WKB-methods in [22] and simultaneously via the field-theoretic technique of spectral integrals in [28]. From the WKB-point of view the present derivation comes as a welcome verification of the result obtained by an intuitive, but less fundamental approach. The agreement with the field theoretical calculation of [28] on the other hand was to be expected. In [28] it was shown that the integral over the spectral function, defined as a difference of the retarded and advanced Green func- tions A = (i/2)(G ret − G adv ) , projects test functions onto energy shells (32); however, in the collisionless limit A satisfies the same equation of motion (8) as the Wightman function G < . The trace of γ 0 A in particular satisfies equation (28), and can be obtained from (29) by the replacement n s → 1. We can hence immediately check the sum-rule to the accuracy at which we are working. Indeed, ∞ −∞ dk 0 π Tr γ 0 A s = ± ∞ −∞ dk 0 2Z s± δ(k 0 ∓ ω s± ) = 1.(34) We finally note that equation (30) has additional poles at k 0 ≃ s\|m\| 2 θ ′ /2ω 0 , which we have left out in the decomposition (29). These poles correspond to unphysical, but harmless, tachyonic modes which arise only because our solutions for the constraint equations involve an expansion in inverse powers of k 0 , which breaks down already for k 0 much larger than the value associated with these poles. Note that their contribution to the sum rule (34) vanishes when summed over spins. Kinetic equations We now turn our attention to the kinetic equations. We are primarily interested in the equation for g s 0 which carries information on the particle density in phase space. From (19) we have ∂ t g s 0 + s∂ z g s 3 − m ′ R ∂ kz g s 1 − m ′ I ∂ kz g s 2 = 0,(35) which is correct up to second order in gradients (first order inh). Just as in the previous section we use the constraint equations (25)(26)(27) to express g s 1 , g s 2 and g s 3 in terms of g s 0 and arrive at an equation for g s 0 alone. To second order in gradients (first order inh) it reads k 0 ∂ t g s 0 + k z ∂ z g s 0 − 1 2 \|m\| 2 ′ − s 2k 0 (\|m\| 2 θ ′ ) ′ ∂ kz g s 0 = 0.(36) We have so far used three out of four constraint equations. To find the acceptable solutions satisfying all constraints, we must yet impose the restriction onto the functional space spanned by (29). Because of the δ-function in the decomposition (29) this is of course trivial. All we need to do is to insert (29) into (36) and integrate over the positive and negative frequencies k 0 . We then get the following form for the Liouville equation ∂ t f s± + v s± ∂ z f s± + F s± ∂ kz f s± = 0,(37) where f s+ ≡ n s (ω s+ , k z , z) f s− ≡ 1 − n s (−ω s− , −k z , z)(38) are the distribution functions for particles and antiparticles with spin s, respectively. These definitions are motivated by the equilibrium result n eq s = 1/(e βk 0 + 1), where β = 1/T is the inverse temperature. The quasiparticle group velocity v s± appearing in Eq. (37) is given by v s± = k z ω s± ,(39) where k z is the kinetic momentum. The spin-dependent and CP-violating semiclassical force reads F s± = − \|m\| 2 ′ 2ω s± ± s(\|m\| 2 θ ′ ) ′ 2ω 2 0 .(40) Equations (37)(38)(39)(40) are among the main results of this paper. Incidentally, the form (40) for the semiclassical force F s± = ω s± dv s± /dt was already found by Joyce, Prokopec and Turok [29]. To obtain kinetic equations by WKB-methods the authors of [15] used canonical variables however. The resulting equations are not invariant under reparametrization of the wave functions, and hence care is required when specifying local thermal equilibrium in derivation of transport equations relevant for baryogenesis. Cline, Joyce and Kainulainen [22] introduced the kinetic momentum as a physical variable in the kinetic equations and obtained the unique reparametrization invariant transport equations identical with (37) and (39)(40). The outstanding contribution of the present work is in a controlled first principle derivation of these equations without any a priori assumptions. Let us finally note that equation (36) could have been obtained by taking the bilinear ✸-derivative of the constraint equation (28), where the ✸-derivative is defined by ✸{a}{b} ≡ 1 2 (∂ t a ∂ k 0 b − ∂ z a ∂ kz b − ∂ k 0 a ∂ t b + ∂ kz a ∂ z b) .(41) This is no coincidence, and even more generally, in the collisionless limit the kinetic equation can be obtained by effecting the tan ✸-derivative on the constraint equation. Currents It is instructive to study the expressions for physical currents in order to shed light on various functions we have encountered in our derivation. Of particular relevance for baryogenesis are the vector and axial vector currents. By making use of (6) and (15)(16) one finds j µ ≡ ψ (x)γ µ ψ(x) = s=±1 d 2 k (2π) 2 ( g s 0 , sg s 3 ) j µ 5 ≡ ψ (x)γ µ γ 5 ψ(x) = s=±1 d 2 k (2π) 2 ( g s 3 , sg s 0 ) ,(42) where we have restricted ourselves to 1+1-dimensions so that d 2 k = dk z dk 0 . This shows that g s 0 is the usual number density in phase space, whereas g s 3 represents the axial charge density. An important consequence of the constraint equations is that there is only one independent dynamical function, here chosen to be g s 0 , while all others can be related to g s 0 via the constraint equations (25)(26)(27). In particular, g s 3 can be written as g s 3 = s k z k 0 + 1 2k 2 0 \|m\| 2 θ ′ ∂ kz g s 0 .(43) The nontrivial gradient correction appears as a total derivative and hence vanishes upon the k z -integration. Using the decomposition (29) one finds j µ s± = dk z 8πZ s± (1, v s± ) f s± = s kz =± s kz dω 8π ( 1 v s± , 1) f s± (44) j µ 5s± = s dk z 8πZ s± (v s± , 1) f s± = s s kz =± s kz dω 8π (1, 1 v s± ) f s± ,(45) where s kz denotes the sign of k z , and we discarded the vacuum contribution. In the last step we used ∂ kz ω s± = Z −1 s± k z /ω s± . The lower limit in the ω-integrals is \|m\| ∓ sθ ′ /2. The functions f s± are the correctly normalized distribution functions, and they retain the correct physical interpretation in that the particle flux is not affected by CP-violating effects, while the density may be either enhanced or suppressed, as given by the inverse velocity. The current (44) was computed by WKB-methods in [22]. The correct result for j 0 s± was also found in [28] by field-theoretical methods. However, j 3 s± found in [28] does not agree with (44) because the spinor structure used for iγ 0 G < was too simple. Interactions In all discussions above, we have left out the effects of interactions. This was done to avoid the necessity to use the full machinery of the Schwinger-Keldysh formalism [38], and to keep things as simple as possible. Moreover, unlike the treatment of the flow term of the kinetic equation presented above, including interactions is necessarily a model dependent task. Nevertheless it is a simple matter to write a formally exact equation of motion for G < including the collision terms. Instead of (8) we then have (i ∂ / −m R − im I γ 5 ) iG < − Σ R ⊙ iG < = Σ < ⊙ G R + 1 2 Σ > ⊙ G < − G < ⊙ Σ > ,(46) where A ⊙ B(u, v) ≡ dw A(u, w)B(w, v). G R is the real part of the (retarded) propagator, and the function Σ R contains the real part of the self-energy corrections including the singular (tadpole) interactions which can be resummed to a renormalized mass term. The terms in parentheses give rise to the usual collision term, where the self-energies Σ <,> arise only from nonlocal loop contributions to the Dyson-Schwinger equations. However, as mentioned above, the exact form of these terms depends on the theory considered. While their treatment is not conceptually difficult, their inclusion brings a considerable amount of technical complications. We shall consider the problem of including collisions elsewhere [27]. Mixing fermionic fields In practical applications, such as in supersymmetric models, one needs to consider cases where several fermion flavours are mixed by a spatially varying mass matrix. We therefore consider a theory with the mass lagrangian L mass = −ψ L Mψ R −ψ R M † ψ L ,(47) where M is a complex (in general nonhermitean) N × N matrix with spatially varying components. We denote the flavour degree of freedom by an additional index i to the spinor ψ α,i (x), so the Wightman function becomes a matrix in the product space of spinor and flavour: G < αβ,ij (u, v) = i ψ β,j (v)ψ α,i (u) .(48) The flavour degree of freedom plays no role in the derivation of the equations of motion for G < αβ,ij in the steps analogous to going from (8) to (19)(20)(21)(22) in the single fermion field case, because those steps dealt only with the spinor structure of G < . We can thus immediately write an equation analogous to (18): k 0 − sk z ρ 3 − ρ 1M 0 − ρ 2M 5 g < s = 0.(49) The sole, but significant difference to (18) is that g < s is now an N × N-matrix in the flavour space, and the mass terms have become N × N-matrix operatorŝ M 0 = 1 2 (M +M † )(50)M 5 = − i 2 (M −M † )(51) whereM ≡ Me UMV † = M d ,(52) where U and V are the unitary matrices which diagonalize the hermitean matrices MM † and M † M, respectively. After the rotation we can write (49) in the component form in the diagonal basis as (k 0 + i 2 D − t )g s 0d − s(k z − i 2 D − z )g s 3d −M 0d g s 1d −M 5d g s 2d = 0 (53) (k 0 + i 2 D − t )g s 3d − s(k z − i 2 D − z )g s 0d − iM 0d g s 2d + iM 5d g s 1d = 0 (54) (k 0 + i 2 D + t )g s 1d + is(k z − i 2 D + z )g s 2d −M 0d g s 0d − iM 5d g s 3d = 0 (55) (k 0 + i 2 D + t )g s 2d − is(k z − i 2 D + z )g s 1d + iM 0d g s 3d −M 5d g s 0d = 0.(56) The 'covariant derivatives' appearing in (53-56) are defined as D ± t ≡ ∂ t − i[Σ t , · ] − − is[∆ z , · ] ± (57) D ± z ≡ ∂ z − i[Σ z , · ] − − is[∆ t , · ] ±(58) where the brackets [·, ·] − refer to commutators and [·, ·] + to anticommutators and Σ µ ≡ i 2 (V ∂ µ V † + U∂ µ U † )(59)∆ µ ≡ i 2 (V ∂ µ V † − U∂ µ U † ) .(60) It should be noted that, while the relation (52) allows arbitrary phase redefinitions U → wU and V → wV , where w is any diagonal matrix with \|w ii \| = 1, the operator ∆ µ remains invariant under these transformations. This reparametrization freedom is exactly what leads to the apparent 'gauge' dependence of the results in the WKB-approach [22]. As we show below, only the diagonal elements of ∆ µ contribute to the constraint and kinetic equations to orderh, which then implies that our results are reparametrization invariant. Finally, the new mass operators in the diagonal basis are given bŷ M 0d = 1 2 UM V † + VM † U † (61) M 5d = − i 2 UM V † − VM † U † .(62) The constraint and kinetic equations now correspond to the hermitean and antihermitean parts of (53-56), respectively. Because of the matrix structure, these equations contain a number of commutator and anticommutator terms involving g s a and the various matrixoperators. However, we shall now argue that (53-56) can effectively be taken to be diagonal to the order at which we are working. Indeed, in the propagating basis (52) the off-diagonal terms are obviously suppressed byh when compared to the diagonal elements. On the other hand, they appear in the diagonal equations through the commutatorsh −1 [hΣ z , g s ad ] ≡ [Σ z , g s ad ] and [∆ z , g s ad ], and thus at the same order as the diagonal elements. This then immediately implies that, when the dynamics of CP-violating densities is considered, the offdiagonals contribute at second order inh in the diagonal equations and can be consistently neglected. With this it is now straightforward to show that, to first order accuracy inh, the constraint equations reduce to the following equations for the diagonal entries of g s ad : k 0 g s 0d − sk z g s 3d − m R g s 1d − m I g s 2d = 0 (63) k 0 g s 3d − sk z g s 0d + 1 2m ′ R ∂ kz g s 2d − 1 2m ′ I ∂ kz g s 1d = 0 (64) k 0 g s 1d + s 2 ∂ z g s 2d + s∆ zd g s 1d − m R g s 0d + 1 2m ′ I ∂ kz g s 3d = 0 (65) k 0 g s 2d − s 2 ∂ z g s 1d + s∆ zd g s 2d − m I g s 0d − 1 2m ′ R ∂ kz g s 3d = 0,(66) where m R,I are the real and imaginary parts of the eigenvalues of M d , ∆ zd is the diagonal part of ∆ z andm ′ R ≡ m ′ R − 2m I ∆ zd andm ′ I ≡ m ′ I + 2m R ∆ zd . Similarly, the kinetic equation for g s 0d to first order inh becomes ∂ t g s 0d + s∂ z g s 3d −m ′ R ∂ kz g s 1d −m ′ I ∂ kz g s 2d = 0.(67) Following our treatment in section 2, it is now straightforward to eliminate g s ad (a = 1, 2, 3) from equations (63) and (67) to obtain the constraint equation k 2 0 − k 2 z − \|M d \| 2 + s k 0 \|M d \| 2 Θ ′ g s 0d = 0 (68) and the kinetic equation k 0 ∂ t g s 0d + k z ∂ z g s 0d − 1 2 \|M d \| 2 ′ − s 2k 0 (\|M d \| 2 Θ ′ ) ′ ∂ kz g s 0d = 0 (69) for the number density function g s 0d alone. Eqs. (68) and (69) are analogous to the one field equations (28) and (36). The difference is that here we have N different equations corresponding to N diagonal elements of g s 0d in the mass eigenbasis. The derivative of the effective angle Θ ′ appearing in (68-69) is defined as Θ ′ = Θ ′ d + 2∆ zd (70) where the angles Θ d are the complex phases of the elements in the diagonal mass matrix: M d ≡ \|M d \|e iΘ d . Since the mixing field contribution shows up as a shift in the derivative of the pseudoscalar phase 2∆ zd , it implies that equations (68-69) are manifestly reparametrization invariant. We can convert (70) to an alternative form in terms of the original mass matrix and the rotation matrix U: \|M d \| 2 Θ ′ = − 1 2 Im U(MM ′ † − M ′ M † )U † d ,(71) which is also manifestly reparametrization invariant. This expression will be convenient for discussion of the chargino sector of the MSSM and the NMSSM in sections 3.1 and 3.2. The final steps in going from equations (68) and (69) to kinetic equations for the mass eigenmodes are exactly analogous to the single fermionic field case: each mass eigenmode i gets projected to its own energy shell ω si± given by (68), and the corresponding spectral decomposition density function f si± obeys a semiclassical Boltzmann equation identical to (37) ∂ t f si± + v si± ∂ z f si± + F si± ∂ kz f si± = 0 ,(72) with the corresponding group velocity v si± = k z ω si± (73) and the CP-violating semiclassical force F si± = − \|M i \| 2 ′ 2ω si± ± s(\|M i \| 2 Θ ′ i ) ′ 2ω 0i 2(74) computed from expression (70) or (71). It has been shown in [29,15,22], that the spindependent term in F si± gives rise to a CP-violating source proportional to \|M i \| 2 Θ ′ i in the diffusion equations. We can therefore loosely call this factor the 'source', and proceed to compute it in some special cases. Charginos in the MSSM We first compute the source in the transport equations for charginos in the MSSM. The chargino mass term reads Ψ R M Ψ L + h.c. ,(75) where Ψ R = (W + R ,h + 1,R ) T and Ψ L = (W + L ,h + 2,L ) T are the chiral fields in the basis of winos. The mass matrix reads M = m 2 gH * 2 gH * 1 µ ,(76) where H 1 and H 2 are the Higgs field vacuum expectation values and µ and m 2 are the soft supersymmetry breaking parameters. For a realistic choice of parameters there is no spontaneous CP-violation in the MSSM, so to a good approximation we can take the Higgs vev's to be real [25,23]. The matrix U in (52) can be parametrized as [22] U = √ 2 Λ(Λ + ∆) 1 2 (Λ + ∆) a −a * 1 2 (Λ + ∆) (77) with a = g(m 2 H 1 + µ * H * 2 ) (78) ∆ = \|m 2 \| 2 − \|µ\| 2 + g 2 (h 2 2 − h 2 1 ) (79) Λ = ∆ 2 + 4\|a\| 2 ,(80) where h i ≡ \|H i \| are normalized such that the tree level W -boson mass is M 2 W = g 2 h 2 /2, h 2 = h 2 1 + h 2 2 . The physical chargino mass eigenvalues are given by m 2 ± = 1 2 \|m 2 \| 2 + \|µ\| 2 + g 2 h 2 ± Λ 2 .(81) Upon inserting (76) and (77) into (71) it is straightforward to show that the source term for charginos becomes m 2 ± Θ ′ ± = ∓ g 2 Λ ℑ(µm 2 )(h 1 h 2 ) ′ ,(82) where Θ + (Θ − ) corresponds to the higgsino-like state when \|µ\| > \|m 2 \| (\|µ\| < \|m 2 \|). CP violation is here mediated via the parameters µ, m 2 and may in fact be large [39]. The result (82) is in perfect agreement with the chargino source obtained by a WKB method in Ref. [22]. Charginos in the NMSSM In the NMSSM there is an additional singlet field S in the Higgs sector. The singlet field couples to higgsinos, and hence the higgsino-higgsino component in the chargino mass matrix (76) is generalized in the NMSSM: µ →μ ≡ µ + λS,(83) where λ is the coupling for higgs(ino)-higgs(ino)-singlet interaction. Another consequence of this extension is the possibility to have spontaneous transitional CP-violation [25], so the Higgs fields H i are in general complex. When the parameters a, ∆ and Λ are defined as in equation (80) with µ →μ, the matrix U in equation (77) still diagonalizes MM † . In the NMSSM we must account for the complex dynamical phases of the higgs fields, and hence we have to be more careful with our definition of the higgs doublets. Our choice of writing the mass matrices (76) corresponds to parametrizing the higgs doublets Φ i as [40]: Φ 1 = h 1 e iθ 1 h − 1 , Φ 2 = h + 2 h 2 e iθ 2 ,(84) where h ± i are the charged higgs fields. Only one of the higgs phases θ i is physical, while the other gets eaten by the gauge fields in the unitary gauge. We wish to choose the physical phase in such a way that the corresponding field does not couple to the neutral weak boson. Given the parametrization (84), this condition implies that h 2 1 θ ′ 1 = h 2 2 θ ′ 2 .(85) Using the gauge constraint (85) we can write θ ′ 1 = h 2 2 h 2 θ ′ , θ ′ 2 = h 2 1 h 2 θ ′ ,(86) where h 2 = h 2 1 + h 2 2 , and θ = θ 1 + θ 2 is the physical CP-violating phase. To get the explicit form for the CP-violating term, we insert the NMSSM mass matrix (76) into (71). After some algebra one finds the following three terms giving rise to CP-violating sources in the NMSSM: Θ ′ NMSSM = Θ ′ h 1 h 2 + Θ ′ θ + Θ ′ S ,(87) The first term is the following generalization of the chargino source (82): m 2 ± Θ ′ h 1 h 2 ± = ∓ g 2 Λ ℑ(μm 2 e iθ ) (h 1 h 2 ) ′(88) for the case involving a new scalar field S and possibly complex higgs fields. However, there are two new types of terms in the NMSSM. The term Θ ′ θ is proportional to a derivative of the CP-violating phase θ in the Higgs sector, and reads m 2 ± Θ ′ θ ± = − g 2 θ ′ Λ Λ ± (\|m 2 \| 2 + \|μ\| 2 ) h 2 1 h 2 2 h 2 ∓ ℜ(μm 2 e iθ )h 1 h 2 .(89) Finally, the source Θ ′ S can be written as a derivative of the singlet condensate: m 2 ± Θ ′ S± = ± λg 2 Λ ℑ(m 2 H 1 H 2 S ′ ) + λg 2 2Λ Λ ± (\|μ\| 2 + g 2 h 2 − \|m 2 \| 2 ) ℑ(μ * S ′ ) .(90) In all formulae (88-90) the mass eigenvalues m 2 ± can be read off from equation (81) with the replacement µ →μ, whereμ is given by Eq. (83). Baryogenesis in the NMSSM from the semiclassical force has been studied in Ref. [24]. Mixing bosonic fields Here we first show that, unlike for fermions, the constraint and kinetic equations for mixing bosons acquire no gradient correction to first order inh in the collisionless limit. We then derive the constraint and kinetic equations accurate to second order in gradients which can be used as a starting point for baryogenesis calculations. N mixing bosonic fields with a spatially varying mass matrix M 2 obey the Klein-Gordon equation (✷ u + M 2 (u))φ(u) = 0,(91) where φ is an N-dimensional vector whose components are coupled by the hermitean mass matrix M 2 . Multiplying (91) from the left by −iφ † (v) and taking the expectation value with respect to the initial state we get (✷ u + M 2 (u))G < (u, v) = 0.(92) where the Wightman function G < is defined as G < (u, v) = −i φ † (v)φ(u) .(93) After performing the Wigner transform, equation (92) becomes 1 4 ∂ 2 − k 2 − ik · ∂ + M 2 e − i 2 ← ∂ · ∂ k G < = 0.(94) In the case when N = 1 it is immediately clear that the first quantum correction to the constraint equation (the real part of (94)) is of second order and to the kinetic equation (imaginary part) of third order in gradients (second order inh). To extract the spectral information to second order inh is quite delicate since the constraint equation in (94) contains derivatives [30]. In the case of more than one mixing fields it is convenient to rotate into the mass eigenbasis, just as in the fermionic case: M 2 d = UM 2 U † ,(95) where U is a unitary matrix. In the propagating basis the equation (94) becomes 1 4 D 2 − k 2 − ik · D + M 2 e − i 2 ← D · ∂ k G < d = 0,(96) where G < d ≡ UG < U † and the 'covariant' derivative is defined as: D µ = ∂ µ − i [Ξ µ , · ] , Ξ µ = iU∂ µ U † .(97) Since (G < d ) † = −G < d and D † µ = D µ , we identify the antihermitean part of (96) as the constraint equation: − 2k 2 G < d + M 2 c + 1 4 D 2 , G < d − i k · D +M 2 s , G < d = 0,(98) and the hermitean part is the kinetic equation k · D +M 2 s , G < d − i M 2 c + 1 4 D 2 , G < d = 0,(99) where we definedM 2 c = M 2 d cos 1 2 ← D · ∂ k M 2 s = M 2 d sin 1 2 ← D · ∂ k .(100) We now use the analogous argument as in the fermionic case in section 3. The off-diagonal elements of G < d in (98-99) are sourced by the diagonal elements through the terms involving commutators which are suppressed by at leasth with respect to the diagonal elements. This implies that, in order to capture the leading order nontrivial effect in gradients, we can work in the diagonal (semiclassical) approximation for G < d . By inspection of (98-99) we can now immediately write the constraint and kinetic equations in the diagonal approximation accurate to orderh as follows k 2 − M 2 d G < d = 0 (101) k · ∂ + 1 2 (∂M 2 d ) · ∂ k G < d = 0.(102) In contrast to the fermionic equivalent (68), these equations contain no CP-violating corrections to orderh and display only the usual classical CP-conserving term associated with the mass eigenvalues. This analysis is relevant for example for calculation of the CP-violating force in the stop sectorq = (t L ,t R ) T of the MSSM, in which the mass matrix reads M 2 q = m 2 Q y(A * H 2 + µH 1 ) y(AH 2 + µ * H 1 ) m 2 U ,(103) where m 2 Q and m 2 U denote the sum of the soft susy-breaking masses, including D-terms and m 2 t = y 2 H 2 2 . Our analysis immediately implies that for squarks there is no CP-violating correction to the dispersion relation at first order in gradients, and hence there is no CPviolating semiclassical force in the kinetic equation at orderh. Continuity equations and CP-violating sources The quantities eventually relevant for baryogenesis are CP-violating fluxes. We now investigate how the CP-violating sources appear in the equations for the divergences of the vector and axial vector currents. Let us first consider the divergence of the vector current. We have ∂ µ j µ = ∂ µ ψ (x)γ µ ψ(x) (104) = d 2 k (2π) 2 ∂ µ Tr(−iG < γ µ )(105) where the derivative in the integral expression is taken with respect to the center-of-mass coordinate. Using the decomposition (16) and the constraint equations to write g s 3i in terms of g s 0i (i is the flavour index), Eq. (16) is easily shown to reduce to just a momentum integral over the kinetic equation (69). Hence we get the usual continuity equation ∂ µ j µ si± ≡ ∂ t n si± + ∂ z (n si± u si± ) = 0,(106) showing that the vector current is conserved, and contains no sources. The fluid density n si± and velocity u si± ≡ v si± are defined as n si+ ≡ + d 2 k (2π) 2 g s 0i n si+ v p si+ ≡ + d 2 k (2π) 2 k z k 0 p g s 0i ,(107) where + ≡ k 0 ≥0 denotes integration over the positive frequencies. The fluid density n si± should not be confused with the phase space density n s in (38). To get the density and velocity moments for antiparticles one should integrate over the negative frequencies and make use of (38). Here we have again restricted ourselves to 1+1 dimensions; in 3+1 dimensions the expressions differ in detail, but not in essence [27]. No source appears in (106) simply because the semiclassical force term in (69) reduces to vanishing boundary terms at k z → ±∞. In the case of a single Dirac fermion this result remains valid to any order in gradients. In a general case of mixing fields the dynamics of off-diagonal elements of g s 0 may induce sources, which would however be higher than first order inh. Our proof that there are no CP-violating sources to the continuity equation for the vector current is contrary to the results of Refs. [37,17]. Our derivation is more general than that of [37] in that by treating mass as part of the flow term it includes the "mass resummation" of Refs. [37,17] to infinite order, it is not based on any particular Ansatz for g s 0d , and finally, but most importantly, we took correct account of the constraint equations. The fact that we have not treated collisions terms here does not resolve the differences, because the collisional contributions arise only from nonsingular loop diagrams which were not treated in Ref. [37] either. We next consider the continuity equation for the axial vector current. This can be obtained from Eq. (106) by replacing γ µ by γ µ γ 5 . The presence of γ 5 essentially changes the roles of g s 3 and g s 0 , so that the axial divergence reduces to an integral over the kinetic equation for g s 3 , which can be inferred from (54): ∂ t g s 3 + s∂ z g s 0 + 2 M I − 1 8 M ′′ I ∂ 2 kz g s 1 − 2 M R − 1 8 M ′′ R ∂ 2 kz g s 2 = 0.(108) It is thus easy to see that the divergence of the axial current acquires the expected form: ∂ µ j µ 5s = −2iM R ψ s γ 5 ψ s − 2M I ψ s ψ s ,(109) where ψ s ψ s = k g s 1 denotes the scalar, and ψ s γ 5 ψ s = i k g s 2 the pseudoscalar density, and k ≡ d 2 k/(2π) 2 . We now make use of the constraint equations (53-56) to express g s 1 , g s 2 and g s 3 in (108) in terms of g s 0 (to second order in gradients), diagonalize (108) by rotating into the propagating basis, where we can take g s 0d to be diagonal to orderh accuracy, and finally integrate over the momenta. We thus arrive at the following equation for the axial current divergence: ∂ t (n si± u si± ) + ∂ z n si± − \|M i \| 2 ∂ z + 1 2 \|M i \| 2 ′ I 2si± ± s \|M i \| 2 Θ ′ i ∂ z + 1 2 (\|M i \| 2 Θ ′ i ) ′ I 3si± = 0,(110) where we defined I psi+ = + d 2 k (2π) 2 g s 0i k p 0 = dk z 8πZ si± f si± ω p si± (p = 2, 3),(111) As usual, one has to introduce some truncation scheme to close the equations to two unknown quantities (here n si± and u si± ). There is of course some freedom as to how to do this step, and one might implement the truncation in (110) by replacing the distributions f si± in the I psi± -integrals by the equilibrium ones. However, it is instructive to observe that, by extracting a total derivative from the \|M i \| 2 -terms in (110), the corresponding integrals can be combined with the ∂ z n si± -term to give a second velocity moment term, in terms of which (110) becomes simply ∂ t (n si± u si± ) + ∂ z n si± v 2 si± = S si± ,(112) where the source S si± is given by the average over the semiclassical force (74) divided by k 0 : S si± = − 1 2 \|M i \| 2 ′ I 2si± ± 1 2 s(\|M i \| 2 Θ ′ i ) ′ I 3si± .(113) This equation can be truncated by the method standardly used for moment expansion. One writes v 2 si± → u 2 si± + σ 2 si± and uses the equilibrium distributions for f si± 's when evaluating the variance σ 2 si± ≡ v 2 si± − u 2 si± and the remaining integrals I psi± appearing in the source term (113). Remarkably, expressions (112-113) show that the divergence of the axial current in fact corresponds to the first velocity moment of the kinetic equation for g s 0d . Because a nontrivial CP-violation is tied to nontrivial complex phases in the pseudoscalar, or axial mass term, the fact that the source appears in the axial current nicely explains why the semiclassical source appears at first order in moment expansion in earlier semiclassical treatments [29,15,22]. The first source in (113) is to leading order in gradients spin-independent and does not violate CP. It is important for the phase transition dynamics however, in that it provides the dominant contribution to the friction on bubble walls from fermions [7,9]. The second source is spin-dependent and CP-violating and it is thus responsible for baryogenesis. This is one of the main results of this paper, as it shows how the source from the semiclassical force enters to momentum integrated transport equations used in practical calculations. To promote equations (106) and (112) into transport equations for baryogenesis calculations we still need to generalize them to include collisions as indicated in (46), which will be done elsewhere. Spontaneous source in relaxation time approximation We have so far shown how the source arising from the semiclassical force enters in the equations for currents, or equivalently momentum averaged transport equations. We shall now give a heuristic account on how sources have been modeled elsewhere in literature. ∂ t + v q · ∂ x + F q · ∂ k f q = − f q − f q0 τ q ,(114) where τ q ≡ Γ −1 q is the equilibration time for q (which we expect to be given by the relevant elastic scattering rate), F q the semiclassical force and f q0 is the thermal equilibrium distribution function. In presence of a background field that violates q, one expects f q0 to be shifted with respect to the naive thermal equilibrium, leading to a 'spontaneous' source that violates q. This source is more important for thick walls when the equilibrium f q ≈ f q0 is approximately attained on the wall. The spontaneous baryogenesis source was originally introduced by Cohen, Kaplan and Nelson [35,16] in the context of two Higgs doublet models, and then subsequently refined to include the m 2 -suppression in [41,15]. The derivation was successively reconsidered in [36,19,20]. For example, in [20] the CP-violating vector current j 0 q = (dk z /(2π))f q0 for charginos in the MSSM was computed and inserted into the transport equations written in the relaxation-time approximation. The spontaneous baryogenesis source can be in our formalism obtained simply by integrating (114) over the momenta. The source then becomes the CP-violating contribution to the vector current (44), which to first order in gradients (or equivalentlyh) reads n si ≡ j 0 si+ − j 0 si− ≈ s\|M i \| 2 Θ ′ i ω≥\|M i \| dω 4π 1 ω 2 − \|M i \| 2 f ω ω 2 1 + ω T (1 − f ω ) ,(115) where we approximated f si± by the equilibrium distribution function in plasma frame, f si± → 1/(e ω si± /T + 1) ≈ f ω ∓ (s\|M i \| 2 Θ ′ i /2ω 2 )df ω /dω, where f ω = 1/(e ω/T + 1), and we used k z dk z = ωdω. To make a comparison with literature, note first that the spontaneous source (115) is nonanalytic in \|M i \| 2 . Since earlier attempts [36,37,17,19] used expansions in powers of \|M i \| 2 to compute the spontaneous source, their results are at best incomplete. Consider next the CP-violating source for charginos in the MSSM. According to our equation (82) it is given by \|M d \| 2 Θ ′ = diag(m 2 + Θ ′ + , m 2 − Θ ′ − ), where m 2 ± Θ ′ ± = ∓(g 2 /Λ)ℑ(µm 2 )(h 1 h 2 ) ′ , showing the parametrical dependence (h 1 h 2 ) ′ on the higgs fields. This is in contrast with Refs. [19,20], where a source proportional to h 1 h ′ 2 − h 2 h ′ 1 was found and claimed to be important for baryogenesis. The origin of the difference may be in the fact that we made use of the constraint equations, which is necessary to obtain the correct results. Consider now the axial vector current. The corresponding spontaneous source can be easily obtained from (45): sj 0 5ds± = s kz =± s kz dω 8π f sd± = 0,(116) where we took f si± → 1/(e ω/T + 1). As a consequence, there is no spontaneous baryogenesis source from the axial vector current when computed in the relaxation time approximation. An attempt to compute the spontaneous baryogenesis source was made by Riotto [37], where the divergence of the vector current was computed in the Schwinger-Keldysh formalism [42] and then, based on [36], identified with a spontaneous source [17]. In this way he found an equation which formally reads: ∂ µ j µ q ∼ spontaneous source. According to Eq (106) however no source appears in the continuity equation for the vector current. Instead sources appear in the continuity equation for the axial vector current (110-112), which has not been so far considered in literature. Discussion and summary The question of a first principle derivation of CP-violating fluxes in transport equations has been the main theoretical challenge of recent work on electroweak baryogenesis. In this paper we derive the kinetic equations appropriate for EWBG in a systematic gradient expansion starting from the (Dirac) equation of motion for the two-point Wightman function G < in the collisionless limit. The gradient expansion we use is well controlled and corresponds to an expansion in the de Broglie wave length divided by the wall width. In EWBG applications one typically has ℓ dB /ℓ w ≪ 1, so that such an expansion should be rapidly converging. We have shown that to first order inh the collisionless kinetic equations for both fermions and bosons can be recast as the Liouville equations for a single particle distribution function where the group velocity and the semiclassical force terms contain all quantum information, and in particular the CP-violating terms which source baryogenesis. These results agree with Ref. [22], where the kinetic equations were obtained in the semiclassical WKB picture, originally developed for EWBG problem in [29,15]. The outstanding contribution of this paper is in a first principles derivation of these results in a completely controlled approximation scheme. We also derive the semiclassical force in the general case of N mixing fermions and in particular for the chargino sector in both the MSSM and NMSSM. Finally we prove that there is no CP-violating force at first order inh for scalar fields (102). Let us point out that the fact that the quasiparticle picture of plasma still holds to orderh in gradient expansion is not surprising since the gradient correction for fermions from a pseudoscalar (CP-violating) mass condensate can be equivalently reformulated in terms of a 'classical' axial vector field condensate [29,15]. We have also studied the vector and axial vector current equations, and showed that, while the vector current is conserved, the axial current contains both CP-conserving and CP-violating sources. This is to be expected, as CP-violation in particular is known to be caused by complex pseudoscalar (axial) mass terms. We have then pointed out that the axial current equation corresponds to the first velocity moment of the kinetic equation. This explains why the CP-violating source appears at first order in moment expansion [29,15]. We finally made connection between the present results and literature where the continuity equations were written in the relaxation time approximation. In this context the source has been claimed to appear either as the vector current divergence [37,17] or the time-component of the vector current j 0 [36,19,20]. However, we have shown here that the vector current continuity equation (106) in fact contains no source. We have also computed j 0 , and applied the result to the MSSM, and found a parametrically different result from Refs. [36,19,20]. For simplicity here we consider only the collisionless limit in 1+1 dimensions. One can show [27] that generalization to the 3+1 dimensional case does not affect our discussions in any qualitative way. The question of how to consistently include collisions we postpone to a future publication. defines the physical dispersion relations for particles and antiparticles of a given spin s. Due to the derivative corrections the spin degeneracy is lifted at first order in gradients and hence particles (antiparticles) of different spin experience different accelerations in a spatially varying background, as we shall see in more detail below. kz . The extra flavour structure of course complicates the solution of equation (49) and it turns out to be convenient to perform a rotation to the basis where the lowest order mass matrix is diagonal. Because M in general can be nonhermitean, the diagonalization requires a biunitary transformation The method very often used in EWBG considerations, apart from the WKB-computations,employs the relaxation time approximation for the kinetic equations. 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[ "New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints", "New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints" ]	[ "Matúš Benko \nInstitute of Computational Mathematics\nJohannes Kepler University Linz\nLinzAustria\n", "Helmut Gfrerer \nInstitute of Computational Mathematics\nJohannes Kepler University Linz\nLinzAustria\n" ]	[ "Institute of Computational Mathematics\nJohannes Kepler University Linz\nLinzAustria", "Institute of Computational Mathematics\nJohannes Kepler University Linz\nLinzAustria" ]	[ "OPTIMIZATION" ]	In this paper, we consider a sufficiently broad class of non-linear mathematical programs with disjunctive constraints, which, e.g. include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of Q-stationarity which can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called Q M -stationarity. We show how the property of Q M -stationarity (and thus also of M-stationarity) can be efficiently verified for the considered problem class by computing Q-stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for Q M -stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.ARTICLE HISTORY	10.1080/02331934.2017.1387547	null	8,620,184	1611.08206	390fc553206589c7b11ac0a6f773c4bdca61bebe	New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints 2018 Matúš Benko Institute of Computational Mathematics Johannes Kepler University Linz LinzAustria Helmut Gfrerer Institute of Computational Mathematics Johannes Kepler University Linz LinzAustria New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints OPTIMIZATION 671201810.1080/02331934.2017.1387547Received 22 November 2016 Accepted 16 September 2017OPEN ACCESSMathematical programs with disjunctive constraintsB-stationarityM-stationarityQ M -stationarity In this paper, we consider a sufficiently broad class of non-linear mathematical programs with disjunctive constraints, which, e.g. include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of Q-stationarity which can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called Q M -stationarity. We show how the property of Q M -stationarity (and thus also of M-stationarity) can be efficiently verified for the considered problem class by computing Q-stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for Q M -stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.ARTICLE HISTORY Introduction In this paper, we consider the following mathematical program with disjunctive constraints (MPDC) where the mappings f : R n → R and F i : R n → R l i , i = 1, . . . , m D are assumed to be continuously differentiable and D j i ⊂ R l i , j = 1, . . . , K i , i = 1, . . . , m D are convex polyhedral sets. Denoting m := m D i=1 l i , F := (F 1 , . . . , F m D ) : R n → R m , D := m D i=1 D i(2) we can rewrite the MPDC (1) in the form min x∈R n f (x) subject to F(x) ∈ D.(3) It is easy to see that D can also be written as the union of m D i=1 K i convex polyhedral sets by CONTACT Helmut Gfrerer [email protected] D = ν∈J D(ν) with J := m D i=1 {1, . . . , K i }, D(ν) := m D i=1 D ν i i .(4) As an example for MPDC consider a mathematical program with complementarity constraints (MPCC) given by min x∈R n f (x) subject to g i (x) ≤ 0, i = 1, . . . m I , h i (x) = 0, i = 1, . . . m E , G i (x) ≥ 0, H i (x) ≥ 0, G i (x)H i (x) = 0, i = 1, . . . m C(5) with f : R n → R, g i : R n → R, i = 1, . . . , m I , h i : R n → R, i = 1, . . . , m E , G i , H i : R n → R, i = 1, . . . , m C . This problem fits into our setting (1) with m D = m C + 1, F 1 = (g 1 , . . . , g m I , h 1 . . . , h m E ) T , D 1 1 = R m I − × {0} m E , l 1 = m I + m E , K 1 = 1 F i+1 = ( − G i , −H i ) T , D 1 i+1 = {0} × R − , D 2 i+1 = R − × {0}, l i+1 = K i+1 = 2, i = 1, . . . , m C . MPCC is known to be a difficult optimization problem, because, due to the complementarity constraints G i (x) ≥ 0, H i (x) ≥ 0, G i (x)H i (x) = 0,F i+1 = ( − H i , G i ) T , D 1 i+1 = {0} × R, D 2 i+1 = R 2 − , l i+1 = K i+1 = 2, i = 1, . . . , m V . Similar as in the case of MPCC, many of the standard constraint qualifications of non-linear programming can be violated at a local solution of (6) and a lot of stationarity concepts have been introduced. For a comprehensive overview for MPVC we refer to [10] and the references therein. However, when we do not formulate MPCC or MPVC as a non-linear program but as a disjunctive program MPDC, then first-order optimality conditions can be formulated which are valid under weak constraint qualifications. We know that a local minimizer is always B-stationary, which geometrically means that no feasible descent direction exists, or, in a dual formulation, that the negative gradient of the objective belongs to the regular normal cone of the feasible region, cf. [11,Theorem 6.12]. The difficult task is now to estimate this regular normal cone. For this regular normal cone always a lower inclusion is available, which yields so-called S-stationarity conditions. For an upper estimate, one can use the limiting normal cone which results in the so-called M-stationarity conditions. The notions of S-stationarity and M-stationarity have been introduced in [12] for general programs (3). S-stationarity always implies B-stationarity, but it requires some strong qualification condition on the constraints which is too restrictive. On the other hand, M-stationarity requires only some weak constraint qualification but it does not preclude the existence of feasible descent directions. Further, it is not known in general how to efficiently verify the M-stationarity conditions, since the description of the limiting normal cone involves some combinatorial structure which is not known to be resolved without enumeration techniques. These difficulties in verifying M-stationarity have also some impact for numerical solution procedures. E.g. for many algorithms for MPCC it cannot be guaranteed that a limit point is M-stationary, cf. [13]. In the recent paper [14], we derived another upper estimate for the regular normal cone yielding so-called Q-stationarity conditions. Q-stationarity has the advantage over S-stationarity that it does not require such unnecessarily strong constraint qualification conditions. Q-stationarity can be easily combined with M-stationarity to obtain so-called Q M -stationarity which is stronger than Mstationarity. This is one of the advantages of Q M -stationarity: there are several stationarity notions, in particular in the MPCC literature, like M-, C-, A-and weak stationarity, which are valid under weak constraint qualification conditions. M-stationarity is known to be the strongest stationarity concept and we even improve M-stationarity by Q M -stationarity. For the disjunctive formulations of the problems MPCC and MPVC the Qand Q M -stationarity conditions have been worked out in detail in [14]. In this paper, we extend this approach to the general problem MPDC. We show that under a qualification condition which ensures S-stationarity of local minimizers, Q-stationarity and S-stationarity are equivalent. Further, we prove that under some weak constraint qualification every local minimizer of MPDC is a Q M -stationary solution and we provide an efficient algorithm for verifying Q M -stationarity of some feasible point. More exactly, this algorithm either proves the existence of some feasible descent direction, i.e. the point is not B-stationary, or it computes multipliers fulfilling the Q M -stationarity condition. To this end, we consider quadratic programs with disjunctive constraints (QPDC), i.e. the objective function f in MPDC is a convex quadratic function and the mappings F i , i = 1, . . . , m D are linear. We propose a basic algorithm for QPDC, which either returns a Q-stationary point or proves that the problem is unbounded. Further, we show that M-stationarity for MPDC is related with Q-stationarity of some QPDC and the combination of the two parts yields the algorithm for verifying Q M -stationarity. This algorithm does not rely on enumeration techniques and this is another big advantage of the concepts of Qand Q M -stationarity. Our approach is well suited to the MPDC (1) when all the numbers K i , i = 1, . . . , m D are small or of moderate size. Our disjunctive structure is not induced by integral variables like, e.g. in [15]. It is also not related to the approach of considering the convex hull of a family of convex sets like in [16,17]. The outline of the paper is as follows. In Section 2, we recall some basic definitions from variational analysis and discuss various stationarity concepts. In Section 3, we introduce the concepts of Qand Q M -stationarity for general optimization problems. These concepts are worked out in more detail for MPDC in Section 4. In Section 5, we consider quadratic programs with disjunctive linear constraints. We present a basic algorithm for solving such problems, which either return a Q-stationary solution or prove that the problem is not bounded below. In the next section, we demonstrate how this basic algorithm can be applied to a certain quadratic program with disjunctive linear constraints in order to verify M-stationarity or Q M -staionarity of a point or to compute a descent direction. In the last Section 7, we present some results for numerical methods for solving MPDC which prevent convergence to non M-stationary and non-Q M -stationary points. Our notation is fairly standard. In Euclidean space R n we denote by · and ·, · the Euclidean norm and scalar product, respectively, whereas we denote by u ∞ := max{\|u i \| \| i = 1, . . . , n} the maximum norm. The closed ball around some point x with radius r is denoted by B(x, r). Given some cone Q ⊂ R n , we denote by Q • := {q * ∈ R n \| q * , q ≤ 0∀q ∈ Q} its polar cone. By d(x, A) := inf { x − −y \| y ∈ A} we refer to the usual distance of some point x to a set A. We denote by 0 + C the recession cone of a convex set C. Preliminaries For the reader's convenience, we start with several notions from variational analysis. Given a set ⊂ R n and a pointz ∈ , the cone T (z) = {w \| ∃w k → w, t k ↓ 0 withz + t k w k ∈ } is called the (Bouligand/Severi) tangent/contingent cone to atz. The (Fréchet) regular normal cone to atz ∈ can be equivalently defined either by N (z) := v * ∈ R d \| lim sup z →z v * , z −z z −z ≤ 0 , where z →z means that z →z with z ∈ , or as the dual/polar to the contingent cone, i.e. by N (z) := T (z) • . For convenience, we putN (z) := ∅ forz / ∈ . Further, the (Mordukhovich) limiting/basic normal cone to atz ∈ is given by N (z) := w * ∈ R d \| ∃ z k →z, w * k → w * with w * k ∈ N (z k ) for all k . If is convex, then both the regular and the limiting normal cones coincide with the normal cone in the sense of convex analysis. Therefore, we will use in this case the notation N . Consider now the general mathematical program min x∈R n f (x) subject to F(x) ∈ D(7) where f : R n → R, F : R n → R m are continuously differentiable and D ⊂ R m is a closed set. Let := {x ∈ R n \| F(x) ∈ D}(8) denote the feasible region of the program (7). Then a necessary condition for a pointx ∈ being locally optimal is ∇f (x), u ≥ 0 ∀u ∈ T (x),( 9 ) which is the same as −∇f (x) ∈ N (x),(10) cf. [11,Theorem 6.12]. The main task of applying this first-order optimality condition now is the computation of the regular normal cone N (x) which is very difficult for nonconvex D. We always have the inclusion ∇F(x) T N D (F(x)) ⊂ N (x),(11) but equality will hold in (11) for nonconvex sets D only under comparatively strong conditions, e.g. when ∇F(x) is surjective, see [11,Exercise 6.7]. The following weaker sufficient condition for equality in (11) uses the notion of metric subregularity. Definition 1: A multifunction : R n ⇒ R m is called metrically subregular at a point (x,ȳ) of its graph gph with modulus κ > 0, if there is a neighborhood U ofx such that d(x, −1 (ȳ)) ≤ κd(ȳ, (x)) ∀x ∈ U.T D (F(x)) + L ⊂ T D (F(x))(12) and ∇F(x)R n + L = R m ,(13) then N (x) = ∇F(x) T N D (F(x)). In order to state an upper estimate for the regular normal cone N (x) we need some constraint qualification. Definition 2 [12, Definition 6]: Let be given by (8) and letx ∈ . (1) We say that the generalized Abadie constraint qualification (GACQ) holds atx if T (x) = T lin (x),( 1 4 ) where T lin (x) := {u ∈ R n \| ∇F(x)u ∈ T D (F(x))} denotes the linearized cone. (2) We say that the generalized Guignard constraint qualification (GGCQ) holds atx if (T (x)) • = (T lin (x)) • .(15) Obviously GGCQ is weaker than GACQ, but GACQ is easier to verify by using some advanced tools of variational analysis. E.g. if the mapping x ⇒ F(x) − D is metrically subregular at (x, 0) then GACQ is fulfilled atx, cf. [19,Proposition 1]. Tools for verifying metric subregularity of constraint systems can be found e.g. in [20]. Proposition 1 [14, Proposition 3]: Let be given by (8), letx ∈ and assume that GGCQ is fulfilled, while the mapping u ⇒ ∇F(x)u − T D (F(x)) is metrically subregular at (0, 0). Then N (x) ⊂ ∇F(x) T N T D (F(x)) (0) ⊂ ∇F(x) T N D (F(x)).(16) Note that we always have N T D (F(x)) (0) ⊂ N D (F(x)), see [11,Proposition 6.27]. However, if D is the union of finitely many convex polyhedral sets, then equality N T D (F(x)) (0) = N D (F(x))(17) holds. This is due to the fact that by the assumption on D there is some neighborhood V of 0 such that (D − F(x)) ∩ V = T D (F(x)) ∩ V . Let us mention that metric subregularity of the constraint mapping x ⇒ F(x) − D at (x, 0) does not only imply GACQ and consequently GGCQ, but also metric subregularity of the mapping u ⇒ ∇F(x)u − T D (F(x)) at (0, 0) with the same modulus, see [21,Proposition 2.1]. The concept of metric subregularity has the drawback that, in general, it is not stable under small perturbations. It is well known that the stronger property of metric regularity is robust. Definition 3: A multifunction : R n ⇒ R m is called metrically regular near a point (x,ȳ) of its graph gph with modulus κ > 0, if there are neighborhoods U ofx and V ofȳ such that d(x, −1 (y)) ≤ κd(y, (x)) ∀(x, y) ∈ U × V . The infimum of the moduli κ for which the property of metric regularity holds is denoted by reg (x,ȳ). In the following proposition, we gather some well-known properties of metric regularity: Proposition 2: Letx ∈ F −1 (D) where F : R n → R m is continuously differentiable and D is the union of finitely many convex polyhedral sets and consider the multifunctions x ⇒ (x) := F(x) − D and u ⇒ D (x)(u) := ∇F(x)u − T D (F(x)). Then reg (x,ȳ) = reg D (x)(0, 0) = max 1 ∇F(x) T λ \| λ ∈ N D (F(x)) = N T D (F(x)) (0), λ = 1 . Moreover for every κ > reg (x,ȳ) there is a neighborhood W ofx such that for all x ∈ W the mapping u ⇒ ∇F(x)u − T D (F(x)) is metrically regular near (0, 0) with modulus κ, λ ≤ κ ∇F(x) T λ ∀λ ∈ N D (F(x)) = N T D (F(x)) (0)(18) and d(u, ∇F(x) −1 T D (F(x))) ≤ κd(∇F(x)u, T D (F(x))) ∀u ∈ R n . Proof: The statement follows from [11,Exercise 9.44] together with the facts that by our assumption on D condition (17) holds and that T D (F(x)) is a cone. We now recall some well known stationarity concepts based on the considerations above. Definition 4: Letx be feasible for the program (7). (i) We say thatx is B-stationary, if (9) or, equivalently, (10) hold. (ii) We say thatx is S-stationary, if −∇f (x) ∈ ∇F(x) T N D (F(x)). (iii) We say thatx is M-stationary, if −∇f (x) ∈ ∇F(x) T N D (F(x)). Every local minimizer of (7) is B-stationary and this stationarity concept is considered to be the most preferable one. S-and M-stationarity have been introduced in [12] as a generalization of these notions for MPCC. Using the inclusion (5) it immediately follows, that S-stationarity implies B-stationarity. However the reverse implication only holds true under some additional condition on the constraints, e.g. under the assumptions of Theorem 1. Note that there always hold the inclusions ∇F(x) T N D (F(x)) ⊂ T lin (x) • ⊂ T (x) • = N (x). In order that a B-stationary point is also S-stationarity, both inclusions must be fulfilled with equality, i.e. besides the GGCQ T lin (x) • = T (x) • which allows to replace the tangent cone by the linearized tangent cone, we need another constraint qualification condition ensuring (12) and (13). It is well known that this additional condition is much more restrictive than the usual constraint qualifications allowing the linearization of the problem like metric (sub)regularity of the constraint mapping F( · ) − D. Thus in the general case one cannot expect that a local minimizer is also S-stationary. This is the reason why other stationarity concepts like M-stationarity have also to be considered. A B-stationary point is M-stationary under the very weak assumptions of Proposition 1. However, ∇F(x) T N D (F(x)) = T lin (x) • like the conditionsthe inclusion N (x) ⊂ ∇F(x) T N D (F(x)) can be strict, implying that a M-stationary pointx needs not to be B-stationary. Hence, M-stationarity does eventually not preclude the existence of feasible descent directions, i.e. directions u ∈ T (x) with ∇f (x), u < 0. On Qand Q M -stationarity In this section, we consider an extension of the concept of Q-stationarity as introduced in the recent paper [14]. Q-stationarity is based on the following simple observation. Consider the general program (7), assume that GGCQ holds at the pointx ∈ and assume that we are given K convex cones Q i ⊂ T D (F(x)), i = 1, . . . , K. Then for each i = 1, . . . , K we obviously have T lin (x) = ∇F(x) −1 T D (F(x)) ⊃ ∇F(x) −1 Q i implying N (x) = (T lin (x)) • ⊂ (F(x) −1 Q i ) • . If we further assume that (F(x) −1 Q i ) • = ∇F(x) T Q • i and by taking into account, that by [14, Lemma 1] we have (∇F(x) T S 1 ) ∩ (∇F(x) T S 2 ) = ∇F(x) T S 1 ∩ ( ker ∇F(x) T + S 2 ) for arbitrary sets S 1 , S 2 ⊂ R m , we obtain N (x) ⊂ K i=1 ∇F(x) T Q • i = ∇F(x) T Q • 1 ∩ ( ker ∇F(x) T + Q • 2 ) ∩ K i=3 ∇F(x) T Q • i = ∇F(x) T Q • 1 ∩ ( ker ∇F(x) T + Q • 2 ) ∩ ( ker ∇F(x) T + Q • 3 ) ∩ K i=4 ∇F(x) T Q • i = . . . = ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) . Here, we use the convention that for sets S 1 , . . . , S K ⊂ R m we set K i=l S i = R m for l > K. It is an easy consequence of (11), that equality holds in this inclusion, provided ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) ⊂ ∇F(x) T N D (F(x). Hence, we have shown the following theorem. Theorem 2: Assume that GGCQ holds atx ∈ and assume that Q 1 , . . . , Q K are convex cones contained in T D (F(x)). If (∇F(x) −1 Q i ) • = ∇F(x) T Q • i , i = 1, . . . , K,(19) then N (x) ⊂ ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) = K i=1 ∇F(x) T Q • i .(20) Further, if ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) ⊂ ∇F(x) T N D (F(x)),(21) then equality holds in (20) and N (x) = ∇F(x) T N D (F(x)). Remark 1: Condition (19) is e.g. fulfilled, if for each i = 1, . . . , K either there is a direction u i with ∇F(x)u i ∈ ri Q i or Q i is a convex polyhedral set, cf. [14, Proposition 1]. The proper choice of Q 1 , . . . , Q K is crucial in order that (20) provides a good estimate for the regular normal cone. It is obvious that we want to choose the cones Q i , i = 1, . . . , K as large as possible in order that the inclusion (20) is tight. Further, it is reasonable that a good choice of Q 1 , . . . , Q K fulfills K i=1 Q • i = N D (F(x))(22) because then equation (21) holds whenever ∇F(x) has full rank. We now show that (21) holds not only under this full rank condition but also under some weaker assumption. Theorem 3: Assume that GGCQ holds atx ∈ and assume that we are given convex cones (19), (22) and Q 1 , . . . , Q K ⊂ T D (F(x)) fulfillingker ∇F(x) T ∩ (Q • 1 − Q • i ) = {0}, i = 2, . . . , K.(23) Then N (x) = ∇F(x) T N D (F(x)) = ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) . In particular, (23) holds if there is a subspace L ⊂ K i=1 Q i ∩ ( − Q i )(24) such that (13) holds. Proof: The statement follows from Theorem 2 if we can show that (21) holds. Consider x * ∈ ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) . Then there are elements λ i ∈ Q • i , i = 1, . . . , K and μ i ∈ ker ∇F(x) T such that λ 1 = μ i + λ i , i = 2, . . . , K and x * = ∇F(x) T λ 1 . We conclude μ i = λ 1 − λ i ∈ Q • 1 − Q • i , implying μ i ∈ ker ∇F(x) T ∩ (Q • 1 − Q • i ) = {0} and thus λ 1 = λ 2 = . . . = λ K ∈ K i=1 Q • i = N D (F(x)). Hence, x * ∈ ∇F(x) T N D (F(x)) and (21) is verified. In order to show the last assertion note that from (24), we conclude L ⊂ Q i and consequently Q • i ⊂ L • = L ⊥ . Thus Q • 1 − Q • i ⊂ L ⊥ − L ⊥ = L ⊥ , i = 2, . . . , K. Since ker ∇F(x) T ∩ L ⊥ = ( ker ∇F(x) T ) ⊥ + L ⊥ = (∇F(x)R n + L) ⊥ = {0}, it follows that (23) holds. Corollary 1: Assume that GGCQ holds atx ∈ and assume that we are given convex cones (19) and (22). Further assume that there is some subspace L fulfilling (12) and (13). Then, the setsQ Q 1 , . . . , Q K ⊂ T D (F(x)) fulfillingi := Q i + L, i = 1, . . . , K are convex cones contained in T D (F(x)), (∇F(x) −1Q i ) • = ∇F(x) TQ• i , i = 1, . . . , K(25) and (12). Next, consider z ∈ riQ i . By (13) there exists u ∈ R n and l ∈ L such that ∇F( N (x) = ∇F(x) T N D (F(x)) = ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T +Q • i ) . Proof: Firstly observe thatQ i = Q i + L ⊂ T D (F(x)) + L ⊂ T D (F(x)) byx)u + l = z. Because of −l ∈ L ⊂Q i we have z − 2l ∈Q i and thus ∇F(x)u = z − l = 1 2 z + 1 2 (z − 2l) ∈ riQ i by [22, Theorem 6.1] implying (25) by taking into account Remark 1. Further, from Q i ⊂Q i ⊂ T D (F(x)) it follows that N D (F(x)) = (T D (F(x))) • ⊂ K i=1Q • i ⊂ K i=1 Q • i = N D (F(x)). Finally, note that L ⊂Q i ∩ ( −Q i ), i = 1, . . . , K and the assertion follows from Theorem 3. The following definition is motivated by Theorem 2. Definition 5: Letx be feasible for the program (7) and let Q 1 , . . . , Q K be convex cones contained in T D (F(x)) fulfilling (19). (i) We say thatx is Q-stationary with respect to Q 1 , . . . , Q K , if −∇f (x) ∈ ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) . (ii) We say thatx is Q M -stationary with respect to Q 1 , . . . , Q K , if −∇f (x) ∈ ∇F(x) T N D (F(x)) ∩ Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) . Note that this definition is an extension of the definition of Qand Q M -stationarity in [14], where only the case K = 2 was considered. The following corollary is an immediate consequence of the definitions and Theorem 2. Corollary 2: Assume that GGCQ is fulfilled at the pointx feasible for (7). Further assume that we are given convex cones Q 1 , . . . , Q K ⊂ T D (F(x)) fulfilling (19). Ifx is B-stationary, thenx is Q-stationary with respect to Q 1 , . . . , Q K . Conversely, ifx is Q-stationary with respect to Q 1 , . . . , Q K and (21) is fulfilled, thenx is S-stationary and consequently B-stationary. We know that under the assumptions of Proposition 1 every B-stationary pointx for the problem (7) is both M-stationary and Q-stationary with respect to every collection of cones Q 1 , . . . , Q K ⊂ T D (F(x)) fulfilling (19), i.e. −∇f (x) ∈ ∇F(x) T N D (F(x)) ∩ ∇F(x) T Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) = ∇F(x) T ker ∇F(x) T + N D (F(x)) ∩ Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) . Comparing this relation with the definition of Q M -stationarity we see that Q M -stationarity with respect to Q 1 , . . . , Q K is stronger than the simultaneous fulfilment of M-stationarity and Q-stationarity with respect to Q 1 , . . . , Q K . We refer to [14,Example 2] for an example which shows that Q Mstationarity is strictly stronger than M-stationarity. This is one of the advantages of Q M -stationarity: However, to ensure Q M -stationarity of a B-stationary pointx, some additional assumption has to be fulfilled. Lemma 1: Letx be B-stationary for the program (7) and assume that the assumptions of Proposition 1 are fulfilled atx. Further assume that for every (19) the pointx is Q M -stationary with respect to Q 1 , . . . , Q K . (19). Similar to the derivation of Theorem 2 we obtain λ ∈ N T D (F(x)) (0) there exists a convex cone Q λ ⊂ T D (F(x)) containing λ and satisfying (∇F(x) −1 Q λ ) • = ∇F(x) T Q • λ . Then there exists a convex cone Q 1 ⊂ T D (F(x)) fulfilling (∇F(x) −1 Q 1 ) • = ∇F(x) T Q • 1 such that for every collection Q 2 , . . . , Q K ⊂ T D (F(x)) fulfilling Proof: From the definition of B-stationarity and (16) we deduce the existence of λ ∈ N T D (F(x)) (0) fulfilling −∇f (x) = ∇F(x) T λ. By taking Q 1 = Q λ we obviously have λ ∈ N T D (F(x)) (0) ∩ Q • 1 ⊂ N D (F(x)) ∩ Q • 1 implying −∇f (x) ∈ ∇F(x) T (N D (F(x)) ∩ Q • 1 ). Now consider cones Q 2 , . . . , Q K ⊂ T D (F(x)) fulfilling−∇f (x) ∈ ∇F(x) T (N D (F(x)) ∩ Q • 1 ) ∩ K i=2 ∇F(x) T Q • i = ∇F(x) T N D (F(x)) ∩ Q • 1 ∩ K i=2 ( ker ∇F(x) T + Q • i ) and the lemma is proved. Lemma 2: Letx be feasible for (7) and assume that T D (F(x)) is the union of finitely many closed convex cones C 1 , . . . , C p . Then for every λ ∈ N T D (F(x)) (0) there is someī ∈ {1, . . . , p} satisfying λ ∈ C • i . Proof: Consider λ ∈ N T D (F(x)) (0). By the definition of the limiting normal cone there are sequences t k T D (F(x)) −→ 0 and λ k → λ with λ k ∈ N T D (F(x)) (t k ) = ⎛ ⎝ i:t k ∈C i T C i (t k ) ⎞ ⎠ • = i:t k ∈C i (T C i (t k )) • = i:t k ∈C i N C i (t k ). By passing to a subsequence if necessary we can assume that there is an indexī such that t k ∈ C¯i for all k and we obtain λ k ∈ N C¯i (t k ) = {c * ∈ C • i \| c * , t k = 0} ⊂ C • i . Since the polar cone C • i is closed, we deduce λ ∈ C • i . If T D (F(x)) is the union of finitely many convex polyhedral cones C 1 , . . . , C p , then the mapping u ⇒ ∇F(x)u − T D (F(x)) is a polyhedral multifunction and thus metrically subregular at (0, 0) by Robinson's result [23]. Further we know that for any convex polyhedral cone Q we have (∇F(x) −1 Q) • = ∇F(x) T Q • . Hence, we obtain the following corollary. (7), that GGCQ is fulfilled atx and that T D (F(x)) is the union of finitely many convex polyhedral cones. Then there is a convex polyhedral cone Q 1 ⊂ T D (F(x)) such that for every collection Q 2 , . . . , Q K of convex polyhedral cones contained in T D (F(x)) the pointx is Q M -stationary with respect to Q 1 , . . . , Q K . Corollary 3: Assume thatx is B-stationary for the program Let us notice that in contrast to S-, M-and many other types of stationarity the properties of Qand Q M -stationarity cannot be characterized by some single multiplier. In fact, Q-and Q M -stationarity with respect to Q 1 , . . . , Q K implies the existence of K multipliers λ 1 , . . . , λ K satisfying λ i ∈ Q • i , ∇f (x) + ∇F(x) T λ i = 0, i = 1, . . . , K. In case of Q M -stationarity the multiplier λ 1 also fulfills the M-stationarity conditions. Further, let us note that Q M -stationarity, although it is stronger that M-stationarity, does not imply B-stationarity in general. Thus, in general Q M -stationarity is not a sufficient condition for a local minimizer as well. Application to MPDC It is clear that Q-stationarity is not a very strong optimality condition for every choice of Q 1 , . . . , Q K ⊂ T D (F(x)). As mentioned above the fulfillment of (22) is desirable. For the general problem (7), it can be impossible to choose the cones Q 1 , . . . , Q K such that (22) holds. If T D (F(x)) is the union of finitely many convex cones C 1 , . . . , C p then we obviously have N D (F(x)) = p i=1 C • i . However, to consider Q-stationarity with respect to C 1 , . . . , C p is in general not a feasible approach because p is often very large. We will now work out that the concepts of Qand Q M -stationarity are tailored for the MPDC (1). In what follows let D and F be given by (2). Given a point y = (y 1 , . . . , y m D ) ∈ D, we denote by A i (y) := {j ∈ {1, . . . , K i } \| y i ∈ D j i }, i = 1, . . . , m D the indices of sets D j i which contain y i . Further we choose for each i = 1, . . . , m D some index set J i (y) ⊂ A i (y) such that T D i (y i ) = j∈J i (y) T D j i (y i ).(26) Obviously the choice J i (y) = A i (y) is feasible but for practical reasons it is better to choose J i (y) smaller if possible. E.g. if T D j 2 i (y i ) ⊂ T D j 1 i (y i ) holds for some indices j 1 , j 2 ∈ A i (y) , then we will not include j 2 in J i (y). Such a situation can occur e.g. in case of MPVC when ( − H i (x), G i (x)) = (0, a) with a < 0. Now consider ν ∈ J (y) := m D i=1 J i (y). Since for every i = 1, . . . , m D the set D i is the union of finitely many convex polyhedral sets, for every tangent direction t ∈ T D i (y i ) we have y i + αt ∈ D i for all α > 0 sufficiently small. Hence, we can apply [24, Proposition 1] to obtain T D(ν) (y) = m D i=1 T D ν i i (y i ), ν ∈ J (y) with D(ν) given by (4), and T D (y) = m D i=1 T D i (y i ) = m D i=1 j∈J i (y) T D j i (y i ) = ν∈J (y) T D(ν) (y).(27) We will apply this setting in particular to points y = F(x) withx feasible for MPDC. Lemma 3: Letx be feasible for the MPDC (1) and assume that we are given K elements ν 1 , . . . , ν K ∈ J (F(x)) such that {ν 1 i , . . . , ν K i } = J i (F(x)), i = 1, . . . , m D .(28) Then for each l = 1, . . . , K the cone Q l := T D(ν l ) (F(x)) is a convex polyhedral cone contained in T D (F(x)), ∇F(x) −1 Q l • = ∇F(x) T Q • l , and K l=1 Q • l = N D (F(x)). Proof: Obviously, for every l = 1, . . . , K the cone Q l is convex and polyhedral because it is the product of convex polyhedral cones. This implies ∇F(x) −1 Q l • = ∇F(x) T Q • l and Q l ⊂ T D (F(x)) follows from (27). By taking into account (27) the last assertion follows from (1) with respect to some (ν 1 , . . . , ν K ) ∈ Q(x). N D (F(x)) = T D (F(x)) • = m D i=1 j∈J i (F(x)) T D j i (F i (x)) • = m D i=1 K l=1 T D ν l i i (F i (x)) • = m D i=1 K l=1 T D ν l i i (F i (x)) • = K l=1 m D i=1 T D ν l i i (F i (x)) • = K l=1 m D i=1 T D ν l i i (F i (x)) • = K l=1 Q • l . Definition 6: Letx be feasible for the MPDC (1) and let index sets J i (F(x)) ⊂ A i (x), i = 1, . . . , m D fulfilling (26) be given. Further we denote by Q(x) the collection of all elements (ν 1 , . . . , ν K ) with ν l ∈ J (F(x)) = m D i=1 J i (F(x)), l = 1, . . . , Definition 6 is an extension of the definition of Qand Q M -stationarity made for MPCC and MPVC in [14]. Note that the number K appearing in the definition of Q(x) is not fixed. Denoting K min (x) the minimal number K such that (ν 1 , . . . , ν K ) ∈ Q(x), we obviously have K min (x) = max i=1,...,m D \|J i (F(x))\| ≤ max i=1,...,m D K i .(29) We see from (27) y). Hence, the minimal number K min (x) is much smaller than the number of components of the tangent cone, except when all or nearly all sets J i (F(x)) have cardinality 1. E.g. when K i ≤ 2 holds for all i = 1, . . . , m D as it is the case of the MPCC (5), then we have K min (x) ≤ 2 whereas the number of convex cones building the tangent cone T D (F(x)) grows exponentially with the number of biactive constraints, i.e. complementarity constraints satisfying G i (x) = H i (x) = 0. The concepts of Qand Q M -stationarity for MPDC take advantage of the fact that although the tangent cone is the union of a huge number of cones, its polar cone can be written as the intersection of a small number of polars. Further, it is clear that for every ν 1 ∈ J (F(x)) and every K ≥ K min (x) we can find ν 2 , . . . , ν K ∈ J (F(x)) such that (ν 1 , . . . , ν K ) ∈ Q(x). that the tangent cone T D (F(x)) is the union of the \|J (F(x))\| = m D i=1 \|J i (F(x))\| convex polyhedral cones T D(ν) ( We allow K to be greater than K min (x) for numerical reasons primarily. Recall that for testing Q-stationarity with respect to (ν 1 , . . . , ν K ), we have to check for all l = 1, . . . , K whether −∇f (x) ∈ ∇F(x) T Q • l , or equivalently, that u = 0 is a solution of the linear optimization program min ∇f (x), u subject to ∇F(x)u ∈ Q l with Q l = T D(ν l ) (F(x)). Theoretically the treatment of degenerated linear constraints is not a big problem but the numerical practice tells us the contrary. In [25] we have implemented an algorithm for solving MPVC based on Q-stationarity and the degeneracy of the linear constraints was the reason when the algorithm crashed. The possibility of choosing K > K min (x) gives us more flexibility to avoid linear programs with degenerated constraints. The following theorem follows from Corollaries 2, 3, Theorem 3 and the considerations above. Theorem 4: Letx be feasible for the MPDC (1) and assume that GGCQ is fulfilled atx. (i) Ifx is B-stationary thenx is Q-stationary with respect to every element (ν 1 , . . . , ν K ) ∈ Q(x) and there exists someν 1 ∈ J (F(x)) such thatx is Q M -stationary with respect to every (ν 1 , ν 2 , . . . , ν K ) ∈ Q(x). (ii) Conversely, ifx is Q-stationary with respect to some (ν 1 , . . . , ν K ) ∈ Q(x) and ∇F(x) T Q • 1 ∩ K l=2 ( ker ∇F(x) T + Q • l ) ⊂ ∇F(x) T N D (F(x)),(30) where Q l := T D(ν l ) F(x), l = 1, . . . , K, thenx is S-stationary and consequently B-stationary. In particular, (30) is fulfilled if ker ∇F(x) T ∩ Q • 1 − Q • l = {0}, l = 2, . . . , K.(31) On quadratic programs with disjunctive constraints In this section, we consider the special case of quadratic programs with disjunctive constraints (QPDC) min x∈R n q(x) := 1 2 x T Bx + d T x (32) subject to A i x ∈ D i := K i j=1 D j i , i = 1, . . . , m D , where B is a positive semidefinite n × n matrix, d ∈ R n , A i , i = 1, . . . , m D are l i × n matrices and D j i ⊂ R l i , i = 1, . . . , m D , j = 1, . . . , K j are convex polyhedral sets, i.e. QPDC is a special case of MPDC with f (x) = q(x) and F i (x) = A i x, i = 1, . . . , m D . In what follows, we denote by A the m × n matrix A = ⎛ ⎜ ⎝ A 1 . . . A m D ⎞ ⎟ ⎠ , where m := m D i=1 l i . We start our analysis with the following preparatory lemma. Lemma 4: Assume that the convex quadratic program min x∈R n 1 2 x T Bx + d T x subject to Ax ∈ C (33) is feasible, where B is some symmetric positive semidefinite n × n matrix, d ∈ R n , A is a m × n matrix and C ⊂ R m is a convex polyhedral set. Then either there exists a direction w satisfying Bw = 0, Aw ∈ 0 + C, d T w < 0,(34) or the program (33) has a global solutionx. Proof: Assume that for every w with Bw = 0, Aw ∈ 0 + C we have d T w ≥ 0, i.e. 0 is a global solution of the program min d T w subject to w ∈ S := w \| B A w ∈ {0} n × 0 + C . Since C is a convex polyhedral set, its recession cone 0 + C is a convex polyhedral cone and so is {0} n × 0 + C as well. Hence, N S (0) = S • = (B T . . .A T )({0} n × 0 + C) • = B T R n + A T (0 + C) • and from the first-order optimality condition −d ∈ N S (0) we derive the existence of multipliers μ B ∈ R n and μ C ∈ (0 + C) • such that −d = B T μ B + A T μ C . The convex polyhedral set C is the sum of the convex hull of its extreme points and its recession cone. Hence, for every x feasible for (33) there is some c 1 ∈ and some c 2 ∈ 0 + C such that Ax = c 1 + c 2 and, by taking into account μ T C c 2 ≤ 0, we obtain 1 2 x T Bx + d T x = 1 2 x T Bx − μ T B Bx − μ T C Ax = 1 2 (x − μ B ) T B(x − μ B ) − 1 2 μ T B Bμ B − μ T C c 1 − μ T C c 2 (35) ≥ − 1 2 μ T B Bμ B − μ T C c 1 . The set is compact and we conclude that the objective of (33) is bounded below on the feasible domain A −1 C by − 1 2 μ T B Bμ B − max c 1 ∈ μ T C c 1 . Thus α := inf 1 2 x T Bx + d T x \| Ax ∈ C is finite and there remains to show that the infimum is attained. Consider some sequence x k ∈ A −1 C with lim k→∞ 1 2 x T k Bx k + d T x k = α. We conclude from (35) that (x k − μ B ) T B(x k − μ B ) is bounded which in turn implies that the sequence B 1/2 x k is bounded. Hence, the sequence x T k Bx k = B 1/2 x k 2 is bounded as well and we can conclude also the boundedness of d T x k . By passing to a subsequence we can assume that the sequence (B 1/2 x k , d T x k ) converges to some (z, β) and it follows that α = In what follows, we assume that we have at hand an algorithm for solving (33), which either computes a global solutionx or a descent direction w fulfilling (34). Such an algorithm is, e.g. the active set method as described in [26], where we have to rewrite the constraints equivalently in the form A T a i , x ≤ b i , i = 1, . . . , p using the representation of C as the intersection of finitely many half-spaces, C = {c \| a i , c ≤ b i , i = 1, . . . , p}. Consider now the following algorithm. Algorithm 1 (Basic algorithm for QPDC): Input: starting point x 1 feasible for the QPDC (32). (1) Set the iteration counter k := 1. (2) Select (ν k,1 , . . . , ν k,K ) ∈ Q(x k ) and consider for l = 1, . . . , K the quadratic programs (QP k,l ) min q(x) subject to Ax ∈ D(ν k,l ). If one of these programs is unbounded below, stop the algorithm and return the current iterate x k together withν := ν k,l and the descent direction w fulfilling (34). Otherwise let x k,l , l = 1, . . . , K denote the global solutions of (QP k,l ). (3) If q(x k ) = q(x k,l ), l = 1, . . . , K, stop the algorithm and return x k together withν := ν k,1 . (4) Choose l ∈ {1, . . . , K} with q(x k,l ) < q(x k ), set x k+1 = x k,l , increase the iteration counter k := k + 1 and go to step (2) Algorithm 1 can be considered as a kind of active index set strategy. The set (ν k,1 , . . . , ν k,K ) ∈ Q(x k ) chosen in step (2) acts as a working set and is a subset of the active pieces of the disjunctive constraints. The number K will also depend on x k and for practical reasons it is desirable to keep K small to have small numerical effort in each iteration. Recall that we can always choose K equal to the number K min (x k ) given by (29) which is bounded by max i=1,...,m D K i . The working set is used for testing for unboundedness of the problem and Q-stationarity, respectively, by investigating the quadratic subproblems (QP k,l ). If one of these subproblem's problem appears unbounded, we stop the algorithm because of unboundedness of the whole program. If x k is a solution for every quadratic subproblem, we stop the algorithm because x k is Q-stationary. On the other hand, if x k is not a solution of one of these subproblems, then we take the point x k+1 as the solution of this subproblem, yielding a smaller objective function value. Then, we repeat the whole procedure by generating a new working set and testing for termination. In the next theorem, we show that Algorithm 1 is finite. However, we do not know any nontrivial bound on the number of iterations needed, as usual for active set strategies. Theorem 5: Algorithm 1 terminates after a finite number of iterations either with some feasible point and some descent direction w indicating that QPDC is unbounded below or with some Q-stationary solution. Proof: If Algorithm 1 terminates in step (2) the output is a feasible point together with some descent direction showing that QPDC is unbounded below. If the algorithm does not terminate in step (2), the computed sequence of function values q(x k ) is strictly decreasing. Moreover, denoting ν k := ν k−1,l where l is the index chosen in step (4), we see that for each k ≥ 2 the point x k is global minimizer of the problem min q(x) subject to Ax ∈ D(ν k ). This shows that all the vectors ν k must be pairwise different and since there is only a finite number of possible choices for ν k , the algorithm must stop in step (3). We will now show that the final iterate x k is Q-stationary with respect to (ν k,1 , . . . , ν k,K ). Since for each l = 1, . . . , K the point x k is a global minimizer of the subproblem (Q k,l ), it also satisfies the first order optimality condition ∇q(x k ), u ≥ 0 for every u ∈ R n satisfying Au ∈ T D(ν k,l ) (Ax k )). This shows Q-stationarity of x k and the theorem is proved. On verifying Q M -stationarity for MPDC The following theorem is crucial for the verification of M-stationarity. Theorem 6: (i) Letx be feasible for the general program (7). If there exists a B-stationary solution of the program min (u,v)∈R n ×R m ∇f (x), u + 1 2 v 2 subject to ∇F(x)u + v ∈ T D (F(x)),(36) thenx is M-stationary. (ii) Letx be B-stationary for the MPDC (1) and assume that GGCQ holds atx. Then the program (36) has a global solution. Proof: (F(x)). Since the matrix (∇F(x) . . . I) obviously has full rank, we have N (ū,v) = (∇F(x) . . . (i) Let (ū,v) denote a B-stationary solution, i.e. −(∇f (x),v) ∈ N (ū,v), where = (∇F(x) . . . I) −1 T D I) T N T D (F(x)) (∇F(x)ū +v) by [11, Exercise 6.7]. Thus there exists a multiplier λ ∈ N T D (F(x)) (∇F(x)ū+v) such that −∇f (x) = ∇F(x) T λ and −v = λ. Using [11, Proposition 6.27] we have N T D (F(x)) (∇F(x)ū +v) ⊂ N T D (F(x)) (∇F(x)ū +v) ⊂ N T D (F(x)) (0) ⊂ N D (F(x)) establishing M-stationarity ofx. (ii) Consider for arbitrarily fixed ν ∈ J (F(x)) the convex quadratic program min (u,v)∈R n ×R m ∇f (x), u + 1 2 v 2 subject to ∇F(x)u + v ∈ T D(ν) (F(x)).(37) Assuming that this quadratic program does not have a solution, by Lemma 4 we could find a direction (w u , w v ) satisfying 0 0 0 I w u w v = 0, ∇F(x)w u + w v ∈ 0 + T D(ν) (F(x)), ∇f (x), w u + 0, w v < 0. This implies w v = 0, ∇F(x)w u ∈ 0 + T D(ν) (F(x)) = T D(ν) (F(x)) ⊂ T D (F(x) ) and ∇f (x), w u < 0 and thus, together with GGCQ, −∇f (x) ∈ (T lin (x)) • = N D (F(x)) contradicting our assumption thatx is B-stationary for (1). Hence, the quadratic program (37) must possess some global solution (u ν , v ν ). vν) is a global solution of (36). By choosingν ∈ J (F(x)) such that ∇f (x), uν = min{ ∇f (x), u ν \| ν ∈ J (F(x))} it follows from (27) that (uν, We now want to apply Algorithm 1 to the problem (36). Note that the point (0, 0) is feasible for (36) and therefore we can start Algorithm 1 with (u 1 , v 1 ) = (0, 0). . . . I) obviously has full rank. Now the statement follows from Theorem 6. We now want to analyse how the output of Algorithm 1 can be further utilized. Recalling that T D (F(x)) has the disjunctive structure T D (F(x)) = m D i=1 j∈J i (F(x)) T D j i (F i (x)) , we define for y = (y 1 , . . . , y m D ) ∈ T D (F(x)) the index sets A T D i (y) := {j ∈ J i (F(x)) \| y i ∈ T D j i (F i (x))}, i = 1, . . . , m D . Further we choose for each i = 1, . . . , m D some index set J T D i (y) ⊂ A T D i (y) such that T T D i (F i (x)) (y i ) = j∈J T D i (y) T T D j i (F i (x)) (y i )(38) and set J T D (y) := m D i=1 J T D i (y). Note that we always have J T D (y) ⊂ J (F(x)). In order to verify Q-stationarity for the problem (36) at some feasible point (u, v), we have to consider the set Q T D (u, v) consisting of all (ν 1 , . . . , ν K ) with ν l ∈ J T D (∇F(x)u + v), l = 1, . . . , K such that {ν 1 i , . . . , ν K i } = J T D i (∇F(x)u + v), i = 1, . . . , m D . At the k-th iterate (u k , v k ) we have to choose (ν k,1 , . . . , ν k,K ) ∈ Q T D (u k , v k ) and then for each l = 1, . . . , K we must analyse the convex quadratic program (QP k,l ) min u,v ∇f (x), u + 1 2 v 2 subject to ∇F(x)u + v ∈ T D(ν k,l ) (F(x)). If for somel ∈ {1, . . . , K} this quadratic program is unbounded below then Algorithm 1 returns the indexν := ν k,l together with a descent direction (w u , w v ) fulfilling, as argued in the proof of Theorem 6(ii), w v = 0, ∇F(x)w u ∈ 0 + T D(ν) (F(x)) = T D(ν) (F(x)), ∇f (x), w u < 0. Therefore, w u constitutes a feasible descent direction, provided GACQ holds atx, i.e. for every α > 0 sufficiently small the projection ofx + αw u on the feasible set F −1 (D) yields a point with a smaller objective function value thanx. If GACQ also holds for the constraint F(x) ∈ D(ν) atx, then we can also project the pointx + αw u on F −1 (D(ν)) in order to reduce the objective function. Now, assume that the final iterate (u k , v k ) of Algorithm 1 is Q-stationary for (36) and consequentlȳ x is M-stationary for the MPDC (1). Setting λ := −v k , the first order optimality conditions for the quadratic programs (QP k,l ) result in 1 is the index vector returned from Algorithm 1. Now choosing ν 2 , . . . , ν K such that (ν, ν 2 , . . . , ν K ) ∈ Q(x) we can simply check by testing −∇f (x) ∈ N D(ν l ) (F(x)), l = 2, . . . , K, whetherx is Q M stationary orx is not B-stationary. −∇f (x) = ∇F(x) T λ, λ ∈ K l=1 N T D(ν k,l ) (F(x)) (∇F(x)u k + v k ) = N T D (F(x)) (∇F(x)u k + v k ) ⊂ N D (F(x)). From this we conclude −∇f (x) ∈ ∇F(x) T (Q • 1 ∩N D (F(x)) with Q 1 := T D(ν) (F(x)) ⊂ T T D(ν) (F(x)) (∇F(x)u k + v k ) whereν = ν k, Further, we have the following corollary. Corollary 5: Letx be B-stationary for the MPDC (1) and assume that GGCQ is fulfilled atx. Letν be the index vector returned by Algorithm 1 applied to (36). Thenν ∈ J (F(x)) and for every ν 2 , . . . , ν K with (ν, ν 2 , . . . , ν K ) ∈ Q(x) the pointx is Q M -stationary with respect to (ν, ν 2 , . . . , ν K ). Numerical aspects In practice, the pointx which should be checked for M-stationarity and Q M -stationarity, respectively, often is not known exactly. E.g.x can be the limit point of a sequence generated by some numerical method for solving MPDC. Hence, let us assume that we are given some pointx close tox and we want to state some rules when we can considerx as approximately M-stationary or Q M -stationary. Let us assume that the convex polyhedral sets D j i have the representation D j i = {y \| a i,j l , y ≤ b i,j l , l = 1, . . . , p i,j }, i = 1, . . . , m D , j = 1, . . . , K i , where without loss of generality we assume a i,j l = 1. We use here the following approach. T j i (x, ) = {v \| a i,j l , v ≤ 0, l ∈Ĩ j i (x, )}, i = 1, . . . , m D , j ∈Ã i (x, ). Assume thatÃ i (x, ) = ∅, i = 1, . . . , m D . (2) Consider QPDC(x, , σ ) min u,v ∇f (x), u + σ 2 u 2 + 1 2 v 2 subject to ∇F(x)u + v ∈ m D i=1 j∈Ã i (x, ) T j i (x, ) . Let (ũ,ṽ) andν denote the output of Algorithm 1 applied to QPDC(x, , σ ). (3) If σ ũ > η consider the nonlinear programming problem min f (x) subject to F(x) ∈ D(ν) in order to improvex. (4) Otherwise considerx as approximately M-stationary and compute ν 2 , . . . , ν K ∈ m D i=1Ã i (x, ) such that Tν i i (x, ) ∪ K l=2 T ν l i i (x, ) = j∈Ã i (x, ) T j i (x, ), i = 1, . . . , m D . If min ∇f (x)u \| ∇F(x)u ∈ m D i=1 T ν l i i (x, ), −1 ≤ u i ≤ 1, i = 1, . . . , n ≥ −η, l = 2, . . . , K,(39) acceptx as approximately Q M -stationary. Otherwise consider the nonlinear programming problem min f (x) subject to F(x) ∈ D(ν¯l) in order to improvex, wherel ∈ {2, . . . , K} denotes some index violating (39). In the first step of Algorithm 2, we want to estimate the tangent cone T D (F(x)). In fact, to calculate T D (F(x)) we need not to know the point F(x), we only need the index sets of constraints active atx and these index sets are approximated by -active constraints. Note that wheneverÃ i (x, ) =Ã i (x, 0) = A i (F(x)) andĨ j i (x, ) =Ĩ j i (x, 0), i = 1, . . . , m D , j ∈ A i (F(x)) this approach yields the exact tangent cones T D j i (F(x)) = T j i (x, ) for all i = 1, . . . , m D , j ∈ A i (F(x)). To be consistent with the notation of Section 4 we make the convention that in this case the index vectorν computed in step (2) belongs to J (x) and also, whenever we determine ν 2 , . . . ν K is step (4), we have (ν, ν 2 , . . . , ν K ) ∈ Q(x). The regularization term σ 2 u 2 in QPDC(x, , σ ) forces the objective to be strictly convex and therefore Algorithm 1 will always terminate with a Q-stationary solution. Further note that the verification of (39) requires the solution of K − 1 linear optimization problems. The following theorem justifies Algorithm 2. In the sequel, we denote by M(x) (M sub (x)) the set of all ν ∈ J (x) such that the mapping F( · ) − D(ν) is metrically regular near (x, 0) (metrically subregular at (x, 0)). Theorem 7: Letx be feasible for the MPDC (1) and assume that ∇f and ∇F are Lipschitz nearx. Consider sequences x t →x, t ↓ 0, σ t ↓ 0 and η t ↓ 0 with lim t→∞ x t −x t = lim t→∞ σ t η t + x t −x η t = 0 and let (ũ t ,ṽ t ),ν t and eventually ν t,2 . . . , ν t,K t andl t denote the output of Algorithm 2 with input data (x t , t , σ t , η t ). (i) For all t sufficiently large and for all i ∈ {1, . . . , m D } we havẽ A i (x t , t ) = A i (F(x)),Ĩ j i (x t , t ) =Ĩ j i (x, 0), j ∈ A i (F(x)).(40)l t ∈ M sub (x) we have min{f (x) \| F(x) ∈ D(ν t,l t )} < f (x). implying − ∇f (x t ),ũ t = σ t ũ t 2 + ṽ t 2 . Hence, σ t ũ t ≤ − ∇f (x t ),ũ t ũ t ≤ ∇f (x t )(43) and from (41) we obtain ṽ t = λ t ≤ 2(κ + 1) ∇f (x t ) .(44) (a) Assume on the contrary thatx is B-stationary but for infinitely many t the point x t is not accepted as approximately M-stationary and hence ũ t ≥ η t /σ t . This implies d(∇F(x)ũ t ũ t , T D (F(x))) ≤ d(∇F(x t )ũ t ũ t , T D (F(x))) + L x t −x ≤ ṽ t ũ t + L x t −x ≤ 2(κ + 1) f (x t ) σ t η t + L x t −x and by the metric regularity of u ⇒ ∇F(x)u−T D (F(x)) near (0, 0) we can findû t ∈ ∇F(x) −1 T D (F(x)) with û t −ũ t ũ t ≤ κ 2(κ + 1) f (x t ) σ t η t + L x t −x . Our choice of the parameters σ t , η t together with (43) ensures that for t sufficiently large we have ∇f (x),û t ≤ ∇f (x),ũ t ũ t + ∇f (x) û t −ũ t ũ t ≤ ∇f (x t ),ũ t ũ t + L x t −x + ∇f (x) û t −ũ t ũ t ≤ −σ t ũ t + L x t −x + ∇f (x) û t −ũ t ũ t ≤ −η t + L x t −x + ∇f (x) κ 2(κ + 1) f (x t ) σ t η t + L x t −x < 0 which contradicts B-stationarity ofx. Hence, for all t sufficiently large the point x t must be accepted as approximately M-stationary. To prove the statement that x t is also accepted as approximately Q M -stationary for all t sufficiently large we can proceed in a similar way. Assume on the contrary thatx is B-stationary but for infinitely many t the point x t is not accepted as approximately Q M -stationary. For those t, let w t denote some element fulfilling ∇F(x t )w t ∈ T D(ν t,l t ) ⊂ T D (F(x)), w t ∞ ≤ 1 and ∇f (x t ), w t ≤ −η t . Then, similar as before we can findŵ t ∈ ∇F(x) −1 T D (F(x)) such that ŵ t − w t ≤ κ ∇F(x) − ∇F(x t ) w t ≤ κL √ n x t −x and for large t we obtain ∇f (x),ŵ t ≤ ∇f (x t ), w t + ∇f (x) − ∇f (x t ) w t + ∇f (x) ŵ t − w t ≤ −η t + L √ n(1 + κ ∇f (x) ) x t −x < 0 contradicting B-stationarity ofx. (b) By passing to a subsequence we can assume that for all t the point x t is accepted as approximately M-stationary and hence σ t u t ≤ η t → 0. By (44) we have that the sequence λ t ∈ N T D (F(x)) (0) is uniformly bounded and by passing to a subsequence once more we can assume that it converges to someλ ∈ N T D (F(x)) (0). By [11,Proposition 6.27] we haveλ ∈ N D (F(x)) and together with 0 = lim t→∞ ∇f (x t ) + ∇F(x t ) T λ t = ∇f (x) + ∇F(x) Tλ M-stationarity ofx is established. (c) By passing to a subsequence we can assume that for all t the point x t is accepted as approximately Q M -stationary and {ν t , ν t,2 , . . . , ν t,K t } ⊂ M(x). Hence, for all t the point x t is also accepted as Mstationary and by passing to a subsequence and arguing as in (b) we can assume that λ t converges to someλ ∈ N D (F(x)) fulfilling ∇f (x) + ∇F(x) Tλ = 0. Since the set M(x) is finite, by passing to a subsequence once more we can assume that there is a number K and elementsν, ν 2 , . . . , ν K such that K t = K ,ν t =ν and ν t,l = ν l , l = 2, . . . , K holds for all t. Since we assume that (40) holds we have (ν, ν 2 , . . . , ν K ) ∈ Q(x) and we will now show thatx is Q M -stationary with respect to (ν, ν 2 , . . . , ν K ). Since (ũ t ,ṽ t ) also solves (42), it follows that λ t = −v t ∈ N T D(ν) (F(x)) (∇F(x t )ũ t +ṽ t ) ⊂ N D(ν) (F(x)) and thusλ ∈ N D (F(x)) ∩ N D(ν) (F(x)) implying −∇f (x) ∈ ∇F(x) T N D (F(x)) ∩ T D(ν) (F(x)) • . There remains to show −∇f (x) ∈ T D(ν l ) (F(x)) • = N D(ν l ) (F(x)), l = 2, . . . , K. Assume on the contrary that −∇f (x) ∈ T D(ν¯l) (F(x)) • for some indexl ∈ {2, . . . , K}. Then there is some u ∈ ∇F(x) −1 T D(ν¯l) (F(x)), u ∞ = 1 2 such that ∇f (x), u =: −γ < 0 and since ν¯l ∈ M(x), for each t there is someû t ∈ ∇F(x t ) −1 T D(ν¯l) (F(x)) with u −û t ≤ (κ + 1) ∇F(x) − ∇F(x t ) u ≤ √ n 2 (κ + 1)L x t −x . It follows that for all t sufficiently large we have û t ∞ ≤ 1 and ∇f (x t ),û t ≤ ∇f (x), u + ∇f (x t ) − ∇f (x) û t + ∇f (x) u −û t ≤ −γ + L √ n 1 + κ + 1 2 x t −x < −η t contradicting our assumption that x t is accepted as approximately Q M -stationary. (d), (e) We assume that κ is chosen large enough such that the mappings F(·)−D(ν), ν ∈ M sub (x) are metrically subregular at (x, 0) with modulus κ. Then by [21, Proposition 2.1] the mappings u ⇒ ∇F(x)u − T D(ν) (F(x)), ν ∈ M sub (x) are metrically subregular at (0, 0) with modulus κ as well. Taking into account that (ũ t ,ṽ t ) solves (42), we can copy the arguments from part (a) with T D (F(x)) replaced by T D(ν t ) (F(x)) to show the existence ofû t ∈ ∇F(x) −1 T D(ν t ) (F(x)) with ∇f (x),û t < 0 whenever x t is not accepted as approximately M-stationary and t is sufficiently large. In doing so, we also have to recognize that metric regularity of u ⇒ ∇F(x)u − T D(ν t ) (F(x)) can be replaced by the weaker property of metric subregularity. Sinceν t ∈ M sub (x),û t is a feasible descent direction and for sufficiently small α > 0 the projection ofx + αû t on F −1 (D(ν t )) yields a point with a smaller objective function value thanx. This proves (d). In order to show (e), we can proceed in a similar way. Using the same arguments as in part (a), we can prove the existence of a feasible directionŵ t ∈ T D(ν t,l t ) with ∇f (x),ŵ t < 0, whenever t is sufficiently large and x t is not accepted as approximately Q M -stationary. Together with ν t,l t ∈ M sub (x) the assertion follows. Disclosure statement No potential conflict of interest was reported by the authors. i = 1, . . . , m D , many of the standard constraint qualifications of nonlinear programming are violated at any feasible point. Hence, it is likely that the usual Karush-Kuhn-Tucker conditions fail to hold at a local minimizer and various first-order optimality conditions such as Abadie (A-), Bouligand (B-), Clarke (C-), Mordukhovich (M-) and Strong (S-) stationarity conditions have been studied in the literature [1-9]. Another prominent example is the mathematical program with vanishing constraints (MPVC) min x∈R n f (x) subject to g i (x) ≤ 0, i = 1, . . . m I ,h i (x) = 0, i = 1, . . . m E , H i (x) ≥ 0, G i (x)H i (x) ≤ 0, i = 1, . . . m V(6)with f : R n → R, g i : R n → R, i = 1, . . . , m I , h i : R n → R, i = 1, . . . , m E , G i , H i : R n → R, i = 1, . . . , m V . Again, the problem MPVC can be written in the form (1) with m D = m V + 1, F 1 , D 1 1 as in the case of MPCC and K such that (28) holds. (1) We say thatx is Q-stationary (Q M -stationary) for (1) with respect to (ν 1 , . . . , ν K ) ∈ Q(x), ifx is Q-stationary (Q M -stationary) with respect to Q 1 , . . . , Q K in the sense of Definition 5 with Q l := T D(ν l ) (F(x)), l = 1, . . . , K. (2) We say thatx is Q-stationary (Q M -stationary) for (1) ifx is Q-stationary (Q M -stationary) for Corollary 4 : 4Letx be feasible for the MPDC (1) and apply Algorithm 1 to the QPDC (36). If the algorithm returns an iterate together with some descent direction indicating that (36) is unbounded below and if GGCQ is fulfilled atx, thenx is not B-stationary. On the other hand, if the algorithm returns a Q-stationary solution, thenx is M-stationary. Proof: Observe that in case when Algorithm 1 returns a Q-stationary solution, by Theorem 4(ii) this solution is B-stationary because the Jocobian of the constraints (∇F(x) Algorithm 2 : 2Input: A pointx and small positive parameters , σ , η.(1) Calculate the index setsA i (x, ) := {j ∈ {1, . . . , K i } \| d(F i (x), D j i ) ≤ }, i = 1, . . . , m D I j i (x, ) := {l ∈ {1, . . . , p i,j } \| a i,j l , F i (x) ≥ b i,j l − }, i = 1, . . . , m D , j ∈Ã i (x, )and the convex polyhedral cones ( ii ) iiAssume that the mapping x ⇒ F(x) − D is metrically regular near (x, 0). (a) Ifx is B-stationary then for all t sufficiently large the point x t is accepted as approximately M-stationary and approximately Q M -stationary. (b) If for infinitely many t the point x t is accepted as approximately M-stationary thenx is M-stationary. (c) If for infinitely many t the point x t is accepted as approximately Q M -stationary and {ν t , ν t,2 , . . . , ν t,K t } ⊂ M(x) thenx is Q M -stationary. (d) For every t sufficiently large such that the point x t is not accepted as approximately Mstationary andν t ∈ M sub (x) we have min{f (x) \| F(x) ∈ D(ν t )} < f (x). (e) For every t sufficiently large such that the point x t is not accepted as approximately Q Mstationary and ν t, FundingProof: (i) Let R > 0 be chosen such that f , F and their derivatives are Lipschitz on B(x, R) with constant L. It is easy to see that we can choose > 0 such that for all i ∈ {1, . . . , m D } we havẽ A i (x, ) =Ã i (x, 0) = A i (F(x)) and such that for every j ∈ A i (F(x)) we haveĨ0). Because of our assumptions we have x t −x < R and L x t −x < t < /2 for all t sufficiently large and this proves (40).(ii) In view of Proposition 2, we can choose κ large enough such that the mappingsF(x)), ν ∈ M(x) are metrically regular near (x, 0) with modulus κ. By eventually shrinking R we can assume that for every x ∈ B(x, R) the mappings u ⇒ ∇F(x)u−T D (F(x)), u ⇒ ∇F(x)u−T D(ν) (F(x)), ν ∈ M(x) are metrically regular near (0, 0) with modulus κ + 1.Without loss of generality we can assume that x t ∈ B(x, R) and (40) holds for all t implying that(F(x)). In fact, then the problem QPDC(x t , t , σ t ) is the same asThe point (ũ t ,ṽ t ) is Q-stationary for this program and thus also S-stationary by Theorem 4(ii) and the full rank property of the matrix (∇F(x t ) . . . I). Hence, there is a multiplierfrom(18). By Q-stationarity of (ũ t ,ṽ t ) we know that (ũ t ,ṽ t ) is the unique solution of the strictly convex quadratic program min ∇f (x t ), u + σ t 2 u 2 + 1 2 v 2 subject to ∇F(x t )u + v ∈ T D(ν t ) (F(x)).For every α ≥ 0, the point α(ũ t ,ṽ t ) is feasible for this quadratic program and thus α = 1 is solution of min α≥0 α ∇f (x t ),ũ t + α 2 σ t 2 ũ t 2 + 1 2 ṽ t John approach to first order optimality conditions for mathematical programs with equilibrium constraints. M L Flegel, C Kanzow, Fritz, Optimization. 52Flegel ML, Kanzow C. A Fritz John approach to first order optimality conditions for mathematical programs with equilibrium constraints. Optimization. 2003;52:277-286. Complementarity constraint qualifications and simplified B-stationary conditions for mathematical programs with equilibrium constraints. M Fukushima, J S Pang, Comput Optim Appl. 13Fukushima M, Pang JS. Complementarity constraint qualifications and simplified B-stationary conditions for mathematical programs with equilibrium constraints. Comput Optim Appl. 1999;13:111-136. Mathematical programs with equilibrium constraints: enhanced Fritz John conditions, new constraint qualifications and improved exact penalty results. C Kanzow, A Schwartz, SIAM J Optim. 20Kanzow C, Schwartz A. Mathematical programs with equilibrium constraints: enhanced Fritz John conditions, new constraint qualifications and improved exact penalty results. SIAM J Optim. 2010;20:2730-2753. Optimality conditions for a class of mathematical programs with equilibrium constraints. J V Outrata, Math Oper Res. 24Outrata JV. Optimality conditions for a class of mathematical programs with equilibrium constraints. Math Oper Res. 1999;24:627-644. A generalized mathematical program with equilibrium constraints. J V Outrata, SIAM J Control Optim. 38Outrata JV. A generalized mathematical program with equilibrium constraints. SIAM J Control Optim. 2000;38:1623-1638. Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. H Scheel, S Scholtes, Math Oper Res. 25Scheel H, Scholtes S. Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math Oper Res. 2000;25:1-22. Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. J J Ye, SIAM J Optim. 10Ye JJ. Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J Optim. 2000;10:943-962. Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J J Ye, J Math Anal Appl. 307Ye JJ. Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J Math Anal Appl. 2005;307:350-369. Necessary optimality conditions for optimization problems with variational inequality constraints. J J Ye, X Y Ye, Math Oper Res. 22Ye JJ, Ye XY. Necessary optimality conditions for optimization problems with variational inequality constraints. Math Oper Res. 1997;22:977-997. Mathematical programs with vanishing constraints. T Hoheisel, Julius-Maximilians-Universität WürzburgPhD thesisHoheisel T. Mathematical programs with vanishing constraints [PhD thesis], Julius-Maximilians-Universität Würzburg; 2009. Variational analysis. R T Rockafellar, Rj-B Wets, SpringerBerlinRockafellar RT, Wets RJ-B. Variational analysis. Berlin: Springer; 1998. Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. M L Flegel, C Kanzow, J V Outrata, 15Flegel ML, Kanzow C, Outrata JV. Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 2007;15:139-162. The price of inexactness: convergence properties of relaxation methods for mathematical programs with equilibrium constraints revisited. C Kanzow, A Schwartz, Math Oper Res. 40Kanzow C, Schwartz A. The price of inexactness: convergence properties of relaxation methods for mathematical programs with equilibrium constraints revisited. Math Oper Res. 2015;40:253-275. On estimating the regular normal cone to constraint systems and stationary conditions. M Benko, H Gfrerer, Optimization. 66Benko M, Gfrerer H. On estimating the regular normal cone to constraint systems and stationary conditions. Optimization. 2017;66:61-92. Representability in mixed integer programming, I: characterization results. R Jeroslow, Discrete Appl Math. 17Jeroslow R. Representability in mixed integer programming, I: characterization results. Discrete Appl Math. 1977;17:223-243. Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. 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[ "Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19 Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19", "Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19 Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19" ]	[ "MdSorwer Alam ", "Parvez 1& \nDepartment of Genetic Engineering and Biotechnology\nShahjalal University of Science and Technology\nSylhetBangladesh\n", "Kazi Faizul Azim \nFaculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh\n\nDepartment of Microbial Biotechnology\nSylhet Agricultural University\nSylhetBangladesh\n", "3&Abdus Shukur Imran \nFaculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh\n\nDepartment of Pharmaceuticals and Industrial Biotechnology\nSylhet Agricultural University\nSylhetBangladesh\n", "4&Topu Raihan \nDepartment of Genetic Engineering and Biotechnology\nShahjalal University of Science and Technology\nSylhetBangladesh\n", "Aklima Begum \nFaculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh\n", "Tasfia Saiyara Shammi \nFaculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh\n", "Sabbir Howlader \nDepartment of Applied Chemistry and Chemical Engineering\nUniversity of Chittagong\nChittagongBangladesh\n", "Farhana Rumzum Bhuiyan \nDepartment of Botany\nUniversity of Chittagong\nChittagongBangladesh\n\nDepartment of Botany\nDepartment of Pharmaceuticals and Industrial Biotechnology Faculty of Biotechnology and Genetic Engineering\nLaboratory of Biotechnology and Molecular biology\nUniversity of Chittagong\nChittagongBangladesh\n\nSylhet Agricultural University\nSylhetBangladesh\n", "Mahmudul Hasan \nFaculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh\n\nDepartment of Pharmaceuticals and Industrial Biotechnology\nSylhet Agricultural University\nSylhetBangladesh\n", "Assistant ProfessorMahmudul Hasan " ]	[ "Department of Genetic Engineering and Biotechnology\nShahjalal University of Science and Technology\nSylhetBangladesh", "Faculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh", "Department of Microbial Biotechnology\nSylhet Agricultural University\nSylhetBangladesh", "Faculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh", "Department of Pharmaceuticals and Industrial Biotechnology\nSylhet Agricultural University\nSylhetBangladesh", "Department of Genetic Engineering and Biotechnology\nShahjalal University of Science and Technology\nSylhetBangladesh", "Faculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh", "Faculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh", "Department of Applied Chemistry and Chemical Engineering\nUniversity of Chittagong\nChittagongBangladesh", "Department of Botany\nUniversity of Chittagong\nChittagongBangladesh", "Department of Botany\nDepartment of Pharmaceuticals and Industrial Biotechnology Faculty of Biotechnology and Genetic Engineering\nLaboratory of Biotechnology and Molecular biology\nUniversity of Chittagong\nChittagongBangladesh", "Sylhet Agricultural University\nSylhetBangladesh", "Faculty of Biotechnology and Genetic Engineering\nSylhet Agricultural University\nSylhetBangladesh", "Department of Pharmaceuticals and Industrial Biotechnology\nSylhet Agricultural University\nSylhetBangladesh" ]	[]	Covid-19, a serious respiratory complications caused by SARS-CoV-2 has become one of the global threat to human healthcare system. The present study evaluated the possibility of plant originated approved 117 therapeutics against the main protease protein (MPP), RNA-dependent RNA polymerase (RdRp) and spike protein (S) of SARS-CoV-2 including drug surface analysis by using molecular docking through drug repurposing approaches. The molecular interaction study revealed that Rifampin (-16.3 kcal/mol) were topmost inhibitor of MPP where Azobechalcone were found most potent plant therapeutics for blocking the RdRp (-15.9 kcal /mol) and S (-14.4 kcal/mol) protein of SARS-CoV-2. After the comparative analysis of all docking results, Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine and Fangchinoline were exhibited as the most potential inhibitory plant compounds for targeting the key proteins of SARS-CoV-2. However, amino acid positions; H41, C145, and M165 of MPP played crucial roles for the drug surface interaction where F368, L371, L372, A375, W509, L514, Y515 were pivotal for RdRP. In addition, the drug interaction surface of S proteins also showed similar patterns with all of its maximum inhibitors. ADME analysis also strengthened the possibility of screened plant therapeutics as the potent drug candidates against SARS-C with the highest drug friendliness.	null	null	218,863,112	2005.11254	36d6e068e62b358929ccba0b849a51cbc4f2a736	Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19 Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19 MdSorwer Alam Parvez 1& Department of Genetic Engineering and Biotechnology Shahjalal University of Science and Technology SylhetBangladesh Kazi Faizul Azim Faculty of Biotechnology and Genetic Engineering Sylhet Agricultural University SylhetBangladesh Department of Microbial Biotechnology Sylhet Agricultural University SylhetBangladesh 3&Abdus Shukur Imran Faculty of Biotechnology and Genetic Engineering Sylhet Agricultural University SylhetBangladesh Department of Pharmaceuticals and Industrial Biotechnology Sylhet Agricultural University SylhetBangladesh 4&Topu Raihan Department of Genetic Engineering and Biotechnology Shahjalal University of Science and Technology SylhetBangladesh Aklima Begum Faculty of Biotechnology and Genetic Engineering Sylhet Agricultural University SylhetBangladesh Tasfia Saiyara Shammi Faculty of Biotechnology and Genetic Engineering Sylhet Agricultural University SylhetBangladesh Sabbir Howlader Department of Applied Chemistry and Chemical Engineering University of Chittagong ChittagongBangladesh Farhana Rumzum Bhuiyan Department of Botany University of Chittagong ChittagongBangladesh Department of Botany Department of Pharmaceuticals and Industrial Biotechnology Faculty of Biotechnology and Genetic Engineering Laboratory of Biotechnology and Molecular biology University of Chittagong ChittagongBangladesh Sylhet Agricultural University SylhetBangladesh Mahmudul Hasan Faculty of Biotechnology and Genetic Engineering Sylhet Agricultural University SylhetBangladesh Department of Pharmaceuticals and Industrial Biotechnology Sylhet Agricultural University SylhetBangladesh Assistant ProfessorMahmudul Hasan Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19 Virtual Screening of Plant Metabolites against Main protease, RNA-dependent RNA polymerase and Spike protein of SARS-CoV-2: Therapeutics option of COVID-19 & authors contributed equally to this work *Corresponding Author Mobile: 008801723698461Plant TherapueticsSARS-CoV-2COVID-19Molecular DockingDrug Repurposing Covid-19, a serious respiratory complications caused by SARS-CoV-2 has become one of the global threat to human healthcare system. The present study evaluated the possibility of plant originated approved 117 therapeutics against the main protease protein (MPP), RNA-dependent RNA polymerase (RdRp) and spike protein (S) of SARS-CoV-2 including drug surface analysis by using molecular docking through drug repurposing approaches. The molecular interaction study revealed that Rifampin (-16.3 kcal/mol) were topmost inhibitor of MPP where Azobechalcone were found most potent plant therapeutics for blocking the RdRp (-15.9 kcal /mol) and S (-14.4 kcal/mol) protein of SARS-CoV-2. After the comparative analysis of all docking results, Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine and Fangchinoline were exhibited as the most potential inhibitory plant compounds for targeting the key proteins of SARS-CoV-2. However, amino acid positions; H41, C145, and M165 of MPP played crucial roles for the drug surface interaction where F368, L371, L372, A375, W509, L514, Y515 were pivotal for RdRP. In addition, the drug interaction surface of S proteins also showed similar patterns with all of its maximum inhibitors. ADME analysis also strengthened the possibility of screened plant therapeutics as the potent drug candidates against SARS-C with the highest drug friendliness. Introduction COVID-19 pandemic situation has tremendously turned the entire world into a place of horrible death tragedy. SARS-CoV-2, initially named as 2019 novel coronavirus (2019-nCoV) by the World Health Organization (WHO) is the causative agent of the recent serious respiratory complications resulting the COVID-19 pandemic [1,2,3,4]. Though the symptoms of COVID-19 infection appear after an incubation period of approximately 5.2 days, the period from the onset of COVID-19 symptoms to death ranged from 6 to 41 days with a median of 14 days [5,6]. It has already completed its world tour, and around 213 countries are now experiencing the deadly scene occurred by COVID-19 including 41, 52,670 infected patients and 2, 84,536 global death cases till 12th May, 2020 [7]. The scientific community is racing to explore the effective remedy against this severe health complications, but till to date there are no any potential therapeutics have been approved for clinical use [8]. There have been few key proteins of SARS-CoV-2 that could be targeted as the vaccine or drug surface [9]. Similar to SARS and MERS, non-structural proteins (e.g. 3-chymotrypsin-like protease coronavirus main protease, papain-like protease, helicase, and RNA-dependent RNA polymerase), structural proteins (e.g. spike glycoprotein) and accessory proteins were investigated in the genome of SARS-CoV-2 where non-structural proteins constitute two-thirds of the entire genome [10]. Among the structural proteins, Nucleocapsid (N) protein is prerequisite for RNA genome assembly where Membrane (M) and Envelope (E) proteins are associated in viral assembly in the host environment [11]. Moreover, Spike (S) protein is mainly responsible for the viral entry into the host cell, and that is why spike protein is now being considered as a major therapeutic target for drug and vaccines against SARS-CoV-2 [12]. Again, The S protein interaction with the human ACE2 interface has been revealed at the atomic level, and the efficiency of ACE2 usage was found to be a main factor of coronavirus transmissibility in human to human [13]. On the contrary, coronavirus main protease (M pro) or 3C-like proteinase (3CLP) was reported for their ability to cleave the polyproteins into individual polypeptides that are required for replication and transcription [14]. The 3CLP is autocleaved initially from the polyproteins to become a mature enzyme leading the translation of the messenger RNA [15]. Then the 3CLP cleaves all the 11 downstream non-structural proteins. As 3CLP plays a vital role in the replication cycle of virus in the host, it has been reported as the attractive target against the human SARS virus [16]. RNA-dependent RNA polymerase, other key target protein of SARS-CoV-2 catalyses the synthesis of viral RNA possibly with the support of other non-structural proteins as co-factors [17,18,19]. The computational drug repurposing method could allow the immediate search of potential antiviral therapy in case of re-emergence of viral infections as like as COVID-19 pandemic situation [20,21,22]. Computational drug repurposing has already been used to identify promising drug candidates for other virus associated diseases like Dengue, Ebola, ZIKA, and influenza infections [23,24]. Most importantly, the SARS-CoV-2 has shown evolutionary convergent relations with SARS-CoV and MERS-CoV, and the drug repurposing methods were also applied to SARS-CoV and MERS-CoV [25,26,27]. Hence, extensive in silico studies were performed to identify potential drug candidates, for example, Prulifloxacin, Bictegravir, Nelfinavir, and Tegobuvi, were identified as repurposing candidates against COVID-19 by looking for drugs with high binding capacity with SARS-CoV main protease [28]. Again, Nelfinavir, an HIV-1 protease inhibitor was also predicted to be a potential inhibitor of COVID-19 main protease by another computational-based study [29]. However, secondary metabolites from plant origin are found to show effective defence mechanism against different deadly pathogens, and they have been widely used for conventional remedy to treat a wide range of human diseases since the ancient period of human civilization [30,31,32]. In this pandemic situation, researchers are trying find out the effective solution against COVID-19 where plant metabolites could be a promising wings for screening out potential drug candidates. Even few plant secondary metabolites have already been reported as effective against other coronaviruses [33,34]. In the present study, a total of 117 plant based drug compound were screened out to check their potentiality for blocking the three important key proteins of SARS-CoV-2. The main protease proteins, RNA-dependent RNA polymerase and spike protein of SARS-CoV-2 were employed to molecular docking study with the repurposed drug candidates from plant origin for find out the better drug option towards the COVID-19 pandemic. Materials and Methods Retrieval of SARS-CoV-2 main protease proteins, RNA-dependent RNA polymerase, spike protein and acquisition of potential natural therapeutics PDB structures of SARS-CoV-2 main protease proteins (6LU7, 6Y2E), RNA-dependent RNA polymerase (6M71) and spike protein (6VYB) were retrieved from RCSB Protein Data Bank [35]. Moreover, a total 117 plant based drugs were collected from PubChem database (Supplementary Table 1). Alpha-ketoamide (CID 6482451) were also retrieved from the PubChem database database (https://pubchem.ncbi.nlm.nih.gov/) of NCBI [36]. Screening of natural therapeutics against the key viral proteins Molecular docking is an effective approach for screening out potential therapeutics against specific drug-targets of deadly pathogens [37,38]. The crystal structure of retrieved SARS-CoV-2 proteins (complexed with inhibitors) were refined by PyMOL v2.0 software [39]. Unwanted molecules i.e. water, ions, inhibitors were removed from the viral retrieved viral protein, and further employed to molecular docking experiment with 117 natural therapeutics. AutoDock Vina software [40] to analyse the binding affinity and interactive amino acids. Alpha-ketoamide, an inhibitor SARS-CoV-2 main protease protein were used as a positive control in this study [41] and also docked against the target proteins of SARS-CoV-2. The default parameters for grid box were set to 62 A° x 71 A° x 60 A° (x, y and z) and center -25.389 A° x 15.333 A° x 56.865 A° (x, y and z) to perform the action. Moreover, 2D ligand-protein interaction diagrams were generated by LigPlot+ find out the involved amino acids with their interactive position were identified in the docked complexes [42]. The ligand molecules' interactions with the viral proteins were visualize and analyzed by Discovery Studio [43] and PyMOL v2.0 software [44]. Structural insights of drug surface hotspot in the viral proteins The drug surface hotspot of SARS-CoV-2 proteins were identified by analysing the docked structures of each protein with the top most natural therapeutics by LigPlot+, PyMOL v.2.0 and Discovery Studio software. Binding patterns of Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine, Fangchinoline with SARS-CoV-2 proteins and the results were allowed for the comparative structural analysis of screened natural therapeutics. Furthermore, interactions of Alpha-ketoamide with the studied proteins were also investigated. Analysis of drug likeness property of top drug candidates The ADME (Absorption, Distribution, Metabolism and Excretion) properties of top drug candidates were analysed by SwissADME server [45]. The pharmacokinetics, drug-likeness property and medicinal chemistry were assessed [46]. Default parameters were used to evaluate various physiochemical parameters (Molar Refractivity, Molecular weight, TPSA), lipophilicity (Log Po/w (WLOGP), Log Po/w (MLOGP), Log Po/w (XLOGP3), Log Po/w, (SILICOS-IT), Log Po/w (iLOGP), Consensus Log Po/w), pharmacokinetics parameters (Log Kp; skin permeation) and water solubility of the probable drug candidates [47]. The inhibition effects of these natural therapeutics with different cytochromes P450s (CYP2C9, CYP2C19 CYP1A2, CYP3A4, CYP2D6) were also studied. In addition, admetSAR and OSIRIS Property Explorer were employed to evaluate the toxic or undesired effects (i.e. mutagenicity, tumerogenecity) of the compounds [48,49,50]. Results Screening of natural therapeutics against the key viral proteins All of the retrieved natural therapeutics were employed for molecular docking against MPP, RdRp and Spike protein of SARS-CoV-2 (Supplementary Table 2). The scoring function of AutoDock Vina was utilized to predict the interaction between the ligands (therapeutics) and the proteins. The top five inhibitors for each protein was identified based on their free binding energy. Results showed that Rifampin had the highest negative binding energy (-16.3 kcal/mol) among top MPP inhibitors (Table 1). Azobechalcone (-14.6 kcal/mol), Isolophirachalcone (-13 kcal/mol), Amentoflavone (-12.8 kcal/mol) and Cepharanthine (-12.7 kcal/mol) docked with the MPP of SARS-CoV-2 were also exhibited topmost place with a higher negative binding energy (<-12.7 kcal/mol) as well (Table 1). While interacting with RdRp of SARS-CoV-2, the most negative binding energy was scored by Azobechalcone (-15.9 kcal /mol) following by Rifampin (-15.6 kcal/mol), Tetrandrine (-13.9 kcal/mol), Biflavone (-13.7 kcal/mol), Biflavone (-13.7 kcal/mol) ( Table 2). Moreover, Azobechalcone, Rifampin, Isolophirachalcone, Fangchinoline and Tetrandrine were found to be top most natural inhibitors for the spike protein of SARS-CoV-2. Azobechalcone required lowest energy (-14.4 kcal/mol) to interact with the spike protein, while Rifampin, Isolophirachalcone, Fangchinoline and Tetrandrine scored -13.7 kcal/mol , -12.8 kcal/mol, -12.6 kcal/mol and -12.5 kcal/mol respectively (Table 3). However, Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine and Fangchinoline were found to be most effective inhibitory natural compounds when the docking results were compared for all three SARS-CoV-2 proteins (Figure 2 and Table 4). All of these plant based natural inhibitors required minimum energy (not more than -12.3 kcal/mol) to interact with the studied protein molecules. 3.2.Structural insights of drug surface hotspot in the viral proteins The docking pattern and interacting amino acid residues with their respective position were analyzed to unravel the binding sites of studied SARS-CoV-2 proteins. Rifampin were involved with the amino acid H41, N142, S144, C145, H163, M165, E166, D187, R188, Q189 of MPP of SARS-CoV-2 ( Figure 2). The position of H41, C145, and M165 were also crucial for the binding of Amentoflavone However, an additional residue T430 were involved in case of Isolophirachalcone (Table 3). P426, D428, T430, P463, F464 and F515 were also critical for binding pattern of Rifampin and SARS-CoV-2 spike protein. 3.3.Analysis of drug likeness property of top drug candidates The most potent MPP, RdRp and spike protein inhibitors (Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine and Fangchinoline) were investigated in the spheres of physicochemical parameters, lipophilicity, pharmacokinetics and water solubility (Table 5). Lipophilicity, partition coefficient between n-octanol and water (log Po/w) were also calculated by using five widely available predictive models (XLOGP3, WLOGP, MLOGP, SILICOS-IT, iLOGP). GI absorption was lower for all the drug candidates. Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine and Fangchinoline molecules had no repressive action with the P450 (CYP) isoforms (CYP1A2, CYP2C19, CYP2C9, CYP2D6, CYP3A4). Moreover, no permeant BBB exists among the protein inhibitors of MPP, RdRp and Spike. The compounds sowed water solubility from moderate to high level i.e. 1.30e-16 mg/ml, 1.50e-02 mg/ml, 1.18e-10 mg/ml, 9.78e 09 mg/ml, 4.61e-08 mg/ml, respectively ( Table 5). The toxicity analysis of these inhibitors showed that there were no carcinogenic effect and organ toxicity. However, Rifampin, Tetrandrine and Fangchinoline were slightly positive in terms of mutagenesis though Azobechalcone and Isolophirachalcone inhibitors were completely negative. Among the top five inhibitors Azobechalcone was listed in the acute oral toxicity category 2 and rest of them were listed in the category 3. Discussion Global pandemic caused by SARS-CoV-2 has become a major concern due to its excessive infection rate and lethality [51,52,53,54]. Despite huge research regarding the pathogen, no drugs or vaccine has proven satisfactory to combat infections caused by SARS-CoV-2 [55,56]. Several investigational drugs exist, however none of these could treat the patients unquestionably. Moreover, lack of rapid detection procedures made SARS-CoV-2 diagnosis troublesome [57]. Computational approach and drug repurposing strategies hold promise to face such challenges caused by SARS-CoV-2. Hence, in the present study, attempts were taken to suggest probable drug candidates by checking the efficacy of natural inhibitors to inhibit the key proteins of SARS-CoV-2. The race against the COVID-19 pandemic has allowed the drug repurposing through virtual for finding drugs that could be used for the treatment of COVID-19. Recent studies prioritized MPP inhibitors of SARS-CoV-2 i.e. alpha-ketoamide, Hydroxy, Remdesivir, Chloroquine and Favipiravir to evaluate their potency as drug [58,59,60]. Several in silico approach was also adopted to screen putative drug candidates against SARS-CoV-2 [61,62]. However, all these experiments used either main protease proteins or RNA-dependent RNA polymerase of SARS-CoV-2 as probable drug targets. In this study, we screened potential natural therapeutics against SARS-CoV-2 MPP, RdRp and spike protein by molecular docking approach. Here, Rifampin, Azobechalcone and Azobechalcone were determined as top most drug candidates as they interacted with SARS-CoV-2 MPP, RdRp and spike protein with lowest negative binding energy had the highest negative binding energy (-16.3 kcal/mol, -15.9 kcal /mol and -14.4 kcal/mol respectively. However, comparative analysis revealed the superiority of 5 drug candidates i.e. Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine, Fangchinoline against SARS-CoV-2 ( Table 4). The common drug surface hotspots were analyzed along with modeling of pharmacophore which is very important step for drug discovery. Three amino acid residues i.e. H41, C145, and M165 played the crucial role for the interaction of MPP with its inhibitors (i.e. Rifampin, Amentoflavone, Cepharanthine) (Table 1). Azobechalcone, the top scorer among RdRp inhibitors, Again, the position of F368, L371, L372, A375, W509, L514, Y515 were vital for binding of RdRP with Tetrandrine, Biflavone and Fangchinoline (Table 2). Most importantly, the binding patterns of spike protein with Isolophirachalcone, Fangchinoline and Tetrandrine were significantly similar. This study revealed the possibility of these amino acids to efficiently interact with drugs, though requires validation in wet lab trials. ADME analysis of top drug candidates reveled no undesirable consequences by these compounds. Various physico-chemical parameters, lipophilicity, pharmacokinetics properties and water solubility were determined (Table 5). In addition, no BBB permeant were identified among the top most inhibitors of MPP, RdRp and spike protein. The consequence of association of the natural inhibitory drugs of three key proteins with the cytochrome P450 (CYP) suggests that no substantial inhibition can occur. However, toxicity analysis revealed that Rifampin, Tetrandrine and Fangchinoline can be slightly mutagenic, though there was no possibility for organ toxicity. The results suggest that Azobechalcone, Rifampin, Isolophirachalcone and Tetrandrine, Fangchinoline could be an option to treat SARS-CoV-2 infections. However, the study employed various computational approaches to screen the potent natural therapeutics and does not involve in-vivo assay. Currently investigational drugs of SARS-CoV-2 are under immense experimental evaluation. Therefore, we suggest clinical trials for the experimental validation of our findings. Conclusion COVID-19 pandemic situation is going to be a worst condition throughout the world. Rapid detection and social distancing are being encouraged at this stage, but we need to search for immediate therapeutic options and effective vaccine candidates for battling this serious health crisis. Drug repurposing approaches could screened out the already approved drugs for reusing against any serious causative agents that are causing health complications. Plant metabolites based repurposed drug molecules could be a promising options against SARS-CoV-2. In the present study, five plantr based therapeutics such as Azobechalcone, Rifampin, Isolophirachalcone, Tetrandrine and Fangchinoline were suggested for potential inhibitors for the Main Protease protein, RNA dependent RNA polymerase and Spike protein of SARS-CoV-2 by using molecular docking based virtual screening study. The study initiated the window towards the thinking of plant based therapy against COVID-19, though extensive research and wet lab validation needs to make it usable for patient. ( H41, C145, H164, M165, E166, D187, R188, Q189) and Cepharanthine (H41, N142, C145, M165, E166, D187, R188, Q189) (Table 1). Azobechalcone, the top scorer among RdRp inhibitors, were engaged by Y32, K47, L49, Y129, H133, N138, D140, T141, S709, T710, D711, K714, K780, N781 in the docked complex. Moreover, results revealed that 6 amino acid positions i.e. F368, L371, L372, A375, W509, L514, Y515 were crucial for binding pattern of RdRP with Tetrandrine, Biflavone and Fangchinoline ( Figure 3 and Table 2). Remarkably, binding patterns of spike protein with Isolophirachalcone (R355, Y396, P426, D428, F429, T430, K462, P463, F464, F515), Fangchinoline (R355, Y396, P426, N428, P463, F464, S514, F515, E516) and Tetrandrine (R355, Y396, P426, N428, P463, F464, S514, F515, E516) were exactly similar as all 9 amino acid residues i.e. R355, Y396, P426, N428, P463, F464, S514, F515, E516 were critical for binding with the protein (Figure 4). Figure 1 : 2 Figure 2 :Figure 3 :Figure 4 : 12234Top screened plant metabolites for targeting the MPP, RdRp and S protein of SARS-CoV-Molecular interaction of Rifampin with MPP of SARS-Molecular interaction of Azobechalcone with RdRp of SARS-CoV-2 by molecular docking (Binding Energy -15.9 kcal/mol) Molecular interaction of Azobechalcone with S protein of SARS-CoV-2 by molecular Table 1 : 1Top screened plant metabolites against Main Protease Protein of SARS-CoV-2 Table 2 : 2Top screened plant metabolites against RdRp of SARS-CoV-2No Pub Chem ID Name Binding Energy (Kcal/mol) Involved Amino Acids 1 135398735 Rifampin -16.3 H41, N142, S144, C145, H163,M165, E166, D187, R188, Q189 2 16148290 Azobechalcone -14.6 R131, K137, T196, D197, T198, T199, K236, Y237, N238, L272, G275, M276, L286, L287 3 101630349 Isolophirachalcone -13 K137, D197, T199, Y237, N238, L271, L272, G275, L286, L287, E288, D289, E290 4 5281600 Amentoflavone -12.8 H41, C145, H164, M165, E166, D187, R188, Q189 5 10206 Cepharanthine -12.7 H41, N142, C145, M165, E166, D187, R188, Q189 No Pub Chem ID Name Binding Energy (kcal/mol) Involved amino acids 1 16148290 Azobechalcone -15.9 Y32, K47, L49, Y129, H133, N138, D140, T141, S709, T710, D711, K714, K780, N781 2 135398735 Rifampin -15.6 L270, L271, Y273, T324, S325, F326, P328, A282, A283, F396, V398 3 73078 Tetrandrine -13.9 F368, L371, L372, A375, W509, Y515, S518 4 9980790 Biflavone -13.7 F368, L371, L372, A375, W509, L514, Y515 5 73481 Fangchinoline -13.7 F368, L371, L372, A375, W509, L514, Y515, S518 Table 3 : 3Top screened plant metabolites against Spike Protein of SARS-CoV-2 Table 4 : 4Top screened suggested plant metabolites against Main Protease, RdRpl and Spike Proteinof SARS-CoV-2 for battling COVID-19 pandemic No Pub Chem ID Name Main Protease RdRpl Spike protein 1 16148290 Azobechalcone -14.6 -15.9 -14.4 2 135398735 Rifampin -16.3 -15.6 -13.7 3 101630349 Isolophirachalcone -13 -13.3 -12.8 4 73078 Tetrandrine -12.6 -13.9 -12.5 5 73481 Fangchinoline -12.3 -13.7 -12.6 Table 5 : 5ADME analysis of top screened suggested plant metabolites against Main Protease, RdRpland Spike Protein of SARS-CoV-2 Parameter Name Azobechalc one Rifampin Isolophirachal cone Tetrandri ne Fangchino line PubChem ID 16148290 135398735 101630349 73078 73481 Physicochem ical parameters Formula C90H70O2 2 C43H58N4 O12 C60H48O15 C38H42N 2O6 C37H40N2 O6 Molecular weight 1503.51 g/mol 822.94 g/mol 1009.01 g/mol 622.75 g/mol 608.72 g/mol Molar Refractivity 415.5 234.22 277.97 186.07 181.6 TPSA 402.58 Å² 220.15 Å² 275.13 Å² 61.86 Å² 72.86 Å² Lipophilicity Log Po/w (iLO GP) 3.63 4.58 2.32 5.16 5.02 Log Po/w (XLO GP3) 15.45 5.46 10.17 6.66 6.34 Log Po/w (WLO GP) 14.52 3 9.48 5.75 5.45 Log Po/w (MLO GP) 1.77 0.14 2.02 3.73 3.55 Log Po/w (SILI COS-IT) 12.03 2.07 7.79 6.06 5.5 Consensus Log Po/w 9.48 3.05 6.36 5.47 5.17 Log Kp (skin permeation) -4.50 cm/s -7.44 cm/s -5.23 cm/s -5.37 cm/s -5.51 cm/s Water Solubility Log S (SILICO S-IT) -19.06 -4.74 -12.93 -10.8 -10.12 Solubility 1.30e-16 mg/ml ; 8.67e-20 mol/l 1.50e-02 mg/ml ; 1.83e-05 mol/l 1.18e-10 mg/ml ; 1.17e- 13 mol/l 9.78e-09 mg/ml ; 1.57e-11 mol/l 4.61e-08 mg/ml ; 7.57e-11 mol/l AcknowledgementsAuthors would like to acknowledge the Faculty of Biotechnology and Genetic Engineering, SylhetAgricultural University for the technical support of the project.Funding informationThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.Conflict of interestAuthors declare that they have no conflict of interests. 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[ "Kepler photometry of the prototypical Blazhko star RR Lyr: An old friend seen in a new light", "Kepler photometry of the prototypical Blazhko star RR Lyr: An old friend seen in a new light" ]	[ "K Kolenberg \nInstitut für Astronomie\nUniversity of Vienna\nTürkenschanzstrasse 17A-1180ViennaAustria\n", "S Bryson \nNASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA\n", "R Szabó \nKonkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary\n", "D W Kurtz \nJeremiah Horrocks Institute of Astrophysics\nUniversity of Central Lancashire\nPR1 2HEPrestonUK\n", "R Smolec \nInstitut für Astronomie\nUniversity of Vienna\nTürkenschanzstrasse 17A-1180ViennaAustria\n", "J M Nemec \nDepartment of Physics & Astronomy\nCamosun College\nV8P 5J2VictoriaBritish ColumbiaCanada\n", "E Guggenberger \nInstitut für Astronomie\nUniversity of Vienna\nTürkenschanzstrasse 17A-1180ViennaAustria\n", "P Moskalik \nCopernicus Astronomical Center\nul. Bartycka 1800-716WarsawPoland\n", "J M Benkő \nKonkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary\n", "M Chadid \nUMR 6525\nObservatoire de la Côte dAzur\nUniversité Nice Sophia-Antipolis\nParc Valrose06108, Cedex 02NiceFrance\n", "Y.-B Jeon \nKorea Astronomy and Space Science Institute\n305-348DaejeonKorea\n", "L L Kiss \nKonkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary\n\nSydney Insitute for Astronomy\nSchool of Physics\nUniversity of Sydney\n2006NSWAustralia\n", "G Kopacki \nInstytut Astronomiczny Uniwersytetu Wroclawskiego\nKopernika 1151-622WroclawPoland\n", "J Nuspl \nKonkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary\n", "M Still \nNASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA\n", "J Christensen-Dalsgaard \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark\n", "H Kjeldsen \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark\n", "W J Borucki \nNASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA\n", "D A Caldwell \nSETI Institute\n94043Mountain ViewCAUSA\n", "J M Jenkins \nSETI Institute\n94043Mountain ViewCAUSA\n", "D Koch \nNASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA\n" ]	[ "Institut für Astronomie\nUniversity of Vienna\nTürkenschanzstrasse 17A-1180ViennaAustria", "NASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA", "Konkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary", "Jeremiah Horrocks Institute of Astrophysics\nUniversity of Central Lancashire\nPR1 2HEPrestonUK", "Institut für Astronomie\nUniversity of Vienna\nTürkenschanzstrasse 17A-1180ViennaAustria", "Department of Physics & Astronomy\nCamosun College\nV8P 5J2VictoriaBritish ColumbiaCanada", "Institut für Astronomie\nUniversity of Vienna\nTürkenschanzstrasse 17A-1180ViennaAustria", "Copernicus Astronomical Center\nul. Bartycka 1800-716WarsawPoland", "Konkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary", "UMR 6525\nObservatoire de la Côte dAzur\nUniversité Nice Sophia-Antipolis\nParc Valrose06108, Cedex 02NiceFrance", "Korea Astronomy and Space Science Institute\n305-348DaejeonKorea", "Konkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary", "Sydney Insitute for Astronomy\nSchool of Physics\nUniversity of Sydney\n2006NSWAustralia", "Instytut Astronomiczny Uniwersytetu Wroclawskiego\nKopernika 1151-622WroclawPoland", "Konkoly Observatory\nHungarian Academy of Sciences\nQuarter Thege Miklósút 15-17H-1121BudapestHungary", "NASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA", "Department of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark", "Department of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark", "NASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA", "SETI Institute\n94043Mountain ViewCAUSA", "SETI Institute\n94043Mountain ViewCAUSA", "NASA Ames Research Center\n244-30, 94035Moffet FieldMS, CAUSA" ]	[ "Mon. Not. R. Astron. Soc" ]	We present our analysis of the long cadence Kepler data for the well-studied Blazhko star RR Lyr, gathered during the first two quarters of the satellite's observations and covering a total of 127 d. Besides being of great importance for our understanding of RR Lyrae stars in general, these RR Lyr data can be regarded as a case study for observations of bright stars with Kepler. Kepler can perform high-precision photometry on targets like RR Lyr, as the saturated flux is conserved to a very high degree. The Kepler data on RR Lyr are revolutionary in several respects. Even with longcadence sampling (one measurement per 29.4 min), the unprecedented precision ( < mmag) of the Kepler photometry allows the study of the star's extreme light curve variations in detail. The multiplet structures at the main frequency and its harmonics, typical for Blazhko stars, are clearly detected up to the quintuplets. For the first time, photometric data of RR Lyr reveal the presence of half-integer frequencies, linked to a period doubling effect. This phenomenon may be connected to the still unexplained Blazhko modulation. Moreover, with three observed Blazhko cycles at our disposal, we observe that there is no exact repetition in the light curve changes from one modulation cycle to the next for RR Lyr. This may be due to additional periodicities in the star, or to transient or quasi-periodic changes.	10.1111/j.1365-2966.2010.17728.x	[ "https://arxiv.org/pdf/1011.5908v1.pdf" ]	73,597,094	1011.5908	b2d3a75e0a9b6d0504d4a8d4f0a6e3c97604882f	Kepler photometry of the prototypical Blazhko star RR Lyr: An old friend seen in a new light Printed 30 November 2010 K Kolenberg Institut für Astronomie University of Vienna Türkenschanzstrasse 17A-1180ViennaAustria S Bryson NASA Ames Research Center 244-30, 94035Moffet FieldMS, CAUSA R Szabó Konkoly Observatory Hungarian Academy of Sciences Quarter Thege Miklósút 15-17H-1121BudapestHungary D W Kurtz Jeremiah Horrocks Institute of Astrophysics University of Central Lancashire PR1 2HEPrestonUK R Smolec Institut für Astronomie University of Vienna Türkenschanzstrasse 17A-1180ViennaAustria J M Nemec Department of Physics & Astronomy Camosun College V8P 5J2VictoriaBritish ColumbiaCanada E Guggenberger Institut für Astronomie University of Vienna Türkenschanzstrasse 17A-1180ViennaAustria P Moskalik Copernicus Astronomical Center ul. Bartycka 1800-716WarsawPoland J M Benkő Konkoly Observatory Hungarian Academy of Sciences Quarter Thege Miklósút 15-17H-1121BudapestHungary M Chadid UMR 6525 Observatoire de la Côte dAzur Université Nice Sophia-Antipolis Parc Valrose06108, Cedex 02NiceFrance Y.-B Jeon Korea Astronomy and Space Science Institute 305-348DaejeonKorea L L Kiss Konkoly Observatory Hungarian Academy of Sciences Quarter Thege Miklósút 15-17H-1121BudapestHungary Sydney Insitute for Astronomy School of Physics University of Sydney 2006NSWAustralia G Kopacki Instytut Astronomiczny Uniwersytetu Wroclawskiego Kopernika 1151-622WroclawPoland J Nuspl Konkoly Observatory Hungarian Academy of Sciences Quarter Thege Miklósút 15-17H-1121BudapestHungary M Still NASA Ames Research Center 244-30, 94035Moffet FieldMS, CAUSA J Christensen-Dalsgaard Department of Physics and Astronomy Aarhus University DK-8000Aarhus CDenmark H Kjeldsen Department of Physics and Astronomy Aarhus University DK-8000Aarhus CDenmark W J Borucki NASA Ames Research Center 244-30, 94035Moffet FieldMS, CAUSA D A Caldwell SETI Institute 94043Mountain ViewCAUSA J M Jenkins SETI Institute 94043Mountain ViewCAUSA D Koch NASA Ames Research Center 244-30, 94035Moffet FieldMS, CAUSA Kepler photometry of the prototypical Blazhko star RR Lyr: An old friend seen in a new light Mon. Not. R. Astron. Soc 0000000Printed 30 November 2010Accepted 2010, ? Received 2010, August(MN L A T E X style file v2.2)stars: oscillations -stars: variable: RR Lyrae -stars: individual: RR Lyr, KIC7198959 -techniques: photometric We present our analysis of the long cadence Kepler data for the well-studied Blazhko star RR Lyr, gathered during the first two quarters of the satellite's observations and covering a total of 127 d. Besides being of great importance for our understanding of RR Lyrae stars in general, these RR Lyr data can be regarded as a case study for observations of bright stars with Kepler. Kepler can perform high-precision photometry on targets like RR Lyr, as the saturated flux is conserved to a very high degree. The Kepler data on RR Lyr are revolutionary in several respects. Even with longcadence sampling (one measurement per 29.4 min), the unprecedented precision ( < mmag) of the Kepler photometry allows the study of the star's extreme light curve variations in detail. The multiplet structures at the main frequency and its harmonics, typical for Blazhko stars, are clearly detected up to the quintuplets. For the first time, photometric data of RR Lyr reveal the presence of half-integer frequencies, linked to a period doubling effect. This phenomenon may be connected to the still unexplained Blazhko modulation. Moreover, with three observed Blazhko cycles at our disposal, we observe that there is no exact repetition in the light curve changes from one modulation cycle to the next for RR Lyr. This may be due to additional periodicities in the star, or to transient or quasi-periodic changes. their "simple" radial pulsations. RR Lyrae stars have typical periods of ∼0.2 d to ∼1 d, amplitudes in the optical of 0.3 mag up to 2 mag, and spectral types of A2 to F6. They pulsate in the radial fundamental mode (RRab stars), the radial first overtone (RRc stars) and, in some cases, in both modes simultaneously (RRd stars). A few RR Lyrae stars are suspected to be pulsating in higher-order radial overtone modes. A large fraction of RR Lyrae stars shows the Blazhko effect (Blazhko 1907), a periodic amplitude and/or phase modulation with a period of typically tens to hundreds of times the pulsation period. The origin of the Blazhko effect remains a matter of controversy. Rotational modulation of the distorted radial mode invoked by a strong dipole-like magnetic field (Shibahashi 2000) appears to have been ruled out as a cause (Chadid et al. 2004;Kolenberg & Bagnulo 2009). Several models proposing a resonance between the main radial mode and another mode have been proposedwith another radial mode (higher overtone: Borkowski 1980, Moskalik 1986), or with a nonradial mode of low spherical degree (Van Hoolst et al. 1998;Dziembowski & Mizerski 2004) -but their validity remains to be proven. Stothers (2006Stothers ( , 2010 proposes a scenario that attributes the Blazhko variation to variable turbulent convection caused by, e.g., transient magnetic fields in the star (that would be hard to detect). Thus far, all models presently proposed for the Blazhko effect have shortcomings in explaining the variety of features shown by modulated stars (Kovács 2009). The RRab star RR Lyr is the brightest star, eponym and prototype of its class, and it shows very strong Blazhko modulation. It has been extensively studied over the course of the past century, through photometric (e.g., Szeidl & Kolláth 2000), spectroscopic (e.g., Preston et al. 1965), and spectropolarimetric (e.g., Chadid et al. 2004) data. However, several aspects of its pulsation remain poorly understood, and to accurately model the star, we have to take into account complex physics that we are only beginning to uncover. By a fortunate coincidence, RR Lyr (α(J2000): 19 h 25 m 28 s and δ(J2000): +42 • 47 04 ) lies in the field of the Kepler space telescope (Kepler ID: KIC7198959), though its brightness is well above the saturation limit for the Kepler CCDs. The unprecedented quality of the Kepler data of the star leads us to several new findings that spur further modelling. To explore theoretical models of RR Lyr, it is necessary to know the physical parameters of the star with the highest accuracy possible. For this reason, a self-consistent atmospheric and abundance analysis from high-resolution, high signal-to-noise spectra of the star, with the aim of parameter determination, was recently published by Kolenberg et al. (2010b). According to the General Catalogue of Variable Stars (Samus 2004) its V brightness changes within the range 7.06 -8.12 mag and its period is P = 0.56686776 d or about 13 h 36 min, corresponding to a frequency of f0 = 1.7640799 d −1 . Kolenberg et al. (2006) found a (mean) frequency of f0 = 1.764170 ± 0.000005 d −1 from data gathered in 2003 and 2004. In Section 2 we present the new observations of RR Lyr obtained with the Kepler satellite and describe the challenges of the data reduction. Section 3 presents the data analysis and its results. In Section 4 we discuss our results in the framework of the current understanding of RR Lyrae pulsations and the Blazhko effect. Finally, some concluding remarks and the outlook for future investigations are given in Section 5. OBSERVATIONS The Kepler space telescope ) was designed to detect transits of Earth-like planets around Sunlike stars. Technical details on the mission can be found in Koch et al. (2010) and Jenkins et al. (2010a,b). In order to exploit the potential of the Kepler asteroseismic data (Gilliland et al. 2010), the Kepler Asteroseismic Science Consortium (KASC) was set up, grouping more than 350 scientists from all over the world. KASC working group 13 is dedicated to the study of the Kepler RR Lyrae stars. At the time of writing, over 30 RR Lyrae stars have been identified in the Kepler field and are now observed by the satellite. About half of the Kepler RR Lyrae targets have turned out to be modulated (Kolenberg et al. 2010a;Benkő et al. 2010). With Kepler's unprecedented precision, we will find additional clues to solving the Blazhko riddle. The Kepler Mission offers two options for observations: either long cadence (29.4 minutes) or short cadence (1 minute). The spacecraft "rolls" every three months to allow for continuous illumination of Kepler's solar arrays. RR Lyr was observed in long-cadence during the first roll of the Kepler survey phase between HJD 2454964. 0109 and HJD 2454997.4812 (2009 May 12 -June 14;Q1 data, 33.5 d, 1628 useful data points), and during the second roll between HJD 55002. 01748 and HJD 55090.96492 (2009 June 19 -September 16;Q2 data 88.8 d, 4097 useful data points). The total time span of the data set is nearly 127 d, more than 3 complete Blazhko cycles, as can be seen in Fig. 1. There are a few small gaps in the data set, due to unplanned safe mode, loss of fine pointing events as well as regular data downlink periods. Note that except for around those gaps, nearly every single Kepler measurement is of excellent quality. Long-cadence sampling is just sufficient to obtain a good coverage of the light curve, but rapid changes and "glitches" in the light curve are missed by this sampling, and can be investigated with forthcoming short-cadence Kepler data. As a consequence of the sampling rate, the Nyquist frequency lies at 24.5 d −1 . Fig. 1 shows the quality of the Kepler data on RR Lyr compared to one of the most precise published data sets on the star obtained from ground-based observatories (Kolenberg et al. 2006). Despite the effort of organizing a multi-site campaign and combining all the standardized observations, the latter data set had a point-to-point scatter of over 0.005 mag at best, and was characterized by nightly and weatherrelated gaps, typical for Earth-based observations. As a consequence, the Fourier spectra of such data sets obtained from the ground have a noise level 10-50 times higher than Kepler Fourier spectra and are subject to aliasing. Early Kepler photometry of RR Lyr Kepler's CCDs saturate between Kepler magnitudes 11 and 12, depending on which CCD a target falls, as well as the target location on that CCD. This saturation spills up and Figure 1. Comparison of the ground-based RR Lyr data, gathered from six different observatories, published by Kolenberg et al. (2006) and the Kepler Q1 and Q2 data of the star transformed to the magnitude scale (top panels). Individual cycles are shown in Fig. 3. Bottom panels: Fourier transform of the data; the insert shows the window function. down the CCD column, and saturated flux is conserved to a very high degree (see Fig. 2). Therefore, Kepler can perform high-precision photometry on saturated targets like RR Lyr. Due to bandwidth constraints, Kepler downlinks pixels selected for each target star (Haas et al. 2010). These pixels are selected algorithmically to maximize the signal-to-noise ratio for the target , based on stellar properties in the Kepler Input Catalog (KIC) and various models of sky and spacecraft characteristics, including saturation behavior. In the case of RR Lyr, the Kepler magnitude in the KIC did not reflect maximum brightness. This, combined with the early and approximate state of the saturation model at the time of Q1 and Q2 observations, resulted in the size of RR Lyr's saturation being underestimated. Therefore, not enough pixels were allocated to RR Lyr to fully capture all flux at its brightest maxima: the central saturation column exceeded the assigned aperture. In Q1 the consequent loss of flux was relatively small but in Q2 the loss of flux at maxima was significant. Fortunately, columns adjacent to the central saturation column also saturated, though to a lesser extent, and these columns were fully captured in all of Q1 and Q2. This allows us to use those measurements where the central column is fully captured (away from maxima) to determine the ratio of the central columns flux to the flux in the adjacent columns, and use this ratio as a predictor of the flux in the central column when the central column is not fully captured. This ratio, r0, is sensitive to various known pointing and focus systematics, so we determine r0 for every measurement, and detrend r0 using a robust piecewise-polynomial fit to those measurements where the central column's flux is fully captured. The resulting polynomial trend rt was consistent with known motion and focus systematics. We define a correction c = rt/r0 which multiplies the central column's flux for those data points in which some of the flux falls outside RR Lyrs pixels, as determined by the detrended ratio r d falling below a cutoff value. In Q1 only minor corrections on the order of 10 per cent were required for the central column flux, and these were at the beginning and end of Q1 (from MJD 54964.25 to 54972.76 and 54995.47 to 54997.17). In Q2 corrections to the central column's flux were required for every maximum and ranged from 30 to 60 per cent. The uncertainty in the correction is estimated, from the variance of r d , to be 0.45 per cent in Q1 and 0.36 per cent in Q2 (neglecting six measurements with large outliers in Q2). Once the central column flux is corrected, it is summed with the other columns in the aperture to create the estimated flux correction for each measurement. This flux correction is applied to the output of the Kepler Science Operations Center pipeline, which corrects light curves by removing cosmic ray hits, background and correlations with known systematics (Jenkins et al. 2010a,b). An example of the original and corrected light curves in Q2 is shown in Fig. 3. In Q1 the final flux corrections were on the order of at most 5 per cent. In Q2 the flux corrections range from 15 to 35 per cent. The uncertainty in the total corrected flux is about 0.25 per cent in Q1 and Q2 for measurements in which the correction was applied, compared with the flux uncertainty of 8 × 10 −6 in measurements where the correction was not applied because the flux was completely captured. RR Lyr's Kepler Magnitude Kepler processing does not provide calibrated Kepler magnitudes Kp. We produced a rough estimate of RR Lyr's showing the saturation spill in the three central columns. The first cadence in Q2 is shown, when RR Lyr was near a minimum. For brighter cadences the central column saturation exceeded these pixels, but the two adjacent columns remained completely captured. Note the differing row and column scales. magnitude in Q1 and Q2 using 5 nearby quiet (95 per cent variation under 1 part per thousand) stars of known Kp. Because this method did not account for colour or uncertainties in the comparison star's magnitudes and measured flux, the Kp magnitude values here are only approximate. We emphasize, however, that the relative precision within a given quarter is unprecedentedly high, the approximate na-ture and ambiguity is only in tying the RR Lyr light curve to the absolute Kepler photometry. As can be seen in Fig. 1's upper right panel, RR Lyr's total light curve amplitude varies significantly over the Blazhko cycle. The total variation (from lowest minimum to highest maximum) spans 0.86 mag in Kp, with 0.84 mag for the largest amplitude light curves, and 0.47 mag for the smallest amplitude light curves. Hence, the total reduction of the light curve amplitude due to the Blazhko effect is about 44 per cent as determined from our Kepler data of RR Lyr so far. These values will be refined with the short cadence data to come. The GCVS lists a 7.06 -8.12 (1.06 mag; see also total amplitude in Fig. 1's left upper panel) variation in V magnitude. Hence, with the wide (white-light) passband of Kepler we observe an amplitude reduction with respect to the V magnitudes by a factor of 1.23. The same factor is found when comparing the amplitude A1 of f0 (the first Fourier component) in the V data published by Kolenberg et al. (2006) to the one obtained from our data ( Table 1). The possible deviations from the "real" Kepler magnitudes introduced by the corrections and their associated uncertainties have a negligible effect on the frequency values. For Q2, the quarter covering more than two Blazhko cycles, and for which the largest corrections were needed, the frequency values derived from the non-corrected and corrected data are identical to within the error bars. For the Q1 data this is also the case but the comparison is less convincing since they cover less than one Blazhko cycle. Q1 and Q2 to the longest data set (Q2). The results are summarized in Table 1, and discussed below. Optimum values for the frequencies, amplitudes and phases were obtained by minimizing the residuals of the fit. Errors were determined through Monte Carlo simulations. Typical Blazhko patterns RR Lyr's high-amplitude (nonlinear) pulsation gives rise to a non-sinusoidal light curve, described in Fourier space by contributions from the main frequency f0 and its harmonics. The Blazhko modulation manifests itself as equidistantly spaced multiplets around the main frequency and its harmonics Benkő et al. 2009). All these frequencies were found through successive prewhitening, and the triplet components are strongly present in the data. Therefore, in a first stage, we fit the data with a Fourier sum of the form: f (t) = A0 + n k=1 [A k sin(2π(kf0t + φ k )) +A + k sin(2π((kf0 + fB)t + φ + k )) +A − k sin(2π((kf0 − fB)t + φ − k ))] +B0 sin(2π(fBt + φB)),(1) From the combined Q1 and Q2 data, we find f0=1.76416 ± 0.00001 d −1 for the main frequency, and its detected harmonics are significant up to the 14th order. From the side peaks (triplet components) we find fB=0.0256 ± 0.0002 d −1 for the Blazhko frequency, or a Blazhko period of 39.1 ± 0.3 d. From the Q2 data alone we find f0 = 1.76422 ± 0.00001 d −1 , and fB = 0.0252 ± 0.0001 d −1 , or a Blazhko period of 39.6 ± 0.3 d. As mentioned above, we also fit the Q2 data (Table 1) with the values derived from Q1 and Q2 because of the longer time base. The main pulsation frequency we obtain is an average value of the actual, varying pulsation frequency over the time interval considered. In fact, the amplitude and period of RR Lyr's pulsation change significantly over the star's Blazhko cycle(s), as is shown in Fig. 4, obtained with the analytical signal method (Kolláth et al. 2002). The pulsation period varies between 0.5652 d (1.7693 d −1 ) and 0.5699 d (1.7547 d −1 ). This corresponds to a period change (δP/P ) of about 0.83 per cent, or nearly 12 minutes. The period change is not identical from one Blazhko cycle to the next (see thick line in Fig. 4 for consecutive Blazhko cycles). The Blazhko frequency may in fact also be variable, and the value we obtain depends on the coverage (see Section 4.1). Note that the harmonics of the Blazhko frequency (2fB, 3fB, etc.) do not reach the significance level in this data set, as they do in several other Blazhko stars (see, e.g., Chadid et al. 2010). This may change with additional data of the star. Quintuplet components After subtracting a fit of the form of Equation 1, we see many additional peaks in the spectrum. First of all, we see clear evidence for quintuplet frequencies. Fig. 5 shows the residual spectrum after prewhitening with the main frequency, its harmonics and the triplet components (top panel), and zooms around some of the quintuplet components (bottom panels). The quintuplet components are added to Table 1. They become more easily discernible in the frequency spectrum with increasing order. At low order, e.g., around f0 and 2f0, other close frequencies have higher amplitudes (see Fig. 5). There are no clear indications for the occurrence of multiplet components of order higher than the quintuplet in the Fourier spectrum of the Kepler data of RR Lyr so far. Period doubling It is not only around the main frequency and its harmonic components where we see additional peaks. After prewhitening with the main frequency f0 = 1.76416 d −1 , its harmonics and the significant multiplet components, we clearly detect other significant frequencies in the Fourier spectrum. Fig. 6 shows a zoom in the frequency range [1.5 -10] d −1 (f0 -5f0). We clearly see additional signal around the half-integer multiples of f0. The occurrence of such half-integer components is connected to the period doubling phenomenon (see Section 4.2). As a consequence of period doubling we see the previously mentioned alternating higher and lower maxima, as can be seen in Fig. 4 (most striking in the middle panel), in fig. 4 of Kolenberg et al. (2010a) and in figs 1, 2 and 4 of Szabó et al. (2010). The highest peak is detected near 3/2f0, followed by 5/2f0 and 1/2f0. The frequency at 2.6643 ± 0.0026 d −1 is the highest among the peaks around 3/2f0. Its ratio to the main frequency is 1.510±0.001, so there is a significant deviation from the exact half-integer ratio. We also observe several other frequency peaks around the 3/2f0 component, and around the other half-integer frequency components, very close to each other, as shown in Fig. 6. They form asymmetric bunches that are more peaked towards the right (higher frequency) side. . Relationship between the period change and amplitude modulation over the Blazhko cycle for RR Lyr. The crosses show the Kepler data for the three monitored Blazhko cycles, the connecting line our multi-frequency fit to the data. The thick line shows an instantaneous period determined using the analytical signal method (units to the right side of the panel). The period is longest just before the lowest amplitude phases and shortest just after the highest amplitude phases. Note the small changes in the period variation between consecutive Blazhko cycles, and the alternating high and low maxima at certain Blazhko phases. Figure 6. Fourier spectrum of the Kepler RR Lyr data in the range [1.5 -10] d −1 , after subtraction of the main frequency, its harmonics, and the triplet and quintuplets components. The positions of the mean pulsation frequency and its harmonics kf 0 , and that of the half-integer frequencies 2k−1 2 f 0 , are indicated by vertical lines. Additional peaks After prewhitening with the main frequency, its harmonics, the observed triplet/quintuplet side peaks and the halfinteger components, we observe many residual peaks around their positions (see Fig. 6). Their spacing to the prewhitened frequency peaks is not equal or suspiciously close to the Blazhko frequency, so we cannot directly connect them with the Blazhko modulation. A changing Blazhko effect, due to longer cycles, secular trends, or transient phenomena would give rise to frequencies close to the typical pattern. Further Kepler data to come will clarify what is the origin of the additional peaks. Light curve modulation Modulation components The properties of the modulation components, specifically the triplet components kf0 + fB and kf0 − fB, in the frequency spectra of a Blazhko star can provide constraints for the models proposed to explain the modulation. Alcock et al. (2003) found that the relative amplitudes of the first order modulation components are usually in the Table 2 lists the amplitude ratios, phase differences and their errors for the first ten modulation component pairs, Table 1. Amplitudes and phases of the pulsation and modulation frequency components of RR Lyr for the best fit to the Q2 data. The values displayed in italics correspond to combination frequencies not exceeding a signal-to-noise level of 3.5. The residuals of the fit to the data are 0.0127 mag. We also list the most prominent additional period doubling components that are found in the data set. The uncertainty on f 0 is 1 × 10 −5 d −1 and the uncertainty on range 0.1 < A ± 1 /A1 < 0.3. From the Kepler data we find A + 1 /A1 = 0.267 ± 0.010 and A − 1 /A1 = 0.049 ± 0.007.defined as R k = A + k /A − k and ∆φ k = φ + k − φ − k , with A ± k andf B is 2 × 10 −4 d −1 . f [d −1 ] A Kp [mag] φ [ rad 2π ] σ(φ) f [d −1 ] A Kp [mag] φ [ rad 2π ] σ(φ) ±0.0006 ±0 Alcock et al. (2003) to quantify the degree of asymmetry in the peaks. The distribution of the Q parameter for the Blazhko stars from the MACHO data base (Alcock et al. 2003) peaks at +0.3. The positive Q points to an asymmetry with higher amplitudes at the higher frequency lobes in the triplets, as is mostly the case in Blazhko stars. Finally, we list the power difference of the side peak amplitudes, ∆A 2 k = (A + k ) 2 − (A − k ) 2 . As showed in their mathematical description of periodically modulated sinusoidal oscillation, this quantity depends on the phase difference between amplitude and phase modulation components. Hence, it is the more physically meaningful quantity to measure the asymmetry of the triplet than the amplitude ratios R k . If the phase difference between the amplitude and phase modulations lies between 0 and π, then A + 1 > A − 1 and the plot showing amplitude variation versus phase of maximum light has an anticlockwise progression, and vice versa (see Fig. 7). Table 2 shows, through its values of R k > 1 and ∆A 2 k , that the triplet components in RR Lyr always show asymmetry to the right side. However, the asymmetry of the side peak amplitudes decreases towards higher orders, reflected in increasingly smaller R k and ∆A 2 k values. Also their phase difference ∆φ k , after peaking at order k = 2, decreases towards higher orders. Jurcsik et al. (2005) pointed out that generally the side peaks show a less steep decrease in amplitude towards higher orders than the successive harmonics of f0. Fig. 8 (left panel) shows the amplitude ratios A k /A1, A + k /A + 1 , and A − k /A − 1 . We confirm the less steep, though not "linear" decrease for the modulation components in RR Lyr. The decrease at higher orders is less steep for the left sidepeak components. For the first time, we also show a similar figure for the quintuplet components (Fig, 8, right panel). Their decrease, especially at higher orders, is similar to that of the triplet components. Q k = A + k −A − k A + k +A − k , introduced by Fourier parameters RR Lyr's light curve dramatically changes its shape over the Blazhko cycle. Fig. 9 shows the variation of RR Lyr's light Table 2. Side lobe amplitude ratios R k , phase differences ∆φ k (×2π), asymmetry parameters Q k , and power differences ∆A 2 k as defined in the text, and their respective errors (based on Monte Carlo errors) for the Kp data of RR Lyr. k denotes the multiplet order. Values with a significance below 3σ are given in italics. We applied a time-dependent Fourier analysis (Kovács et al. 1987) to derive the Fourier parameters for subsets of data. The data set was subdivided into segments consisting of either one pulsation cycle (26-28 points in a group), or four pulsation cycles (109-111 points in a group), where the pulsation cycle was defined starting from minimum brightness. The data were phased with a constant period, the average period P = 0.566843 d obtained from our frequency analysis. A seventh-order harmonic fit, applied to these subsets of data, proved to yield the best fits for our sampling. The so derived Fourier parameters, i.e., the amplitude ratios R k1 = A k /A1 and the epoch-independent phase differences ϕ k1 = ϕ k −kϕ1, offer a way to quantify the shape of the light curves. We also used the analytical signal method (Kolláth et al. 2002) to analyze the data, yielding consistent results, but due to the gaps in the data we used the time-dependent Fourier analysis to present the continuous variation over the whole Blazhko cycle. k R k σ R k ∆φ k σ ∆φ k Q k σ Q k ∆A 2 k σ Time-dependent Fourier parameters derived from segments four periods long show a smooth time variation revealing the Blazhko time scale; this procedure averages out the period-to-period oscillation caused by the period doubling phenomenon which is seen when segmenting on single periods (Fig. 10). To show the light curve changes over the Blazhko cycle we use the four-cycle groups (Fig. 11), thus averaging out the period doubling effect. The interrelations between A1 and R21 (amplitude -amplitude ratio) and A1 and ϕ21 (amplitude -phase difference) describe loops with particular shapes, which models for the Blazhko effect should be able to reproduce. The A1 -R21 interrelation shows a double loop, of which the largest part has a counter-clockwise progression. For A1 -ϕ21 the (single) loop is wider towards lower A1 values, and its progression is clockwise. The loops for consecutive Blazhko cycles do not completely overlap: there is no exact repetition from one Blazhko cycle to the next. Fig. 12 shows the variation of the Fourier parameters during the modulation. The variability of A1 is almost sinusoidal, which shows that the modulation itself is not very nonlinear. R21 reaches its maximum when A1 is at minimum. This is an observation that imposes constraints on the theoretical models for the Blazhko effect. Generally, stars with smaller amplitudes have more sinusoidal light curves. If dur- ing the modulation pulsation energy is removed from the mode, the star is expected to behave as if being excited to a lower amplitude: its variability should be more sinusoidalcloser to the linear regime -and R21 should decrease. We observe the opposite: the pulsation is more nonlinear at smaller amplitudes. This may suggest that the modulation is caused by a mode resonance. Detailed hydrodynamical models are needed to investigate the origin of this nonlinearity. DISCUSSION Blazhko modulation For many decades, RR Lyr's Blazhko period was known to be 40.8 d (see, e.g., Szeidl & Kolláth 2000). Kolenberg et al. (2006, their modulation results in equidistantly spaced multiplets Benkő et al. 2009 As shown by Jurcsik et al. (2009), the mean parameters of a Blazhko RR Lyrae star vary over the course of the Blazhko cycle, such as the star's mean temperature, luminosity and radius. Due to these changes, also the star's intrinsic period, determined from the Kp light variations, varies between 0.5652 d and 0.5699 d (Fig. 4) for the Q2 data. As the Blazhko modulation appears to vary from one Blazhko cycle to the next, it is likely that the Blazhko period itself is also variable. Period doubling With the Kepler photometry, the period doubling phenomenon in RR Lyrae stars was reported for the first time (Kolenberg et al. 2010a). Even for a star as well-studied as RR Lyr, it had never been detected before. This is not only due to the much higher accuracy of the Kepler data, but also to the fact that from the mostly single-site observations of the star, consecutive pulsation cycles cannot be observed for stars with periods of typically half a day. The alternating higher and lower maxima would be easily missed by singlesite observations due to daily gaps. Also recent multi-site observing campaigns of the star (e.g., Kolenberg et al. 2006) did not lead to a detection of the effect. The period doubling and alternating heights of maxima are securely detected in two other Kepler stars, namely V808 Cyg (KIC4484128) and V355 Lyr (KIC755345) (see also Szabó et al. 2010). A further four Blazhko RR Lyrae stars in the Kepler field possibly exhibit the effect: V2178 Cygni (KIC3864443), V354 Lyrae (KIC6183128), V445 Lyrae (KIC6186029) and V360 Lyrae (KIC9697825) (see Benkő et al. 2010). So far the period doubling phenomenon has only been found in Blazhko stars and, though not all of our Kepler Blazhko stars seem to show it, there may be a connection between its occurrence and the Blazhko effect . The period doubling does not maintain the same strength along the Blazhko cycle. In some parts of the Blazhko cycle, the alternating light curve shapes are very obvious, in others indiscernible (see Fig. 4). It is also of interest that period doubling occurs in RV Tauri stars and in many hydrodynamical models of Pop.I (e.g., Moskalik & Buchler 1991;Aikawa 2001) and Pop.II cepheids (e.g., Buchler & Kovacs 1987;Buchler & Moskalik 1992). We refer to Szabó et al. (2010) for a more detailed study of the period doubling phenomenon from the Kepler data. They also investigate and simulate the deviation of the ratio from exact half-integer values. The latter is due to a combined effect of the Blazhko modulation and the temporal onset and disappearance of the half-integer frequencies.They suggest that a 9:2 resonance between the radial mode and the 9th-order radial overtone is responsible for the period doubling. Moskalik & Buchler (1990) studied the period doubling found in Pop.I and Pop.II cepheid models in detail. Through the analysis of the limit cycle stability they showed that the 3:2 (in Pop.I) and 5:2 (in Pop.II) resonances are responsible. They demonstrated the coincidence of the period doubling and the resonance center, and provided a description of the mechanism in which the resonance destabilizes the limit cycle and causes the bifurcation. They showed that every half integer resonance, in principle, is capable of causing period doubling. The detailed modeling of the period doubling in Blazhko RR Lyrae stars will be discussed in a forthcoming paper by Kolláth et al.. Residual frequencies and additional cycles The Kepler data of RR Lyr indicate that there is no strict repetition from one Blazhko cycle to the next, as is also reflected in Figs. 5 and 11. The meticulous efforts to reconstruct the light variation of RR Lyr described in Section 2.2 corroborate the reality of this variation. In their recent highquality CoRoT data of V1127 Aql, Chadid et al. (2010) found a small but significant shift of the maximum phase over five consecutive Blazhko cycles. On a longer time scale, irregularities in RR Lyr's Blazhko cycle were mentioned earlier by, e.g., Preston et al. (1965). Szeidl (1988) reported on weaker and stronger cycles in the star. From the Blazhko cycles covered by the data presented in this paper we also see variation, even from one Blazhko cycle to the next. Such a variation may be due to additional cycles in the star that are yet to be unraveled, or to non-periodic changes. The fact that we see residual signal around the position of the main frequency and its harmonics after subtracting the average frequency and the multiplet components is most likely due to these variations (periodic or transient). Finally, the large corrections needed to adjust the flux measurements during some phases (Fig. 3), particularly in Q2, are probably not critical with regard to the detected additional variations. As pointed out in Section 2.2, these corrections do not introduce significant "scatter" in the frequency results. Hence we can safely claim that most of the "scatter" can be attributed to variations in the Blazhko effect from cycle to cycle. RR Lyr itself is reported to have a longer cycle, of about four years, as first mentioned by Detre & Szeidl (1973). At the end of such a four-year cycle, the star's Blazhko effect is reported to weaken in strength, after which a new cycle starts with increased strength and with a shift (of about 0.25) in the Blazhko phase ψ. Kolenberg et al. (2006Kolenberg et al. ( , 2008 list ephemerides for the Blazhko phase based on data gathered over the past decade. However, with a changing Blazhko period and large gaps in the data, it is nearly impossible to determine whether there has been a phase shift in the Blazhko effect at a distinct moment in time. If the abrupt phase shift is a real phenomenon, we hope to be able to observe it in the Kepler data to come. Due to the period doubling it is not trivial to establish precise ephemerides for RR Lyr, especially with long-cadence data. Forthcoming data gathered in short cadence will allow us to do so. Fig. 9 also shows the periodic behaviour of the so-called "bump" in the light curve that appears just before minimum light. The bump occurs at an earlier phase around Blazhko minimum and later around Blazhko maximum. Also its strength shows a dependence on the Blazhko phase: it is most distinct during Blazhko minimum and becomes weaker around Blazhko maximum. A similar behaviour of the bump was also observed by in the Blazhko star SS For. Guggenberger & Kolenberg (2007) investigated it more quantitatively for both SS For and RR Lyr. Light curve changes and bump behaviour The bump has been explained by a collision between layers in the deep atmosphere, leading to shock waves that, depending on their direction, are explained by the "infall model" or the "echo model" (Hill 1972;Gillet & Crowe 1988). In their seminal work on spectroscopic data of RR Lyr, Preston et al. (1965) concluded that a displacement of the shock forming region occurs over the Blazhko cycle. This induces an earlier or later occurrence of the bump in the light curves. Our observations suggest that this collision may be stronger and earlier when the star is at minimum Blazhko phase. According to the findings by Jurcsik et al. (2009) for MW Lyr, this is also the phase when the star's mean radius is smaller (see their fig. 14), which could explain the earlier occurrence of the collision. Although the radiative hydrodynamical models of Stothers (2006) are very crude, especially regarding the amplitudes of small features, the phase of the bump in his models also occurs as observed when the pulsation amplitude is low. So it may be that this phasing is a consequence of an overall lower light amplitude, with the "echo" arriving at the surface sooner because it is running through a more compact star. Moreover, the so-called "hump", the shoulder on the rising branch of the light curve, only occurs at certain phases in the Blazhko cycle. In another early paper, Struve (1948) reported striking hydrogen emission line profiles in RR Lyr at particular, always "rising Blazhko" phases (increasing light amplitude of the cycle light curves). This happens to be the Blazhko phase interval when we see a more pronounced hump on the rising branch. The sudden slope change of the rising branch may be connected with layers colliding (hence emission in the spectra) and thus slowing down the motions (hence slope change in the light curve). With long cadence data the rising branch is covered with too few data points (often only three). Both the bump and hump characteristics can be investigated in detail with short-cadence data of RR Lyr. Constraints on the existing Blazhko models How do the Kepler observations of RR Lyr provide constraints for improving the existing Blazhko models? The most striking features we found, for the first time, in this data set are the following: (i) We found additional frequencies: quintuplet components and, most importantly, the period doubling phenomenon. (ii) We detected extreme light curve variations measured with high accuracy, e.g., the hump and bump feature. (iii) We observed that there is no strict repetition from one Blazhko cycle to the next, and that the Blazhko period may also vary on this short time scale. In addition, the most important findings so far from long cadence data of all 29 Kepler RR Lyrae stars are ): (i) About half of the stars, 14 out of 29 studies RR Lyrae stars in our Kepler sample, are modulated. (ii) Period doubling and additionally excited higher-order radial overtones detected in seven stars, all of them Blazhko stars. (iii) All 14 Blazhko stars show both period modulation and amplitude modulation, in differing degrees. With increasingly impressive data sets and subsequent new and detailed findings, it is clear that the models so far proposed for the Blazhko effect are lagging behind. Any model for the Blazhko effect has to take into account both amplitude and phase modulation, as pointed out by Benkő et al. (2010). Moreover, explanations that strictly impose a single modulation period, (e.g., the star's rotation period) not allowing for variation of the modulation, are definitely no longer viable. The moving light curve features (hump, bump) can provide additional clues about the stellar dynamics varying over the Blazhko cycle. Moreover, any plausible model has to be able to account for the (transient) occurrence of period doubling and possible excitation of higher order overtones. The recent findings by Szabó et al. (2010) in connection with the period doubling suggest a 9:2 resonance between the radial mode and the 9th-order radial overtone (a strange mode) to be responsible for the period doubling. In the scenario proposed by Stothers (2006) a variation of the star's convection is the cause for the Blazhko modulation. As the star's mean stellar parameters (temperature, radius and luminosity) vary over the Blazhko cycle (see Jurcsik et al. 2009), one can reasonably assume that its turbulent/convective structure also changes. It is unclear, however, what could lie behind this change. We note that transient magnetic fields at the origin of the variation in convective turbulence, as postulated by Stothers (2006), are most probably undetectable with present-day instrumentation, and thus their existence would be hard to prove. It is clear that it is time to revise or expand the existing models for the Blazhko effect, as well as explore alternative explanations, using the constraints described above. Perhaps it is also time to revisit previously discarded scenarios, such as models based on radial mode resonances (see, e.g., Goranskij et al. 2009;Kovács 2009), or explore radial hydrodynamical models in much more detail. After the models by Borkowski (1980) and Moskalik (1986) were dismissed by lack of confirmation from hydrodynamical models, most attention has been on resonances involving nonradial modes. CONCLUSIONS AND OUTLOOK The first releases of the Kepler data of the prototypical star RR Lyr are revolutionary because: • These data prove that bright stars such as RR Lyr can be observed with Kepler, as the saturated flux is conserved to a high degree (less than 2.5×10 −3 fractional error even at maximum flux). Generally the philosophy is not to waste any pixels on a star of which one cannot capture all the useful flux. RR Lyr is an unusual case due to its high variability. Because there are measurements in which the star is well captured and there is a lot of saturated flux in columns captured in every measurement, we can make good estimates of missing flux and thus recover photometry. Such unusual cases were not considered earlier in the Kepler Mission. • From the three Blazhko cycles covered by the data set (Q1+Q2 data), we obtain f0 = 1.76416±0.00001 d −1 for the average radial pulsation frequency and fB = 0.0256 ± 0.0002 d −1 for the Blazhko frequency. The Blazhko period is PB = 39.1 ± 0.3 d, which confirms the earlier findings that the star's modulation period has gradually become shorter over the past decades (Kolenberg et al. 2006). • There are, however, also indications that the Blazhko period is variable on shorter time scales, because our data clearly indicate that there is no strict repetition from one Blazhko cycle to the next. • We clearly detect quintuplet components in the frequency spectrum of the Kepler RR Lyr data. It is the first time these are found in the star. Their behaviour is similar to that of the triplet components. • Moreover, we find clear evidence for additional frequencies in the data. The most striking new frequency pattern is given by the so-called half-integer frequencies, that occur at 2k−1 2 f0, with the highest peaks around 3/2f0, 5/2f0, 1/2f0, etc. Their occurrence is connected to the observation of alternating higher and lower maxima throughout (certain phase intervals of) the Blazhko cycle. This phenomenon, called period doubling, has been observed for the first time in Kepler data (Kolenberg et al. 2010a). Meanwhile, we have observed it in several Blazhko stars observed with Kepler, and its presence may be connected to the Blazhko effect. The period doubling is variable over the course of the Blazhko cycle, and also does not repeat identically from one Blazhko cycle to the next . • The Fourier parameters derived from individual cycles oscillate around the Fourier parameters derived from several consecutive cycles, for which the period doubling effect is smoothed out. • The increase of the Fourier parameter R21 with decreasing pulsation amplitude A1 is counterintuitive, because pulsations with lower amplitudes (lower A1) are expected to be less nonlinear (lower R21). This observation provides constraints for modelling the physics of the Blazhko modulation. • The position and strength of the bump and hump vary over the Blazhko cycle. A detailed study of these features in the pulsation cycle, in photometric as well as spectroscopic data, may shed new light upon the pulsation and shock propagation in RR Lyrae stars and the understanding of the Blazhko effect. After the Q2 observations, it was realized that RR Lyr's Kepler KIC magnitude was not bright enough to match the observations, and it was adjusted upwards. As a consequence, the standard aperture assignment algorithms assigned it to an aperture that went off silicon. Thus the target was rejected, and was not observed in Q3 and Q4. The first Kepler findings described in Kolenberg et al. (2010a), however, triggered further interest in the star. Therefore, a custom aperture was devised for RR Lyr (by Steve Bryson), reducing the amount of originally assigned pixels from 433 to about 150. Thanks to the calibration work described in this paper, RR Lyr will likely be scheduled to be a target of Kepler for the rest of the mission. At the time of writing of this paper, RR Lyr is being observed through the Kepler Guest Observer (GO) program (http://keplergo.arc.nasa.gov/). As further Kepler data of RR Lyr become available, we will be able to perform more focused analyses: • A longer time base can lead to a better resolved frequency spectrum. However, due to the period variation and the transient phenomena, the frequency spectrum will be smeared out. Hence, in reality we will probably not obtain "sharper" peaks with a longer time base, but several closelyspaced peaks. New and additional Kepler data will enable us to analyse the temporal variation of the frequencies. They will also help uncover whether the variation between consecutive Blazhko cycles is due to additional cycles in the star, or to transient or quasi-periodic changes. • Kepler short-cadence data will allow us to study shorttime effects and light curve features such as the hump and bump. Simultaneous ground-based data are planned for the period of the short-cadence Kepler observations. • With Kepler's potential coverage of RR Lyr's light variations over 3.5-5 y, many mysteries still surrounding this prototypical star can be solved. We may witness the start of a new 4-year cycle (Detre & Szeidl 1973), as RR Lyr is known to have. • With Kepler we can anticipate a dramatic overhaul in the models for the Blazhko effect. The constraints provided by the ultra-precise data of RR Lyr motivate us to revisit, revise or expand the existing models for the Blazhko effect, as well as explore alternative explanations. Figure 2 . 2RR Lyr's downlinked pixels during quarter 2 (Q2), Figure 3 . 3Comparison between the captured flux from RR Lyr (dashed line) and the corrected flux (full line) during Q2, with ±1 σ error bars. Figure 5 . 5Residual spectrum of the Kepler RR Lyr data after prewhitening with the main frequency, its harmonics and the triplet components (top panel). Zooms around some of the quintuplet components are shown in the bottom panels. Figure 4 4Figure 4. Relationship between the period change and amplitude modulation over the Blazhko cycle for RR Lyr. The crosses show the Kepler data for the three monitored Blazhko cycles, the connecting line our multi-frequency fit to the data. The thick line shows an instantaneous period determined using the analytical signal method (units to the right side of the panel). The period is longest just before the lowest amplitude phases and shortest just after the highest amplitude phases. Note the small changes in the period variation between consecutive Blazhko cycles, and the alternating high and low maxima at certain Blazhko phases. Figure 7 . 7Amplitude variation versus phase of maximum light. The big dot marks the beginning of the trajectory. Figure 8 . 8Left panel: amplitude ratios A k /A 1 , A + k /A + 1 , and A − k /A − 1 of the detected harmonic and triplet frequencies in RR Lyr, with error bars. Right panel: amplitude ratios A ++ k /A ++ 1 , and A −− k /A −− 1 of the detected quintuplet frequencies in RR Lyr, with error bars. Figure 9 . 9Variation of RR Lyr's Kepler light curve (magnitude versus phase) for the second full Blazhko cycle in Q2, showing 12 snapshots (connected points) roughly equally spaced in time; each sixth pulsation cycle is plotted. Blazhko phase ψ = 0 corresponds to maximum light amplitude. The dots are all the data in the Blazhko cycle folded with the main period. Figure 10 . 10Comparison of the Fourier parameters (FP) A 1 derived from one-cycle and from four-cycle segments. The effect of period doubling is clear in the parameters derived from one-cycle segments. The oscillation of the parameter A 1 reflects the alternating height of successive light curves. When taking four-cycle segments this effect is smoothed out. Figure 11. Interrelations between the Fourier parameters for successive Blazhko cycles, with error bars. The Q1 data are connected with a dashed line, the Q2 data with a solid line. Arrows show the plot's progression. In addition, big dots mark the beginning of trajectories. There is no exact repetition from one Blazhko cycle to the next.table 6) list a summary of the Blazhko peri- ods of RR Lyr quoted in previous papers devoted to the star. Unfortunately, error bars are rarely given and hard to recon- struct. In the course of the past decades, the Blazhko period appears to have shortened. From the Q1 and Q2 Kepler data we find a value for the Blazhko period PB = 39.1 ± 0.3 d, corresponding to a Blazhko frequency of fB = 0.0256 d −1 . From the Q2 Kepler data alone, we find PB = 39.6 ± 0.3 d, corresponding to a Blazhko frequency of fB = 0.0252 d −1 . It is well established that periodic amplitude and phase 0.35 0.40 0.45 0.50 0.55 0.60 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 R 21 A 1 Q2 Q1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 21 A 1 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 A 1 0.40 0.45 0.50 0.55 0.60 R 21 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 55000 55010 55020 55030 55040 55050 55060 55070 55080 55090 HJD 2400000 21 Figure 12. Variation of the Fourier parameters during the Blazhko modulation. DATA ANALYSIS AND RESULTSA detailed visual inspection of the Kepler data of RR Lyr has already revealed changes between consecutive Blazhko cycles, as well as alternating higher and lower light curve maxima at certain phases in the Blazhko cycle (seefig. 4inKolenberg et al. 2010a). Because of this, we can expect a more complex frequency spectrum than the "classical" "main frequency + harmonics + equidistant multiplet structures" predicted for a monoperiodic star undergoing amplitude and phase modulationBenkő et al. 2009). Moreover, the frequency solution we obtain partly depends on the coverage, as long-term modulation, secular trends, or transient phenomena cannot be fully captured by the Fourier techniques we adopt. The Fourier analyses presented here were performed with Period04(Lenz & Breger 2005).Without adjustments, there is a small shift between the Q1 and Q2 data even after the calibrations described in Section 2. To create the best match between the two data sets (as shown inFig. 1), we fitted the complete frequency solution (described in the next sections) to each set separately and subsequently shifted by the zero point of the obtained fit. We used the combined data set (Q1 and Q2) to determine frequencies, as a longer time base and better coverage of the Blazhko cycle yields more accurate (mean) frequency values. Though we do not have sufficient information to reliably combine the Q1 and Q2 data regarding their amplitudes, the obtained frequency values are not significantly influenced by the small zero point shifts. Subsequently, we applied the harmonic fit with the frequencies derived from ACKNOWLEDGMENTSThe authors kindly thank the anonymous referee for constructive comments. Funding for this Discovery mission is provided by NASA's Science Mission Directorate. The authors gratefully acknowledge the entire Kepler team, whose outstanding efforts have made these results possible.KK and EG acknowledge support from the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project number T359-N16 and P19962. 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[ "Provably Efficient Lifelong Reinforcement Learning with Linear Function Approximation", "Provably Efficient Lifelong Reinforcement Learning with Linear Function Approximation" ]	[ "Sanae Amani [email protected] ", "Lin F Yang [email protected] \nUniversity of California\nLos Angeles\n", "Ching-An Cheng [email protected] \nMicrosoft Research\n\n" ]	[ "University of California\nLos Angeles", "Microsoft Research\n" ]	[]	We study lifelong reinforcement learning (RL) in a regret minimization setting of linear contextual Markov decision process (MDP), where the agent needs to learn a multi-task policy while solving a streaming sequence of tasks. We propose an algorithm, called UCB Lifelong Value Distillation (UCBlvd), that provably achieves sublinear regret for any sequence of tasks, which may be adaptively chosen based on the agent's past behaviors. Remarkably, our algorithm uses only sublinear number of planning calls, which means that the agent eventually learns a policy that is near optimal for multiple tasks (seen or unseen) without the need of deliberate planning. A key to this property is a new structural assumption that enables computation sharing across tasks during exploration. Specifically, for K task episodes of horizon H, our algorithm has a regret boundÕ( (d 3 + d ′ d)H 4 K) based on O(dH log(K)) number of planning calls, where d and d ′ are the feature dimensions of the dynamics and rewards, respectively. This theoretical guarantee implies that our algorithm can enable a lifelong learning agent to accumulate experiences and learn to rapidly solve new tasks.	10.48550/arxiv.2206.00270	[ "https://arxiv.org/pdf/2206.00270v1.pdf" ]	249,240,563	2206.00270	054f555d1a414172dfbd60e8bfe71129c2ddf860	Provably Efficient Lifelong Reinforcement Learning with Linear Function Approximation 1 Jun 2022 Sanae Amani [email protected] Lin F Yang [email protected] University of California Los Angeles Ching-An Cheng [email protected] Microsoft Research Provably Efficient Lifelong Reinforcement Learning with Linear Function Approximation 1 Jun 2022 We study lifelong reinforcement learning (RL) in a regret minimization setting of linear contextual Markov decision process (MDP), where the agent needs to learn a multi-task policy while solving a streaming sequence of tasks. We propose an algorithm, called UCB Lifelong Value Distillation (UCBlvd), that provably achieves sublinear regret for any sequence of tasks, which may be adaptively chosen based on the agent's past behaviors. Remarkably, our algorithm uses only sublinear number of planning calls, which means that the agent eventually learns a policy that is near optimal for multiple tasks (seen or unseen) without the need of deliberate planning. A key to this property is a new structural assumption that enables computation sharing across tasks during exploration. Specifically, for K task episodes of horizon H, our algorithm has a regret boundÕ( (d 3 + d ′ d)H 4 K) based on O(dH log(K)) number of planning calls, where d and d ′ are the feature dimensions of the dynamics and rewards, respectively. This theoretical guarantee implies that our algorithm can enable a lifelong learning agent to accumulate experiences and learn to rapidly solve new tasks. Introduction Recently there has been a surging interest in designing lifelong learning agents that can continuously learn to solve multiple sequential decision making problems in its lifetime [Thrun and Mitchell, 1995, Silver et al., 2013, Xie and Finn, 2021. This scenario is in particular motivated by building multi-purpose embodied intelligence [Roy et al., 2021], such as robots working in a weakly structured environment. Typically, curating all tasks beforehand for such problems is nearly infeasible, and the problems the agent is tasked with may be adaptively selected based on the agent's past behaviors. Consider household robot as an example. Because each household is unique, it is difficult to anticipate upfront all scenarios the robot would encounter. Moreover, the tasks the robot faces are not independent and identically distributed (i.i.d.). Instead, what the robot has done before can affect the next task and its starting state; e.g., if the robot fails to bring a glass of water and breaks it, then the user is likely to command the robot to clean up the mess. Therefore, it is critical that the agent can continuously improve and generalize learned abilities to different tasks, regardless of their order. In this work, we study lifelong reinforcement learning (RL) theoretically in a regret minimization setting [Thrun andMitchell, 1995, Ammar et al., 2015], where the agent needs to solve a sequence of tasks using rewards in an unknown environment while balancing exploration and exploitation. Motivated by the embodied intelligence scenario, we suppose that tasks differ in rewards, but share the same state and action spaces and the transition dynamics [Xie and Finn, 2021]. To be realistic, we make no assumptions on how the tasks and initial states are selected. 1 Generally, we allow them to be chosen from a continuous set potentially by an adversary based on the agent's past behaviors. Once a task is specified, the agent has one chance to complete the task and then the next task is revealed. The goal of the agent is to perform near optimally for the tasks it faces, despite the online nature of the problem. For simplicity, we assume that there is no memory constraint; this is usually the case for robotics applications where real-world interactions are the main bottleneck [Xie and Finn, 2021]. Nonetheless, the agent should eventually learn to make decisions without frequent deliberate planning, because planning is time consuming and creates undesirable wait time for user-interactive scenarios. In other words, the agent needs to learn a multi-task policy, generalizing from not only past samples but also past computation, to solve new tasks. Formally, we consider an episodic setup based on the framework of contextual Markov decision process (MDP) [Abbasi-Yadkori andNeu, 2014, Hallak et al., 2015]. It repeats the following steps: 1) At the beginning of an episode, the agent is set to an initial state and receives a context specifying the task reward, both of which can be arbitrarily chosen. 2) When needed, the agent uses its past experiences to plan for the current task. 3) The agent runs a policy in the environment for a fixed horizon in an attempt to solve the assigned task and gains experiences from its policy execution. The agent's performance is measured as the regret with respect to the optimal policy of the corresponding task. We require that, for any task sequence, both the agent's overall regret and number of planning calls to be sublinear in the number of episodes. While lifelong RL is not new, the need of simultaneously achieving 1) sublinear regret and 2) sublinear number of planning calls for 3) a potential adversarial sequence of tasks and initial states makes the setup considered here particularly challenging. To our knowledge, existing works only address a strict subset of these requirements; especially, the computation aspect is often ignored. Most provable works in lifelong RL make the assumption that the tasks are finitely many [Ammar et al., 2015, Ammar et al., 2014, Zhan et al., 2017, Brunskill and Li, 2015, or are i.i.d. [Brunskill and Li, 2014, Abel et al., 2018a, Abel et al., 2018b, Lecarpentier et al., 2021, while others considering similar setups to ours do not provide regret guarantees [Isele et al., 2016, Xie andFinn, 2021]. On the technical side, we found that the closest works are [Modi and Tewari, 2020, Abbasi-Yadkori and Neu, 2014, Hallak et al., 2015, Modi et al., 2018, Kakade et al., 2020 for contextual MDP and[Wu et al., 2021, Abels et al., 2019] for the dynamic setting of multi-objective RL, which study the sample complexity of learning for arbitrary task sequences; however, they either assume the problem is tabular or require a model-based planning oracle with unknown complexity. Importantly, none of the existing works properly address the need of sublinear number of planning calls, which creates a large gap between the abstract setup and practice need. In this paper, we propose the first provably efficient lifelong RL algorithm, UCB Lifelong Value Distillation (UCBlvd, pronounced as "UC Boulevard"), that possesses all three desired qualities. Under the assumption of contextual MDP with linear features [Yang andWang, 2019, Jin et al., 2020] and a new completeness-style assumption, UCBlvd achieves sublinear regret for any online sequence of tasks while using sublinear number of planning calls. Specifically, for K episodes of horizon H, we prove a regret boundÕ( (d 3 + d ′ d)H 4 K) based onÕ(dH log(K)) number of planning calls, where d and d ′ are the feature dimensions of the dynamics and rewards, respectively. From a high-level viewpoint, UCBlvd uses the linear structure to identify what to transfer and operates by interleaving 1) independent planning for a set of representative task contexts and 2) distilling the planned results into a multi-task value-based policy. UCBlvd also constantly monitors whether the new experiences it gained is sufficiently significant, based on a doubling schedule, to avoid unnecessary planning. The design of UCBlvd is inspired by single-task LSVI-UCB [Jin et al., 2020] but we introduce a novel distillation step, along with a new completeness assumption, to enable computation sharing across tasks; in addition, we extend the low-switching cost technique [Abbasi-Yadkori et al., 2011, Gao et al., 2021 for single-task RL to the lifelong setup to achieve sublinear number of planning calls. Preliminaries Notation. Throughout the paper, we use lower-case letters for scalars, lower-case bold letters for vectors, and upper-case bold letters for matrices. The Euclidean-norm of x is denoted by x 2 . We denote the transpose of a vector x by x ⊤ . For any vectors x and y, we use x, y to denote their inner product. We denote the Kronecker product by A ⊗ B. Let A ∈ R d×d be a positive definite and ν ∈ R d . The weighted 2-norm of ν with respect to A is defined by ν A := √ ν ⊤ Aν. For a positive integer n, [n] denotes the {1, 2, . . . , n}. For a real number α, we denote {α} + = max{α, 0}. Finally, we use the notationÕ for big-O notation that ignores logarithmic factors. Problem Formulation We formulate lifelong RL as a regret minimization problem in contextual MDP [Abbasi-Yadkori andNeu, 2014, Hallak et al., 2015] with adversarial context and initial state sequences. We suppose that a context determines the reward but does not affect the dynamics. Such a context dependency is common for the lifelong learning scenario where an embodied agent consecutively solves multiple tasks. Below we give the formal problem definition. Finite-horizon Contextual MDP. We consider a finite-horizon contextual MDP denoted by M = (S, A, W, H, P, r), where S is the state space, A is the action space, W is the task context space, H is the horizon (length of each episode), P = {P h } H h=1 are the transition probabilities, and r = {r h } H h=1 are the reward functions. We allow S and W to be continuous or infinitely large, while we assume A is finite such that max a∈A can be performed easily. For h ∈ [H], r h (s, a, w) denotes the reward function whose range is assumed to be in [0, 1], and P h (s ′ \|s, a) denotes the probability of transitioning to state s ′ upon playing action a at state s. In short, a contextual MDP can be viewed as an MDP with state space S × W and action space A where the context part of the state remains constant in an episode. 2 To simplify the notation, for any function f , we write P h [f ](s, a) := E s ′ ∼P h (.\|s,a) [f (s ′ )]. Policy and Value Functions. In a finite-horizon contextual MDP, a policy π = {π h } H h=1 is a sequence where π h : S × W → A determines the agent's action at time-step h. Given π, we define its state value function as V π h (s, w) := E[ H h ′ =h r h ′ s h ′ , π h ′ (s h ′ , w) , w)\|s h = s and its action-value function as Q π h (s, a, w) := r h (s, a, w) + P h [V π h+1 (., w)](s, a), where Q π H+1 = 0. We denote the optimal policy as π * h (s, w) := sup π V π h (s, w), and let V * h := V π * h and Q * h := Q π * h denote the optimal value functions. Lastly, we recall the Bellman equation of the optimal policy: Q * h (s, a, w) = r h (s, a, w) + P h [V * h+1 (., w)](s, a), V * h (s, w) = max a∈A Q * h (s, a, w),(1) Interaction Protocol of Lifelong RL. The agent interacts with a contextual MDP M in episodes. For presentation simplicity, we assume that the reward functions r are known, while the transition probabilities P are unknown and must be learned online; we will discuss how reward learning can be naturally incorporated in Section 4.3. At the beginning of episode k, the agent receives a task context w k ∈ W and is set to an initial state s k 1 , both of which can be adversarially chosen. The agent can use past experiences to plan for the current task, if needed. Then the agent executes its policy π k : at each time-step h ∈ [H], it observes the state s k h , plays an action a k h = π k h (s k h , w k ), observes a reward r k h := r h (s k h , a k h , w k ), and goes to the next state s k h+1 according to P h (.\|s k h , a k h ). Let K be the total number of episodes. The agent's goal is to achieve sublinear regret, where the regret is defined as R K := K k=1 V * 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ).(2) As the comparator policy above (namely π * that defines V * 1 ) also knows the task context, achieving sublinear regret implies that the agent would attain near task-specific optimal performance on average. Assumptions Throughout the paper, we rely on the following assumptions. := [µ h (1) , . . . , µ h (d) ] ⊤ over S such that P h (.\|s, a) = µ h (.), φ(s, a) and r h (s, a, w) = η h , ψ(s, a, w) , for all (s, a, w) ∈ S × A × W. Assumption 2 (Boundedness). Without loss of generality, φ(s, a) 2 ≤ 1, ψ(s, a, w) 2 ≤ 1, µ h (S) 2 ≤ √ d, and η h 2 ≤ √ d ′ for all (s, a, w, h) ∈ S × A × W × [H]. Example 1 (Weighted Rewards). An interesting and common special case is ψ(s, a, w) = φ(s, a) ⊗ ρ(w), for some mapping ρ : W → R m . In this case, it holds that d ′ = md and r h (s, a, w) = ρ(w), r h (s, a) , where r h (s, a) = A h φ(s, a) ∈ R m , for some A h ∈ R m×d , is the vector reward functions at time-step h. We can view r h (s, a, w) as a weighted reward with weights ρ(w) that depend on task w. This setting is closely related to Multi-Objective RL studied for tabular case in [Wu et al., 2021], which studies the case where ρ(w) = w ∈ R m along with tabular S and A. A Warm-up Algorithm for Lifelong RL We first present a warm-up algorithm, termed Lifelong Least-Squares Value Iteration (Lifelong-LSVI), in Algorithm 1. Lifelong-LSVI extends the single-task LSVI-UCB algorithm proposed by [Jin et al., 2020] to the lifelong learning setting. It runs LSVI-UCB as a subroutine for each task by leveraging the structure in Assumption 1: as the transition dynamics is context independent, dynamics samples collected during solving other tasks can be relabeled with the current reward to plan for the current task. The motivation of this warm-up algorithm is to give intuitions on how the problem structure in Assumption 1 can be used to achieve small regret in lifelong learning. We will show that Lifelong-LSVI has a sublinear regret bound, which matches the minimax optimal rate in the special case studied by [Wu et al., 2021] in terms of number of objectives, m (see Example 1). However, we will also show that Lifelong-LSVI is not computationally efficient, in the sense that the number of planning calls it requires grows linearly with the number of episodes. This is because the agent never learns to internalize the task solving skills but requires going though all past experiences for planning every time a new task arrives. Moreover, we will discuss why it cannot be made computationally efficient in an easy manner. This drawback motivates our main algorithm, UCBlvd, in Section 4, which is provably efficient in terms of both regret and number of planning calls. Algorithmic Notations To begin, we introduce the template and the notations that will be used commonly in presenting the warmup algorithm, Lifelong-LSVI, and our main algorithm, UCBlvd. For each algorithm, first we will define an algorithm-specific action-value function Q k h : S × A × W → R, which determines the agent's policy at time-step h in episode k; then we present the full algorithm and its analyses using the quantities below, which are defined with respect to each algorithm's definition of Q k h . Given {Q k h } h∈[H] , we define state value functions and their backups as V k h (s, w) := min max a∈A Q k h (s, a, w), H ,(3)θ k h (w) := S V k h+1 (s ′ , w)dµ h (s ′ ),(4) Thanks to the linear MDP structure in Assumption 1, it holds that P h V k h+1 (., w) (s, a) = θ k h (w), φ(s, a) .(5) Let λ > 0 be a constant. We define the λ-regularized least squares estimator of θ k h (w) as θ k h (w) := Λ k h −1 k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) (6) Λ k h := λI d + k−1 τ =1 φ τ h φ τ h ⊤ . (7) whereθ k h (w) is the solution to min θ∈R d k−1 τ =1 ( θ, φ(s τ h , a τ h ) − V k h+1 (s τ h+1 , w)) 2 + λ θ 2 2 , φ τ h := φ(s τ h , a τ h ) , and I d ∈ R d×d is the identity matrix. Details of Lifelong-LSVI and Theoretical Analysis We define the upper confidence bound (UCB) style action-value function of Lifelong-LSVI as follows: Q k h (s, a, w) := r h (s, a, w) + θ k h (w), φ(s, a) + β φ(s, a) (Λ k h ) −1 ,(8) where Q k H+1 (., ., .) = 0 andθ k h (w) and Λ k h are defined in (6) and (7), respectively. Here, β is an exploration factor that is passed as an input of Lifelong-LSVI and will be appropriately chosen in Theorem 1. At episode k, given w k , Lifelong-LSVI first performs planning backward in time based on past data to computeθ k h (w k ) in (6) using Q k h+1 defined in (8) (Lines 4-5). Then, in execution, it usesθ k h (w k ) to compute Q k h (s k h , a, w k ) for the current state and all a ∈ A (Line 7) and executes the action with the highest value (Line 8). We show that Lifelong-LSVI achieves sublinear regret for our lifelong RL setup. The complete proof is reported in Appendix A, which follows the ideas of LSVI-UCB [Jin et al., 2020]. Theorem 1. Let T = KH. Under Assumptions 1 and 2, there exists an absolute constant c > 0 such that for any fixed δ ∈ (0, 0.5), if we set λ = 1 and β = cH d + √ d ′ log(dd ′ T /δ) in Algorithm 1, then with probability at least 1 − 2δ, it holds that R K ≤ 2H T log(dT /δ) + 2Hβ 2dK log(K) ≤Õ (d 3 + dd ′ )H 3 T . Before moving forward to our main algorithm in Section 4, we make a few remarks on the regret and number of planning calls of Lifelong-LSVI. First, Theorem 1 implies that for the special case studied by Compute Qk h (s k h , a, w k ) for all a ∈ A as in (10). 10 Play a k h = arg max a∈A Qk h (s k h , a, w k ) and observe s k h+1 and r k h . [Wu et al., 2021] (summarized in Example 1), the regret bound of Lifelong-LSVI becomesÕ( √ md 3 H 3 T ). This rate is optimal in terms of its dependency on m, as shown in [Wu et al., 2021], for this specific reward structure. Furthermore, this rate matches the regret dependencies on d and H of LSVI-UCB's for the single-task setting [Jin et al., 2020]. While Lifelong-LSVI has a decent regret guarantee, we observe that it requires computingθ k h (w k ) for all h ∈ [H], whenever a distinct new task w k arrives. Since the number of unique tasks may be as large as K, the total number of planning calls required in Lifelong-LSVI is K in the worst case. Unfortunately, the number of planning calls of Lifelong-LSVI cannot be easily improved due to the nonlinear dependency of (8), which could lead to a covering number no less than K in general. In particular, it is also hopeless to employ low switching cost techniques like [Abbasi-Yadkori et al., 2011] to reduce the number of planning calls, because we always need to recalculateθ k h (w) for every new task. In the next section, we discuss how placing a completeness-style assumption would help circumvent the issue of non-linear dependency of the action-value functions on w, and consequently would enable computation sharing to decrease the number of planning calls to O(dH log 1 + K/dλ ). Q k h (s, a, w) on w throughθ k h (w) in UCB Lifelong Value Distillation (UCBlvd) In this section, we present our main algorithm, UCB Lifelong Value Distillation (UCBlvd), in Algorithm 2. Under extra structural assumptions we will introduce in Section 4.1, UCBlvd shares the same regret bound as Lifelong-LSVI but reduces the number of planning calls to be sublinear. In contrast to Lifelong-LSVI which learns individual action-value function for each w k using φ(s, a), UCBlvd learns a single action-value function for all w ∈ W based on ψ(s, a, w) to enable computation sharing across tasks. In general, directly extending Lifelong-LSVI to use feature ψ(s, a, w) ∈ R d ′ with d ′ ≥ d would increase the regret from that with φ(s, a) ∈ R d , because the latter can exploit the context-independent dynamics structure. UCBlvd maintains the same order of regret as Lifelong-LSVI by separating the planning into a novel two-step process: 1) independent planning with φ for a set of representative task contexts and 2) distilling the planned results into a multi-task value function parameterized by ψ. In addition, UCBlvd runs a doubling schedule to decide whether replanning is necessary, which makes the total number of planning calls sublinear. Below we give the details of UCBlvd. Enabling Computation Sharing First, we introduce two extra assumptions needed by UCBlvd to share computation across tasks. We will discuss how these assumptions can be relaxed in Section 4.3. The first assumption is a new completeness-style assumption. Assumption 3 (Completeness). Given feature maps φ : S × A → R d and ψ : S × A × W → R d ′ in Assumption 1, consider the function class F = f : f (s, w) = min max a∈A ν, ψ(s, a, w) + β φ(s, a) Λ −1 + , H , ν ∈ R d ′ , Λ ∈ S d ++ , β ∈ R . For any f ∈ F and h ∈ [H], there exists a vector ξ f h ∈ R d ′ with ξ f h ≤ H √ d ′ such that P h f (., w) (s, a) = ξ f h , ψ(s, a, w) . It says the backup of functions in F should be captured by the feature ψ with bounded parameters. The definition of F models closely the structure of action-value function used by Lifelong-LSVI in (8), except θ k h (w), φ(s, a) there is replaced by functions linear in ψ(s, a, w). We will see that the action-value function used by UCBlvd defined in the next section is contained in F . We introduce an extra structure on ψ inspired by Example 1. Assumption 4 (Mappings). We assume ψ(s, a, w) = φ(s, a) ⊗ ρ(w), for some mapping ρ : W → R m , i.e., d ′ = md. We assume that there is a known set {w (1) , w (2) , . . . , w (n) } of n ≤ m task contexts such that ρ(w) ∈ Span({ρ(w (j) )} j∈[n] ) for all w ∈ W. That is, for any w ∈ W, there exist coefficients {c j (w)} j∈[n] such that ρ(w) = j∈[n] c j (w)ρ(w (j) ).(9) We assume j∈[n] c j (w) ≤ L for all w ∈ W and some L < ∞ 3 . Details of UCBlvd We define the UCB style action-value function of UCBlvd as follows: Q k h (s, a, w) := r h (s, a, w) + ξ k h , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k h ) −1 + ,(10) The parameterξ k h is computed by solving the convex quadratically constrained quadratic program (QCQP) in (11) below, which is defined on a set of representative task contexts {w (1) , w (2) , . . . , w (n) } in Assumption 4 and state-action pairs D := (s, a) : φ(s, a) are d linearly independent vectors. . ξ k h , {θ k(j) h } j∈[n] = arg min ξ,{θ (j) } j∈[n] j∈[n] (s,a)∈D θ (j) , φ(s, a) − ξ, ψ(s, a, w (j) ) 2 (11) s.t. θ (j) −θ k h (w (j) ) Λ k h ≤ β, ∀j ∈ [n] and ξ 2 ≤ H √ md, whereθ k h (w) and Λ k h are defined in (6) and (7), respectively. We will show later in Lemma 3 that the action-value function in (10) is an optimistic estimate of the optimal action-value function.. UCBlvd also uses the linear dependency of Q k h on ψ to reduce calls of the planning step in (11). The agent triggers replanning only when it has gathered enough new information compared to the last update at episodẽ k. This is measured by tracking the variations in the gram matrices {Λ k h } h∈[H] (Line 4 for Algorithm 2). Finally, when executing the policy at episode k, the agent chooses the action according to Qk h in Line 10. Theoretical Analysis of UCBlvd We present the main theoretical result which shows UCBlvd achieves sublinear regret in lifelong RL using sublinear number of planning calls, for any sequence of tasks. Theorem 2. Let T = KH. Under Assumptions 1, 2, 3, and 4, the number of planning calls in Algorithm 2 is at most dH log(1 + K dλ ), and there exists an absolute constant c > 0 such that for any fixed δ ∈ (0, 0.5), if we set λ = 1 and β = cH(d + √ md) log(mdT /δ) in Algorithm 2, then with probability at least 1 − 2δ, it holds that R K ≤ 2H T log(dT /δ) + 8HLβ 2dK log(K) ≤Õ L (d 3 + md 2 )H 3 T . Theorem 2 shows that UCBlvd has the same regret bound as Lifelong-LSVI in Theorem 1, but reduces the number of planning calls from K to dH log(1 + K dλ ). As we discussed before, this is made possible by the unique QCQP-based distillation step of UCBlvd in (11). If we were to simply perform least-squares regression to fit ψ(s, a, w),ξ k h to { φ(s, a),θ k h (w (j) )} j∈[n] for distillation, we cannot guarantee the required optimism, because φ(s, a),θ k h (w) computed based on finite samples can be an irregular function that cannot be modelled by ψ(s, a, w). Remark 1. We can extend our results to learn unknown rewards, i.e., η h in Assumption 1. This can be done by introducing a slightly different completeness assumption with an additional exploration bonus in terms of ψ, and then combining tools from linear bandits [Abbasi-Yadkori et al., 2011] and our analysis for proving Theorem 2. Because reward learning affects the radius of our high probability confidence intervals for θ k h (w), the number of planning calls and regret would increase by factors of O(m) and O( √ m) 4 , respectively, compared to those in Theorem 2. See Appendix C for details. Remark 2. It is possible to eliminate the assumption that ψ(s, a, w) = φ(s, a) ⊗ ρ(w). In this case, our analysis requires a set {w (1) , w (2) , . . . , w (n) } of n tasks such that ψ(s, a, w) ∈ Span({ψ(s, a, w (j) )} j∈ [n] ) for all (s, a, w) ∈ S × A × W. In Appendix D, we provide details of this relaxation, and show that the corresponding modified version of UCBlvd still enjoys planning calls and regret of the same order as those of UCBlvd. Remark 3. We can eliminate Assumptions 1 and 4 and instead design a computation-sharing version of Lifelong-LSVI by a sightly different completeness assumption with an exploration bonus β ψ(s, a, w) Λ −1 . This version would use Q k h (s, a, w) := {r h (s, a, w) + ν k h , ψ(s, a, w) + β ψ(s, a, w) (Λ k h ) −1 } + , whereν k h = (Λ k h ) −1 k−1 τ =1 ψ τ h . min{max a∈A Q k h+1 (s τ h+1 , a, w τ ), H},Λ k h = λI d ′ + k−1 τ =1 ψ τ h ψ τ h ⊤ , ψ τ h = ψ(s τ h , a τ h , w τ ), and β =Õ(d ′ ). In Appendix E, we show how this change results inÕ(mdH) number of planning calls and a regret scaling withÕ( √ m 3 d 3 ) for settings with ψ(s, a, w) = φ(s, a) ⊗ ρ(w). These are worse than the number of planning calls and regret in Theorem 2 of UCBlvd by a factor of O(m). Proof Sketch of Theorem 2 The complete proof of Theorem 2 is reported in Appendix B. Here we provide a sketch. Because the proof for the bound on the number of planning calls follows standard arguments in low switching cost analysis [Abbasi-Yadkori et al., 2011], in this section, we focus on the proof sketch for the regret bound. 4 While for both settings in this remark and Remark 3, the action-value functions contain exploration bonus in terms of ψ, the regret here is better by a factor of √ m and this is because the multiplicative factor β here saves a factor √ m compared to that in Remark 3 We start by introducing the following lemma of a high probability event E 1 , which is the foundation of the analysis. Lemma 1. Follow the setting of Theorem 2. The event E 1 (w) := θ k h (w) −θ k h (w) Λ k h ≤ β, ∀(h, k) ∈ [H] × [K] .(12) holds with probability at least 1 − δ for a fixed w. The following lemma highlights the importance of the carefully designed planning step in (11). In particular, it emphasizes how this step paired with the choice of set D, Assumptions 3 and 4 leads to good estimators for ξ V * h+1 h , without the need of the bonus term ψ(s, a, w) Λ k h −1 that the alternate extension of Lifelong-LSVI in Remark 3 has. This step saves a factor of O(m) in the number of planning calls and regret. Lemma 2. Let W = {w τ : τ ∈ [K]} ∪ {w (j) : j ∈ [n]}. Conditioned on events {E 1 (w)} w∈ W defined in (12), for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds that ξ k h , ψ(s, a, w) − P h [V k h+1 (., w)](s, a) ≤ 2Lβ φ(s, a) (Λ k h ) −1 . As the final step in the regret analysis, we state the following lemma which uses Lemma 2 to prove the optimistic nature of UCBlvd. Then following the standard analysis of single-task LSVI-UCB we derive the regret bound in Theorem 2. (12), and with Q k h computed as in (10) Lemma 3. Let W = {w τ : τ ∈ [K]} ∪ {w (j) : j ∈ [n]}. Conditioned on events {E 1 (w)} w∈ W defined in, it holds that Q k h (s, a, w) ≥ Q * h (s, a, w) for all (s, a, w, h, k) ∈ S × A × W × [H] × [K]. Related Work We consider the regret minimization setup of lifelong RL under the contextual MDP framework, where the agent receives tasks specified by contexts in sequence and needs to achieve a sublinear regret for any task sequence. Below, we contrast our work with related work in the literature. Lifelong RL Generally lifelong RL studies how to learn to solve a streaming sequence of tasks using rewards. While it was originally motivated by the need of endless learning of robots [Thrun and Mitchell, 1995], historically many works on lifelong RL [Brunskill and Li, 2014, Abel et al., 2018a, Abel et al., 2018b, Lecarpentier et al., 2021 assume that the tasks are i.i.d. (similar to multi-task RL; see below). There are works for adversarial sequences, but most of them assume finite number of tasks [Ammar et al., 2014, Brunskill and Li, 2015, Ammar et al., 2015, Zhan et al., 2017 or are purely empirical [Xie and Finn, 2021]. The work by [Isele et al., 2016] uses contexts to enable zero-shot learning like here, but it (as well as most works above) do not provide formal regret guarantees. 5 [Brunskill andLi, 2015, Xie andFinn, 2021] assume the task identity is latent, which requires additional exploration; in this sense, their problem is harder than the setup here where the task context is revealed. Extending the setup here to consider latent context is an important future research direction. Contextual MDP and Multi-objective RL Our setup is closely related to the exploration problem studied in the contextual MDP literature, though contextual MDP is originally not motivated from the lifelong learning perspective. A similar mathematical problem appears in the dynamic setup of multiobjective RL [Wu et al., 2021, Abels et al., 2019, which can be viewed as a special case of contextual MDP where the context linearly determines the reward function but not the dynamics. Most contextual MDP works allow adversarial contexts and initial states, but a majority of them focuses on the tabular setup [Abbasi-Yadkori and Neu, 2014, Hallak et al., 2015, Modi et al., 2018, Modi and Tewari, 2020, Levy and Mansour, 2022, Wu et al., 2021, whereas our setup allows continuous states. [Kakade et al., 2020, Du et al., 2019 allow continuous state and actions, but the former assumes a planning oracle with unclear computational complexity and the latter focuses on only LQG problems. While generally contextual MDP allows both the reward and the dynamics to vary with contexts, we focus on the effects of context-independent dynamics similar to [Kakade et al., 2020, Wu et al., 2021. In particular, the recent work of [Wu et al., 2021] is the closest to ours, but they study the sample complexity in the tabular setup with linearly parameterized rewards. In view of Example 1, their proposed algorithm has a regret boundÕ( min{m,\|S\|}H\|S\|\|A\| K). However, they need linear number of planning calls. On the contrary, our algorithm, UCBlvd, allows continuous states, nonlinear context dependency, and has both sublinear regret and number of planning calls. Multi-Task RL Another closely related line of work is multi-task RL. Compared to our setting, multitask RL assumes that there are beforehand known finite tasks and/or they are i.i.d .samples from a fixed distribution. For example, in [Yang et al., 2020, Hessel et al., 2019, Brunskill and Li, 2013, Fifty et al., 2021, Zhang and Wang, 2021, Sodhani et al., 2021, tasks are assumed to be chosen from a known finite set, and in [Yang et al., 2020, Wilson et al., 2007, Brunskill and Li, 2013, Sun et al., 2021, tasks are sampled from a fixed distribution. By contrast, our setting provides guarantees on regret and number of planning calls for adversarial task sequences. Discussion In this paper, we make a link between lifelong RL and contextual MDPs. We propose UCBlvd, an algorithm that simultaneously satisfies the need of achieving 1) sublinear regret and 2) sublinear number of planning calls for 3) a potential adversarial sequence of tasks and initial states. Specifically, for K task episodes of horizon H, we proved that UCBlvd has a regret boundÕ( (d 3 + d ′ d)H 4 K) based onÕ(dH log(K)) number of planning calls, where d and d ′ are the feature dimensions of the dynamics and rewards, respectively. We believe that our results would inspire several research directions in the literature of CMDP and multiobjective RL, as existing work to our knowledge does not cover the computation complexity sharing aspect. That said, our work's limitations motivate further investigations in the following directions: 1) extension to more general class of MDPs, potentially using general function approximation tools, 2) establishing an information-theoretic lower bound on the number of planning calls/computation complexity. References A Proofs of Section 3 To prove Theorem 1, we will use the high probability event E 2 defined in Lemma 5 to prove the UCB nature of Lifelong-LSVI in Lemma 6, which is the key to controlling the regret. We first state the following lemma that will be used in the proof of Lemma 5. Lemma 4. Under the setting of Theorem 1, let c β be the constant in the definition of β. Then, for a fixed w, there is an absolute constant c 0 independent of c β , such that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ it holds that k−1 τ =1 φ τ h . V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) (Λ k h ) −1 ≤ c 0 H d + √ d ′ log((c β + 1)dd ′ T /δ), where c 0 and c β are two independent absolute constants. Proof. We note that η h 2 ≤ √ d ′ (Assumption 2), θ k h (w) 2 ≤ H √ d (k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) 2 (Λ k h ) −1 ≤ 4H 2   d 2 log k + λ λ + d ′ log(1 + 4d ′ /ǫ) + d log(1 + 4Hd/ǫ) + d 2 log 1 + 8B 2 √ d λǫ 2 + log 1 δ   + 8k 2 ǫ 2 λ . If we let ǫ = dH k and β = c β (d + √ d ′ )H log(dT /δ), then, there exists an absolute constant C > 0 that is independent of c β such that k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) 2 (Λ k h ) −1 ≤ C(d ′ + d 2 )H 2 log (c β + 1)dd ′ T /δ . Lemma 5. Let the setting of Theorem 1 holds. The event E 2 (w) := θ k h (w) −θ k h (w) Λ k h ≤ β, ∀(h, k) ∈ [H] × [K] .(13) holds with probability at least 1 − δ for a fixed w. Proof. θ k h (w) −θ k h (w) = θ k h (w) − Λ k h −1 k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) = Λ k h −1   Λ k h θ k h (w) − k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w)   = λ Λ k h −1 θ k h (w) q1 − Λ k h −1   k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h )   q2 . Thus, in order to upper bound θ k h (w) −θ k h (w) Λ k h , we bound q 1 Λ k h and q 2 Λ k h separately. From Lemma 16, we have q 1 Λ k h = λ θ k h (w) (Λ k h ) −1 ≤ √ λ θ k h (w) 2 ≤ H √ λd.(14) Thanks to Lemma 4, for all (w, h, k), with probability at least 1 − δ, it holds that q 2 Λ k h ≤ k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) (Λ k h ) −1 ≤ c 0 H d + √ d ′ log((c β + 1)dd ′ T /δ),(15) where c 0 and c β are two independent absolute constants. Combining (14) and (15), for all (w, h, k), with probability at least 1 − δ, it holds that θ k h (w) −θ k h (w) Λ k h ≤ cH d + √ d ′ λ log(dd ′ T /δ) for some absolute constant c > 0. Lemma 6. Let W = {w 1 , w 2 , . . . , w K }. Conditioned on events {E 2 (w)} w∈ W defined in (13), and with Q k h computed as in (8) r h (s, a, w) + θ k h (w), φ(s, a) − Q π h (s, a, w) − P h V k h+1 (., w) − V π h+1 (., w) (s, a) = r h (s, a, w) + θ k h (w), φ(s, a) − r h (s, a, w) − P h V k h+1 (., w) (s, a) = θ k h (w), φ(s, a) − P h V k h+1 (., w) (s, a) = θ k h (w) − θ k h (w), φ(s, a) ≤ θ k h (w) − θ k h (w) Λ k h φ(s, a) (Λ k h ) −1 ≤ β φ(s, a) (Λ k h ) −1 ,(Lemma 5) for any policy π. Now, we prove the lemma by induction. The statement holds for H because Q k H+1 (., ., .) = Q * H+1 (., ., .) = 0 and thus conditioned on events {E 2 (w)} w∈ W , defined in (13) 0 ≤ r h (s, a, w) + θ k h (w), φ(s, a) − Q * h (s, a, w) − P h V k h+1 (., w) − V * h+1 (., w) (s, a) + β φ(s, a) (Λ k h ) −1 ≤ r h (s, a, w) + θ k h (w), φ(s, a) − Q * h (s, a, w) + β φ(s, a) (Λ k h ) −1 . (Induction assumption) Therefore, conditioned on events {E 2 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have Q * h (s, a, w) ≤ r h (s, a, w) + θ k h (w), φ(s, a) + β φ(s, a) (Λ k h ) −1 = Q k h (s, a, w). This completes the proof. A.1 Proof of Theorem 1 Let δ k h = V k h (s k h , w k ) − V π k h (s k h , w k ) and ξ k h+1 = E δ k h+1 \|s k h , a k h − δ k h+1 . Conditioned on events {E 2 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have Q k h (s, a, w) − Q π k h (s, a, w) = r h (s, a, w) + θ k h (w), φ(s, a) − Q π k h (s, a, w) + β φ(s, a) (Λ k h ) −1 ≤ P h V k h+1 (., w) − V π k h+1 (., w) (s, a) + 2β φ(s, a) (Λ k h ) −1 .(16)Note that δ k h ≤ Q k h (s k h , a k h , w k ) − Q π k h (s k h , a k h , w k ). Thus, combining (16), Lemma 5, and a union bound over W, we conclude that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ, it holds that δ k h ≤ ξ k h+1 + δ k h+1 + 2β φ(s k h , a k h ) (Λ k h ) −1 . Now, we complete the regret analysis R K = K k=1 V * 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ) ≤ K k=1 V k 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ) (Lemma 6) = K k=1 δ k 1 ≤ K k=1 H h=1 ξ k h + 2β K k=1 H h=1 φ(s k h , a k h ) (Λ k h ) −1 ≤ 2H T log(dT /δ) + 2Hβ 2dK log(1 + K/λ) ≤Õ λ(d 3 + dd ′ )H 3 T . The third inequality is true because of the following: we observe that {ξ k h } is a martingale difference sequence satisfying \|ξ k h \|≤ 2H. Thus, thanks to Azuma-Hoeffding inequality, we have P   K k=1 H h=1 ξ k h ≤ 2H T log(dT /δ)   ≥ 1 − δ. (17) In order to bound K k=1 H h=1 φ k h (Λ k h ) −1 , note that for any h ∈ [H], we have K k=1 φ k h (Λ k h ) −1 ≤ K K k=1 φ k h 2 (Λ k h ) −1 (Cauchy-Schwartz inequality) ≤ 2K log    det Λ K h det Λ 1 h    (18) ≤ 2dK log 1 + K dλ .(19) In inequality (18), we used the standard argument in regret analysis of linear bandits [Abbasi-Yadkori et al., 2011] (Lemma 11) as follows: n t=1 min y t 2 V −1 t , 1 ≤ 2 log det V n+1 det V 1 where V n = V 1 + n−1 t=1 y t y ⊤ t .(20) In inequality (19), we used Assumption 2 and the fact that det(A) = d i=1 λ i (A) ≤ (trace(A)/d) d . B Proofs of Section 4 B.1 Proof of Lemma 1 First, we state the following lemma that will be used in the proof of Lemma 1. Lemma 7. Under the setting of Lemma 1, let c β be a constant in the definition of β. Then, for a fixed w, there is an absolute constant c 0 independent of c β , such that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ it holds that k−1 τ =1 φ τ h . V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) (Λ k h ) −1 ≤ c 0 H d + √ md log((c β + 1)mdT /δ), where c 0 and c β are two independent absolute constants. Proof. We note that η h +ξ k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) 2 (Λ k h ) −1 ≤ 4H 2   d 2 log k + λ λ + md log(1 + 8H √ md/ǫ) + d 2 log 1 + 32L 2 β 2 √ d λǫ 2 + log 1 δ   + 8k 2 ǫ 2 λ . If we let ǫ = dH k and β = c β (d + √ md)H log(dT /δ), then, there exists an absolute constant C > 0 that is independent of c β such that k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) 2 (Λ k h ) −1 ≤ C(md + d 2 )H 2 log (c β + 1)mdT /δ . Now, we begin the formal proof of Lemma 1: θ k h (w) −θ k h (w) = θ k h (w) − Λ k h −1 k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) = Λ k h −1   Λ k h θ k h (w) − k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w)   = λ Λ k h −1 θ k h (w) q1 − Λ k h −1   k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h )   q2 . Thus, in order to upper bound θ k h (w) −θ k h (w) Λ k h , we bound q 1 Λ k h and q 2 Λ k h separately. From Lemma 16, we have q 1 Λ k h = λ θ k h (w) (Λ k h ) −1 ≤ √ λ θ k h (w) 2 ≤ H √ λd.(21) Thanks to Lemma 7, for all (w, h, k), with probability at least 1 − δ, it holds that q 2 Λ k h ≤ k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) (Λ k h ) −1 ≤ c 0 H d + √ md log((c β + 1)mdT /δ),(22) where c 0 and c β are two independent absolute constants. Combining (21) and (22), for all (h, k) ∈ [H] × [K], with probability at least 1 − δ, it holds that θ k h (w) −θ k h (w) Λ k h ≤ cH d + √ md λ log(mdT /δ) for some absolute constant c > 0. B.2 Proof of Lemma 2 Thanks to Assumption 3 and conditioned on events {E 1 (w)} w∈ W , one set of solution for (11) is θ k h w (j) j∈[n] and ξ V k h+1 h with corresponding zero optimal objective value. Therefore, it holds that θ k(j) h , φ(s, a) = ξ k h , ψ s, a, w (j) , ∀(j, (s, a)) ∈ [n] × D.(23) Let s (i) , a (i) be the i-th element of D and {c ′ i (s, a)} i∈ [d] be the coefficients such that φ(s, a) = i∈[d] c ′ i (s, a)φ s (i) , a (i) . For any triple (s, a, j) ∈ S × A × [n], we have ξ k h , ψ s, a, w (j) = ξ k h , φ(s, a) ⊗ ρ w (j) = ξ k h , i∈[d] c ′ i (s, a)φ s (i) , a (i) ⊗ ρ w (j) = i∈[d] c ′ i (s, a) ξ k h , ψ s (i) , a (i) , w (j) (Assumption 4) = i∈[d] c ′ i (s, a) θ k(j) h , φ s (i) , a (i) (Eqn. (23)) = θ k(j) h , φ(s, a) .(24) For any (s, a, w) ∈ S × A × W, it holds that (9) and (25)) P h V k h+1 (., w) (s, a) = θ k h (w), φ(s, a) (Eqn. (5)) = ξ V k h+1 h , ψ(s, a, w) (Assumption 3) = j∈[n] c j (w) ξ V k h+1 h , ψ s, a, w (j) (Eqn. (9)) = j∈[n] c j (w)P h V k h+1 ., w (j) (s, a) (Assumption 3) = j∈[n] c j (w) θ k h w (j) , φ(s, a) .(25)ξ k h , ψ(s, a, w) − P h V k h+1 (., w) (s, a) (26) = ξ k h , ψ(s, a, w) − θ k h (w), φ(s, a) = j∈[n] c j (w) ξ k h , ψ s, a, w (j) − θ k h w (j) , φ(s, a) (Eqns.≤ j∈[n] c j (w) ξ k h , ψ s, a, w (j) − θ k(j) h , φ(s, a) + j∈[n] c j (w) θ k(j) h −θ k h w (j) , φ(s, a) + j∈[n] c j (w) θ k h w (j) − θ k h w (j) , φ(s, a) = j∈[n] c j (w) θ k(j) h −θ k h w (j) , φ(s, a) + j∈[n] c j (w) θ k h w (j) − θ k h w (j) , φ(s, a) (Eqn. (24)) ≤ 2Lβ φ(s, a) (Λ k h ) −1 . (Lemma 1) B.3 Proof of Lemma 3 We first note that conditioned on events {E 1 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds that r h (s, a, w) + ξ k h , ψ(s, a, w) − Q π h (s, a, w) − P h V k h+1 (., w) − V π h+1 (., w) (s, a) = r h (s, a, w) + ξ k h , ψ(s, a, w) − r h (s, a, w) − P h V k h+1 (., w) (s, a) = ξ k h , ψ(s, a, w) − P h V k h+1 (., w) (s, a) ≤ 2Lβ φ(s, a) (Λ k h ) −1 ,(Lemma 2) for any policy π. Now, we prove the lemma by induction. The statement holds for H because Q k H+1 (., ., .) = Q * H+1 (., ., .) = 0 and thus conditioned events {E 1 (w)} w∈ W , defined in (12), for all (s, a, w, k) ∈ S × A × W × [K], we have r H (s, a, w) + ξ k H , ψ(s, a, w) − Q * H (s, a, w) ≤ 2Lβ φ(s, a) (Λ k H ) −1 . Therefore, conditioned on events {E 1 (w)} w∈ W , for all (s, a, w, k) ∈ S × A × W × [K], we have s, a, w), where the first equality follows from the fact that Q * H (s, a, w) ≥ 0. Now, suppose the statement holds at time-step h + 1 and consider time-step h. Conditioned on events , w), where the first equality follows from the fact that Q * h (s, a, w) ≥ 0. This completes the proof. Q * H (s, a, w) ≤ r H (s, a, w) + ξ k H , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k H ) −1 = r H (s, a, w) + ξ k H , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k H ) −1 + = Q k H ({E 1 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have 0 ≤ r h (s, a, w) + ξ k h , ψ(s, a, w) − Q * h (s, a, w) − P h V k h+1 (., w) − V * h+1 (., w) (s, a) + 2Lβ φ(s, a) (Λ k h ) −1 ≤ r h (s, a, w) + ξ k h , ψ(s, a, w) − Q * h (s, a, w) + 2Lβ φ(s, a) (Λ k h ) −1 . (Induction assumption) Therefore, conditioned on events {E 1 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have Q * h (s, a, w) ≤ r h (s, a, w) + ξ k h , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k h ) −1 = r h (s, a, w) + ξ k h , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k h ) −1 + = Q k h (s, a B.4 Proof of Theorem 2 First, we bound the number of times Algorithm 2 updatesξ k h , i.e., number of planning calls. Let P be the total number of updates and k p be the episode at which, the agent did replanning for the p-th time. Note that det Λ 1 h = λ d and det Λ K h ≤ trace(Λ K h /d) d ≤ λ + K d d , and consequently: det Λ K h det Λ 1 h = P p=1 det Λ kp h det Λ kp−1 h ≤ 1 + K dλ d , and therefore H h=1 det Λ K h det Λ 1 h = H h=1 P p=1 det Λ kp h det Λ kp−1 h ≤ 1 + K dλ dH .(27) Since = Vk h (s k h , w k ) − V π k h (s k h , w k ) and ξ k h+1 = E δ k h+1 \|s k h , a k h − δ k h+1 . Conditioned on events {E 1 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have Qk h (s, a, w) − Q π k h (s, a, w) = r h (s, a, w) + ξk h , ψ(s, a, w) − Q π k h (s, a, w) + 2Lβ φ(s, a) (Λk h ) −1 ≤ P h Vk h+1 (., w) − V π k h+1 (., w) (s, a) + 4Lβ φ(s, a) (Λk h ) −1 .(28)Note that δ k h ≤ Qk h (s k h , a k h , w k ) − Q π k h (s k h , a k h , w k ). Thus, combining (28), Lemma 1, and a union bound over W, we conclude that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ, it holds that gives δ k h ≤ ξ k h+1 + δ k h+1 + 4Lβ φ(s k h , a k h ) (Λk h ) −1 . Note that for any positive semi-definite matrices A, B, and C such that A = B + C, we have: det(A) ≥ det(B), det(A) ≥ det(C),(29) and for any x = 0 ([ Abbasi-Yadkori et al., 2011, Lemm. 12]): x 2 A x 2 B ≤ det(A) det(B) and x 2 B −1 x 2 A −1 ≤ det(A) det(B) .(30) Now, we complete the regret analysis following similar steps as those of Theorem 1's proof: R K = K k=1 V * 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ) ≤ K k=1 Vk 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ) (Lemma 3) = K k=1 δ k 1 ≤ K k=1 H h=1 ξ k h + 4Lβ K k=1 H h=1 φ(s k h , a k h ) Λk h −1 ≤ K k=1 H h=1 ξ k h + 4Lβ K k=1 H h=1 φ(s k h , a k h ) (Λ k h ) −1 det Λ k h det Λk h (Eqn. (30)) ≤ 2H T log(dT /δ) + 8HLβ 2dK log(1 + K/λ) ≤Õ L λ(d 3 + md 2 )H 3 T . C UCBlvd with Unknown Rewards In order for our analysis to go through, we need a slightly different completeness assumption as below: Assumption 5. Given feature maps φ : S × A → R d and ψ : S × A × W → R d ′ , consider function class F = f : f (s, w) = min max a∈A ν, ψ(s, a, w) + β φ(s, a) Λ −1 +β ψ(s, a, w) Λ −1 + , H , ν ∈ R d ′ , Λ ∈ S d ++ ,Λ ∈ S d ′ ++ , β,β ∈ R . Algorithm 3: UCBlvd with Unknown Rewards Input: A, λ, δ, H, K, β,β 1 Set : Q k H+1 (., ., .) = 0, ∀k ∈ [K],k = 1 2 for episodes k = 1, . . . , K do 3 Observe the initial state s k 1 and the task context w k . Compute Qk h (s k h , a, w k ) for all a ∈ A as in (31). 4 if ∃h ∈ [H] such that det Λ k h det Λk h > e or 10 Play a k h = arg max a∈A Qk h (s k h , a, w k ) and observe s k h+1 and r k h . Then for any f ∈ F , and h ∈ [H], there exists a vector ξ f h ∈ R d ′ with ξ f h ≤ H √ d ′ such that P h f (., w) (s, a) = ξ f h , ψ(s, a, w) . C.1 Overview Let ψ τ h = ψ(s τ h , a τ h , w τ ). UCBlvd with unknown rewards works with the following action-value functions: (6) and (7), respectively. Q k h (s, a, w) = η k h +ξ k h , ψ(s, a, w) + β φ(s, a) (Λ k h ) −1 +β ψ(s, a, w) (Λ k h ) −1 + ,(31)whereη k h = Λ k h −1 k−1 τ =1 ψ τ h .r τ h andΛ k h = λI md + k−1 τ =1 ψ τ h ψ τ h ⊤ ,(32)andξ k h , θ k(j) h j∈[n] = arg min ξ,{θ (j) } j∈[n] j∈[n] (s,a)∈D θ (j) , φ(s, a) − ξ, ψ s, a, w (j) 2 (33) s.t. θ (j) −θ k h w (j) Λ k h ≤ β, We note that compared to (10), action-value function defined in (31) involves an extra term η k h , ψ(s, a, w) + β ψ(s, a, w) (Λ k h ) −1 . This term is in fact an upper bound on r h (s, a, w). Specifically, from Theorem 2 in [Abbasi-Yadkori et al., 2011], we know that forβ = √ λmd, it holds that η h −η k h Λ k h ≤β, ∀(h, k) ∈ [H] × [K].(34) Theorem 3. Let T = KH. Under Assumptions 1, 2, 4, and 5, the number of planning calls in Algorithm 3 is at most dH log 1 + K dλ + mdH log 1 + K mdλ , and there exists an absolute constant c > 0 such that for any fixed δ ∈ (0, 0.5), if we set λ = 1, β = cH (md) log(mdT /δ) andβ = √ md in Algorithm 3, then with probability at least 1 − 2δ, it holds that R K ≤ 2H T log(dT /δ) + 4H √ K Lβ 2d log(1 + K/λ) +β 2md log(1 + K/λ) ≤Õ L √ m 2 d 3 H 3 T . C.2 Necessary Analysis for the Proof of Theorem 3 Lemma 8. Let c β be a constant in the definition of β. Then, under Assumptions 1, 2, 4, and 5, for a fixed w, there is an absolute constant c 0 independent of c β , such that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ it holds that k−1 τ =1 φ τ h . V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) (Λ k h ) −1 ≤ c 0 mdH log((c β + 1)mdT /δ), where c 0 and c β are two independent absolute constants. Proof. We note that η k h +ξ k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) 2 (Λ k h ) −1 ≤ 4H 2   d 2 log k + λ λ + md log(1 + 8H √ md/ǫ) + d 2 log 1 + 32L 2 β 2 √ d λǫ 2 +m 2 d 2 log 1 + 8β 2 √ md λǫ 2 + log 1 δ   + 8k 2 ǫ 2 λ . If we let ǫ = dH k and β = c β (md)H log(mdT /δ), then, there exists an absolute constant C > 0 that is independent of c β such that k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w) − P h [V k h+1 (., w)](s τ h , a τ h ) 2 (Λ k h ) −1 ≤ C(m 2 d 2 )H 2 log (c β + 1)mdT /δ . Lemma 9. Under Assumptions 1, 2, 4, and 5, if we let β = cmdH λ log(mdT /δ) with an absolute constant c > 0, then the event E 3 (w) := θ k h (w) −θ k h (w) Λ k h ≤ β, ∀(h, k) ∈ [H] × [K] .(35) holds with probability at least 1 − δ for a fixed w. Proof. The proof follows the same steps as those of Lemma 1, except that it uses Lemma 8 instead of Lemma 7 due to different structure of action-value functions Q k h in this section. ξ k h , ψ(s, a, w) − P h V k h+1 (., w) (s, a) ≤ 2Lβ φ(s, a) (Λ k h ) −1 . Proof. The proof follows the exact same steps as those of Lemma 2's proof. (35), and with Q k h computed as in (31) Lemma 11. Let W = {w τ : τ ∈ [K]} ∪ {w (j) : j ∈ [n]}. Conditioned on events {E 3 (w)} w∈ W defined inQ * h (s, a, w) ≤ η k h +ξ k h , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k h ) −1 +β ψ(s, a, w) Λ k h −1 = η k h +ξ k h , ψ(s, a, w) + 2Lβ φ(s, a) (Λ k h ) −1 +β ψ(s, a, w) Λ k h −1 + = Q k h (s, a, w), where the first equality follows from the fact that Q * h (s, a, w) ≥ 0. This completes the proof. C.3 Proof of Theorem 3 First, we bound the number of times Algorithm 3 updatesξ k h , i.e., number of planning calls. Let P be the total number of policy updates and k p be the episode at, the agent did replanning for the p-th time. Note that det Λ 1 h = λ d and det Λ K h ≤ trace(Λ K h /d) d ≤ λ + K d d , and consequently: det Λ K h det Λ 1 h = P p=1 det Λ kp h det Λ kp−1 h ≤ 1 + K dλ d , and therefore H h=1 det Λ K h det Λ 1 h = H h=1 P p=1 det Λ kp h det Λ kp−1 h ≤ 1 + K dλ dH .(37) happens for at most dH log 1 + K dλ + mdH log 1 + K mdλ number of episodes k ∈ [K]. This concludes that number of planning calls in Algorithm 3 is at most dH log 1 + K dλ + mdH log 1 + K mdλ . Now, we prove the regret bound. Let δ k h = Vk h (s k h , w k ) − V π k h (s k h , w k ) and ξ k h+1 = E δ k h+1 \|s k h , a k h − δ k h+1 . Conditioned on events {E 3 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have Qk h (s, a, w) − Q π k h (s, a, w) = ηk h +ξk h , ψ(s, a, w) − Q π k h (s, a, w) + 2Lβ φ(s, a) (Λk h ) −1 +β ψ(s, a, w) (Λk h ) −1 ≤ P h Vk h+1 (., w) − V π k h+1 (., w) (s, a) + 4Lβ φ(s, a) (Λk h ) −1 + 2β ψ(s, a, w) (Λk h ) −1 .(40)Note that δ k h ≤ Qk h (s k h , a k h , w k ) − Q π k h (s k h , a k h , w k ). Thus, combining (40), Lemma 9, and a union bound over W, we conclude that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ, it holds that gives δ k h ≤ ξ k h+1 + δ k h+1 + 4Lβ φ(s k h , a k h ) (Λk h ) −1 + 2β ψ(s k h , a k h , w k ) (Λk h ) −1 . Now, we complete the regret analysis following similar steps as those of Theorem 1's proof: R K = K k=1 V * 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ) ≤ K k=1 Vk 1 (s k 1 , w k ) − V π k 1 (s k 1 , w k ) (Lemma 11) = K k=1 δ k 1 ≤ K k=1 H h=1 ξ k h + 4Lβ K k=1 H h=1 φ(s k h , a k h ) Λk h −1 + 2β K k=1 H h=1 ψ(s k h , a k h , w k ) Λk h −1 ≤ K k=1 H h=1 ξ k h + 4Lβ K k=1 H h=1 φ(s k h , a k h ) (Λ k h ) −1 det Λ k h det Λk h + 2β K k=1 H h=1 ψ(s k h , a k h , w k ) Λ k h −1 detΛ k h detΛk h ((30)) ≤ 2H T log(dT /δ) + 4H √ K Lβ 2d log(1 + K/λ) +β 2md log(1 + K/λ) ≤Õ L √ λm 2 d 3 H 3 T . D Relaxation of Assumption 4 In this section, we replace Assumption 4 with the following assumption: Assumption 6. There is a known set {w (1) , w (2) , . . . , w (n) } of n ≤ d ′ tasks such that ψ(s, a, w) ∈ Span ψ(s, a, w (j) ) j∈[n] for all (s, a, w) ∈ S × A × W. This implies that for any (s, a, w) ∈ S × A × W, c j (s, a, w)ψ s, a, w (j) . Moreover, j∈[n] c j (s, a, w) ≤ L for all (s, a, w) ∈ S × A × W. Define the concatenated mappingψ : S×A×W → R d+d ′ such thatψ(s, a, w) = φ(s, a) ⊤ , ψ(s, a, w) ⊤ ⊤ . For any w ∈ W, define D(w) = (s, a) :ψ(s, a, w) are d + d ′ linearly independent vectors. . Given Assumption 6, we modify the planning step of UCBlvd to the following: ξ k h , θ k(j) h j∈[n] = arg min ξ,{θ (j) } j∈[n] j∈[n] (s,a)∈D(w (j) ) θ (j) , φ(s, a) − ξ, ψ s, a, w (j) 2(42)s.t. θ (j) −θ k h w (j) Λ k h ≤ β, ∀j ∈ [n] and ξ 2 ≤ H √ d ′ . The only change we make in Algorithm 2 is in Line 9, in whichξ k h is now computed as defined in (42). We present this modification in Algorithm 4 for completeness. Compute Qk h (s k h , a, w k ) for all a ∈ A as in (10). 10 Play a k h = arg max a∈A Qk h (s k h , a, w k ) and observe s k h+1 and r k h . Theorem 4. Let T = KH. Under Assumptions 1, 2, 3, and 6, the number or planning calls in Algorithm 4 is at most dH log 1 + K dλ and there exists an absolute constant c > 0 such that for any fixed δ ∈ (0, 0.5), if we set λ = 1 and β = cH d + √ d ′ λ log(dd ′ T /δ) in Algorithm 4, then with probability at least 1 − 2δ, it holds that R K ≤ 2H T log(dT /δ) + 8HLβ 2dK log(K) ≤Õ L (d 3 + dd ′ )H 3 T .(43) Proof of Theorem 4 follows exactly the same steps as those of Theorem 2. The only difference is the proof of Lemma 2, which we clarify in the proof of following lemma. ξ k h , ψ(s, a, w) − P h V k h+1 (., w) (s, a) ≤ 2Lβ φ(s, a) (Λ k h ) −1 . Proof. We letψ i (w) = φ ⊤ i , ψ i (w) ⊤ ⊤ be the i-th element ofD(w) = ψ (s, a, w) : (s, a) ∈ D(w) and for any triple (s, a, w) ∈ S × A × W, we let {c ′ i (s, a, w)} i∈[d+d ′ ] be the coefficients such that ψ(s, a, w) = i∈[d+d ′ ] c ′ i (s, a, w)ψ i (w), which implies that φ(s, a) = i∈[d+d ′ ] c ′ i (s, a, w)φ i and ψ(s, a, w) = i∈[d+d ′ ] c ′ i (s, a, w)ψ i (w).(44) Thanks to Assumption 3 and conditioned on events {E 1 (w)} w∈ W , one set of solution for (42) is θ k h w (j) j∈[n] and ξ V k h+1 h with corresponding zero optimal objective value. Therefore, it holds that θ k(j) h , φ i = ξ k h , ψ i w (j) , ∀(i, j) ∈ [d + d ′ ] × [n].(45) Moreover, for any triple (s, a, j) ∈ S × A × [n], we have ξ k h , ψ s, a, w (j) = i∈[d+d ′ ] c ′ i s, a, w (j) ξ k h , ψ i w (j) (Eqn. (44)) = i∈[d+d ′ ] c ′ i s, a, w (j) θ k(j) h , φ i (Eqn. (45)) = θ k(j) h , φ(s, a) .(46) For any (s, a, w) ∈ S × A × W, it holds that c j (s, a, w) ξ k h , ψ s, a, w (j) − θ k h w (j) , φ(s, a) (Eqns. (41) and (25)) P h V k h+1 (., w) (s, a) = θ k h (w), φ(s, a) (Eqn. (5)) = ξ V k h+1 h , ψ(s, a, w) (Assumption 3) = j∈[n] c j (s, a, w) ξ V k h+1 h , ψ s, a, w (j) (Eqn. (41)) = j∈[n] c j (s, a, w)P h V k h+1 ., w (j) (s, a) (Assumption 3) = j∈[n] c j (s, a, w) θ k h w (j) , φ(s, a) .(47)≤ j∈[n] c j (s, a, w) ξ k h , ψ s, a, w (j) − θ k(j) h , φ(s, a) + j∈[n] c j (s, a, w) θ k(j) h −θ k h w (j) , φ(s, a) + j∈[n] c j (s, a, w) θ k h w (j) − θ k h w (j) , φ(s, a) = j∈[n] c j (s, a, w) θ k(j) h −θ k h w (j) , φ(s, a) + j∈[n] c j (s, a, w) θ k h w (j) − θ k h w (j) , φ(s, a) (Eqn. (24)) ≤ 2Lβ φ(s, a) (Λ k h ) −1 . (Lemma 1) E Standard Lifelong-LSVI with Computation Sharing In this section, we only rely on the following two assumptions: Compute Qk h (s k h , a, w k ) for all a ∈ A as in (51). 10 Play a k h = arg max a∈A Qk h (s k h , a, w k ) and observe s k h+1 and r k h . Assumption 7. Given a feature map ψ : S × A × W → R d ′ , consider function class F = f : f (s, w) = min max a∈A ν, ψ(s, a, w) + β ψ(s, a, w) Λ −1 + , H ν ∈ R d ′ , β ∈ R, Λ ∈ S d ′ ++ .(49) Then for any f ∈ F and h ∈ [H], there exists a vector ν f h ∈ R d ′ with ν f h 2 ≤ H √ d ′ such that P h f (., w) (s, a) = ψ(s, a, w), ν f h .(50 E.1 Overview Let ψ τ h = ψ(s τ h , a τ h , w τ ). Standard Lifelong-LSVI with computation sharing works with the following actionvalue functions: Q k h (s, a, w) = r h (s, a, w) + ν k h , ψ(s, a, w) + β ψ(s, a, w) (Λ k h ) −1 + ,(51)whereν k h = Λ k h −1 k−1 τ =1 ψ τ h . min max a∈A Q k h+1 (s τ h+1 , a, w τ ), H andΛ k h = λI d ′ + k−1 τ =1 ψ τ h ψ τ h ⊤ .(52) Theorem 5. Let T = KH. Under Assumptions 7 and 8, the number of planning calls in 5 is at most d ′ H log 1 + K d ′ λ and there exists an absolute constant c > 0 such that for any fixed δ ∈ (0, 0.5), if we set λ = 1 and β = cd ′ H log(d ′ T /δ) in Algorithm 5, then with probability at least 1 − 2δ, it holds that R K ≤ 2H T log(d ′ T /δ) + 4Hβ 2d ′ K log(K) ≤Õ d ′ 3 H 3 T . E.2 Necessary Analysis for the Proof of Theorem 5 Thanks to Assumption 7, we have P h V k h+1 (., w) (s, a) = ν k h , ψ(s, a, w) , where ν k h = ν , with probability at least 1 − δ it holds that k−1 τ =1 ψ τ h . V k h+1 (s τ h+1 , w τ ) − P h [V k h+1 (., w τ )](s τ h , a τ h ) Λ k h −1 ≤ c 0 d ′ H log((c β + 1)d ′ T /δ), where c 0 and c β are two independent absolute constants. Proof. We note that η h +ν k , with probability at least 1 − δ it holds that k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w τ ) − P h [V k h+1 (., w τ )](s τ h , a τ h ) 2 Λ k h −1 ≤ 4H 2   d ′ 2 log k + λ λ + d ′ log(1 + 8H √ d ′ /ǫ) + d ′ 2 log 1 + 32L 2 β 2 √ d ′ λǫ 2 + log 1 δ   + 8k 2 ǫ 2 λ . If we let ǫ = dH k and β = c β (d ′ + √ d ′ )H log(dT /δ), then, there exists an absolute constant C > 0 that is independent of c β such that k−1 τ =1 φ τ h V k h+1 (s τ h+1 , w τ ) − P h [V k h+1 (., w τ )](s τ h , a τ h ) 2 Λ k h −1 ≤ C(d ′ + d ′ 2 )H 2 log (c β + 1)d ′ T /δ . Lemma 14. Under Assumptions 7 and 8, if we let β = cd ′ H λ log(d ′ T /δ) with an absolute constant c > 0, then the event E 4 := ν k h −ν k h Λ k h ≤ β, ∀(h, k) ∈ [H] × [K] .(54) holds with probability at least 1 − δ. Proof. ν k h −ν k h = ν k h − Λ k h −1 k−1 τ =1 ψ τ h V k h+1 (s τ h+1 , w τ ) = Λ k h −1  Λ k h ν k h − k−1 τ =1 ψ τ h V k h+1 (s τ h+1 , w τ )   = λ Λ k h −1 ν k h q1 − Λ k h −1   k−1 τ =1 ψ τ h V k h+1 (s τ h+1 , w τ ) − P h [V k h+1 (., w τ )](s τ h , a τ h )   q2 . (Eqn. (53)) Thus, in order to upper bound ν k h −ν k h (w) Λ k h , we bound q 1 Λ k h and q 2 Λ k h separately. From Assumption 8, we have q 1 Λ k h = λ ν k h Λ k h −1 ≤ √ λ ν k h 2 ≤ H √ λd ′ .(55) Thanks to Lemma 13, for all (h, k) ∈ [H] × [K], with probability at least 1 − δ, it holds that q 2 Λ k h ≤ k−1 τ =1 ψ τ h V k h+1 (s τ h+1 , w τ ) − P h [V k h+1 (., w τ )](s τ h , a τ h ) (Λ k h ) −1 ≤ c 0 d ′ H log((c β + 1)d ′ T /δ),(56) where c 0 and c β are two independent absolute constants. Combining (55) and (56) Proof. We first note that conditioned on the event E 4 , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds that r h (s, a, w) + ν k h , ψ(s, a, w) − Q π h (s, a, w) − P h V k h+1 (., w) − V π h+1 (., w) (s, a) = r h (s, a, w) + ν k h , ψ(s, a, w) − r h (s, a, w) − P h V k h+1 (., w) (s, a) = ν k h , ψ(s, a, w) − P h V k h+1 (., w) (s, a) = ν k h − ν k h , ψ(s, a, w) ≤ ν k h − ν k h Λ k h ψ(s, a, w) Λ k h −1 ≤ β ψ(s, a, w) Λ k h −1 ,(Lemma 14) = r h (s, a, w) + ν k h , ψ(s, a, w) + β ψ(s, a, w) Λ k h −1 + = Q k h (s, a, w), where the first equality follows from the fact that Q * H (s, a, w) ≥ 0. This completes the proof. E.3 Proof of Theorem 5 First, we bound the number of times Algorithm 5 updatesν k h . Let P be the total number of updates and k p be the episode at which, the agent did replanning for the p-th time. Note that detΛ 1 h = λ d ′ and detΛ K h ≤ trace(Λ K h /d ′ ) d ′ ≤ λ + K d ′ d ′ , Proof. Recall that V k h (s, w) = min max a∈A Q k h (s, a, w), H and θ k h (w) := S V k h+1 (s ′ , w)dµ h (s ′ ). Thus, we have θ k h (w) 2 = S V k h+1 (s ′ , w)dµ h (s ′ ) ≤ H √ d. Lemma 17 (Lemma D.4 in [Jin et al., 2020]). Let {s τ } ∞ τ =1 be a stochastic process on state space S with corresponding filtration {F τ } ∞ τ =0 . Let {φ τ } ∞ τ =0 be an R d -valued stochastic process where φ τ ∈ F τ −1 , and φ τ ≤ 1. Let Λ k = λI d + k−1 τ =1 φ τ φ ⊤ τ . Then with probability at least 1 − δ, for all k ≥ 0 and V ∈ V such that sup s∈S V (s) ≤ H, we have k τ =1 φ τ . V (s τ ) − E V (s τ )\|F τ −1 2 Λ −1 k ≤ 4H 2 d 2 log k + λ λ + log N ǫ (V) δ + 8k 2 ǫ 2 λ . Lemma 18. For any ǫ > 0, the ǫ-covering number of the Euclidean ball in R d with radius R > 0 is upper bounded by (1 + 2R/ǫ) d . Lemma 19. For a fixed w, let V denote a class of functions mapping from S to R with following parametric form V (.) = min max a∈A z, ψ(., a, w) + y, φ(., a) + β φ(., a) ⊤ Yφ(., a), H , where the parameters β ∈ R, z ∈ R d ′ , y ∈ R d , and Y ∈ R d×d satisfy 0 ≤ β ≤ B, z ≤ z, y ≤ y, and Y ≤ λ −1 . Assume φ(s, a) ≤ 1 and ψ(s, a, w) ≤ 1 for all (s, a, w) ∈ S × A × W. Then log N ǫ (V) ≤ d ′ log(1 + 4z/ǫ) + d log(1 + 4y/ǫ) + d 2 log 1 + 8B 2 √ d λǫ 2 . Proof. First, we reparametrize V by lettingỸ = β 2 Y. We have V (.) = min max a∈A z, ψ(., a, w) + y, φ(., a) + φ(., a) ⊤Ỹ φ(., a), H , for z ≤ z, y ≤ y, and Ỹ ≤ B 2 λ . For any two functions V 1 , V 2 ∈ V with parameters z 1 , y 1 ,Ỹ 1 and z 2 , y 2 ,Ỹ 2 , respectively, we have dist(V 1 , V 2 ) ≤ sup (s,a)∈S×A z 1 , ψ(s, a, w) + y 1 , φ(s, a) + φ(s, a) ⊤Ỹ1 φ(s, a) − z 2 , ψ(s, a, w) + y 2 , φ(s, a) + φ(s, a) ⊤Ỹ2 φ(s, a) ≤ sup ψ: ψ ≤1,φ: φ ≤1 z 1 , ψ + y 1 , φ + φ ⊤Ỹ 1 φ − z 2 , ψ + y 2 , φ + φ ⊤Ỹ 2 φ ≤ sup ψ: ψ ≤1 z 1 − z 2 , ψ + sup φ: φ ≤1 y 1 − y 2 , φ + sup φ: φ ≤1 φ ⊤ Ỹ1 −Ỹ 2 φ (because √ a − √ b ≤ \|a − b\| for a, b ≥ 0) = z 1 − z 2 + y 1 − y 2 + Ỹ1 −Ỹ 2 ≤ z 1 − z 2 + y 1 − y 2 + Ỹ1 −Ỹ 2 F .(59) [ Abbasi-Yadkori and Neu, 2014] Abbasi-Yadkori, Y. andNeu, G. (2014). Online learning in mdps with side information. arXiv preprint arXiv:1406.6812.[Abbasi-Yadkori et al., 2011] Abbasi-Yadkori, Y.,Pál, D., and Szepesvári, C. (2011). Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems, pages 2312-2320.[Abel et al., 2018a] Abel, D., Arumugam, D., Lehnert, L., and Littman, M. (2018a). 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Then, under Assumption 8, there is an absolute constant c 0 independent of c β , such that for all (h, k) ∈ [H] × [K] Algorithm 1: Lifelong-LSVI Input: A, λ, δ, H, K, β 1 Set: Q k H+1 (., ., .) = 0, ∀k ∈ [K] 2 for episodes k = 1, . . . , K do Observe the initial state s k 1 and the task context w k .3 4 for time-steps h = H, . . . , 1 do 5 Computeθ k h (w k ) as in (6) using Q k h+1 defined in (8). 6 for time-steps h = 1, . . . , H do 7 Compute Q k h (s k h , a, w k ) for all a ∈ A as in (8). 8 Play a k h = arg max a∈A Q k h (s k h , a, w k ) and observe s k h+1 and r k h . Assumption 1 (Linear MDP). M = (S, A, H, P, r, W) is a linear MDP with feature maps φ : S × A → R d and ψ : S × A × W → R d ′ . 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[Yang et al., 2020] Yang, R., Xu, H., Wu, Y., and Wang, X. (2020). Multi-task reinforcement learning with soft modularization. Advances in Neural Information Processing Systems, 33:4767-4777. [Zhan et al., 2017] Zhan, Y., Ammar, H. B., and Taylor, M. E. (2017). Scalable lifelong reinforcement learning. Pattern Recognition, 72:407-418. [Zhang and Wang, 2021] Zhang, C. and Wang, Z. (2021). Provably efficient multi-task reinforcement learning with model transfer. Advances in Neural Information Processing Systems, 34. Lemmas 17 and 19 together imply that for all (h, k) ∈ [H] × [K], with probability at least 1 − δ it holds thatLemma 16), and Λ k h −1 ≤ 1 λ . Thus, , it holds that Q k h (s, a, w) ≥ Q * h (s, a, w) for all (s, a, w, h, k) ∈ S × A × W × [H] × [K].Proof. We first note that conditioned on events {E 2 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds that , for all (s, a, w, k) ∈ S × A × W × [K], we have Now, suppose the statement holds at time-step h + 1 and consider time-step h.r H (s, a, w) + θ k H (w), ψ(s, a) − Q * H (s, a, w) ≤ β φ(s, a) (Λ k H ) −1 . Therefore, conditioned on events {E 2 (w)} w∈ W , for all (s, a, w, k) ∈ S × A × W × [K], we have Q * H (s, a, w) ≤ r H (s, a, w) + θ k H (w), φ(s, a) + β φ(s, a) (Λ k H ) −1 = Q k H (s, a, w). Conditioned on events {E 2 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have Finally, conditioned on events {E 1 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds that Therefore, conditioned on events {E 3 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], we have, it holds that Q k h (s, a, w) ≥ Q * h (s, a, w) for all (s, a, w, h, k) ∈ S × A × W × [H] × [K]. there exist coefficients {c j (s, a, w)} j∈[n] such that ψ(s, a, w) =j∈[n] Algorithm 4 : 4Modified UCBlvdInput: A, λ, δ, H, K, β 1 Set : Q k H+1 (., ., .) = 0, ∀k ∈ [K],k = 1 2 for episodes k = 1, . . . , K do Observe the initial state s k 1 and the task context w k .for time-steps h = H, . . . , 1 do for time-steps h = 1, . . . , H do3 4 if ∃h ∈ [H] such that det Λ k h det Λk h > e then 5k = k 6 7 Computeξ k h as in (42). 8 9 Lemma 12. Let W = {w τ : τ ∈ [K]} ∪ {w (j) : j ∈ [n]}. Under Assumptions 1, 2, 3, and 6, if we let β = cH d + √ d ′ λ log(dd ′ T /δ) with an absolute constant c > 0, then for all (s, a, w, h, k) ∈ S × A × W × [H] × [K] with probability at least 1 − δ, it holds that Finally, conditioned on events {E 1 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds thatξ k h , ψ(s, a, w) − P h V k h+1 (., w) (s, a) (48) = ξ k h , ψ(s, a, w) − θ k h (w), φ(s, a) = j∈[n] Algorithm 5 : 5Standard Lifelong-LSVI with Computation Sharing Input: A, λ, δ, β, H, K 1 Set : Q k H+1 (., ., .) = 0, ∀k ∈ [K],k = 1 2 for episodes k = 1, . . . , K do Observe the initial state s k 1 and the task context w k . if ∃h ∈ [H] such that for time-steps h = H, . . . , 1 do Compute Computeνk h as in(52).for time-steps h = 1, . . . , H do3 4 detΛ k h detΛk h > e then 5k = k 6 7 8 9 ) Moreover, for every h ∈ [H], there exists a vector η h such that r h (s, a, w) = η h , ψ(s, a, w) .Assumption 8. Without loss of generality, ψ(s, a, w) 2 ≤ 1 for all (s, a, w) ∈ S × A × W, and η h 2 ≤ √ d ′ for all h ∈ [H]. ≤ 1 λ . Thus, Lemmas 17 and 22 together imply that for all (h, k) ∈ [H] × [K]h 2 ≤ (1 + H) √ d ′ and Λ k h −1 , for all (h, k) ∈ [H] × [K], with probability at least 1 − δ, it holds that ≤ cd ′ H λ log(d ′ T /δ) for some absolute constant c > 0. Lemma 15. Let the setting of Lemma 14 holds. Conditioned on events E 4 defined in (54), and with Q k h computed as in (51), it holds that Q k h (s, a, w) ≥ Q * h (s, a, w) for all (s, a, w, h, k) ∈ S × A × W × [H] × [K].ν k h −ν k h Λ k h and consequently:detΛ K h detΛ 1 h = P p=1 detΛ kp h detΛ kp−1 h ≤ 1 + K d ′ λ d ′ , and therefore H h=1 detΛ K h detΛ 1 h = H h=1 P p=1 detΛ kp h detΛ kp−1 h ≤ 1 + K d ′ λ d ′ H . We adopt a stricter definition of lifelong RL here to distinguish it from multi-task RL, while there are existing works on lifelongRL (e.g. [Brunskill and Li, 2014, Lecarpentier et al., 2021]) assuming i.i.d. tasks. In general, a context-dependent dynamics would take the form P h (s ′ \|s, a, w). Such set {ρ(w (j) )} j∈[n] always exists for finite-dimensional problems. We assume that this set is known to the algorithm. [Ammar et al., 2015] give regret bounds but only for linearized value difference;[Brunskill and Li, 2015] show regret bounds only for finite number of tasks. d λ }, with respect to the Frobenius norm. By Lemma 18, we know AcknowledgementThis work was supported in part by DARPA grant HR00112190130.Proof. We first note that conditioned on events {E 3 (w)} w∈ W , for all (s, a, w, h, k) ∈ S × A × W × [H] × [K], it holds thatfor any policy π.Now, we prove the lemma by induction. The statement holds for H because Q k H+1 (., ., .) = Q * H+1 (., ., .) = 0 and thus conditioned events {E 3 (w)} w∈ W , defined in (35), for all (s, a, w, k) ∈ S × A × W × [K], we haveTherefore, conditioned on events {E 3 (w)} w∈ W , for all (s, a, w, k) ∈ S × A × W × [K], we have, where the first equality follows from the fact that Q * H (s, a, w) ≥ 0. Now, suppose the statement holds at time-step h + 1 and consider time-step h. Conditioned on events(Induction assumption) for any policy π. Now, we prove the lemma by induction. The statement holds for H because Q k H+1 (., ., .) = Q * H+1 (., ., .) = 0 and thus conditioned on the event E 4 , defined in (54), for all (s, a, w, k) ∈ S × A × W × [K], we haveTherefore, conditioned on the event E 4 , for all (s, a, w, k) ∈ S × A × W × [K], we havewhere the first equality follows from the fact that Q * H (s, a, w) ≥ 0. Now, suppose the statement holds at timestep h + 1 and consider time-step h. Conditioned on events. This concludes that number of planning calls in Algorithm 5 is at most(58)and Lemma 14 imply that for allNow, we complete the regret analysis following similar steps as those of Theorem 1's proof:F Auxiliary LemmasNotations. N ǫ (V) denotes the ǫ-covering number of the class V of functions mapping S to R with respect to the distance dist(V,According to (59), it holds that N ǫ (V) ≤ \|C z \| C y \|C Y \|, and thereforeLemma 20. For a fixed w, let V denote a class of functions mapping from S to R with following parametric formwhere the parameters β ∈ R, z ∈ R d ′ and Y ∈ R d×d satisfy 0 ≤ β ≤ B, z ≤ z, and Y ≤ λ −1 . Assume φ(s, a) ≤ 1 and ψ(s, a, w) ≤ 1 for all (s, a, w) ∈ S × A × W. ThenProof. First, we reparametrize V by lettingỸ = β 2 Y. We have V (.) = min max a∈A z, ψ(., a, w) + φ(., a) ⊤Ỹ φ(., a), H , for z ≤ z, and Ỹ ≤ B 2 λ . For any two functions V 1 , V 2 ∈ V with parameters z 1 ,Ỹ 1 and z 2 ,Ỹ 2 , respectively, we haveLet C z be an ǫ/2-cover of {z ∈ R d ′ : z ≤ z} with respect to the 2-norm, and C Y be an ǫ 2 /4-cover of {Y ∈ R d×d : Y F ≤ B 2 √ d λ }, with respect to the Frobenius norm. By Lemma 18, we knowAccording to (60), it holds that N ǫ (V) ≤ \|C z \|\|C Y \|, and thereforeLemma 21. For a fixed w, let V denote a class of functions mapping from S to R with following parametric form V (.) = min max a∈A z, ψ(., a, w) + 2Lβ φ(., a) ⊤ Yφ(., a) +β φ(., a, w) ⊤Ỹ φ(., a, w)Y ≤ λ −1 and Ỹ ≤ λ −1 . Assume φ(s, a) ≤ 1 and ψ(s, a, w) ≤ 1 for all (s, a, w) ∈ S × A × W. ThenProof. First, we reparametrize V by letting Z = β 2 Y andZ =β 2Ỹ . We have V (.) = min max a∈A z, ψ(., a, w) + φ(., a) ⊤ Zφ(., a) + φ(., a) ⊤Z φ(., a), H , for z ≤ z, Z ≤ B 2 λ , and Z ≤B 2 λ . For any two functions V 1 , V 2 ∈ V with parameters z 1 , Z 1 ,Z 1 and z 2 , Z 2 ,Z 2 , respectively, we have dist(V 1 , V 2 ) ≤ sup (s,a)∈S×A z 1 , ψ(s, a, w) + φ(s, a) ⊤ Z 1 φ(s, a) + ψ(s, a, w) ⊤Z1 ψ(s, a, w)− z 2 , ψ(s, a, w) + φ(s, a) ⊤ Z 2 φ(s, a) + ψ(s, a, w) ⊤Z2 ψ(s, a, w)Let C z be an ǫ/2-cover of {z ∈ R d ′ : z ≤ z} with respect to the 2-norm, C Z be an ǫ 2 /4-cover of {Z ∈ R d×d :According to (61), it holds that N ǫ (V) ≤ \|C z \|\|C Y \|, and thereforeLemma 22. Let V denote a class of functions mapping from S to R with following parametric form V (., .) = min max a∈A z, ψ(., a, .) + 2Lβ ψ(., a, .) ⊤ Yψ(., a, .)Proof. First, we reparametrize V by lettingỸ = β 2 Y. We have V (., .) = min max a∈A z, ψ(., a, .) + ψ(., a, .) ⊤Ỹ ψ(., a, .), H , for z ≤ z, and Ỹ ≤ B 2 λ . For any two functions V 1 , V 2 ∈ V with parameters z 1 ,Ỹ 1 and z 2 ,Ỹ 2 , respectively, we have dist(V 1 , V 2 ) ≤ sup (s,a,w)∈S×A×W z 1 , ψ(s, a, w) + ψ(s, a) ⊤Ỹ1 ψ(s, a)− z 2 , ψ(s, a, w) + ψ(s, a, w) ⊤Ỹ2 ψ(s, a, w) ≤ sup ψ: ψ ≤1 z 1 , ψ + ψ ⊤Ỹ 1 ψ − z 2 , ψ + ψ ⊤Ỹ 2 ψ ≤ sup ψ: ψ ≤1 z 1 − z 2 , ψ + supLet C z be an ǫ/2-cover of {z ∈ R d ′ : z ≤ z} with respect to the 2-norm, and C Y be an ǫ 2 /4-cover of}, with respect to the Frobenius norm. By Lemma 18, we knowAccording to (62), it holds that N ǫ (V) ≤ \|C z \|\|C Y \|, and therefore log N ǫ (V) ≤ d ′ log(1 + 4z/ǫ) + d ′ 2 log 1 + 8B 2 √ d ′ λǫ 2 . /2-covers of {z ∈ R d ′ : z ≤ z} and {y ∈ R d : y ≤ y}, respectively, with respect to the 2-norm, and C Y be an ǫ 2 /4-cover of {Y ∈ R d×d. C Let, Y F ≤ B 2 √Let C z and C y be ǫ/2-covers of {z ∈ R d ′ : z ≤ z} and {y ∈ R d : y ≤ y}, respectively, with respect to the 2-norm, and C Y be an ǫ 2 /4-cover of {Y ∈ R d×d : Y F ≤ B 2 √	[]
[]	[]	[]	[]	In this paper, we give explicit descriptions of the centers and cocenters of 0-Hecke algebras associated to finite Coxeter groups.	10.1007/978-3-319-23443-4_8	[ "https://arxiv.org/pdf/1502.02183v2.pdf" ]	119,563,915	1502.02183	f24fe80b53a410ee89d1a14935d166a30970efec	31 Mar 2015 31 Mar 2015CENTERS AND COCENTERS OF 0-HECKE ALGEBRAS XUHUA HE Dedicated to David Vogan on his 60th birthdayand phrases finite Coxeter groups0-Hecke algebrasConjugacy classes In this paper, we give explicit descriptions of the centers and cocenters of 0-Hecke algebras associated to finite Coxeter groups. Introduction 0.1. Iwahori-Hecke algebras H q are deformations of the group algebras of finite Coxeter groups W (with nonzero parameters q). They play an important role in the study of representations of finite groups of Lie type. In 1993, Geck and Pfeiffer [4] discovered some remarkable properties of the minimal length elements in their conjugacy classes in W (see Theorem 1.2). Based on these properties, they defined the "character table" for Iwahori-Hecke algebras. They also gave a basis of the cocenter of Iwahori-Hecke algebras, using minimal length elements. Later, Geck and Rouquier [6] gave a basis of the center of Iwahori-Hecke algebras. It is interesting that both centers and cocenters of Iwahori-Hecke algebras are closely related to minimal length elements in the finite Coxeter groups and their dimensions both equal the number of conjugacy classes of the finite Coxeter groups. 0.2. The 0-Hecke algebra H 0 was used by Carter and Lusztig in [2] in the study of p-modular representations of finite groups of Lie type. It is a deformation of the group algebras of finite Coxeter groups (with zero parameter). In this paper, we study the center and cocenter of 0-Hecke algebras H 0 . We give a basis of the center of H 0 in Theorem 4.4 and a basis of the cocenter of H 0 in Theorem 5.5. 0.3. It is interesting to compare the (co)centers of H q and H 0 . Let W min be the set of minimal length elements in their conjugacy classes in W . There are two equivalence relations ∼ and ≈, on W min (see §1.2 for the precise definition). Hence we have the partition of W min into ∼-equivalence classes and ≈-equivalence classes. The second partition is finer than the first one. The center and cocenter of H q have basis sets indexed by the set of conjugacy classes of W , which are in natural bijection with W min / ∼. The cocenter of H 0 has a basis set indexed by W min / ≈ and the center of H 0 has a basis set indexed by W max / ≈. Here W max / ≈ is defined using maximal length elements instead and there is a natural bijection between W max / ≈ with the set of ≈-equivalence classes of minimal length elements in their "twisted" conjugacy classes in W . In general, the number of elements in W max / ≈ is different from the number of elements in W min / ≈. 0.4. The paper is organized as follows. In section 1, we recall some properties of the minimal length and maximal length elements. In section 2, we recall the results on the center and cocenter of H q . We give parameterizations of W min / ≈ and W max / ≈ in section 3. In section 4, we give a basis of the center of H 0 and in section 5, we give a basis of the cocenter of H 0 . In section 6, we describe the image of a standard element t w in the cocenter of H 0 and discuss some applications to the class polynomials of H q . Finite Coxeter groups 1.1. Let S be a finite set. A Coxeter matrix (m s,s ′ ) s,s ′ ∈S is a matrix with entries in N ∪ {∞} such that m ss = 1 and m s,s ′ = m s ′ ,s 2 for all s = s ′ in S. The Coxeter group W associated to the Coxeter matrix (m s,s ′ ) is the group generated by S with relations (ss ′ ) m s,s ′ = 1 for s, s ′ ∈ S with m s,s ′ < ∞. The Coxeter group W is equipped with the length function ℓ : W → N and the Bruhat order . For any J ⊆ S, let W J be the subgroup of W generated by elements in J. Then W J is also a Coxeter group. 1.2. Let δ be an automorphism of W with δ(S) = S. We say that the elements w, w ′ ∈ W are δ-conjugate if there exists x ∈ W such that w ′ = xwδ(x) −1 . Let cl(W ) δ be the set of δ-conjugacy classes of W . We say that a δ-conjugacy class O is elliptic if O ∩ W J = ∅ for any J = δ(J) S. For any w ∈ W , let supp(w) be the set of simple reflections that appear in some (or equivalently, any) reduced expression of w. Set supp δ (w) = ∪ i 0 δ i (supp(w)). Then O ∈ cl(W ) δ is elliptic if and only if supp δ (w) = S for any w ∈ O. For w, w ′ ∈ W and s ∈ S, we write w s − → δ w ′ if w ′ = swδ(s) and ℓ(w ′ ) ℓ(w). We write w → δ w ′ if there exists a sequence w = w 0 , w 1 , · · · , w n = w ′ of elements in W such that for any k, w k−1 s − → δ w k for some s ∈ S. We write w ≈ δ w ′ if w → δ w ′ and w ′ → δ w. We say that the two elements w, w ′ ∈ W are elementarily strongly δ-conjugate if ℓ(w) = ℓ(w ′ ) and there exists x ∈ W such that w ′ = xwδ(x) −1 , and ℓ(xw) = ℓ(x) + ℓ(w) or ℓ(wδ(x) −1 ) = ℓ(x) + ℓ(w). We say that w, w ′ are strongly δ-conjugate if there exists a sequence w = w 0 , w 1 , · · · , w n = w ′ such that for each i, w i−1 is elementarily strongly δ-conjugate to w i . We write w ∼ δ w ′ if w and w ′ are strongly δconjugate. It is easy to see that Lemma 1.1. If w, w ′ ∈ W with w ≈ δ w ′ , then w ∼ δ w ′ . Note that ∼ δ and ≈ δ are both equivalence relations. For any O ∈ cl(W ), let O min be the set of minimal length elements in O and O max be the set of maximal length elements in O. Since ∼ δ and ≈ δ are compatible with the length function, both O min and O max are unions of ∼ δ -equivalence classes and unions of ≈ δ -equivalence classes. Let W δ,min = ⊔ O∈cl(W ) δ O min and W δ,max = ⊔ O∈cl(W ) δ O max . Let W δ,min / ∼ δ be the set of ∼ δ -equivalence classes in W min . We define W δ,min / ≈ δ , W δ,max / ∼ δ and W δ,max / ≈ δ in a similar way. If δ is the identity map, then we may omit δ in the subscript. The following result is proved in [4, Theorem 1.1], [3, Theorem 2.6] and [7, Theorem 7.5] (see also [9] for a case-free proof). Theorem 1.2. Let W be a finite Coxeter group and O be a δ-conjugacy class of W . Then (1) For any w ∈ O, there exists w ′ ∈ O min such that w → δ w ′ . (2) O min is a single strongly δ-conjugate class. (3) If O is elliptic, then O min is a single ≈ δ -equivalence class. As a consequence of Theorem 1.2, it is proved in [7, Corollary 4.5] that the set of minimal length elements in O coincides with the set of minimal elements in O with respect to the Bruhat order . w ∈ O, there exists w ′ ∈ O max such that w ′ → δ w. (2) O max = {w ∈ O; w ≮ w ′ for any w ′ ∈ O}. Finite Hecke algebras In the rest of this paper, we assume that W is a finite Coxeter group. Let q be an indeterminate and Λ = C[q] . The generic Hecke algebra (with equal parameters) H of W is the Λ-algebra generated by {T w ; w ∈ W } subject to the relations: (1) T w · T w ′ = T ww ′ , if ℓ(ww ′ ) = ℓ(w) + ℓ(w ′ ). (2) (T s + 1)(T s − q) = 0 for s ∈ S. Given q ∈ C, let C q be the Λ-module where q acts by q. Let H q = H ⊗ Λ C q be a specialization of H. In particular, H 1 = C[W ] is the group algebra. The algebra H 0 is called the 0-Hecke algebra. We will discuss it in details in the next section. For any w ∈ W , we denote by T w,q = T w ⊗ 1 ∈ H q . We simply write t w for T w,0 . Let [H, H] δ be the δ-commutator of H, that is, the Λ-submodule of H spanned by [h, h ′ ] = hh ′ − h ′ δ(h) for h, h ′ ∈ H. Let H δ = H/[H, H] δ be the δ-cocenter of H. For any q ∈ C, we define the δ-cocenter H q,δ in the same way. Notice that if q = 0, then T w,q is invertible in H q for any w ∈ W . However, if q = 0, then t w is invertible in H q if and only if w = 1. This makes a difference in the study of the cocenter of H q (for q = 0) and the cocenter of H 0 . Proposition 2.1. Let w, w ′ ∈ W . If w ≈ δ w ′ , then the image of T w and T w ′ in H δ are the same. Proof. It suffices to prove the case where w s − → δ w ′ and ℓ(w) = ℓ(w ′ ). Without loss of generality, we may assume furthermore that sw < w. Then T w = T s T sw and T w ′ = T sw T δ(s) . Hence the image of T w and T w ′ are the same. For q = 0, a similar argument shows that if w ∼ δ w ′ , then the image of T w,q and T w ′ ,q in H q,δ are the same. By Theorem 1.2 (2), for any δ-conjugacy class O of W , O min is a single strongly δ-conjugacy class. Thus Proposition 2.2. ([4, §1] and [3, 7.2]) If q = 0, then for any O ∈ cl(W ) δ and w, w ′ ∈ O min , the image of T w,q and T w ′ ,q in H q,δ are the same. Remark. We denote this image by T O,q . Let q = 0. Let Z(H q ) δ = {h ∈ H q ; h ′ h = hδ(h ′ ) for any h ′ ∈ H q } be the δ-center of H q . For any O ∈ cl(W ) δ , set z O = w∈W q −ℓ(w) f w,O T w −1 . Then {z O } O∈cl(W ) δ form a basis of Z(H q ) δ . As a consequence of the results above, we have Corollary 2.6. If q = 0, then dim Z(H q ) δ = dim H q,δ = ♯cl(W ) δ . 3. Parameterizations of W δ,min / ≈ δ and W δ,max / ≈ δ 3.1. Notice that for q = 0, both H q,δ and Z(H q ) δ have basis sets indexed by cl(W ) δ , which is in natural bijection with W δ,min / ∼ δ . As we will see later in this paper, for H 0,δ and Z(H 0 ) δ , we need to use W δ,min / ≈ δ and W δ,max / ≈ δ instead. We give parameterizations of these sets here. Let Γ δ = {(J, C); J = δ(J) ⊆ S, C ∈ cl(W J ) δ is elliptic}. There is a natural map f : Γ δ → cl(W ) δ , (J, C) → O, where O is the unique δ-conjugacy class of W that contains C. We say that (J, C) is equivalent to (J ′ , C ′ ) if there exists x ∈ W δ and the conjugation by x sends J to J ′ and sends C to C ′ . By [1, Proposition 5.2.1], f induces a bijection from the equivalence classes of Γ δ to cl(W ) δ . Proposition 3.1. Let O ∈ cl(W ) δ . Then O min = ⊔ (J,C)∈Γ δ with f (J,C)=O C min . Proof. If (J, C) ∈ Γ δ with f (J, C) = O, we have C min ⊆ O min by [7, Lemma 7.3]. Let w ∈ O min . Let J = supp δ (w) and C ∈ cl(W J ) δ with w ∈ C. By [7, Theorem 7.5 (P1)], C is an elliptic δ-conjugacy class of W J . Since w ∈ O min and w ∈ C, w ∈ C min . Proof. Let (J, C) ∈ Γ δ and w ∈ C min . If w s − → δ w ′ , then w ′ = w or sw < w or wδ(s) < w. In the latter two cases, s ∈ J. Therefore w ′ ∈ C. Since w ∈ C min and ℓ(w ′ ) ℓ(w), w ′ ∈ C min . By definition of ≈ δ , v ∈ C min for any v ∈ W with w ≈ δ v. On the other hand, by Theorem 1.2, C min is a single ≈ δ -equivalence class. Hence the map (J, C) → C min ∈ W δ,min / ≈ δ is well-defined. It is obvious that this map is injective. The surjectivity follows from Proposition 3.1. Using the argument in §1.3, we also obtain (1) min{x, sx} min{y, sy} and max{x, sx} max{y, sy}. Corollary 3.3. Set δ ′ = Ad(w 0 ) • δ. The map Γ δ ′ → W δ,max / ≈ δ , (J, C) → C min w 0 is a bijection. (2) min{x, xs} min{y, ys} and max{x, xs} max{y, ys}. Lemma 4.2. Let Σ ∈ W δ,max / ≈ δ and s ∈ S. Then {x ∈ W ; x / ∈ W Σ , sx ∈ W Σ } = {x ∈ W ; x / ∈ W Σ , xδ(s) ∈ W Σ }. Proof. Let x ∈ W with x / ∈ W Σ , sx ∈ W Σ . By definition, sx w for some w ∈ Σ. Since x w, we have sx < x and sw > w by Lemma 4.1. Thus ℓ(swδ(s)) ℓ(sw) − 1 = ℓ(w). Since w ∈ W δ,max , ℓ(swδ(s)) = ℓ(w) and sws ∈ Σ. Moreover, swδ(s) < sw. Since sx w and w < sw, x sw. By Lemma 4.1, min{x, xδ(s)} swδ(s). Since x / ∈ W Σ , xδ(s) ∈ W Σ . Lemma 4.3. Let Σ ∈ W δ,max / ≈ δ . Then t Σ ∈ Z(H 0 ) δ . Proof. Let s ∈ S. Then t s t Σ = x∈W Σ t s t x = x,sx∈W Σ t s t x + y∈W Σ ,sy / ∈W Σ t s t x . If x, sx ∈ W Σ , then t s t x + t s t sx = 0. If y ∈ W Σ , sy / ∈ W Σ , then y < sy and t s t y = t sy . Therefore t s t Σ = x∈W ;x / ∈W Σ ,sx∈W Σ t x . Similarly, t Σ t δ(s) = x∈W ;x / ∈W Σ ,xδ(s)∈W Σ t x . By Lemma 4.2, t s t Σ = t Σ t δ(s) for any s ∈ S. Thus t Σ ∈ Z(H 0 ) δ . Theorem 4.4. The elements {t Σ } Σ∈W δ,max /≈ δ form a basis of Z(H 0 ) δ . Proof. For any h = w∈W a w t w ∈ H 0 , we write supp(h) = {w ∈ W ; a w = 0}. Let supp(h) max be the set of maximal length elements in supp(h). We show that (a) If h ∈ Z(H 0 ) δ and w ∈ supp(h) max , then swδ(s) ∈ supp(h) max and a swδ(s) = a w for any s ∈ S with sw > w or ws > w. Without loss of generality, we assume that sw > w. Then sw ∈ supp(t s h) = supp(ht δ(s) ) and supp(t s h) max = {sx; x ∈ supp(h) max , sx > x}, supp(ht δ(s) ) max = {yδ(s); y ∈ supp(h) max , yδ(s) > y}. Therefore swδ(s) ∈ supp(h) max and ℓ(swδ(s)) = ℓ(w). The coefficient of t sw in t s h is a w and the coefficient of t sw = t (swδ(s))δ(s) in ht δ(s) is a swδ(s) . Thus a w = a swδ(s) . (a) is proved. Now we show that (b) If h ∈ Z(H 0 ) δ , then supp(h) max ⊆ W δ,max . If w / ∈ W δ,max , then by Theorem 1.4, there exists w ′ with ℓ(w ′ ) = ℓ(w) + 2 and s ∈ S with w ′ → δ sw ′ δ(s) ≈ δ w. By (a), sw ′ δ(s) ∈ supp(h) max since sw ′ δ(s) ≈ δ w. Since sw ′ < w ′ , by (a) again, w ′ ∈ supp(h) max . That is a contradiction. (b) is proved. Now suppose that ⊕ Σ∈W δ,max /≈ δ Ct Σ Z(H 0 ) δ . Let h be an element in Z(H 0 ) δ −⊕ Σ∈W δ,max /≈ δ Ct Σ and max w∈supp(h) ℓ(w) max w∈supp(h ′ ) ℓ(w) for any h ′ ∈ Z(H 0 ) δ − ⊕ Σ∈W δ,max /≈ δ Ct Σ . By (a) and (b), supp(h) max is a union of Σ with Σ ∈ W δ,max / ≈ δ . By (a), if Σ ⊆ supp(h) max , then a w = a w ′ for any w, w ′ ∈ Σ. We set a Σ = a w for any w ∈ Σ. Set h ′ = h − Σ⊆supp(h)max a Σ t Σ . Then h ′ ∈ Z(H 0 ) δ − ⊕ Σ∈W δ,max /≈ δ Ct Σ . But max w∈supp(h ′ ) ℓ(w) < max w∈supp(h) ℓ(w). That is a contradiction. 5. Cocenters of 0-Hecke algebras 5.1. For any Σ ∈ W δ,min / ≈ δ , we denote by T Σ the image of T w in H δ for any w ∈ Σ. By Proposition 2.1, the element T Σ is well-defined. Similar to the proof of Theorem 2.3, we have Proposition 5.1. The set {T Σ } Σ∈W δ,min /≈ δ spans H δ . Via the natural bijection f : Γ δ → W δ,min / ≈ δ in Corollary 3.2, we may write T (J,C) for T f (J,C) . We also write t (J,C) = t f (J,C) for T (J,C) ⊗1 ∈ H 0,δ = H δ ⊗ Λ C 0 . It is worth mentioning that H δ is not a free module over Λ by Theorem 2.3 and Theorem 5.5 we will prove later. This is because dim H q,δ = ♯cl(W ) δ for any q = 0 and dim H 0,δ = ♯W δ,min / ≈ δ . These numbers do not match in general (see Example 3.4). Now we come to the cocenter of 0-Hecke algebras. We first recall the Demazure product. By [8], for any x, y ∈ W , the set {uv; u x, v y} contains a unique maximal element. We denote this element by x * y and call it the Demazure product of x and y. It is easy to see that supp(x * y) = supp(x) ∪ supp(y). The following result is proved in [8,Lemma 1]. Lemma 5.2. Let x, y ∈ W . Then t x t y = (−1) ℓ(x)+ℓ(y)−ℓ(x * y) t x * y . Lemma 5.3. For any J = δ(J) ⊆ S, set H supp δ =J 0 = ⊕ supp δ (w)=J Ct w . Then [H 0 , H 0 ] δ = ⊕ J=δ(J)⊆S H supp δ =J 0 ∩ [H 0 , H 0 ] δ . Proof. By Lemma 5.2, for any x, y ∈ W , t x t y = (−1) ℓ(x)+ℓ(y)−ℓ(x * y) t x * y , t y t δ(x) = (−1) ℓ(x)+ℓ(y)−ℓ(y * (δ(x)) t y * δ(x) . Moreover, supp δ (x * y) = supp δ (x) ∪ supp δ (y) = supp δ (y * (δ(x)). Thus t x t y , t y t δ(x) ∈ H supp δ =supp δ (x * y) 0 and t x t y −t y t δ(x) ∈ H supp δ =supp δ (x * y) 0 . Another result we need here is that the elliptic conjugacy classes never "fuse". C∈cl(W J ) δ is elliptic a (J,C) t (J,C) = 0. Fix J = δ(J) ⊆ S. We show that (a) The set {T (J,C) } C∈cl(W J ) δ is elliptic is a linearly independent set in H δ . Suppose that C∈cl(W J ) δ is elliptic b C T (J,C) = 0 ∈ H δ for some b C ∈ Λ. Then C∈cl(W J ) δ is elliptic b C \| q=q T (J,C) = 0 ∈ H q,δ for any q = 0. By Theorem 2.3, the set {T (J,C),q } C∈cl(W J ) δ is elliptic is a linearly independent set in H q,δ for any q = 0. Hence b C \| q=q = 0 for any q = 0. Thus b C = 0. (a) is proved. In other words, C∈cl(W J ) δ is elliptic ΛT (J,C) is a free submodule of H with basis T (J,C) . Thus C∈cl(W J ) δ is elliptic Ct (J,C) is a free submodule of H 0,δ with basis t (J,C) . Therefore a J,C = 0. For any J ⊆ S, let λ J be the one-dimensional representation of H 0 defined by λ J (t s ) = −1, if s ∈ J; 0, if s / ∈ J. By [13], the set {λ J } J⊆S is the set of all the irreducible representations of H 0 . Let R(H 0 ) be the Grothendieck group of finite dimensional representations of H 0 . Then R(H 0 ) is a free group with basis {λ J } J⊆S . Consider the trace map T r : H 0 → R(H 0 ) * , h → (V → tr(h, V )). It is easy to see that for any (J, C) ∈ Γ and K ⊆ S, tr(t J,C , λ K ) = (−1) ℓ(C) , if J ⊆ K; 0, otherwise. Here ℓ(C) is the length of any minimal length element in C. By [10, Proposition 6.10], for any J ⊆ S and any two elliptic conjugacy classes C and C ′ of W J , ℓ(C) ≡ ℓ(C ′ ) mod 2. Therefore, Proposition 5.6. The trace map T r : H 0 → R(H 0 ) * is surjective and the kernel equals ⊕ J⊆S,C,C ′ ∈cl(W J ) are elliptic C{t (J,C) − t (J,C ′ ) }. 6. A partial order on W δ,min / ≈ δ 6.1. Let w ∈ W and Σ ∈ W δ,min / ≈ δ , we write Σ w if there exists w ′ ∈ Σ with w ′ w. For w ∈ W and O ∈ cl(W ) δ , we define O w in the same way. We define a partial order on W δ,min / ≈ δ as follows. For Σ, Σ ′ ∈ W δ,min / ≈ δ , we write Σ ′ Σ if Σ ′ w for some w ∈ Σ. By [7,Corollary 4.6], Σ ′ Σ if and only if Σ ′ w for any w ∈ Σ. In particular, is transitive. This defines a partial order on W δ,min / ≈ δ . We define a partial order on cl(W ) δ in a similar way. (1) For any w ∈ O min , there exists w ′ ∈ O ′ min such that w ′ w. (2) There exists w ∈ O min and w ′ ∈ O ′ min such that w ′ w. Remark. We write O ′ O if the conditions above are satisfied. Then the map W δ,min / ≈ δ → cl(W ) δ is compatible with the partial orders . Proof. Let w, w 1 ∈ O min and w ′ ∈ O ′ min with w ′ w. Let J = supp δ (w), J 1 = supp δ (w 1 ) and J ′ = supp δ (w ′ ). Let C ∈ cl(W J ) δ and C 1 ∈ cl(W J 1 ) δ with w ∈ C and w ′ 1 ∈ C 1 . By §3.2, there exists x ∈ W δ and the conjugation of x sends J to J 1 and sends C to C 1 . Since w ′ w, J ′ ⊆ J. As the conjugation by x sends simple reflections in J to simple reflections in J 1 , we have xw ′ x −1 xwx −1 . Moreover, xwx −1 ∈ C 1 is a minimal length element. By Theorem 1.2, xwx −1 ≈ δ w ′ . By [ Corollary 1 . 3 . 13Let W be a finite Coxeter group and O be a δ-conjugacy class of W . Then O min = {w ∈ O; w ′ ≮ w for any w ′ ∈ O}. 1.3. One may transfer the results on minimal length elements to results on maximal length elements via the trick in [3, §2.9]. Let w 0 be the longest element in W and δ ′ = Ad(w 0 ) • δ be the automorphism on W . Then the map W → W, w → ww 0 reverses the Bruhat order and sends a δ-conjugacy class O to a δ ′conjugacy class O ′ . Moreover, w 1 → δ w 2 if and only if w 2 w 0 → δ ′ w 1 w 0 . Thus Theorem 1.4. Let W be a finite Coxeter group and O be a δ-conjugacy class of W . Then (1) For any If q = 0, then {T O,q } O∈cl(W ) δ form a basis of H q,δ . Proposition 2.4. ([4, §1.2] and [3, Theorem 7.4 (2)]) If q = 0, then there exists a unique polynomial f w,O ∈ Z[q] for any w ∈ W and O ∈ cl(W ) δ such that the image of T w in H q,δ equals Remark. The polynomials f w,O are called the class polynomials. They play an important role in the study of characters of Hecke algebras. Theorem 2.5. ([6, Theorem 5.2]) Corollary 3 . 2 . 32The mapf : Γ δ → W δ,min / ≈ δ , (J, C) → C min is a bijection. Example 3. 4 . 4Let W = S 3 . Then ♯cl(W ) = 3, ♯Γ = 4 and ♯Γ Ad(w 0 ) = 3. Therefore ♯(W min / ≈) = ♯cl(W ) and ♯(W min / ≈) = ♯(W max / ≈) for W = S 3 .4. Centers of 0-Hecke algebras 4.1. Let Σ ∈ W δ,max / ≈ δ . Set W Σ = {x ∈ W ; x w for some w ∈ Σ}, Now we recall the following known result on the Bruhat order (see, for example, [12, Lemma 2.3]). Lemma 4 . 1 . 41Let x, y ∈ W with x y. Let s ∈ S. Then 4. 2 . 2In fact, Theorem 4.4 also holds for the 0-Hecke algebras associated to any affine Weyl group and the proof is similar (the only difference is that one use [14, Main Theorem 1.1] instead of Theorem 1.4). On the other hand, there are other explicit descriptions of the centers of finite and affine Hecke algebras. • Geck and Rouquier [6, Theorem 5.2] gave a basis of the centers of finite Hecke algebras with parameter q = 0. • Bernstein, and Lusztig [11, Proposition 3.11] gave a basis of the centers of affine Hecke algebras with parameter q = 0. • Vignéras [15, Theorem 1.2] gave a basis of the centers of affine 0-Hecke algebras and pro-p Hecke algebras. It is interesting to compare Theorem 4.4 (for finite and affine 0-Hecke algebras) with the above results. Theorem 5.4. ([5, Theorem 3.2.11] and [1, Theorem 5.2.2])1 Let J = δ(J) ⊆ S. Let C, C ′ be two distinct elliptic δ-conjugacy classes of W J . Then C and C ′ are not δ-conjugate in W .Now we come to the main theorem of this section.Theorem 5.5. The elements {t (J,C) } (J,C)∈Γ δ form a basis of H 0,δ . Proof. Suppose that (J,C)∈Γ δ a (J,C) t (J,C) = 0 in H 0,δ for some a (J,C) ∈ C. Then by Lemma 5.3, for any J = δ(J) ⊆ S, 5. 4 . 4Now we relate the cocenter of H 0 to the representations of H 0 . Proposition 6. 1 . 1Let O, O ′ ∈ cl(W ) δ .The following conditions are equivalent: 7 , 7Lemma 4.4], there exists w ′ 1 ∈ O ′ min with w ′ Proposition 6.2. Let w ∈ W . Then (1) The set {Σ ∈ W δ,min / ≈ δ ; Σ w} contains a unique maximal element Σ w .Therefore for any O ∈ cl(W ) δ , p(Σ)=O a w,Σ = f w,O . By Proposition 6.2,a w,Σ ∈ (−1) ℓ(w)−ℓ(Σw) + qΛ, if Σ = Σ w ; qΛ, otherwise.1 w 1 . Therefore f w,O ∈ (−1) ℓ(w)−ℓ(Σw) + qZ[q], if Σ w ⊆ O; qZ[q], otherwise. O∈cl(W ) δ f w,O T O,q . The proof in[5] and[1] are based on a characterization of elliptic conjugacy classes using characteristic polynomials [5, Theorem 3.2.7 (P3)] and [7, Theorem 7.5 (P3)], which is proved via a case-by-case analysis. It is interesting to find a case-free proof of these results. AcknowledgementWe thank D. Ciubotaru, G. Lusztig and S. Nie for helpful discussions. We thank the referee for his/her valuable suggestions.Proof. We argue by induction on ℓ(w).If w ∈ W δ,min , we denote by Σ w the ≈ δ -equivalence class that contains w. By definition, for any Σ ∈ W δ,min / ≈ δ with Σ w, Σ Σ w . Also by definition, the image of t w in H 0,δ is t Σw . Now suppose that w ∈ W δ,min . By Theorem 1.2 (1), there exists w ′ ∈ W and s ∈ S such that w ≈ w ′ and ℓ(sw ′ δ(s)) < ℓ(w ′ ). LetWe also haveBy inductive hypothesis, the image of t swNow we discuss some application to class polynomials. Let w ∈ W . By Proposition 2.4, for any q = 0,By the same argument as in Proposition 6.2, f w,O = 0 unless O O w . Moreover, by Proposition 5.1, there exists a w,Σ ∈ Λ such thatLet p : W δ,min / ≈ δ → cl(W ) δ be the natural map. Then for any q = 0, D Ciubotaru, X He, arXiv:1208.0914The cocenter of graded affine Hecke algebra and the density theorem. D. Ciubotaru and X. He, The cocenter of graded affine Hecke algebra and the density theorem, arXiv:1208.0914. Modular representations of finite groups of Lie type. R W Carter, G Lusztig, Proc. London Math. Soc. 3R. W. Carter and G. Lusztig, Modular representations of finite groups of Lie type, Proc. London Math. Soc. (3) 32 (1976), 347-385. Minimal length elements in twisted conjugacy classes of finite Coxeter groups. M Geck, S Kim, G Pfeiffer, J. Algebra. 2292M. Geck, S. Kim, and G. Pfeiffer, Minimal length elements in twisted conju- gacy classes of finite Coxeter groups, J. Algebra 229 (2000), no. 2, 570-600. On the irreducible characters of Hecke algebras. M Geck, G Pfeiffer, Adv. Math. 1021M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), no. 1, 79-94. Characters of finite Coxeter groups and Iwahori-Hecke algebras. M Geck, G Pfeiffer, New Series. 21The Clarendon Press Oxford University PressM. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori- Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press Oxford University Press, New York, 2000. Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups (Luminy, 1994). M Geck, R Rouquier, Progr. Math. 141251272BirkhäuserM. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups (Luminy, 1994), Progr. Math., 141, pp. 251272, Birkhäuser, Boston, 1997. Minimal length elements in some double cosets of Coxeter groups. X He, Adv. Math. 2152X. He, Minimal length elements in some double cosets of Coxeter groups, Adv. Math. 215 (2007), no. 2, 469-503. A subalgebra of 0-Hecke algebra. X He, J. Algebra. 322X. He, A subalgebra of 0-Hecke algebra, J. Algebra 322 (2009), 4030-4039. Minimal length elements of finite Coxeter group. X He, S Nie, Duke Math. J. 161X. He and S. Nie, Minimal length elements of finite Coxeter group, Duke Math. J. 161 (2012), 2945-2967. G Lusztig, Characters of reductive groups over a finite field. Princeton, NJPrinceton University Press107G. Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. Affine Hecke algebras and their graded version. G Lusztig, J. Amer. Math. Soc. 2G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635. Hecke algebras with unequal parameters. G Lusztig, CRM Monograph Series. 18American Mathematical SocietyG. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Se- ries, vol. 18, American Mathematical Society, Providence, RI, 2003. 0-Hecke algebras. P N Norton, J. Austral. Math. Soc. Ser. A. 273P. N. Norton, 0-Hecke algebras, J. Austral. Math. Soc. Ser. A 27 (1979), no. 3, 337-357. S Rostami, arXiv:1306.5255Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras. Amer. Math. Socto appear in TransS. Rostami, Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras, arXiv:1306.5255, to appear in Trans. Amer. Math. Soc. The pro-p-Iwahori-Hecke algebra of a reductive p-adic group. M.-F Vignéras, II, Münster J. Math. 7M.-F. Vignéras, The pro-p-Iwahori-Hecke algebra of a reductive p-adic group, II, Münster J. Math. 7 (2014), 363-379.	[]
[ "Multiwavelength observations of the γ-ray emitting narrow-line Seyfert 1 PMN J0948+0022 in 2011", "Multiwavelength observations of the γ-ray emitting narrow-line Seyfert 1 PMN J0948+0022 in 2011" ]	[ "F D'ammando \nINAF -Istituto di Radioastronomia\nVia Gobetti 101I-40129BolognaItaly\n\nDipartimento di Fisica\nUniversità degli Studi di Perugia\nVia A. PascoliI-06123PerugiaItaly\n", "J Larsson \nDepartment of Physics\nAlbaNova\nKTH\nOskar Klein Centre\nSE-106 91StockholmSweden\n", "M Orienti \nINAF -Istituto di Radioastronomia\nVia Gobetti 101I-40129BolognaItaly\n\nDipartimento di Astronomia\nUniversità di Bologna\nVia Ranzani 1I-40127BolognaItaly\n", "C M Raiteri \nINAF -Osservatorio Astrofisico di Torino\nVia Osservatorio 20I-10025\n\nPino Torinese (TO)\nItaly\n", "E Angelakis \nMax-Planck-Institute für Radioastronomie\nAuf dem Hügel 69D-53121BonnGermany\n", "A Carramiñana \nInstituto National de Astrofísica\nÓptical\n", "L Carrasco \nInstituto National de Astrofísica\nÓptical\n", "A J Drake \nCalifornia Institute of Technology\n1200 E. California Blvd91225CAUSA\n", "L Fuhrmann \nMax-Planck-Institute für Radioastronomie\nAuf dem Hügel 69D-53121BonnGermany\n", "M Giroletti \nINAF -Istituto di Radioastronomia\nVia Gobetti 101I-40129BolognaItaly\n", "T Hovatta \nCahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA\n", "W Max-Moerbeck \nCahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA\n", "A Porras \nInstituto National de Astrofísica\nÓptical\n", "A C S Readhead \nCahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA\n", "E Recillas \nInstituto National de Astrofísica\nÓptical\n", "J L Richards \nDepartment of Physics\nPurdue University\n525 Northwestern Avenue47907West LafayetteINUSA\n", "\n72840Electrónica, Tonantzintla, PueblaMexico\n" ]	[ "INAF -Istituto di Radioastronomia\nVia Gobetti 101I-40129BolognaItaly", "Dipartimento di Fisica\nUniversità degli Studi di Perugia\nVia A. PascoliI-06123PerugiaItaly", "Department of Physics\nAlbaNova\nKTH\nOskar Klein Centre\nSE-106 91StockholmSweden", "INAF -Istituto di Radioastronomia\nVia Gobetti 101I-40129BolognaItaly", "Dipartimento di Astronomia\nUniversità di Bologna\nVia Ranzani 1I-40127BolognaItaly", "INAF -Osservatorio Astrofisico di Torino\nVia Osservatorio 20I-10025", "Pino Torinese (TO)\nItaly", "Max-Planck-Institute für Radioastronomie\nAuf dem Hügel 69D-53121BonnGermany", "Instituto National de Astrofísica\nÓptical", "Instituto National de Astrofísica\nÓptical", "California Institute of Technology\n1200 E. California Blvd91225CAUSA", "Max-Planck-Institute für Radioastronomie\nAuf dem Hügel 69D-53121BonnGermany", "INAF -Istituto di Radioastronomia\nVia Gobetti 101I-40129BolognaItaly", "Cahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA", "Cahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA", "Instituto National de Astrofísica\nÓptical", "Cahill Center for Astronomy and Astrophysics\nCalifornia Institute of Technology\n1200 E. California Blvd91125PasadenaCAUSA", "Instituto National de Astrofísica\nÓptical", "Department of Physics\nPurdue University\n525 Northwestern Avenue47907West LafayetteINUSA", "72840Electrónica, Tonantzintla, PueblaMexico" ]	[ "Mon. Not. R. Astron. Soc" ]	We report on radio-to-γ-ray observations during 2011 May-September of PMN J0948+0022, the first narrow-line Seyfert 1 (NLSy1) galaxy detected in γ-rays by Fermi-LAT. Strong variability was observed in γ-rays, with two flaring periods peaking on 2011 June 20 and July 28. The variability observed in optical and near-infrared seems to have no counterpart in γ-rays. This different behaviour could be related to a bending and inhomogeneous jet or a turbulent extreme multi-cell scenario. The radio spectra showed a variability pattern typical of relativistic jets. The XMM spectrum shows that the emission from the jet dominates above ∼2 keV, while a soft X-ray excess is evident in the low-energy part of the X-ray spectrum. Models where the soft emission is partly produced by blurred reflection or Comptonisation of the thermal disc emission provide good fits to the data. The X-ray spectral slope is similar to that found in radio-quiet NLSy1, suggesting that a standard accretion disc is present, as expected from the high accretion rate. Except for the soft X-ray excess, unusual in jet-dominated AGNs, PMN J0948+0022 shows all characteristics of the blazar class.	10.1093/mnras/stt2464	[ "https://arxiv.org/pdf/1312.5522v1.pdf" ]	1,013,231	1312.5522	565b045764b17567e5b71dacc74225083f9d63b6	Multiwavelength observations of the γ-ray emitting narrow-line Seyfert 1 PMN J0948+0022 in 2011 19 Dec 2013 20 December 2013 F D'ammando INAF -Istituto di Radioastronomia Via Gobetti 101I-40129BolognaItaly Dipartimento di Fisica Università degli Studi di Perugia Via A. PascoliI-06123PerugiaItaly J Larsson Department of Physics AlbaNova KTH Oskar Klein Centre SE-106 91StockholmSweden M Orienti INAF -Istituto di Radioastronomia Via Gobetti 101I-40129BolognaItaly Dipartimento di Astronomia Università di Bologna Via Ranzani 1I-40127BolognaItaly C M Raiteri INAF -Osservatorio Astrofisico di Torino Via Osservatorio 20I-10025 Pino Torinese (TO) Italy E Angelakis Max-Planck-Institute für Radioastronomie Auf dem Hügel 69D-53121BonnGermany A Carramiñana Instituto National de Astrofísica Óptical L Carrasco Instituto National de Astrofísica Óptical A J Drake California Institute of Technology 1200 E. California Blvd91225CAUSA L Fuhrmann Max-Planck-Institute für Radioastronomie Auf dem Hügel 69D-53121BonnGermany M Giroletti INAF -Istituto di Radioastronomia Via Gobetti 101I-40129BolognaItaly T Hovatta Cahill Center for Astronomy and Astrophysics California Institute of Technology 1200 E. California Blvd91125PasadenaCAUSA W Max-Moerbeck Cahill Center for Astronomy and Astrophysics California Institute of Technology 1200 E. California Blvd91125PasadenaCAUSA A Porras Instituto National de Astrofísica Óptical A C S Readhead Cahill Center for Astronomy and Astrophysics California Institute of Technology 1200 E. California Blvd91125PasadenaCAUSA E Recillas Instituto National de Astrofísica Óptical J L Richards Department of Physics Purdue University 525 Northwestern Avenue47907West LafayetteINUSA 72840Electrónica, Tonantzintla, PueblaMexico Multiwavelength observations of the γ-ray emitting narrow-line Seyfert 1 PMN J0948+0022 in 2011 Mon. Not. R. Astron. Soc 000000019 Dec 2013 20 December 2013Accepted. Received; in original formarXiv:1312.5522v1 [astro-ph.HE] Printed (MN L A T E X style file v2.2)galaxies: active -galaxies: nuclei -galaxies: Seyfert -galaxies: individual (PMN J0948+0022) -gamma-rays: general We report on radio-to-γ-ray observations during 2011 May-September of PMN J0948+0022, the first narrow-line Seyfert 1 (NLSy1) galaxy detected in γ-rays by Fermi-LAT. Strong variability was observed in γ-rays, with two flaring periods peaking on 2011 June 20 and July 28. The variability observed in optical and near-infrared seems to have no counterpart in γ-rays. This different behaviour could be related to a bending and inhomogeneous jet or a turbulent extreme multi-cell scenario. The radio spectra showed a variability pattern typical of relativistic jets. The XMM spectrum shows that the emission from the jet dominates above ∼2 keV, while a soft X-ray excess is evident in the low-energy part of the X-ray spectrum. Models where the soft emission is partly produced by blurred reflection or Comptonisation of the thermal disc emission provide good fits to the data. The X-ray spectral slope is similar to that found in radio-quiet NLSy1, suggesting that a standard accretion disc is present, as expected from the high accretion rate. Except for the soft X-ray excess, unusual in jet-dominated AGNs, PMN J0948+0022 shows all characteristics of the blazar class. INTRODUCTION Narrow-line Seyfert 1 (NLSy1) galaxies, first suggested as a distinct class of active galactic nuclei (AGNs) by Osterbrock & Pogge (1985), are characterized in optical by their narrow permitted emission lines (full-width at half maximum FWHM 2000 km s −1 ), weak [OIII]λ5007 emission line ([OIII]/Hβ < 3), and strong Fe ii emission lines. They also exhibit strong X-ray variability, steep X-ray spectra, substantial soft X-ray excess and relatively high lumi-⋆ E-mail: [email protected] nosity (Boller et al. 1996;Zhou et al. 2006). These characteristics suggest that NLSy1s have smaller masses of the central black hole (MBH = 10 6 -10 8 M⊙) and higherṀ /Ṁ Edd values (up to the Eddington limit or above) than those observed in blazars and radio galaxies. Only a small percentage (<7%) of NLSy1 are radio-loud (Komossa et al. 2006) compared to ∼15% of the quasars. In the radio-loud NLSy1s, flat radio spectra and flux density variability suggest that several of them could host relativistic jets (Zhou et al. 2003;Doi et al. 2006;Yuan et al. 2008). PMN J0948+0022 shows optical properties typical of a NLSy1 (i.e. FWHM(Hβ) = 1432±87 km s −1 , [OIII]/Hβ∼0.1, a strong Fe ii bump), and a radio loudness of R = 355 (Yuan et al. 2008). High brightness temperature and a compact structure have been observed for PMN J0948+0022 (Doi et al. 2006), in addition to a possible core-jet structure (Giroletti et al. 2011). This source was the first radio-loud NLSy1 to be detected in γ rays by the Large Area Telescope (LAT) on board the Fermi Gammaray Space Telescope (Abdo et al. 2009a). After that, other 4 radio-loud NLSy1s were detected with high significance in γ rays (Abdo et al. 2009b;D'Ammando et al. 2012a), suggesting the radio-loud NLSy1s as a third class of γ-ray emitting AGN with relativistic jets. On the contrary, no radio-quiet Seyfert galaxies have been detected in γ rays (Ackermann et al. 2012). Three γ-ray flares were observed from PMN J0948+0022 during 2010-2013, reaching daily peak fluxes of (1-2)×10 −6 photons cm −2 s −1 (Foschini et al. 2011;D'Ammando & Ciprini 2011;D'Ammando & Orienti 2013). This indicates that radio-loud NLSy1s also can host powerful relativistic jets such as blazars. These findings pose intriguing questions about the nature of these γ-ray emitting NLSy1s, the onset of production of their relativistic jets, and the cosmological evolution of radio-loud AGNs. After the discovery of γ-ray emission from PMN J0948+0022, this source has been the target of different multifrequency campaigns with the aim of understanding its nature. The first spectral energy distributions (SEDs) collected for this object, as well as for the other three γ-ray NLSy1s detected in the first year of Fermi operation, showed clear similarities with blazars: a double-humped shape with a first peak in the IR/optical band due to synchrotron emission, a second peak in the MeV/GeV band likely due to inverse Compton emission, and an accretion disk component in UV. The physical parameters of these NLSy1s are blazar-like, and jet powers are in the average range of blazars (Abdo et al. 2009b). In addition, the comparison of the SED of PMN J0948+0022 during the 2010 July flaring activity with that of 3C 273, a typical flat spectrum radio quasar (FSRQ), showed a more extreme Compton dominance in the NLSy1. The disagreement between the two SEDs can be due to the difference in black hole (BH) masses and Doppler factor of the two jets (Foschini et al. 2011). The radio-toγ-ray light curves of the source collected over years showed correlated variability, with a delay of a few months of the radio emission with respect to the γ-ray, as usually observed in FSRQs (Abdo et al. 2009c;Foschini et al. 2012). In this paper, we discuss the results of the analysis of the multiwavelength data of PMN J0948+0022 collected during 2011 May-September. Part of the data presented here has been already published in Foschini et al. (2012). XMM-Newton and Catalina Real-time Transient Survey data are presented here for the first time. Fermi-LAT, Swift, and MOJAVE data are re-analyzed. The paper is organized as follows. In Section 2, we report the LAT data analysis and results, while in Sections 3 and 4 we present the Xray and optical/UV results of the Swift and XMM-Newton observations, respectively. Near-infrared (NIR) and optical data from ground-based observatories are presented and discussed in Section 5. Radio data collected by the Very Long Baseline Array (VLBA) interferometer, the 40 m Owens Valley Radio Observatory (OVRO), the 32 m Medicina, the 13.7 m Metsähovi, and 100 m Effelsberg single-dish telescopes are presented and discussed in Section 6. In Section 7, we discuss the properties of the source and draw our conclusions. Throughout the paper the quoted uncertainties are given at the 1σ level, unless otherwise stated, and the photon indices are parameterized as dN/dE ∝ E −Γ , where Γ is the photon index. We adopt a Λ cold dark matter (Λ-CDM) cosmology with H0 = 71 km s −1 Mpc −1 , ΩΛ = 0.73, and Ωm = 0.27. The corresponding luminosity distance at z = 0.5846 is dL = 3413 Mpc, and 1 arcsec corresponds to a projected size of 6.590 kpc. 2 FERMI-LAT DATA: ANALYSIS AND RESULTS Observations and data reduction The Fermi-LAT is a pair-conversion telescope operating from 20 MeV to > 300 GeV. It has a large peak effective area (∼8000 cm 2 for 1 GeV photons), an energy resolution of typically ∼10%, and a field of view of about 2.4 sr with an angular resolution (68% containment radius) better than 1 • for energies above 1 GeV. Further details about the Fermi-LAT are given in Atwood et al. (2009). The LAT data reported in this paper were collected from 2011 May 1 (MJD 55682) to September 30 (MJD 55834). During this time, the Fermi spacecraft operated almost entirely in survey mode. The analysis was performed with the ScienceTools software package version v9r27p1. The LAT data were extracted within a region of 20 • radius centred at the location of PMN J0948+0022. Only events belonging to the 'Source' class were used. The time intervals when the rocking angle of the LAT was greater than 52 • were rejected. In addition, a cut on the zenith angle (< 100 • ) was applied to reduce contamination from the Earth limb γ rays, which are produced by cosmic rays interacting with the upper atmosphere. The spectral analysis was performed with the instrument response functions P7SOURCE_V6 using an unbinned maximum-likelihood method implemented in the Science tool gtlike. A Galactic diffuse emission model and isotropic component, which is the sum of an extragalactic and residual cosmic-ray background, were used to model the background 1 . The normalizations of both components in the background model were allowed to vary freely during the spectral fitting. We analysed a region of interest of 10 • radius centred at the location of PMN J0948+0022. We evaluated the significance of the γ-ray signal from the sources by means of a maximum-likelihood test statistic T S = 2 (logL1 -logL0), where L is the likelihood of the data given the model with (L1) or without (L0) a point source at the position of PMN J0948+0022 (Mattox et al. 1996). Following the second Fermi LAT source catalogue (2FGL; Nolan et al. 2012), the spectral model used for PMN J0948+0022 is a log- (Landau et al. 1986;Massaro et al. 2004), where the parameter α is the spectral slope at the energy E0 and the parameter β measures the curvature around the peak. We fixed the reference energy E0 to 272 MeV as in the 2FGL catalogue. The source model used in gtlike includes all of the point sources from the 2FGL catalogue that fall within 15 • of PMN J0948+0022. The spectra of these sources were parametrized by power law functions, dN/dE ∝ (E/E0) −Γ , except for 2FGL J0909.1+0121, for which we used a log-parabola as in the 2FGL catalogue. A first maximum-likelihood analysis was performed to remove from the model the sources having T S < 25 and/or a predicted number of counts based on the fitted model N pred < 3. A second maximum-likelihood analsyis was performed on the updated source model. The fitting procedure has been performed with the sources within 10 • of PMN J0948+0022 included with the normalization factors and the photon indices left as free parameters. For the sources located between 10 • and 15 • from our target, we kept the normalizations and the photon indices fixed to the values of the 2FGL catalogue. parabola, dN/dE ∝ (E/E0) −α−β log(E/E 0 ) Results Integrating over the period 2011 May 1-September 30 (MJD 55682-55835) the fit yielded a T S = 674 in the 0.1-100 GeV energy range, with an average integral flux of (21.6 ± 1.5) ×10 −8 ph cm −2 s −1 , a spectral slope α = 2.41 ± 0.10, and a curvature parameter around the peak β = 0.20 ± 0.06. The corresponding apparent isotropic γ-ray luminosity is ∼1.8×10 47 erg s −1 . As a comparison during the first two years of Fermi operation (2008 August 4-2010 August 4) the average integral flux was (9.2 ± 0.5) ×10 −8 ph cm −2 s −1 , with a spectral slope α = 2.26 ± 0.08, and a curvature parameter around the peak β = 0.26 ± 0.06 (Nolan et al. 2012). Thus the average 0.1-100 GeV flux over 2011 May-September is about a factor of 2 higher than the 2FGL flux, but no significant spectral changes were observed. Fig. 1 shows the γ-ray light curve for the period considered, using a log-parabola spectral model and 7-day time bins. For each time bin, the spectral parameters of PMN J0948+0022 and all sources within 10 • of it were frozen to have the parameter values resulting from the likelihood analysis over the entire period. For the highest significance periods, we also reported as open circles the fluxes in 1day time intervals. If T S < 10, 2σ upper limits were calculated. All uncertainties in measured γ-ray flux and index reported throughout the paper are statistical only. The systematic uncertainty in the effective area is energy dependent: it amounts to 10% at 100 MeV, decreasing to 5% at 560 MeV, and increasing to 10% above 10 GeV (Ackermann et al. 2012). A daily peak flux of (101 ± 20) ×10 −8 ph cm −2 s −1 in the 0.1-100 GeV energy range was detected on 2011 June 20 (MJD 55732), representing an increase of a factor of 5 with respect to the 2011 May-September average flux and more than an order of magnitude above the average 2FGL flux of the source. The corresponding apparent isotropic γ-ray luminosity is ∼8.8×10 47 erg s −1 . Preliminary results about this γ-ray flaring activity were reported by D'Ammando & Ciprini (2011) and Lucarelli et al. (2011). A second γ-ray flaring activity was observed at the end of 2011 July, peaking on July 28 (MJD 55770) with a flux of (87 ± 25) ×10 −8 ph cm −2 s −1 in the 0.1-100 GeV energy range. By means of the gtsrcprob tool, we estimated that the highest energy photon emitted from PMN J0948+0022 (with probability > 80% of being associated with the source) was observed on 2011 September 13 (MJD 55807), at a distance of 0.09 • from the source and with an energy of 3.2 GeV. This suggests that this NLSy1 emits mainly at E < 10 GeV, even during flaring activity (see e.g., Foschini et al. 2011). SWIFT DATA: ANALYSIS AND RESULTS The Swift satellite (Gehrels et al. 2004) performed six observations of PMN J0948+0022 between 2011 April 29 and July 2. The observations were performed with all three on board instruments: the Burst Alert Telescope (BAT; Barthelmy et al. 2005, 15-150 keV), the X-ray Telescope (XRT; Burrows et al. 2005, 0.2-10.0 keV), and the Ultraviolet/Optical Telescope (UVOT; Roming et al. 2005, 170-600 nm). The hard X-ray flux of this source is below the sensitivity of the BAT instrument for the short exposures of these observations and therefore the data from this instrument were not used. Moreover, the source is not present in the Swift-BAT 70-month hard X-ray catalogue (Baumgartner et al. 2013). Swift-XRT The XRT data were processed with standard procedures (xrtpipeline v0.12.6), filtering, and screening criteria by using the HEASoft package (v6.12). The data were collected in photon counting mode for all of the observations. The source count rate was low (< 0.5 counts s −1 ); thus pile-up correction was not required. Source events were extracted from a circular region with a radius of 20 pixels (1 pixel ∼ 2.36 arcsec), while background events were extracted from a circular region with radius of 50 pixels away from the source region or other bright sources. Ancillary response files were generated with xrtmkarf and account for different extraction regions, vignetting and point spread function corrections. We used the spectral redistribution matrices in the Calibration data base maintained by HEASARC. When the number of photons collected was too low (< 200 counts), the spectra were rebinned with a minimum of 1 count per bin and the Cash statistic (Cash 1979) was used. Since the effective area of Swift-XRT is a factor of ∼10 lower than that of the XMM-Newton EPIC cameras, detailed spectral modeling was not performed with the XRT observations. The spectrum was fitted with an absorbed power law using the photoelectric absorption model TBABS (Wilms et al. 2000), with a neutral hydrogen column density fixed to its Galactic value (5.07×10 20 cm −2 ; Kalberla et al. 2005). The fit results are reported in Table 1. The relatively harder X-ray spectrum observed for PMN J0948+0022 with respect to the other NLSy1s (e.g., Grupe et al. 2010;Zhou & Zhang 2010) could be due to the contribution of inverse Compton radiation from a relativistic jet, similarly to what is found in FS-RQs. A photon index 1.4-1.5 was observed in X-rays also for another γ-ray NLSy1, SBS 0846+513 (D' Ammando et al. 2012aAmmando et al. , 2013b. Swift-UVOT During the Swift pointings the UVOT instrument observed PMN J0948+0022 in its optical (v, b, and u) and UV (w1, m2, and w2) photometric bands. The data analysis was performed using the uvotsource task included in the HEASoft package (v6.12). Source counts were extracted from a circular region of 5 arcsec radius centred on the source, while background counts were derived from a circular region of 10 arcsec radius in the source neighbourhood. We calculated the effective wavelengths, count rate to flux conversion factors, and Galactic extinctions for the UVOT bands according to the procedure explained in Raiteri et al. (2010) and D'Ammando et al. (2012a. The observed magnitudes are reported in Table 2, and flux densities for the v, u, and w2 filters are shown in Fig. 7. The optical u-band magnitude ranged from 17.67 to 17.11, with the peak detected on 2011 June 4 (MJD 55716). No significant variability was observed in UV, but we noted that no UV observations were available during the optical peak in u-band. The EPIC pn was operated in the large window mode and the EPIC MOS cameras (MOS1 and MOS2) were operated in the prime partial mode. The data were reduced using the XMM-Newton Science Analysis System (SAS v11.0.0), applying standard event selection and filtering. Inspection of the background light curves showed that strong flaring was present during the whole observation. The flaring time intervals were removed for the spectral analysis, leaving good exposure times of 36, 58 and 60 ks for the pn, MOS1 and MOS2, respectively. For each of the detectors the source spectrum was extracted from a circular region of radius 34 arcsec centred on the source, and the background spectrum from a nearby region on the same chip. All the spectra were binned to contain at least 20 counts per bin to allow for χ 2 spectral fitting. The Optical Monitor (OM) and Reflection Grating Spectrometers (RGS1 and RGS2) were also operating during the observation. No spectral lines were detected in the RGS spectra. This could be because the signal was rather low. The RGS data are not discussed further in this work. The data from the OM are discussed in Sect. 4.4. X-ray spectral analysis All spectral fits were performed over the 0.3-10 keV energy range using XSPEC v.12.7.1. The energies of spectral features are quoted for the rest frame of the source while plots are in the observed frame, unless otherwise stated. All errors are quoted for 90% confidence for one interesting parameter (∆χ 2 = 2.7). The data from the three EPIC cameras were initially fitted separately, but since good agreement was found (< 5%) we proceeded to fit them together. As for the Swift spectra, Galactic absorption was included in all fits using the TBABS model. The results of all the fits are presented in Table 3. It is clear that a simple power law model was insufficient to describe the data, while a broken power law yielded an acceptable fit (the improvement between the models is ∆χ 2 = 969 for two additional free parameters). The broken power law has a break at observed energy E break = 1.72 +0.09 −0.11 keV, with photon indices below and above the break of Γ1 = 2.14 +0.03 −0.02 and Γ2 = 1.48 +0.04 −0.03 , respectively. In order to check for intrinsic absorption, a neutral absorber at the redshift of the source was added to this model, but it was found not to be required. The power law slope above the break energy is significantly harder than those observed in radio-quiet NLSy1s (e.g., Grupe et al. 2010;Zhou & Zhang 2010), but similar to the slopes found in FSRQs (e.g., Donato et al. 2001). As already noted based on the Swift-XRT observations above, this makes highly probable that the emission from the jet dominates the X-ray spectrum at these energies. This conclusion is further supported by the fact that there was no detection of an Fe line in the spectrum. The 90% upper limit on the equivalent width (EW) of a narrow emission line at 6.4 keV is EW< 19 eV, or EW< 29 eV if the energy is allowed to vary between 6.4 and 7 keV. On the other hand, the steeper slope found at low energies may be associated with the corona and the accretion disc of the system. In radio-quiet NLSy1s, the low-energy part of the spectrum is composed of a steep power law, originating in the corona, as well as a strong so-called soft excess, the origin of which is debated. Fig. 2 shows the spectrum of PMN J0948+0022 as a ratio to the power law with Γ2 = 1.48 +0.04 −0.03 , which fits the spectrum above the break energy, illustrating the strength of the soft component in this source. The soft component can be fitted with a blackbody model with kT ∼ 0.18 keV, which is within the typical range of temperatures for the soft excess (Gierlínski & Done 2004). However, it is well known that such a high temperature is inconsistent with standard accretion disc theory, which predicts that the thermal emission from the disc around a super-massive BH should peak in the ultraviolet band. Indeed, from the SED of the source (see e.g., Foschini et al. 2011Foschini et al. , 2012 the observed emission from the accretion disc seems to peak at 20 eV, corresponding to a disc temperature of 11 eV in the rest frame of the source. The soft X-ray emission may instead be due to Comptonisation of the disc emission by a population of electrons with low temperature and large optical depth, which may exist in a transition region between the disc and the corona (Done et al. 2012). To investigate this model we proceeded to fit the spectrum with a power law and a Comptonised blackbody, using the COMPTT model (Titarchuk 1994) and fixing the seed temperature at 11 eV. As seen in Table 3 and the left panel of Fig. 3, this model provides a good fit to the spectrum, with best-fit electron temperature kTe = 0.50 +0.16 −0.09 keV and optical depth τ = 10.2 +0.3 −0.1 . We note that allowing a higher seed temperature of 20 eV changes only the normalisation of the Comptonised component, without significantly affecting any of the other best fit values presented in Table 3. The photon index of the power law in this model is Γ2 = 1.44±0.03. In principle the model should also contain a second power law to account for any emission from the corona, but such a component was not required in the fits. An alternative explanation for the soft excess is that it is due to relativistically blurred reflection from the accretion disc. Such a model can explain the fact that the energy of the soft excess is observed to be fairly constant for black holes spanning several orders of magnitude in mass (Gierlínski & Done 2004;Bianchi et al. 2009), and has been successfully applied for a large number of sources (Crummy et al. 2006). In this picture the X-ray spectrum of PMN J0948+0022 would be composed of a steep power law associated with the corona, a reflection component resulting from irradiation of the disc by this power law, as well as a hard power law associated with the jet, dominating at high energies. The large number of parameters in such a model are impossible to constrain with the available data, especially given that there is no Fe line or Compton hump for the reflection model to anchor to. To set some constraints on such a scenario we constructed a model made of two power laws (with the photon indices constrained to be steeper and flatter than 2, for the corona and jet, respectively) together with a relativistically blurred reflection component with parameters fixed at 'standard' values. The reflection was modelled with KDBLUR acting on the RE- FLIONX model by Ross & Fabian (2005), with the photon index of the illuminating power law tied to the steep power law. The inner and outer radii of the disc were fixed at 6 gravitational radii (rg) and 400 rg, respectively. Solar abundances were assumed and the emissivity index was fixed at q = 3 (i.e. assuming a standard emissivity profile of r −3 from a central point source). The inclination was fixed at i = 3 • , which is the angle of the jet inferred by Foschini et al. (2011). The disc may not be exactly perpendicular to the jet, but we note that allowing a larger value, i.e. up to around i = 15 • , did not change the results. After optimisation the ionisation state of the disc was fixed at ξ = 3000 erg cm s −1 . As seen in Table 3 this model results in a fit of quality comparable to the Comptonised blackbody. However, it should be noted that it is the steep power law and not the reflection component that provides most of the flux at low energies, as seen in Fig. 3, which shows the model components. In fact, a fit with two power laws but no reflection is only marginally worse (∆χ 2 = 6 for one more degree of freedom, see Table 3). Allowing each of the parameters of the reflection model to vary did not significantly change these results. X-ray variability The EPIC pn light curve is shown in Fig. 4, with the time intervals with low background that were used for the spectral analysis marked in red. The source was only mildly variable during the XMM observation, as suggested by the count rate light curve. The observed fluxes corrected for Galactic absorption are F 2−10 keV = (2.56 ± 0.05) × 10 −12 erg cm −2 s −1 and F 0.3−10 keV = 4.59 +0.03 −0.05 × 10 −12 erg cm −2 s −1 , in agreement with the Swift-XRT results (see Table 1). The root-mean-square (RMS) variability spectrum of the observation is shown in Fig. 5, and it is calculated following Vaughan et al. (2003) using light curves with 0.5 ks bins. The RMS spectrum shows the variability amplitude of the source as a function of energy, corrected for the variance due to measurement errors and normalised by the mean count rate in each energy band. The error bars represent the uncertainty expected from the Poisson noise (Vaughan et al. 2003). Since the flaring background becomes increasingly dominant with energy we plot the RMS spectrum only up to 5 keV (at which energy the ratio of signal to background is about 4). The variability clearly decreases with energy up Table 3. Summary of fits to the 0.3-10 keV XMM-Newton spectra. All fits also included absorption fixed at the Galactic value. Superscript f indicates that a parameter was kept fixed. See text for a description of the models. to around 1.7 keV, but then starts to increase again. It is interesting to note that this break coincides with the break energy of the broken power law model (see Table 3), since it is consistent with our interpretation that the Seyfert and jet components dominate at low and high energies, respectively (see further Section 7.3). The RMS spectrum presented in Bhattacharyya et al. (2013) has a different energy binning, but exhibits an overall shape that is consistent with our results. While we find that the increase in RMS variability above ∼ 2 keV is not affected by our choice of energy binning, it will be important to confirm these results with an observation where the variability analysis can be extended to higher energies. We further investigated the spectral variability by fitting the broken power law model to spectra extracted in 15 ks intervals (corresponding to 3-7.5 ks of good exposure time per spectrum). Fig. 6 shows the evolution of the bestfit parameters and the flux as a function of time. There are no systematic trends with time and only the flux is found to vary significantly above the 3σ level. However, the photon index and E break are found to be marginally variable (> 90% CL). Optical Monitor data The Angelakis et al. 2008). Corrections for pointing offset, gain, atmospheric opacity, and sensitivity have been applied to the data. The different spectra collected by Effelsberg are represented in Fig. 8. Radio spectra and fluxes indicate that the source was highly active already in 2011 May 24 (MJD 55705), before the first peak of the γ-ray activity, followed by a flux decrease and a flattening of the spectrum in 2011 August-October. The spectral index calculated between 8.4 and 32 GHz changes between −0.4±0.1 and 0.2±0.1 from 2011 May to October. No significant flux changes were observed at frequencies below 8.4 GHz. This is likely due to opacity effects at the low frequencies. Flux densities at 32 GHz and 14.6 GHz are also shown in Fig. 7. Metsähovi Observations at 37 GHz were made with the 13.7 m Metsähovi radio telescope, which is a radome enclosed paraboloid antenna situated in Finland. The measurements were made with a 1 GHz-band dual beam receiver centred at 36.8 GHz. The observations are ON-ON measurements, alternating between the source and the sky in each feed horn. A typical integration time to obtain one flux density data point is between 1200 and 1400 s. The detection limit at 37 GHz is on the order of 0.2 Jy under optimal conditions. Data points with a signal-to-noise ratio < 4 are handled as non-detections. The flux density scale is set by observations of DR 21. Sources NGC 7027, 3C 274 and 3C 84 are used as secondary calibrators. A detailed description of the data reduction and analysis is given in Teräsranta et al. (1998). The error on the flux density includes the contribution from the measurement RMS and the uncertainty of the absolute calibration. Flux densities at 37 GHz are shown in Fig. 7. OVRO 40 m As part of an ongoing blazar monitoring program, the OVRO 40 m radio telescope has observed PMN J0948+0022 at 15 GHz regularly since the end of 2007 (Richards et al. 2011). This monitoring program includes about 1700 known and likely γ-ray loud blazars above declination −20 • . The sources in this programme are observed in total intensity twice per week with a 4 mJy (minimum) and 3% (typical) uncertainty on the flux densities. Observations were performed with a dual-beam (each 2.5 arcmin FWHM) Dickeswitched system using cold sky in the off-source beam as the reference. Additionally, the source is switched between beams to reduce atmospheric variations. The absolute flux density scale is calibrated using observations of 3C 286, adopting the flux density (3.44 Jy) from Baars et al. (1977). This results in about a 5% absolute scale uncertainty, which is not reflected in the plotted errors. During the OVRO monitoring, the flux density varied from 671 mJy (on MJD 55725; 2011 June 13) to 335 mJy (on MJD 55836; 2011 October 2), as shown in Fig. 7. Medicina PMN J0948+0022 was observed with the 32 m Medicina radio telescope eight times between 2011 May and October. The new enhanced single-dish control acquisition system, which provides enhanced sensitivity and supports observations with the cross scan technique, was used. Observations were performed at both 5 and 8.4 GHz; the typical on-source time was 1.5 minutes and the flux density was calibrated with respect to 3C 286. Since the signal-to-noise ratio in each scan across the source was low (typically∼ 3), we performed a stacking analysis of the scans, which allowed us to significantly improve the signal-to-noise ratio and the precision of the measurement. The flux densities at 5 and 8.4 GHz are reported in Table 5 MOJAVE: data analysis and results We investigated the parsec-scale morphology and flux density variability at 15 GHz by means of 5-epoch VLBA data from the MOJAVE programme (Lister et al. 2009). The data sets span the time interval between 2011 February and December, in order to overlap with the Fermi-LAT data. We imported the calibrated uv data into the National Radio Astronomy Observatory AIPS package. In addition to the total intensity images, we produced the Stokes Q and U images, to derive information on the polarized emission. The flux density was derived by means of the AIPS task JMFIT which performs a Gaussian fit on the image plane. Total intensity flux density and polarisation information are reported in Table 6. During some observing epochs we detected a hint of the jet emerging from the core component with a position angle of ∼30 • (see Fig. 9 and 10), in agreement with previous works (e.g., Doi et al. 2006;Giroletti et al. 2011). In accordance with the flux density observed by OVRO at 15 On the image we provide the restoring beam, plotted in the bottom-left corner, the peak flux density in mJy/beam, and the first contour (f.c.) intensity in mJy/beam, which is three times the off-source noise level. Contour levels increase by a factor of 2. The vectors superimposed on the total intensity contours show the percentage polarization and the position angle of the electric vector. DISCUSSION AND CONCLUSIONS PMN J0948+0022 is the best studied γ-ray NLSy1 (see e.g., Abdo et al. 2009c;Foschini et al. 2012). Simultaneous multiwavelength observations presented here allow a broad-band characterization of this source, including the first XMM-Newton observation of this NLSy1. Behaviour of the light curves In Fig. 7, we compare the γ-ray light curve collected by Fermi-LAT in the 0.1-100 GeV energy range to the X-ray June 14 (MJD 55726) and July 2 (MJD 55744), after the γ-ray flare observed by LAT. The flux increase was accompanied by a hardening of the X-ray spectrum. However, the lack of X-ray data during the peaks of the γ-ray activity does not allow us to draw conclusions regarding a contemporaneous increase of the activity in X-ray and γ rays. The source increased its V -band flux by a factor of ∼3 from 2011 May 15 (MJD 55696) to May 26 (MJD 55707). An increse of flux density between 2011 May 24 and 25 of about 50% in R-band was reported by Eggen et al. (2013), with a significant decrease of a factor of ∼3.5 in two days. These rapid changes are in agreement with the optical intraday variability observed in 2011 March 27-31, with a total amplitude of ∼ 0.9 mag during ∼4 hours on April 1 . Similar intraday variability has been reported for this source by Liu et al. (2010); Maune et al. (2011);Paliya et al. (2013) and Itoh et al. (2013), indicating a relativistic jet as the most likely origin for the optical emission in PMN J0948+0022. If this variability is due to the accretion disc, the amplitude variability should be higher during faint states, when the jet activity is lower, and minimised during bright states, as already discussed in Maune et al. (2011). At the time of the optical peak on 2011 May 25 a high optical polarisation percentage (P=12.31%) was observed, with a significant increase with respect to the observation performed in May 24 (P=1.35%) ). An increase of polarisation percentage from 0.8% on 2011 February 20 to 2.0% on May 26 was also observed in the MOJAVE data (see Section 6.5). The sparse sampling of the optical and NIR light curves does not allow us to make a detailed comparison with the γ-ray and radio light curves. However, it is worth noting that an increase of flux density was also observed in NIR in 2011 mid-May (about MJD 55700), with no counterpart in γ rays. In fact, PMN J0948+0022 was quite active in Hband on 2011 April 26 (MJD 55677), and after a decrease in the following 3 weeks, we observed on 2011 May 22 (MJD 55703) a significant increase by a factor of ∼4.5 in 24 hours. A similar behaviour seems to be observed in J-band, with an increase by a factor of ∼2.5 on May 22 (MJD 55703). On the other hand, no significant variability was observed in UV during the Swift-UVOT observations, probably due to the accretion disc emission that dilutes the variability in that part of the spectrum. During the OVRO monitoring the flux density at 15 GHz changed more smoothly than at higher energies, with an amplitude variability of ∼2 between 670 mJy (2011 June 13; MJD 55725) and 335 mJy (2011 October 2; MJD 55836) and a gradual decrease. A flaring activity seems to start before the first γ-ray flare, with a peak at about MJD 55700, close to the optical and NIR flare, and a monotonic decreasing trend in the following months at 15 and 37 GHz. In the same way the radio spectra collected by Effelsberg, Medicina, and OVRO showed a significant spectral evolution with a high frequency component dominating the emission in 2011 May. A similar spectral evolution was already reported in Angelakis et al. (2013) for PMN J0948+0022 and other γ-ray emitting NLSy1s with variability patterns typical of relativistic jets and thus similar to blazars. The different behaviour observed in the radio-to-optical and the γ-ray energy bands could be related to a bending and inhomogeneous jet, as proposed for some blazars (e.g., Raiteri et al. 2010Raiteri et al. , 2011, with a variable misalignment between the zone responsible for the radio-to-optical emission and the zone responsible for γ rays. The change of the viewing angle of the different emitting regions may produce a change of the Doppler factor, with the Doppler boosting of the radio-to-optical emission increasing first, followed by an increase of the Doppler boosting of the γ-ray emission. This complex behaviour also could be in agreement with the turbulent extreme multi-cells scenario proposed by Marscher (2012). Alternatively, the radio activity could be related to the γ-ray flaring activity observed in 2010 July-August, but the large delay of about 1 year makes this hypothesis unlikely. The sparse and irregular sampling, especially from NIR to X-rays, does not allow us to test the different scenarios. Energetics The high apparent isotropic γ-ray luminosity observed for PMN J0948+0022 in 2011 May-September (∼ 1.8×10 47 erg s −1 ) should reflect a small viewing angle with respect to the jet axis and thus high beaming factors, similarly to what is observed for the FSRQs and also for the NLSy1 SBS 0846+0513 (D'Ammando et al. 2012a(D'Ammando et al. , 2013b. This is consistent with the viewing angle of 3 • used for modeling the SEDs of this source in Foschini et al. (2011Foschini et al. ( , 2012. On the contrary, most of the radio galaxies detected by the LAT have an apparent isotropic γ-ray luminosity lower than 10 46 erg s −1 , suggesting a smaller beaming factor and possibly a different structure of the jet (Abdo et al. 2010). Assuming a BH mass of 10 8 M⊙ (but see Section 7.5) we obtain an Eddington luminosity of 1.3×10 46 erg s −1 . During the 2011 June flare PMN J0948+0022 reached an apparent isotropic γ-ray luminosity of Lγ = 8.8×10 47 erg s −1 , making the radiative power L rad = Lγ /Γ 2 = 3.5×10 45 erg s −1 , assuming a quite typical value for this source of Γ = 16 (Foschini et al. 2011(Foschini et al. , 2012. This is about 25% of the Eddington luminosity, comparable to the values observed for bright FSRQs detected by LAT (see e.g., Nemmen et al. 2012). X-ray spectrum Thanks to the first high quality XMM-Newton observation of PMN J0948+0022, we are able to study in detail its Xray spectrum (see Section 4.2). The spectral modelling of the XMM data of PMN J0948+0022 showed that emission from the jet most likely dominates the spectrum above ∼ 2 keV, while the emission from the underlying Seyfert galaxy can be seen at lower energies. Interestingly, the observation of such a component, typical in the X-ray spectra of radio-quiet NLSy1s, is quite unusual in jet-dominated AGNs, even if not unique (e.g., the FSRQ PKS 1510−089; Kataoka et al. 2008). Contrary to what is observed in some blazars (e.g., BL Lacertae; Raiteri et al. 2010), no excess absorption above the Galactic column density was necessary by the fit for modelling the low-energy part of the spectrum (Grupe et al. 2010). As well as for PKS 1510−089, we hypothesize that the emission below 2 keV was mainly due to the soft Xray excess. However, we cannot distinguish between different models for the soft X-ray emission on a statistical basis. Models where the soft emission is partly produced by blurred reflection, or Comptonisation of the thermal disc emission, or simply a steep power law, all provide good fits to the data. Our reflection model differs substantially from that presented by Bhattacharyya et al. (2013). While we confirm that their model is a good fit to the spectrum, we note that it relies on a high inclination angle of i = 74 • , which is inconsistent with the blazar-like properties of the source, as well as the inclusion of a warm absorber, which is not required in any of the other models. Furthermore, their model does not allow the power law, which illuminates the disc, to also contribute directly to the spectrum. A blackbody model also gives a comparable fit, but a temperature of kT = 0.18 keV is necessary. Such a high temperature is inconsistent with the standard accretion disc theory (see e.g., Shakura & Sunyaev 1973). Further long-duration X-ray observations with XMM-Newton, Suzaku and NuS-TAR, preferably when the source is in a low γ-ray state, will be needed to place stronger constraints on these models. Our interpretation that the X-ray emission above 2 keV is produced by the jet is also supported by the RMS spectrum (Fig. 5), which shows a break at 1.7 keV, above which the variability increases with energy. While Seyfert galaxies typically exhibit RMS spectra decreasing with energy above ∼ 1 − 2 keV (Markowitz et al. 2004;Ponti et al. 2007;Chitnis et al. 2009), the opposite behaviour is often observed in blazars (e.g., Ravasio et al. 2004;Gliozzi et al. 2006). It should be noted, however, that there is no one-to-one correlation between the RMS spectrum and AGN type and that RMS spectra often change substantially with time (as seen in e.g. Gliozzi et al. 2006;Larsson et al. 2008). The spectrum of PMN J0948+0022 changed from a steep slope of Γ ∼2.1 to a much harder one of Γ ∼1.5 above 1.7 keV. A similar spectrum has also been observed in the XMM observation of the γ-ray NLSy1 PKS 2004-447 ). An X-ray spectrum unusually hard for a NLSy1 was observed by Swift-XRT in SBS 0846+513 (D'Ammando et al. 2012a(D'Ammando et al. , 2013b and PKS 1502+036 (D'Ammando et al. 2013a), two other NLSy1s detected by LAT. Models more complex than the simple power law were not applicable in these cases due to insufficient statistics, in particular below 1-2 keV. Thus the 0.3-10 keV spectra collected by Swift-XRT seem to be dominated by the jet emission. The small variability amplitude (∼2) observed in Xrays with respect to the γ rays (∼10) could be an indication that the X-ray emission is produced by the low-energy tail of the same electron distribution. On the other hand, the peak flux in X-rays was observed on 2011 July 2 during a period of low γ-ray activity, suggesting that different mechanisms could be at work in the X-ray and γ-ray bands (e.g., synchrotron self-Compton and external Compton, respectively). The presence of the soft X-ray excess below 2 keV could dilute the X-ray variability over the 0.3-10 keV energy range, but an amplitude variability of a factor of ∼2 (with fluxes between 2.1-4.1×10 −12 erg cm −2 s −1 ) was also observed considering only the 2-10 keV energy range. In this context no obvious relation exists between the soft X-ray excess and the γ-ray emission. An intriguing possibility is that the excess observed below 2 keV is a signature of the bulk Comptonisation process by a cold relativistic plasma accelerating along the jet and scattering on disc photons reprocessed by the broad-line region (BLR) (Celotti et al. 2007). A dedicated modeling of the source's SED including XMM-Newton and Fermi-LAT data will be presented in a forthcoming paper. Host galaxy The discovery of a relativistic jet in a class of AGN thought to be hosted in spiral galaxies such as the NLSy1s, as opposed to blazars and radio galaxies hosted in elliptical galaxies (Blandford & Rees 1978), was a great surprise challenging the current knowledge on how the jet structures are generated and developed (see e.g., Böttcher & Dermer 2002;Marscher 2010). Unfortunately only very sparse observations of the host galaxies of radio-loud NLSy1s are available and the sample of objects studied by Deo et al. (2006) and Zhou et al. (2006) have redshifts z < 0.03 and z < 0.1, respectively, while four out five of the NLSy1s detected in γ rays have z > 0.2. Among the radio-loud NLSy1s detected up to now by LAT only for the closest one, 1H 0323+342, was the host galaxy clearly detected. Observations with the Hubble Space Telescope and Nordic Optical Telescope revealed a one-armed galaxy morphology or a circumnuclear ring, respectively, suggesting two possibilities: the spiral arm of the host galaxy (Zhou et al. 2007) or the residual of a galaxy merger (Anton et al. 2008). These observations, together with the lack of information about the host galaxy of the other γ-ray emitting NLSy1s, leaves room for the hypothesis that the NLSy1s detected in γ rays by LAT could have peculiar host galaxies with respect to the other NLSy1s. Therefore the possibility that the development of relativistic jets in these objects occurs in hosts undergoing strong merger activity, or with non-spiral morphology, cannot be ruled out. Further high-resolution observations of the host galaxies of PMN J0948+0022 and the other γ-ray NLSy1s will be fundamental to obtain insights into the onset of production of relativistic jets in these sources. BH mass and jet formation The mechanism for producing a relativistic jet is still unclear. In particular the physical parameters that drive the jet formation are still under debate. One of the key parameters should be the BH mass, with only large masses allowing an efficient jet formation (see e.g., Sikora et al. 2007). Therefore one of the most surprising facts related to the discovery of PMN J0948+0022 was the development of a relativistic jet in an object with a relatively small BH mass, 3.2×10 7 M⊙ (Yuan et al. 2008). Recently, Chiaberge & Marconi (2011) suggested that a BH mass higher than 10 8 M⊙ is necessary for producing a radio-loud AGN and that the merger history together with the subsequent galaxy morphology plays a fundamental role. In any case, the estimated mass of this source, as well as for the other NLSy1s, has large uncertainties. By means of the broad band SED modeling a BH mass of 1.5×10 8 M⊙ was estimated for PMN J0948+0022 in Foschini et al. (2011). Marconi et al. (2008) suggested that BLR clouds are subjected to radiation pressure from the absorption of ionizing photons, and by applying a correction to the virial relation we have higher masses for the NLSy1s. Recently, also Calderone et al. (2013) pointed out that the BH masses of the NLSy1s estimated by the modelling of optical/UV data with a Shakura & Sunyaev disc spectrum could be significantly higher than those derived on the basis of single epoch virial methods. In particular, for PMN J0948+0022 they found a value of 10 9 M⊙ in agreement with the typical BH mass of blazars. This may solve the problem of the minimum BH mass predicted in different scenarios of relativistic jet formation and development, but introduces a possible new one. For spiral galaxies, the BH mass typically ranges between 10 6 and 10 8 M⊙ (see e.g., Woo & Urry 2002). If the BH mass is on the larger side of the estimated values, how is it possible to reconcile such a large BH mass with a spiral galaxy? A second fundamental parameter for the efficiency of relativistic jet production should be the BH spin, with super-massive black holes (SMBH) in elliptical galaxies having on average much larger spins than SMBHs in discspiral galaxies, as proposed in the "modified spin paradigm" (Sikora et al. 2007). This is because the spin evolution of BHs in spiral galaxies seems to be limited by multiple accretion events with random orientation of the angular momentum vectors and small increments of mass, while elliptical galaxies underwent at least one major merger with large matter accretion triggering an efficient spin-up of the SMBHs. Thus, the mass and the spin of the BH seem to be related to the host galaxies, leading to the hypothesis that relativistic jets can efficiently develop only in elliptical galaxy (e.g. Böttcher & Dermer 2002;Marscher 2010). However, the presence of a rapidly spinning BH was inferred by means of X-ray reflection spectroscopy in a few radio-quiet AGNs hosted by spiral/disc galaxies, suggesting that the BH spin is not the only parameter that drives the radioquiet/radio-loud dichotomy (see e.g., Reynolds 2013). We noted that BH masses of radio-loud NLSy1s reported in Komossa et al. (2006) and Yuan et al. (2008) are generally larger than those in the entire sample of NLSy1s (MBH ≈(2-10)×10 7 M⊙), even if still smaller than those in radio-loud quasars. The larger BH masses of radio-loud NLSy1s could be related to higher mass accretion events that can spin up the BHs. In the same way the smaller fraction of radio-loud NLSy1s with respect to the radio-loud quasars could be because the high accretion rate regime does not last sufficiently long in all NLSy1s to substantially spin up the central BH (Sikora 2009). Another consideration which is likely to be important for jet formation is the nature of the accretion flow. In particular, a geometrically thick accretion flow is needed in order to create large-scale poloidal magnetic fields, which may play a dominant role in the launching of jets (Reynolds et al. 2006;Sikora et al. 2013). In cases where standard thin discs are present, the jet activity may be due to the dissipation of coronal magnetic fields (Sikora et al. 2013). For PMN J0948+0022 it is clear that emission from the jet dominates the X-ray spectrum above 2 keV, while we are likely seeing the accretion disc plus corona of the AGN at lower X-ray energies. Although we cannot constrain detailed models for the soft X-ray emission with the current observations, we note that the spectral slope is similar to that found in radio-quiet NLSy1, indicating that a standard disc is present as also expected from the high accretion rate. Future deep X-ray observations with the jet in different states are needed to explore in detail the connection between the disc and jet in this source. The presence of a relativistic jet in some radio-loud NLSy1 galaxies, first suggested by their variable radio emission and flat spectra, is now confirmed by the Fermi-LAT detection of five NLSy1s in γ rays. PMN J0948+0022 showed all characteristics of the blazar phenomenon with a BH mass of 10 8 -10 9 M⊙, not much less than those of blazars. The impact on the γ-ray emission mechanisms of the properties of the central engine in radio-loud NLSy1s, derived from their peculiar optical characteristics, is still under debate. In addition, the detection of relativistic jets in a class of AGN thought to be hosted in spiral galaxies is very intriguing, challenging the theoretical scenario of relativistic jet formation proposed so far. Further multifrequency observations of this object and other γ-ray emitting NLSy1s will be fundamental for investigating in detail their characteristics over the entire electromagnetic spectrum. Figure 1 . 1Integrated flux light curve of PMN J0948+0022 obtained by Fermi-LAT in the 0.1-100 GeV energy range during 2011 May 1 -September 30 (MJD 55682-55835) with 7-day time bins. Arrows refer to 2σ upper limits on the source flux. Upper limits are computed when T S < 10. Open circles represent daily fluxes reported for the periods of high activity. for a total duration of 93 ks (observation ID 067370101, PI: D'Ammando). A simultaneous observation was performed by Swift on 2011 May 28. Figure 2 . 2XMM-Newton EPIC pn (black), MOS1 (red) and MOS2 (green) data shown as a ratio to a power law with Γ = 1.48. Figure 3 . 3Components of the best-fitting models from Figure 4 . 40.2-10 keV EPIC pn light curve on a 200 s time scale, using only fully exposed bins. The time periods used for the spectral analysis are marked in red. The remaining time intervals were severely affected by background flaring. Figure 5 . 5RMS variability spectrum of the XMM-Newton observation of PMN J0948+0022, calculated using 0.5 ks bins. Figure 6 . 6Results of fits to time-resolved spectra of the XMM-Newton observation using a broken power law model. The top panel shows the temporal evolution of the photon indices (the indices below and above the break energy are shown as filled circles and filled triangles, respectively), the middle panel shows the break energy, and the bottom panel shows the 0.3 − 10 keV unabsorbed flux derived from the fits. Optical Monitor (OM; Mason et al. 2011) on board XMM-Newton is a 30 cm telescope carrying six optical/UV filters, and two grisms. Observations of PMN J0948+0022 in 2011 May 28-29 consisted of seven subsequent exposuresin v-band, followed by ten in b-band, nine in u-band, ten in w1-band and m2-band, then five in w2-band. All optical exposures were 800 s long, while UV exposures were 1600 s and 2700 s long. We used the SAS task omichain to reduce the data and the tasks omsource and omphotom to derive the source magnitude. Average observed magnitudes are: v = 18.28±0.06, b = 18.63±0.02, u = 17.64±0.02, w1 = 17.27±0.02, m2 = 17.24±0.03, and w2 = 17.22±0.11. The difference of 0.2-0.3 mag in the optical filters with respect to the Swift-UVOT observations performed on 2011 May 28 at least partially could be due to the source variability. Figure 7 . 7Multifrequency light curve of PMN J0948+0022 for the period 2011 May 1-September 30 (MJD 55682-55834) collected (from top to bottom) in: γ rays by Fermi-LAT (0.1-100 GeV; in units of 10 −8 ph cm −2 s −1 ); X-rays by Swift-XRT (filled circles) and XMM-Newton (open square) (0.3-10 keV; in units of 10 −12 erg cm −2 s −1 ); w2 band by Swift-UVOT (filled circles) and XMM-OM (open square); u band by Swift-UVOT (filled circles) and XMM-OM (open square); V band by Swift-UVOT (filled circles), XMM-OM (open square) and CRTS (open pentagons), and R band taken from Eggen et al. (2013) (filled triangles); J (open squares) and H (filled circles) bands by INAOE; 37 GHz by Metsähovi (filled circles) and 32 GHz by Effelsberg (open squares); 15 GHz by OVRO (filled circles) and Effelsberg (open squares). The flux densities collected from w2 to 15 GHz are reported in units of mJy. In the top panel daily integrated γ-ray fluxes are reported as open circles. . The peak of the flux density was observed by Medicina first at 8.4 GHz on 2011 August 1 (MJD 55774), and then at 5 GHz on August 10 (MJD 55783), about 7-8 weeks after the peak observed at 15 GHz by OVRO. Using the Medicina data at 5 GHz and 8.4 GHz together with the nearest OVRO observation spectral indices of −0.54±0.04, −0.36±0.03, and −0.04±0.06 were measured for 2011 June 12, August 1, and August 10, respectively, suggesting a radio spectral evolution in agreement with the behaviour observed by Effelsberg. Figure 8 . 8Radio spectra of PMN J0948+0022 collected from 2.64 and 32 GHz by Effelsberg in 5 epochs: 2011 May 24 (MJD 55705; empty circles), 2011 June 5 (MJD 55717; empty triangles), 2011 August 6 (MJD 55779; empty squares), 2011 September 17 (MJD 55821; filled circles), and 2011 October 1 (MJD 55835; filled stars). Figure 9 .Figure 10 . 910VLBA image at 15.3 GHz of PMN J0948+0022 collected on 2011 May 26. On the image we provide the restoring beam, plotted in the bottom-left corner, the peak flux density in mJy/beam, and the first contour (f.c.) intensity in mJy/beam, which is three times the off-source noise level. Contour levels increase by a factor of 2. The vectors superimposed on the total intensity contours show the percentage polarization and the position angle of the electric vector.GHz, a clear decrease of the flux density was observed by MOJAVE from 657 mJy on 2011 May 26 (MJD 55707) to 378 mJy on 2011 December 12 (MJD 55907). In addition, a higher polarized emission (S pol ) and polarisation percentage was observed on May 26 with respect to December 12 (Table 6). On the other hand, the electric vector position angle (EVPA) of the core does not change significantly, ranging between 25 • and 67 • . Center at RA 09 48 57.32008896 DEC 00 22 25.5593100 ICONT:0946+006 IPOL 15357.490 MHZ 0948 SEP11.ICL001.1 PLot file version 1 created 29-FEB-2012 17:28:46 Cont peak flux = 4.5351E-01 JY/BEAM Levs = 1.764E-04 * (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) Pol line 1 milli arcsec = 4.7619E-03 JY/VLBA image at 15.3 GHz of PMN J0948+0022 collected on 2011 September 12. ( 0 . 3 - 0310 keV), UV (w2 filter), optical (V , R, and u filters), NIR (J and H filters), and radio (37 GHz and 15 GHz) light curves collected by Swift, XMM, CRTS, INAOE, Metsähovi, Effelsberg, and OVRO. Strong variability was observed in γ rays, with two flaring periods peaked on 2011 June 20 and July 28 and a variability amplitude (calculated as the ratio of maximum to minimum flux) of ∼10. Such a variability amplitude as well as the rapid flaring episodes and high apparent isotropic γ-ray luminosity (∼10 48 erg s −1 at the peak) observed in 2011 are typical of FSRQs. During the Swift observations, PMN J0948+0022 was observed in an intermediate X-ray state (0.3-10 keV flux of 3.5-5.9×10 −12 erg cm −2 s −1 ) between the highest flux observed from this source on 2012 December 30 dur-ing a γ-ray flaring activity (12.6×10 −12 erg cm −2 s −1 ; see D'Ammando & Orienti 2013) and the low flux observed on 2008 December 8 (2.2×10 −12 erg cm −2 s −1 ; Abdo et al. 2009a). A clear increase of the flux was observed between 2011 Table 1 . 1Log and fitting results of Swift-XRT observations of PMN J0948+0022 using an absorbed power law model with an absorbing column density of N H = 5.07×10 20 cm −2 .Date Date Net exposure time Photon index Flux 0.3-10 keV a (UT) (MJD) (sec) (Γ) (×10 −12 erg cm −2 s −1 ) 2011-Apr-29 55680 1978 1.81 ± 0.19 4.3 ± 0.4 2011-May-15 55696 4657 1.75 ± 0.15 4.5 ± 0.3 2011-May-28 55709 3629 1.80 ± 0.16 4.8 ± 0.5 2011-June-04 55716 2020 1.76 ± 0.20 3.5 ± 0.4 2011-June-14 54726 5160 1.65 ± 0.16 3.7 ± 0.3 2011-July-02 54744 2008 1.44 ± 0.17 5.9 ± 0.6 a Unabsorbed flux Table 2 . 2Results of the Swift-UVOT observations of PMN J0948+0022.Date (UT) Date (MJD) v b u w1 m2 w2 2011-Apr-29 55680 - - 17.27±0.03 - - - 2011-May-15 55696 18.20±0.13 18.43±0.08 17.67±0.07 17.46±0.05 17.44±0.06 17.46±0.04 2011-May-28 55709 18.05±0.15 18.31±0.09 17.48±0.07 17.31±0.05 17.35±0.06 17.33±0.04 2011-June-04 55716 - - 17.11±0.03 - - - 2011-June-14 55726 17.86±0.11 18.31±0.08 17.34±0.06 17.34±0.05 17.35±0.05 17.29±0.04 2011-July-02 55744 - - 17.64±0.30 - - - 1 10 0.5 2 5 1 1.5 2 2.5 Ratio Energy (keV) Table 3 3together with the residuals of the fits. Left: Comptonised blackbody (blue, dashed line), power law (red, dotted line) and their sum (black, solid line). Right: Steep power law (blue, dashed line), its associated reflection spectrum (green, dashed-dotted line), hard power law (red, dotted line) and their sum (black, solid line). In the lower panels the black, red, and green points represent data from pn, MOS1 and MOS 2, respectively.0 2×10 4 4×10 4 6×10 4 8×10 4 0.5 1 1.5 2 Count rate (s −1 ) Time (s) Table 4 . 4Results of the INAOE observations of PMN J0948+0022in J, H, and K bands. AZ, and an unfiltered CCD. The typical cadence is four exposures separated by 10 min in a given night; this may be repeated up to four times per lunation, over a period of ∼6-7 months each year, while the field is observable. Photometry is obtained using the stanand Ks filters are reported inTable 4. The flux densities collected in J and H band are reported also inFig. 7.Date J H Ks (MJD) (mJy) (mJy) (mJy) 55677.219 0.864 ± 0.026 1.248 ± 0.075 1.234 ± 0.136 55689.243 - 0.721 ± 0.065 - 55693.171 - 0.400 ± 0.020 - 55694.139 0.469 ± 0.028 0.601 ± 0.024 0.929 ± 0.084 55696.181 - 0.380 ± 0.030 55702.163 0.316 ± 0.016 0.331 ± 0.020 0.762 ± 0.084 55703.139 0.866 ± 0.069 1.530 ± 0.138 5 GROUND-BASED OPTICAL AND INFRARED OBSERVATIONS 5.1 CRTS The source has been monitored by the Catalina Real- time Transient Survey (CRTS) 2 (Drake et al. 2009; Djorgovski et al. 2011), using the 0.68 m Schmidt telescope at Catalina Station, dard Source-Extractor package (Bertin & Arnouts 1996), and transformed from the unfiltered instrumental magni- tude to Cousins V by V = VCSS + 0.31(B − V ) 2 + 0.04 with a scatter of 0.056 mag 3 . The flux densities collected by CRTS in V band are reported in Fig. 7. 5.2 INAOE NIR observations of PMN J0948+0022 were performed dur- ing 2011 April-May, at the 2.1 m telescope "Guillermo Haro", with the NIR camera "CANICA" equipped with a Rockwell 1024 × 1024 pixel Hawaii infrared array, working at 75.4 K, with standard J(1.164-1.328 µm), H(1.485-1.781 µm), and Ks (1.944-2.294 µm) filters. The plate scale is 0.32 arc- sec pixel −1 . Observations were carried out in series of 10 dithered frames in each filter. Data sets were co-added after correcting for bias and flat fielding. Flats were obtained from sky frames derived from the dithered ones. Magnitudes in J, H, 6 RADIO OBSERVATIONS 6.1 Effelsberg 100 m The radio spectrum of PMN J0948+0022 was observed with the Effelsberg 100 m telescope between 2011 May 24 and October 1 within the framework of a Fermi-related moni- toring programme of γ-ray blazars (F-GAMMA programme; Fuhrmann et al. 2007). The measurements were conducted with the secondary focus heterodyne receivers at 2.64, 4.85, 8.35, 10.45, 14.60, 23.05, and 32.00 GHz. The observations were performed quasi-simultaneously with cross-scans, that is, slewing over the source position, in azimuth and elevation directions, with adaptive numbers of sub-scans for reaching the desired sensitivity (for details, see Fuhrmann et al. 2008; Table 5 . 5Resultsof the Medicina radio observations at 5 GHz and 8.4 GHz of PMN J0948+0022. Date Date S 5 GHz S 8.4 GHz (UT) (MJD) (mJy) (mJy) 2011-June-12 55724 0.37 ± 0.02 0.38 ± 0.02 2011-July-29 55771 - 0.35 ± 0.02 2011-Aug-01 55774 0.35 ± 0.02 0.45 ± 0.02 2011-Aug-10 55783 0.45 ± 0.05 0.39 ± 0.05 2011-Sep-08 55812 0.35 ± 0.02 - 2011-Sep-22 55826 0.34 ± 0.02 - 2011-Oct-13 55847 0.33 ± 0.05 - 2011-Nov-16 55881 0.25 ± 0.05 - Table 6 . 6Flux density and polarisation of PMN J0948+0022 from 15 GHz MOJAVE data.Date Date S Core S Jet S pol EVPA (UT) (MJD) (mJy) (mJy) (mJy) (%) (deg) 2011-02-20 55612 622 3 5 (0.8%) 25 2011-05-26 55707 659 6 13 (2.0%) 51 2011-06-24 55736 665 6 14 (2.1%) 49 2011-09-12 55816 458 - 6 (1.3%) 44 2011-12-12 55907 380 - 2 (0.5%) 67 http://fermi.gsfc.nasa.gov/ssc/data/access/lat/Background Models.html http://crts.caltech.edu 3 http://nesssi.cacr.caltech.edu/DataRelease/FAQ2.html#improve ACKNOWLEDGMENTSThe Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.We thank the Swift team for making these observations possible, the duty scientists, and science planners. The OVRO 40 m monitoring program is supported in part by NASA grants NNX08AW31G and NNX11A043G, and NSF grants AST-0808050 and AST-1109911. This paper is partly based on observations with the 100 m telescope of the (numbers 212656, 210338, 121148, and others). This research made use of data from MOJAVE database that is maintained by the MOJAVE team(Lister et al. 2009). This work is based on observations obtained with XMM-Newton, an ESA science mission with intrument and contributions directly funded by ESA Member States and the USA (NASA). JL acknowledges financial support from the Swedish National Space Board. FD thanks A. Breeveld and P. Roming for useful discussion about OM and UVOT cross-calibration. JL thanks Andy Fabian for useful discussion. We thank the anonymous referee, S. Cutini, S. Digel, and D. Thompson for useful comments and suggestions. . A A Abdo, ApJ. 699976Abdo, A. A., et al. 2009a, ApJ, 699, 976 . A A Abdo, ApJ. 707142Abdo, A. A., et al. 2009b, ApJ, 707, L142 . 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[ "A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions", "A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions" ]	[ "Richard C Kraaij ", "I Frank ", "Redig Ii ", "Willem B Van ", "Zuijlen Iii " ]	[]	[]	We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions.The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time dependent rate function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential.We hereby create a unifying framework for the treatment of mean-field Gibbsnon-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations.In this paper, as the initial model at time t = 0 we consider two mean-field models. In the next two sections we will describe the dynamics to which this initial model will be subjected. Namely, we consider one with spins that attain values in R, which we refer to Lemma A.4. Let H satisfy Assumption 2.10(a) for the R-space-model, then H satisfies Assumption 4.1. Proof. The proof follows the same lines as the proof of Lemma A.1 using θ(r) = c max{\|r\| 2 , 1}. The calculations in this setting are significantly easier. Lemma A.5. Let H satisfy Assumption 2.10(a) for the R-space-model, then H satisfies Assumption 4.11.Proof. By [CK17, Lemma 3.4], the one-sided Lipschitz property of W implies that Υ(x) = log(1 + 1 2 x 2 ) is appropriate.	10.1090/tran/8408	[ "https://arxiv.org/pdf/1711.03489v1.pdf" ]	54,810,842	1711.03489	cefbe4828927a13a73684f8d5420e45e4370c345	A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions September 17, 2018 Richard C Kraaij I Frank Redig Ii Willem B Van Zuijlen Iii A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions September 17, 2018AMS 2010 subject classification: 49L9960F1082C2282C27 Keywords: Hamiltonian dynamicsHamilton-Jacobi equationMean-field modelslarge deviation principleGibbs versus non-Gibbsdynamical transitionglobal minimisers of rate functions We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions.The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time dependent rate function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential.We hereby create a unifying framework for the treatment of mean-field Gibbsnon-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations.In this paper, as the initial model at time t = 0 we consider two mean-field models. In the next two sections we will describe the dynamics to which this initial model will be subjected. Namely, we consider one with spins that attain values in R, which we refer to Lemma A.4. Let H satisfy Assumption 2.10(a) for the R-space-model, then H satisfies Assumption 4.1. Proof. The proof follows the same lines as the proof of Lemma A.1 using θ(r) = c max{\|r\| 2 , 1}. The calculations in this setting are significantly easier. Lemma A.5. Let H satisfy Assumption 2.10(a) for the R-space-model, then H satisfies Assumption 4.11.Proof. By [CK17, Lemma 3.4], the one-sided Lipschitz property of W implies that Υ(x) = log(1 + 1 2 x 2 ) is appropriate. Introduction The large deviation approach to dynamical Gibbs-non-Gibbs transitions, initiated in van Enter, Fernández, den Hollander and Redig [Ent+10] characterizes the emergence of 'bad configurations' via the non-uniqueness of optimal starting configurations corresponding to a given arrival configuration. 'Bad configurations' have to be interpreted as points of essential discontinuity of conditional probabilities and 'optimal' has to be interpreted here in the sense of minimizing a large deviation cost, I(γ) = I 0 (γ(0)) + ∞ 0 L (γ(s),γ(s)) ds, (1) which is the sum of an initial cost I 0 corresponding to the starting measure and a pathspace cost in the form of a Lagrangian action. In the mean-field context one considers trajectories of the magnetization and the dynamical Gibbs-non-Gibbs transitions are to be interpreted in the sense of Gibbsianness for mean-field models, a notion introduced in Külske and le Ny [KLN07] and studied in [EK10;FHM13;HRZ15]. It is widely believed that for a large variety of models the following three statements are equivalent: (a) Mean-field Gibbsianness at time t; (b) Unique optimal trajectories: for all arrival points b at time t the optimal trajectory arriving at b is unique, i.e., (c) Differentiability of the rate function at time t, i.e., I t given by I t (b) := inf γ,γ(t)=b I 0 (γ(0)) + t 0 L (γ(s),γ(s))ds,(2) is differentiable as a function of b. In this paper, we prove, for a broad class of models in the one-dimensional setting, the equivalence of (b) and (c) and introduce methods to investigate the differentiability of the rate function at time t. The proofs are based on techniques from the theory of calculus of variations. By using this general approach, we do not use specific information of the considered models and therefore our methods are applicable in a large variety of models . This in contrast to [EK10;FHM13;HRZ15;KLN07], which approaches rely on explicit information about the specific models, in which they treat the relation between (a) and (b) . Let H(x, p) = sup v pv −L (x, v) be the Hamiltonian corresponding to the Lagrangian in (1). Following classical mechanics, if the characteristics of the Hamilton-Jacobi equation ∂ t u(t, x) + H(x, ∂ x u(t, x)) = 0, u(0, x) = I 0 (x),(3) do not intersect, then u(t, x) := I t (x), with I t defined in (2), is continuously differentiable and a classical solution of (3). Following the theory of calculus of variations, even if the characteristics intersect; then u(t, x) = I t (x) still solves (3) as a viscosity solution. In addition, one can show that u is locally semi-concave. The characteristics of (3) are exactly the Hamilton trajectories, i.e., they solve the Hamilton equations Ẋ (s) P (s) = ∂ p H(X(s), P (s)) −∂ x H(X(s), P (s)) . (4) Moreover, every optimal trajectory as in (b) above has an associated characteristic. A rigorous analysis shows that these observations can be turned into a proof that (b) is equivalent to (c). In addition, a study of solutions to (4) can be used to prove or disprove that I t is differentiable. A major difficulty that we overcome in this paper is that the mean-field models generally considered have Hamiltonians for which the solutions to (4) have a finite time of existence. In comparison to e.g. [CS04], this introduces various new problems that we solve. These problems are not merely technical: they are responsible for the notion of 'recovery of Gibbsianness' in the sense of 'recovery of differentiability' (which for measures on the lattice has been shown in [Ent+02] and for infinite temperature meanfield dynamics in [FHM13]). We proceed with giving three techniques based on the analysis of Hamiltonian flows that either guarantee differentiability or non-differentiability of I t : order preservingness, linearization and rotation. We illustrate these methods to the natural examples of Glauber dynamics for the Curie-Weiss model and that of mean-field interacting Brownian particles in a single or double well-potential Overview The rest of our paper is organized as follows. In Section 2 we introduce Gibbs-non-Gibbs transitions, path-space large deviations, examples of models giving rise to Hamiltonians that fall within our framework and some definitions and preleminaries from the theory of calculus of variations. Then we give our main result on the equivalence between (b) and (c), the relation between uniqueness of optimisers and the regularity of the rate function and finally we present a relation between this regularity and the push-forward of the graph of the derivative of the initial rate function. In Section 3 we prescribe conditions for which the regularity is preserved or broken and apply this to obtain different scenarios for the models introduced in Section 2. One of the scenarios treated here it the one of recovery, as mentioned before. Techniques and proofs for theorems of Section 2 can be found in Section 4, 5 and 6. Techniques and proofs for theorems of Section 3 can be found in Section 7, 8 and 9. as the R-space-model, and one with spins that attain values in {−1, 1}, which we refer to as the ±1-space-model. We will write K for the space in which empirical averages will take their values, in particular we have that K equals R for the R-space-model, and [−1, 1] for the ±1-space-model. We start in both cases from an initial measure µ N,0 of the form µ N,0 (dσ 1 , · · · , dσ N ) = e −N V (m N (σ 1 ,...,σ N )) Z N λ N (dσ 1 , · · · , dσ N ), where m N (σ 1 , . . . , σ N ) = 1 N N i=1 σ i ,(6) λ N is the N -fold product of λ, Z N the normalizing constant, (i) for the R-space-model, V : R → [0, ∞) is continuous and λ is a standard normal distribution on R. (ii) for the ±1-space-model, V : [−1, 1] → R is continuous and λ is the uniform measure on {−1, 1}. The "potential" V determines in both cases uniquely the rate function for the large deviation principle of the magnetization m N under µ N,0 , which is the function x → V (x) + i(x) − inf x∈K (V (x) + i(x)),(7) where (i) for the R-space-model, i(x) = 1 2 x 2 , (ii) for the ±1-space-model, i(x) = 1−x 2 log(1 − x) + 1+x 2 log(1 + x). We consider the spins to evolve according to the following dynamics (i) for the R-space-model; interacting diffusions as described in Section 2.3. (ii) for the ±1-space-model; Glauber dynamics as described in Section 2.2. The initial measure µ N,0 is transformed by the dynamics to the measure µ N,t at time t > 0. Definition 2.1. Let t ≥ 0. α ∈ K is called a good magnetization for (µ N,t ) N ∈N if there exists a probability measure γ t (·\|α) such that 1 µ N,t (dσ 1 \|σ N 2 , . . . , σ N N ) weakly − −−− → γ t (dσ 1 \|α),(8) for all σ N 2 , . . . , σ N N such that m N −1 (σ N 2 , . . . , σ N N ) → α. If α is not a good magnetization, it is call a bad magnetization. The sequence (µ N,t ) N ∈N is called sequentially Gibbs if α is a good magnetisation for all α ∈ K. If (µ n,t ) N ∈N is sequentially Gibbs, then α → γ t (·\|α) is weakly continuous (see [Zui16,Theorem 3. A.1] or [HRZ15, Lemma 1.3]). Remark 2.2. The definition of sequentially Gibbs follows those in [KLN07], [EK10], [FHM13], [HRZ15]. We refer to [KLN07] for the explanation of the definition with regards to Gibbs measures on the lattice. Glauber dynamics In this section, we describe the dynamics for the ±1-space-model. For each N , we consider a continuous-time Markov process (X 1 (t), · · · , X N (t)) ∈ {−1, 1} N of mean-field interacting spins with mean-field jump rates c N . The law of (X 1 (0), · · · , X N (0)) is µ N,0 and the Markov generator of these spin-flip systems are of the form A N f (σ 1 , . . . , σ N ) := N i=1 c N (σ i , m N (σ)) f (σ i ) − f (σ) , where c N ≥ 0 and where the configuration σ i ∈ {−1, 1} N is given by σ i j = −σ j if i = j, σ j if i = j. We denote by M N (t) := 1 N N i=1 X i (t) the empirical magnetization at time t. Due to the mean-field character of this dynamics, also the dynamics of the empirical magnetization is Markovian, and an elementary computation shows that the generator of the process (M N (t)) t≥0 on m N ({−1, 1} N ) ⊆ [−1, 1] is given by A N f (x) = N 1 − x 2 c N (−1, x) f (x + 2N −1 ) − f (x) + N 1 + x 2 c N (+1, x) f (x − 2N −1 ) − f (x) ,(9)as it satisfies A N (f • m N ) = (A N f ) • m N . For later purposes, we assume the following assumptions. Assumption 2.3. There exist functions v + , v − : [−1, 1] → [0, ∞) such that lim N →∞ sup x∈m N ({−1,1} N ) 1 − x 2 c N (−1, x) − v + (x) + 1 + x 2 c N (+1, x) − v − (x) = 0,(10) for which the following properties hold: (a) v − (−1) = 0, v − (x) > 0 for x = 1, and v + (1) = 0 and v + (x) > 0 for x = 1, (b) v − , v + have(c) v + (1) < 0 and v − (−1) > 0. In concrete examples, we consider v − , v + of the form v − (x) = 1 + x 2 e −βx−h , v + (x) = 1 − x 2 e βx+h ,(11) which correspond to the rates obtained from Glauber spin-flip dynamics reversible with respect to the Curie-Weiss measure in (5) at inverse temperature β ≥ 0 and external magnetic field h ∈ R, i.e., for V (x) = −βx 2 − hx . Interacting diffusion processes In this section, we describe the dynamics for the R-space-model. For each N , we consider N mean-field interacting diffusions (X 1 (t), . . . , X N (t)) ∈ R N in a potential landscape W N : R → R, where W N is continuously differentiable. We assume that −W N is one- sided Lipschitz: there is some M ≥ 0 such that for all x > y −(W N (x) − W N (y)) ≤ M (x − y). The law of (X 1 (0), . . . , X N (0)) is given by µ N,0 and the dynamics are given by dX i (t) = −W N (M N (t))dt + dB i (t) where M N (t) := 1 N N i=1 X i (t) is the empirical magnetization at time t as above and where B 1 , . . . , B N are independent standard Brownian motions. Note that there exists a unique solution to this stochastic differential equation by [PR14,Proposition 3.38] and the one-sided Lipschitz property of −W N . The empirical magnetization is also Markovian and satisfies dM N (t) = −W N (M N (t))dt + 1 √ N dB 1 (t). Again by [PR14,Proposition 3.38] this equation has a unique solution, and additionally, its generator A N with domain C 2 b (R) is given by A N f (x) = −W N (x)f (x) + 1 2N f (x). Also in this case we assume to have the following assumptions. Assumption 2.4. We assume that there is some three times continuously differentiable function W : R → R for which −W is one-sided Lipschitz and such that for every compact set K ⊆ R lim N →∞ sup x∈K W N (x) − W (x) = 0.(12) W (x) = 2k i=1 a i x i with a i ∈ R and a 2k > 0 is an example of such a function for which −W is one-sided Lipschitz. In the examples that we will consider, we will use W (x) = 1 4 x 4 − 1 2 dx 2 with d ∈ R. This function is strictly convex for d ≤ 0 ('high temperature') and has a double well for d > 0 ('low temperature'). Path-space large deviations In various works, see e.g. [CK17; Com89; DPH96; FK06; FW98; Kra16b; Léo95], it has been shown that if the initial magnetization M N (0) satisfies a large deviation principle with rate function I 0 , then the Markov process t → M N (t) satisfies the large deviation principle on 2 D K ([0, ∞)), i.e., P [(M N (t)) t≥0 ≈ γ] ≈ e −N I (γ) , with rate function I (γ) = I 0 (γ(0)) + ∞ 0 L (γ(s),γ(s))ds if γ ∈ A C , ∞ otherwise.(13) Here A C is the space of absolutely continuous trajectories γ : [0, ∞) → K and L : K × R → [0, ∞] is the Lagrangian obtained by taking the Legendre transform L (x, v) := sup p∈R (pv − H(x, p)) , of the Hamiltonian H : K × R → R given (i) for the R-space-model, by H(x, p) = 1 2 p 2 − pW (x).(14) (ii) for the ±1-space-model, by H(x, p) = v + (x) e 2p − 1 + v − (x) e −2p − 1 .(15) This Hamiltonian in turn is for example obtained by an operator approximation procedure introduced by Feng and Kurtz [FK06]. This procedure is explained informally in Redig and Wang [RW12] and rigorously for R d valued processes in Kraaij [Kra16b] and Collet and Kraaij [CK17]. H is derived from an operator H by the relation H(x, f (x)) = H f (x), where H satisfies H f = lim N →∞ H N f where H N is the opera- tor defined by H N f = N −1 e −N f A N e N f . For any two points a, b ∈ K and time t, denote by S t (a, b) = inf γ∈A C :γ(0)=a,γ(t)=b t 0 L (γ(s),γ(s))ds.(16) S t (a, b) is the minimal Lagrangian action of a trajectory starting at a and arriving at time t at b. By the contraction principle, the rate function for the large deviation principle for the magnetization M N (t) at time t > 0 is given by (for I see (13)) I t (b) = inf x∈K (I 0 (x) + S t (x, b)) = inf γ∈A C :γ(t)=b I (γ).(17) Definition 2.5. We call γ ∈ A C an optimal trajectory for S t (a, b) (see (16)) if γ(0) = a, γ(t) = b and t 0 L (γ(s),γ(s))ds = S t (a, b). Analogously, γ is called an optimal trajectory for I t (b) (see (17)) if γ(t) = b and I t (b) = I 0 (γ(0)) + ∞ 0 L (γ(s),γ(s))ds. Finally, x ∈ K is called a optimal starting point for I t (b) if I t (b) = I 0 (x) + S t (x, b). In the following definition we define another way to say that (c) of the conjecture in the introduction does not hold. Definition 2.6. We will say that α ∈ K • is a point of non-differentiability of I t , when I t is not differentiable at α. Preliminaries from the theory of calculus of variations We follow the route of studying the optimal trajectories and non-differentiabilities in the rate function, by introducing techniques from the theory of calculus of variations. The first observation from classical mechanics is that optimal trajectories are known to solve the second order Euler-Lagrange equation, which is the point of view taken in [EK10]. On equal footing is that dual variables satisfy the first-order Hamilton equations. Definition 2.7. Let t > 0. Let A be either one of the intervals [0, t], [0, t), (0, t] or (0, t). Let γ : A → K • be absolutely continuous. If (γ(s),γ(s)) is in the domain where L is C 1 for all s ∈ A, then the trajectory η defined by η(s) = ∂ v L (γ(s),γ(s)),(18) is called the dual trajectory to γ. Let γ ∈ C 1 (A, K) and η ∈ C 1 (A). We say that (γ, η) satisfies the Hamilton equations, if they solve γ(s) η(s) = ∂ p H(γ(s), η(s)) −∂ x H(γ(s), η(s)) .(19) If (γ, η) satisfies the Hamilton equations and γ ∈ C 1 (A, K), then η is the dual trajectory to γ (see [CS04,Corollary A.2.7, equation (A.28)]). Moreover, there exists a c ∈ R such that H(γ(s), η(s)) = c for all s ∈ A. In addition, we use the following definitions of Cannarsa and Sinestrari [CS04]. Let d ∈ N and A ⊆ R d be open. Let v : A → R. Superdifferential [CS04, Definition 3.1.1] The superdifferential of v at x ∈ A is defined as D + v(x) := {p ∈ R d : lim sup y→x v(y)−v(x)− p,y−x \|y−x\| ≤ 0}. Similarly, we define a subgradient D − v(x). If v is differentiable at x ∈ A, we write Dv(x) for the derivative of v at x. Note that in that case D + v(x) = D − v(x) = {Dv(x)}. Reachable gradient [CS04, Definition 3.1.10] Let v be locally Lipschitz. A p ∈ R d is called a reachable gradient of v at x if there exists a sequence (x k ) k∈N in A \ {x} such that v is differentiable at x k for all k and x k → x, Dv(x k ) → p. We write D * v(x) for the set of all reachable gradients of v at x. Viscosity solutions [CS04, Definition 5.2.1] Let F ∈ C(A × R × R d ). Consider the equation F (x, v, Dv) = 0. (20) v ∈ C(A) is called a viscosity subsolution to (20) if for all x ∈ B, we have F (x, v(x), p) ≤ 0, for all p ∈ D + v(x), u ∈ C(A) is called a viscosity supersolution to (20) if for all x ∈ B, we have F (x, v(x), p) ≥ 0, for all p ∈ D − v(x). v is called a viscosity solution to (20) it it is both a sub-and a supersolution. Local semi-concavity [CS04, Definition 1.1.1 and Proposition 1. 1.3] Let B be a sub- set of R d . Let K ⊆ B be compact. We say that v is semi-concave on K if there is some C ∈ R such that λv(x) + (1 − λ)v(y) − v (λx + (1 − λ)y) ≤ C λ(1 − λ) 2 \|x − y\| 2(21) for all x, y ∈ K such that the line from x to y is contained in K and for all λ ∈ [0, 1]. We call v locally semi-concave on B if it is semi-concave on each compact set K ⊆ B. Note, in [CS04] these functions are called semi-concave with linear modulus, to distinguish them from a broader class of semi-concave functions. We do not need this generality here, and therefore will call "semi-concave functions with linear modulus" simply "semi-concave functions". Remark 2.8. With Φ 2 V the second difference quotient of V (see [RS82, Section 1.2] for a definition), Φ 2 describes the convexity and concavity of a function in the sense that V is convex if and only if Φ 2 V ≥ 0 and V is concave if and only if Φ 2 V ≤ 0. But it also relates to semi-concavity, as one has that if V is continuous, then V is semi-concave with constant C > 0 if and only if Φ 2 V ≤ C. In [HRZ15] the Φ 2 V is used to describe the Gibbsiannity of the R-space-model with the dynamic of independent Brownian motions, see also Remark 3.13. Regularity of the rate-function In this section we establish the announced equivalence between the uniqueness of optimizing trajectories and differentiability of the rate function (Theorem 2.13). The main issue, setting our problems apart from the ones considered in [CS04], is that the maximal time of existence of solutions of the Hamilton equations, contrary to [CS04], may be finite. This causes certain divergence of the momentum at the boundary of K. To extend the techniques to our setting, we will work under the assumptions in Assumption 4.1. In our setting it is natural to start with rate functions whose superdifferential is close to −∞ and close to ∞ near the left or right boundary of K respectively as this property is preserved for I t (see Theorem 2.11 and (22)). Moreover, we assume our initial rate function to be C 1 . The space of functions that combines these two properties is called C 1,∂ (see Definition 2.9). The main result (Theorem 2.13) then shows that such a rate function under the time evolution is again in C 1,∂ (K) if and only if there is a unique optimizing trajectory. Definition 2.9. For K = R we write ∂ − = −∞ and ∂ + = ∞, while for K = [−1, 1] we write ∂ − = −1 and ∂ + = 1. We write C k,∂ (K) for the set of functions g : K → R such that g is k times continuously differentiable on K • , continuous on K and lim x→∂ + g (x) = ∞, lim x→∂ − g (x) = −∞.(22) Note that for V ∈ C 1 [−1, 1] the function (7) is an element of C 1,∂ [−1, 1]. Moreover, note that I 0 ∈ C 1,∂ (K) implies that I 0 is bounded from below and has compact sublevel sets. Assumption 2.10. We will assume that I 0 ∈ C 1,∂ (K) and for H : K × R → R we assume The examples that we consider satisfy condition (a), however the proofs of the theory is based on the more general condition (b). In Appendix A we show that (a) indeed implies (b). inf b≥a inf D + I t (b) = ∞, inf a∈(∂ − ,0) sup b≤a sup D + I t (b) = −∞.Then D + v(x) = ∅ and (a) D * v(x) ⊆ ∂D + v(x). (b) D + v(x) = coD * v(x), where co(S) denotes the closed convex hull of a set S ⊆ R d . (c) D + v(x) is a singleton if and only if v is differentiable at x. (d) If D + v(y) is a singleton for every y ∈ A, then v ∈ C 1 (A). Theorem 2.13. Assume Assumption 2.10. Let u : [0, ∞) × K → [0, ∞] be given by u(t, x) = I t (x). Let b ∈ K • , t > 0. The following are equivalent. (a) x → I 0 (x) + S t (x, b) has a unique optimizer (I t (b) has a unique optimal point). (b) γ → I 0 (γ(0)) + ∞ 0 L (γ(s),γ(s))ds has an unique optimal trajectory with γ(t) = b. (c) I t is differentiable at b. (d) u is differentiable at (t, b), (e) D * u(t, b) is a singleton, Moreover, the following are equivalent. (A) x → I 0 (x) + S t (x, b) has a unique optimizer for all b ∈ K • . (B) γ → I 0 (γ(0)) + t 0 L (γ(s),γ(s))ds has an unique optimal trajectory with γ(t) = b for all b ∈ K • . (C) I t is differentiable on K • . (D) I t ∈ C 1,∂ (K). Proof. (b) ⇒ (a). If γ is an optimal trajectory, then γ(0) is an optimal point for I t (b). (a) ⇒ (b). Suppose x is a unique optimal starting point for I t (b), and γ 1 , γ 2 are optimal trajectories for I t (b) that start in x. To both trajectories, we can associate dual trajectories η 1 , η 2 so that the pairs (γ 1 , η 1 ) and (γ 2 , η 2 ) solve the Hamilton equations. Because their starting points are the same, the starting condition implies that η i (0) = I 0 (x). This, however, implies that the Hamilton equations are initialized with the same starting data, implying that γ 1 = γ 2 . We conclude there is a unique optimal trajectory. By Proposition 5.2 the optimal trajectories for I t (b) are in one to one correspondence with the elements of D * u(t, b). Whence (b) ⇐⇒ (e) are equivalent. Theorem 2.11 implies that u and I t are locally semi-concave. By Lemma 2.12 we have (d) ⇐⇒ (e). (d) ⇒ (c) as differentiability of u at (t, b) implies differentiability of I t at b. On the other hand, if there exists two distinct optimal trajectories for I t (b), then the corresponding end momenta are different. By Lemma 2.12 and [CS04, Theorems 6.4.8] this implies that D + I t (b) consists of at least two elements, i.e., I t is not differentiable at b by Lemma 2.12. Hence (c) ⇒ (b). (A) ⇐⇒ (B) ⇐⇒ (C) follow from (a) ⇐⇒ (b) ⇐⇒ (c). As I t is locally semi-concave I t is differentiable if and only if I t ∈ C 1 on K • (see Lemma 2.12). Whence (C) ⇐⇒ (D) by Theorem 2.11. Regularity via the push-forward of the graph of I 0 Our next step is to relate optimal trajectories to the solutions of the Hamilton equations. In the following definition we present the push forward under the Hamiltonian flow of the graph of I 0 . In Proposition 2.15, we show that if this push forward at time t is a graph then I t is differentiable. Additionally, we show that overhangs in the push forward are indications for existence of points of non-differentiability. Definition 2.14. Assume Assumption 2.10. (a) We write G := {(x, I 0 (x)) : x ∈ K • }, for the graph of the derivative of I 0 . (c) For all t > 0 we define the push-forward of G to be the set (b) For all (x, p) ∈ K × R let (X x,p t , P x,p t ) beG t := {(X x,p t , P x,p t ) : (x, p) ∈ G , t < t x,p } . (d) Fix t > 0. We say that G t has an overhang at x ∈ K • , if there exist y 1 , y 2 ∈ R with y 1 = y 2 such that (x, y 1 ), (x, y 2 ) ∈ G t . Hence if G t has an overhang, then it is not a graph (of a function). Proposition 2.15. Assume Assumption 2.10. Fix t > 0. Then {(x, p) : x ∈ K • , p ∈ D * I t (x)} ⊆ G t .(23) Consequently, if G t has no overhang at x, then I t is differentiable at x. On the other hand, if there are x 1 , x 2 ∈ K • , x 1 < x 2 such that G t has no overhang at x 1 and x 2 , then (a) I t is differentiable on [x 1 , x 2 ] =⇒ G t has no overhang at x for all x ∈ (x 1 , x 2 ). (b) G t has an overhang at some x ∈ (x 1 , x 2 ) =⇒ I t is not differentiable on [x 1 , x 2 ] Proof. (23) is proved in Proposition 5.2(c). If G t has no overhang at x, then D * I t (x) is a singleton and so I t is differentiable at x by Theorem 2.13. We prove (a) as (b) is equivalent to (a). The interval (a, b) as in Proposition 6.2(e) is such that Φ t (a, b) contains (x i , I t (x i )) for i ∈ {1, 2}. As Φ t (a, b) is connected, even {(x, I t (x)) : x ∈ [x 1 , x 2 ]} ⊆ Φ t (a, b) . By continuity and injectivity of Φ t , see Proposition 6.2(a), and the fact hat {y : (x i , y) ∈ G t } are singletons for i ∈ {1, 2}, it follows for all x ∈ (x 1 , x 2 ) that {y ∈ R : (x, y) ∈ G t } = {y ∈ R : (x, y) ∈ Φ t (a, b )} and that this set cannot contain multiple elements. Applications of analyzing the Hamiltonian flow In this section we study the differentiability of I t by analysing the push-forward G t . We will give a description of the results of this section in terms of the informal -but more familiar-notions high-and low-temperature. We say that a rate function I 0 is hightemperature if it is strictly convex, whereas we say that I 0 is low-temperature if it has at least two strict local minima. We say that the dynamics is high-and low-temperature if there is a high-and low-temperature I 0 such that H(x, I 0 (x)) = 0, respectively. This means that I 0 is the rate function of the stationary distributions of the dynamics with Hamiltonian H. In Section 3.1 we give two general results on the preservation of differentiability, and two general results on the creation of overhangs: Preservation at high temperature We show that for certain types of high-temperature dynamics, combined with high-temperature starting rate functions, we have differentiability of I t for all t ≥ 0. Short-time preservation We show that 'order preserving' behaviour of the dynamics close to the boundary, combined with a starting rate-function I 0 that is strictly convex close to the boundary, implies short-time differentiability of I t . Large-time loss, heating We consider linearizations of the Hamiltonian flow around stationary points, which in combination with low-temperature rate-functions, create overhangs for sufficiently large times. Large-time loss, cooling We considers areas in phase-space where the energy is negative and where the Hamiltonian flow 'rotates'. If the graph of the gradient of the rate function I 0 crosses this 'rotating' region, an overhang is created for sufficiently large times. Rotating regions occur for low-temperature Hamiltonians. We proceed in Section 3.2 by applying these results to two sets of well known examples: Glauber dynamics for the Curie-Weiss model and interacting diffusions in a potential. Before starting with the various applications, we introduce some notation. Definition 3.1. For (y, q) ∈ K • × R, we write y,q := {(x, p) ∈ K × R : x ≥ y, p ≥ q}, y,q := {(x, p) ∈ K × R : x ≤ y, p ≤ q}, and • y,q , • y,q for the interiors of y,q , y,q in K × R. E.g., for K = [−1, 1], y,q = [y, 1] × [q, ∞) and • y,q = (y, 1] × (q, ∞). Definition 3.2. Assume Assumption 2.10. (a) Let A ⊆ K × R. We say that H preserves A, or A is preserved under H, if all trajectories (γ, η) satisfying the Hamilton equations with (γ(0), η(0)) ∈ A stay in A during their life-time, i.e., (γ(s), η(s)) ∈ A for all s < t γ(0),η(0) if (γ(0), η(0)) ∈ A. (b) If (x 0 , p 0 ) ∈ K × R is such that H preserves the set {(x 0 , p 0 )}, then we call (x 0 , p 0 ) stationary under H. Mostly we consider stationary points of the form (x 0 , 0), in such case we will also say that x 0 is stationary. (c) We say that H : K×R → R preserves order on a subset A ⊆ K×R if A is preserved under H and if for (x 1 , p 1 ), (x 2 , p 2 ) ∈ K × R and t < t x 1 ,p 1 ∧ t x 2 ,p 2 , x 1 < x 2 , p 1 < p 2 =⇒ X x 1 ,p 1 t < X x 2 ,p 2 t , P x 1 ,p 1 t < P x 2 ,p 2 t . (d) We say that H preserves order at infinity if there are (y − , q − ), (y + , q + ) ∈ K • × R such that H preserves order on y − ,q − and y + ,q + . (e) We say that I 0 is strictly convex at infinity if there is some compact set K ⊆ K • such that I 0 is strictly increasing on the complement of K. Two effective methods to determine that G t has overhangs are via rotating area's in the Hamiltonian flow and via linearizations of the flow, of which the definition follows. U, V ⊆ R 2 of (x 0 , 0), and a C 1 -diffeomorphism Ψ : U → V such that Ψ(x 0 , 0) = (x 0 , 0) and DΨ(x 0 , 0) = 1 (the identity matrix) and such that a trajectory (γ, η) with values in U solves the Hamilton equations (19) if and only if (ξ, ζ) = Ψ(γ, η) solves ξ (s) ζ(s) = ∇ 2 H(x 0 , 0) ξ(s) − x 0 ζ(s) ,(24) where ∇ 2 H(x 0 , 0) denotes the Hessian of H at (x 0 , 0), so that with m := ∂ x ∂ p H(x 0 , 0), c := ∂ 2 p H(x 0 , 0),(25) it has the form ∇ 2 H(x 0 , 0) = ∂ x ∂ p H(x 0 , 0) ∂ 2 p H(x 0 , 0) −∂ 2 x H(x 0 , 0) −∂ x ∂ p H(x 0 , 0) = m c 0 −m , note that −∂ 2 x H(x 0 , 0) = 0 because H(x, 0) = 0 for all x and that c > 0. Remark 3.4. Note that we include in the definition that DΨ(x 0 , 0) = 1 which is not a standard assumption. However, we need this assumption to connect the dynamics of the graph G under the Hamiltonian flow, to that of the dynamics of the tangent of the push-forward of G . Preservation of differentiability and the creation of overhangs The following theorem relates preservation of order under H and strict convexity of I 0 to differentiability of I t . The proof can be found in Section 7. Theorem 3.5. Assume Assumption 2.10. (a) Suppose that I 0 is strictly convex and that G is contained in a set on which H preserves order. Then I t ∈ C 1,∂ (K) for all t ≥ 0. (b) Assume that I 0 is C 2 on K • , strictly convex at infinity and that H preserves order at infinity. Then there is a t 0 > 0 such that I t ∈ C 1,∂ (K) for all 0 ≤ t ≤ t 0 . The next theorem gives conditions under which overhangs are created. The proofs of (a) and (b) can be found in Section 8. Theorem 3.6. Assume Assumption 2.10. (a) Suppose x 0 ∈ K • is such that I 0 (x 0 ) = 0 and that (x 0 , 0) is a stationary point. Let m, c be as in (25). Suppose I 0 is C 2 in a neighbourhood of x 0 with I 0 (x 0 ) < min{− 2m c , 0}. In addition, assume that the Hamiltonian flow admits a C 1 linearization at (x 0 , 0) (a sufficient condition for this is that H is C ∞ and m = 0 3 ). Let t 0 :=    − 1 2m log 1 + c 2m I 0 (x 0 ) −1 if m = 0, − 1 cI 0 (x 0 ) if m = 0.(26) Then G t contains an overhang at x 0 for all t > t 0 . (b) Suppose that H is C 3 . In addition suppose that m 1 , m 2 ∈ K • are two points such that m 1 < m 2 and (i) ∂ p H(m 1 , 0) = 0 = ∂ p H(m 2 , 0) and ∂ p H(x, 0) = 0 for all x ∈ (m 1 , m 2 ), (ii) ∂ x ∂ p H(m 1 , 0) = 0 and ∂ x ∂ p H(m 2 , 0) = 0. Suppose that G ∩ {(x, p) ∈ (m 1 , m 2 ) × R : H(x, p) < 0} = ∅. Then there is a time t 0 > 0 such that there is a x 0 ∈ (m 1 , m 2 ) such that G t contains an overhang at x 0 for all t ≥ t 0 . Remark 3.7. In (a) t 0 is the time that the line x → x(1, I 0 (x 0 )) is transformed into a vertical line under the linearized flow. Regarding (b), in Section 8 we show that if a set A satisfies certain properties, then the Hamiltonian flow rotates over the boundary ∂A of A, which means that every Hamilton path started on ∂A will move along ∂A and return to its initial point in finite time. We prove that the conditions of (b) imply the existence of such set in Lemma 8.6. Explicit results for two main classes of examples In the present section, we analyze particular canonical instances of the ±1-space-model and the R-space-model: (i) for the R-space-model we consider H and I of the form H(x, p) = 1 2 p 2 − pW b (x), I 0 (x) = W a (x) + C,(27)for a, b ∈ R, where W d (x) = 1 4 x 4 − 1 2 dx 2 for d ∈ R. (ii) for the ±1-space-model we consider H and I of the form H(x, p) = 1 − x 2 e βx+h e 2p − 1 + 1 + x 2 e −βx−h e −2p − 1 ,(28)I 0 (x) = 1 − x 2 log(1 − x) + 1 + x 2 log(1 + x) − 1 2 αx 2 − θx + C,(29) where α, β ≥ 0 and θ, h, C ∈ R is such that inf x∈[−1,1] I 0 (x) = 0. I.e., we consider Curie-Weiss dynamics with inverse temperature β and a starting rate function that corresponds to an inverse temperature α. In Proposition 3.9 we consider conditions sufficient for H to preserve order. In Theorem 3.10 we apply the results of Theorems 3.5 and 3.6 for various choices of a and b or of α, β, h and θ. In Theorem 3.11, we show that for specific choices of α and θ one obtains recovery of differentiability. We restrict our analysis mostly to β = h = 0, i.e., infinite temperature dynamics and zero dynamical external magnetic field. For other parameters we expect that one can obtain similar results at the expense of more involved computations, which go beyond the scope of this text. In the proof of Theorem 3.10 and Theorem 3.11, we use the creation of overhangs proved in Theorem 3.6 to show non-differentiability with the help of Proposition 2.15. Remark 3.8. One can rewrite H as in (28) and compute the following derivatives, which will be used later in proofs. We spare the reader the computations. H(x, p) = 2 sinh(p) [sinh(βx + h + p) − x cosh(βx + h + p)] ,(30)∂ p H(x, p) = 2 sinh(βx + h + 2p) − 2x cosh(βx + h + 2p) (31) ∂ x H(x, p) = −2 sinh(p) [(1 − β) cosh(βx + h + p) + βx sinh(βx + h + p)] ,(32)∂ p ∂ x H(x, p) = 2(β − 1) cosh(βx + h + 2p) − 2βx sinh(βx + h + 2p),(33)∂ 2 x H(x, p) = −2β sinh(p) [(2 − β) sinh(βx + h + p) + βx cosh(βx + h + p)] ,(34)∂ 2 p H(x, p) = 4 cosh(βx + h + 2p) − 4x sinh(βx + h + 2p).(35) Note that H(x, p) = 0 if and only if either p = 0 or h + p = arctanh(x) − βx and that ∂ p H(x, p) = 0 if and only if 2p = arctanh(x) − βx − h. For I 0 as in (29) we have I 0 (x) = arctanh(x) − αx − θ,(36)I 0 (x) = 1 1 − x 2 − α. (37) Let H as in (28). Then H preserves order at infinity for all β ≥ 0 and h ∈ R; for I 0 I 0 I 0 α < 1 α = 1 θ = 0 α > 1 θ < 0 θ < 0h = 0 and β ∈ [0, 2], H preserves order on • 0,0 ∪ {0} ∪ • 0,0 ; and, for β = h = 0, H preserves order on the entire space [−1, 1] × R. (b) Assume Assumption 2.4 for the R-space-model, with H as in (14). Let y, z ∈ R. If W (x) ≤ 0 for all x ≤ y, then H preserves order on • y,0 . If W (x) ≥ 0 for all x ≥ z, then H preserves on • z,0 . If such y and z as above exist, then H is preserves order at infinity. Such y and z exists, e.g., for W of the form W (x) = 2k i=1 a i x i with a 2k > 0 and thus for W as in (27). Proof. (a) By Proposition 7.2 and because Assumption 4.1(e) holds (see Lemma A.1), it is sufficient to show that ∂ 2 x H < 0 on quadrants x,p and x,p for x, p close to (1, ∞) and (−1, −∞), respectively. Those x, p can be found as ∂ 2 x H(x, p) = v + (x)[e 2p − 1] + v − (x)[e −2p − 1] and v is continuous. For H as in (28), i.e., H as in (15) with v − and v + as in (11) we have v + (1) = −βe β+h and v − (−1) = −βe β−h which are both < 0 for all β > 0 and h ∈ R. For β = 0 and h ∈ R it follows that (x, p) → ∂ x H(x, p) is a decreasing function at infinity, whence H preserves at infinity by Proposition 7.2. For β ∈ (0, 2] and h = 0, by (34) we see that For β = h = 0, by (32) ∂ x H(x, p) = −2 sinh(p) cosh(p) which is decreasing as a function of (x, p), so that H preserves on the whole space [−1, 1] × R. ∂ 2 x H(x, p) < 0 on • 0,0 ∪ • 0,0 . As ∂ p H(0, p) = 2 sinh(2p), (b) Note that ∂ 2 x H(x, p) = −pW (x), which immediately establishes the claim. In addition to applying the abstract results of Section 3.1, we use Proposition 2.15 to show that in particular settings overhangs do induce non-differentiabilities of the rate function. The proof of the following results are given in Section 9. Theorem 3.10. For the R-space-model and the ±1-space-model, under the above assumptions, we have the following scenarios for the following a, b and α, β: (a) [a = b], [α = β, θ = h], equilibrium. We have I 0 = I t for all t ≥ 0. (b) [a, b ∈ R], [α, β ≥ 0, θ, h ∈ R], short time differentiability. There is a t 0 > 0 such that for t ≤ t 0 we have I t ∈ C 1,∂ (K). (c) [a ≤ 0], [α ≤ 1, 0 < β ≤ 2, θ ∈ R, h = 0], high-temperature starting-profile. For all t ≥ 0 we have I t ∈ C 1,∂ (K). (d) [a > b ∨ 0, b = 0], [α > β ∨ 1, β = 1, θ = h = 0], heating up a low-temperature starting-profile. There is an overhang at x = 0 for all t ≥ t 1 where (i) for the R-space-model t 1 = − 1 b log a−b a . (ii) for the ±1-space-model t 1 = − 1 4(1−β) log β−α 1−α . (e) [0 < a < b], [1 < α < β, θ = h = 0] , cooling down a low-temperature starting-profile. There is some t 2 such that for all t ≥ t 2 there exits at least two points (a) For α ≥ 0 and θ ∈ R such that I 0 = 0 has a unique solution, there is some time t * such that x − , x + of non-differentiability of I t with m − < x − < 0 < x + < m + , where (i) for the R-space-model m ± = ± √ b. (ii) for the ±1-space-model m ± are the solutions to arctanh(x) = βx. 4I t ∈ C 1,∂ [−1, 1] for t ≥ t * . (b) For all α > 1 there exists a κ > 0 such that for all θ ∈ R with \|θ\| > κ, there are times t 0 < t 1 ≤ t 2 such that (i) I t ∈ C 1,∂ [−1, 1] for t < t 0 , (ii) G t contains an overhang for t ∈ (t 0 , t 2 ) and I t is non-differentiable for t ∈ (t 0 , t 1 ), (iii) I t ∈ C 1,∂ [−1, 1] for t > t 2 . Remarks on the results and comparison with the literature Remark 3.12. Our method to verify non-differentiability for Theorem 3.10 (d) is based on Proposition 2.15 up to the time at which the push forward of G falls apart in three separate curves, cf. Proposition 6.2. Similarly, in our proof of Theorem 3.11(b), we actually have t 1 < t 2 (see also Remark 9.1), i.e., for t ∈ [t 1 , t 2 ) there is an overhang but again Proposition 2.15 cannot be used to conclude that I t is non-differentiable. Remark 3.13. In [HRZ15] the R-space-model with H(x, p) = 1 2 p 2 and I 0 (x) = 1 2 x 2 + V (x), where V ∈ C 1 (R, [0, ∞)) is considered. We show that the existence of an overhang as in Theorem 3.6(a) agrees with the non-differentiability claimed in [HRZ15]. Note that the Hamiltonian flow admits a C 1 linearization by the identity map as the flow itself is linear. In [HRZ15, Corollary 1.12] it is shown that I t (b) has a unique global minimiser for all b ∈ R if and only if Φ 2 V > − 1+t 2t . Whence I t is not differentiable if Φ 2 V > − 1+t 2t , which by [HRZ15, Lemma 5.9] is the case when there exists an x 0 for which V (x 0 ) + 1 2 < − 1 t , which is the same as t > t 0 for t 0 as in (26). Theorem 3.5 also agrees with [HRZ15, Corollary 1.12] in case I 0 (x) = 1 2 x 2 + V (x) is strictly convex (at infinity). However, the setting in [HRZ15] allows I 0 not to be in C 1,∂ . This is the case for, e.g., V (x) = 1 + cos(x 2 ) as in [HRZ15, Example 1.16]. Here one has immediate loss of uniqueness of minimisers and therefore an immediate loss of differentiability, which can be proved by [CS04, Corollary 6.4.4 and Theorem 6.4.8] in the same way as in the proof of Theorem 2.13. In the literature on dynamical systems, the failure of ∂ p ∂ x H(x 0 , 0) = 0 implies that (x 0 , 0) is a non-hyperbolic fixed point of the Hamiltonian flow. This can be considered to be critical behaviour: for a non-hyperbolic fixed point, the first order approximation does not describe the global behaviour of the flow around this point. This is similar to the statement that α = 1 is critical for the Curie-Weiss model: the first order approximation I 0 (0), with I 0 as in (29) of the rate function I 0 at the point 0 vanishes for α = 0 indicating a transition from a convex to a non-convex rate-function. Remark 3.15. Both the idea of linearization and rotation already appeared for the Lagrangian flow in [EK10]. The idea of considering the Hamiltonian flow instead of the Lagrangian flow already appeared in [Kra16c, Chapter 5]. As our methods do not depend on a specific model, we recover part of the results of [EK10], however some of our results are slightly weaker: Using the explicit calculations [EK10, Theorem 1.3] obtains the result in Theorem 3.10(c) for all β instead of β ≤ 2. Our result in (d) for 1 ∨ β < α is sub-optimal. [EK10;KLN07] show that points of non-differentiability occur before the linearized system assures that we have a point of non-differentiability at 0. In this setting, there is a different mode, apart from the rotation around 0 that creates the overhang. This mode can easily be identified by using pictorial analysis based on the Hamiltonian flow. A study of the Hamiltonian dynamics, optimizers, and the time-evolved rate function The extension of calculus of variations to a setting where the Hamiltonian trajectories may have finite maximal times of existence, needs a treatment of the behaviour of the Hamiltonian flow close to the boundary. This analysis will be carried out under general conditions that are introduced in Sections 4.1 and 4.2. We will show that these conditions imply that Hamiltonian trajectories are pushed away from the boundary, and can only arrive at the boundary with infinite momentum at their maximal time of existence. In Section 4.3, we show that optimizers of I t (b) together with their dual trajectories, are solutions of the Hamilton equations. In Section 5 we establish the regularity properties of the rate function that we introduced in Sections 2.6 and 2.7. Conditions on Hamiltonian and properties of the Hamiltonian flow Below, we introduce the main assumption of our paper. The assumptions (a)-(e) fall apart in two natural groups. (i) lim r→∞ r −1 θ K (r) = ∞, (ii) for every M ≥ 0, there is a k M ≥ 0 such that θ K (r + m) ≤ k M (1 + θ K (r)) for all m ∈ [0, M ] and r ≥ 0, (iii) there exists c K such that L (x, v) ≥ θ K (\|v\|) − c K for all x ∈ K and v ∈ R, (iv) there exists C K such that \|∂ x L (x, v)\|+\|∂ v L (x, v)\| ≤ C K θ K (\|v\|) for all x ∈ K and v ∈ R. (c) For each compact K ⊆ K • , we have lim \|p\|→∞ inf x∈K H(x,p) \|p\| = ∞. If K = [−1, 1], then we additionally assume lim p→∞ H(−1, p) p = ∞, lim p→−∞ H(1, p) −p = ∞. (d) We have lim x→∂ − argmin p∈R H(x, p) = −∞, lim x→∂ + argmin p∈R H(x, p) = ∞ (e) There exists a sequence ((y + n , q + n )) n∈N in K • × (0, ∞) with lim n→∞ (y + n , q + n ) = (∂ + , ∞) and ∂ p H(y + n , q) ≥ 0 for q ≥ q + n ,(38)−∂ x H(y, q + n ) ≥ 0 for y ≥ y + n ,(39) and there exists a sequence ((y − n , q − n )) n∈N in K • × (−∞, 0) with lim n→∞ (y − n , q − n ) = (∂ − , −∞) and ∂ p H(y − n , q) ≤ 0 for q ≤ q − n ,(40)−∂ x H(y, q − n ) ≤ 0 for y ≤ y − n ,(41)lim p→−∞ H(−1, p) p = v − ≥ 0, lim p→∞ H(1, p) p = v + ≤ 0,(42) and that the Hamiltonian vector field on the boundary points inwards: ∂ p H(−1, p) > 0, ∂ p H(1, p) < 0.(44) Moreover, these assumptions together with Assumption 4.1(c) imply that for all c ∈ R, for every (a, q) ∈ (−1, 1) × (0, ∞) and (b, r) ∈ (−1, 1) × (−∞, 0) the set (see Figure 4.1(a)) • a,q ∪ • b,r c ∩ H −1 (−∞, c](45q < p =⇒ ∂ p H(x, q) < H(x, q) − H(x, p) q − p < ∂ p H(x, p).(46) Whence (42) implies (43) and (44). Note that We proceed with exploring the behaviour of solutions to the Hamilton equations 'close' to the boundary. This analysis will crucially depend on Assumptions 4.1 (c), (d) and (e). We start with a result that captures the idea that the Hamiltonian flow points away from the 'boundary' points ∂ − and ∂ + , unless one starts with very low and very high momentum, respectively (see also Figure 4.1(b)). Lemma 4.4. Let H satisfy Assumption 4.1. There exists a sequence (z n ) n∈N in (0, ∂ + ) such that −z n < y − n , z n > y + n ,(47)∂ p H(x, p) > 0 if ∂ − ≤ x ≤ −z n , p ≥ q − n ,(48)∂ p H(x, p) < 0 if z n ≤ x ≤ ∂ + , p ≤ q + n .(49)argmin p∈R H(x, p) ≤ q − n , inf x≥zn argmin p∈R H(x, p) ≥ q + n . By the assumed strict convexity of p → H(x, p) we have ∂ p H(x, p 0 ) = 0 for p 0 = argmin p∈R H(x, p) and ∂ p H(x, p) < 0 for p < p 0 and ∂ p H(x, p) > 0 for p > p 0 . Therefore we conclude (48) and (49). We can choose z n large enough such that (47) is satisfied as well. (a, q) Remark 4.5. Note that Lemma 4.4 implies that for x ∈ K and p ∈ R, one has the following implications for t > 0 (b, r) u v (y + n , q + n ) z n y + n ,q + n (y − n , q − n ) −z n y − n ,q − n (a) (b)lim s↑t X x,p t = −1 =⇒ lim s↑t P x,p t = −∞,(50)lim s↑t X x,p t = 1 =⇒ lim s↑t P x,p t = ∞.(51) Note that in both cases t = t x,p . Lemma 4.6. Suppose that (x, p) ∈ K × R are such that t x,p < ∞. Then 1(a)); for any compact set K ⊆ K × R there exists a time β < t x,p such that (X x,p β , P x,p β ) / ∈ K. We will use this fact in two separate ways, depending on the setting, to prove the result. First suppose K = [−1, 1]. Let c = H(x, p) and n ∈ N, (a, q) = (y + n , q + n ) and (b, r) = (y − n , q − n ) (see Assumption 4.1(e)). By Lemma 4.3 the set (45) is compact. Whence there exists a β 0 < t x,p such that (X x,p β 0 , P x,p β 0 ) is either in y + n ,q + n or y − n ,q − n . Assume it is in y + n ,q + n . Then (X x,p β 0 , P x,p β 0 ) is in y + n ,q + n for all β ∈ (β 0 , t x,p ) as y + n ,q + n is preserved (see also Remark 4.2). In a similar way as above, for each n ∈ N there exists a β n such that (X x,p β , P x,p β ) ∈ y + n ,q + n for all β > β n . This implies lim t↑tx,p (X x,p t , P x,p t ) = (1, ∞). Now let K = R. Let c = H(x, p) and let n 0 ∈ N be such that y − n < x < y + n for n ≥ n 0 . Instead of using Lemma 4.3 in the above lines of proof for K = [−1, 1], we can use Remark 4.2 and Lemma 4.4 to obtain the existence of a β 0 and β n as above and follow the same lines. (x, p) ∈ (−v, v) × R, with X x,p t = b ∈ (−u, u) we have X x,p s ∈ (−v, v) for all s ∈ [0, t]. Proof. Let n be such that b ∈ (y − n , y + n ). Let z n be as in Lemma 4.4 and v > z n . Suppose x is in (−v, v) and p ∈ R are such that X x,p t = b. As the end point (X x,p t , P x,p t ) lies outside the quadrants y + n ,q + n and y − n ,q − n the whole trajectory does not enter either of the quadrants as those quadrants are preserved. As the complements of the sets {x : x ≤ z n } × [q − n , ∞) and {x : x ≥ z n } × (−∞, q + n ] are preserved, X x,p s for s ∈ [0, t] is prevented from entering {x : x ≤ v} ∪ {x : x ≥ v}. Additional model dependent conditions and properties In addition to the properties of the Hamiltonian flow that were established above, there are some peculiarities due to the boundaries of both settings that need to be treated separately. In the setting of the ±1-space-model we need these auxiliary technical results in Section 4.3 below to show that optimizers that start at the boundary can be related to solutions of the Hamilton equations. For the R-space-model, we introduce a condition that ensures that the rate function has compact sublevel sets, something that for the ±1-space-model follows from Assumption 4.1 above. The ±1-space-model In this section, we consider K = [−1, 1]. We start by showing that L is twice continuously differentiable on an appropriate domain. We then proceed by extending the regularity of the Hamiltonian flow up to the boundary of [−1, 1] × R, after which we give an additional assumption that is needed to verify Proposition 4.14 (a) below. −1, v). In an analogous way as in [CS04, (v − , ∞). Similarly, the argument of the proof of [CS04, Theorem A.2.5] carries over so that we have that L is C 2 on (v − , ∞). , v) = ∞ for all v ∈ (−∞, v − ) and L (1, v) = ∞ for all v ∈ (v + , ∞), and L is C 2 on (−1, 1) × R ∪ ({−1} × (v − , ∞)) ∪ ({1} × (−∞, v + )). Proof. That L (−1, v) = ∞ for all v ∈ (−∞, v − ) and L (1, v) = ∞ for all v ∈ (v + , ∞) follows from (43). First we prove that v → L (−1, v) is C 2 on (v − , ∞), similarly one proves that v → L (1, v) is C 2 on (−∞, v + ). Write L(v) = L ( That L is C 2 on (−1, 1) × R follows from [CS04, Theorem A.2.7]. Following the lines of the proof of [CS04, Theorem A.2.7], it is sufficient to show that the map q : q(x, v)) = 0, and ∂ 2 p H (see Assumption 4.1(a)) is > 0 in an neighbourhood of (x, p) for all (x, p) ∈ [−1, 1] × R, by the implicit function theorem it follows that q is C 1 . (−1, 1) × R ∪ ({−1} × (v − , ∞)) ∪ ({1} × (−∞, v + )) → R, which assigns to an element (x, v) the unique p such that L (x, v) = pv − H(x, p), is C 1 . As q satisfies v − ∂ p H(x,L (x, 0) − L (−1, 0) S(x) − S(−1) = ∞, lim x↑1 L (x, 0) − L (1, 0) S(x) − S(1) = ∞. The R-space-model In this section, we consider K = R. We assume the existence of a Lyapunov function Υ that is needed to treat the non-compactness of R in the proof of compactness of the sublevel sets of the rate function (see Lemma 4.12). Assumption 4.11. There is a continuously differentiable function Υ with compact sublevel sets, i.e. for all c ∈ R the set {x ∈ R : Υ(x) ≤ c} is compact, with the additional property that sup x∈R H(x, Υ (x)) < ∞. The relation between optimizers and Hamiltonian trajectories and a study of the time-evolved rate function From the discussion above, we understand the behaviour of the Hamiltonian flow. In this section, we show that optimal trajectories exist (Lemma 4.12), and, combined with their dual trajectories, satisfy the Hamilton equations (Propositions 4.14 and 4.13). Then we show that the range of optimal trajectories can be controlled if I 0 ∈ C 1,∂ (K) (Proposition 4.15). In addition, for every t > 0, a ∈ K and b ∈ K • there is an optimal trajectory γ for S t (a, b) and I t (b). Proof. The proof of compactness sublevel sets of I , is generally proven as a part of a large deviation principle. For a proof of this result in the ±1-space-model setting, see [Kra16b,Theorem 2]. A similar proof for the R-space-model can be carried out by using Υ, see the proof of [CK17, Theorem A.14]. Pick a ∈ K, b ∈ K • and t > 0. Pick a C 1 curve γ connecting a to b in time t such that in addition (γ(s),γ(s)) takes its values in the region where L is C 2 , cf. Lemma 4.8 for the case where K = [−1, 1]. This implies that γ has finite cost. Thus, by the contraction principle and the compactness of the sublevel sets of I , where we take as a starting rate function I 0 (a) = 0 and I 0 (x) = ∞ if x = a, there is an optimizer for S t (a, b). Again by the contraction principle, but now with a general starting rate function I 0 , we find that there is an optimizer for I t (b). Proposition 4.13. Let H satisfy Assumption 4.1 in the setting that K = R, let I 0 ∈ C 1,∂ (R), and t > 0. Let a, b ∈ R and suppose that γ is an optimal trajectory for S t (a, b). C 1 and (γ, η) If γ is an optimal trajectory for I t (b), then η(0) = I 0 (γ(0)). Then γ is C 2 on [0, t]. The dual trajectory η defined on [0, t] is Proof. This follows by [CS04, Theorem 6.2.8 and 6.3.3]. (a) Any optimal trajectory γ for S t (a, b) satisfies γ(s) ∈ (−1, 1) for all s ∈ (0, t). (b) Let γ be an optimal trajectory for S t (a, b). If b ∈ (−1, 1), then γ is C 2 on [0, t]. The dual trajectory η is defined on [0, t] is C 1 and (γ, η) satisfies the Hamilton (c) Any optimal trajectory γ for I t (b) satisfies γ(0) ∈ (−1, 1). Let η be the dual trajectory as above, then we have η(0) = I 0 (γ(0)). equations (19) on the interval [0, t]. If b ∈ {−1, 1} then γ is C 2 on [0, t). The dual trajectory η is defined on [0, t) is C 1 and (γ, Proof. (a) follows from [Kra16a, Proposition 4.9] and Assumption 4.10. For (b), first we consider b ∈ (−1, 1). Let γ be an optimizer for S t (a, b). By (a), we have γ(s) ∈ (−1, 1) for s ∈ (0, t). Thus, for any δ ∈ (0, t), the trajectory γ restricted to [δ, t] is contained in (−1, 1) and is the optimal trajectory over all absolutely continuous trajectories ξ on [δ, t] with ξ(t) = γ(t) and ξ(δ) = γ(δ) of the functional ξ → t δ L (ξ(s),ξ(s))ds. By Assumption 4.1 (a), (b) and [CS04, Theorem 6.2.8] (note that L is C 2 on (−1, 1)×R by Lemma 4.8), we find that the restriction of γ to [δ, t] is C 2 and solves the Euler-Lagrange equation classically. This can be done for all δ > 0, so γ is C 2 on (0, t] and solves the Euler-Lagrange equation d ds ∂ v L (γ(s),γ(s)) = ∂ x L (γ(s),γ(s)), s ∈ (0, t].(54) Let η : (0, t] → R be the dual path as in (18) If b ∈ {−1, 1}, we can follow the same lines as above, now restricting to trajectories on [δ, t − δ] instead of [δ, t] with δ ∈ (0, t 2 ) for example. For the limit lim s↑t (γ(s), η(s)) we refer to Remark 4.5. For (c), suppose that there exists an optimal trajectory γ such that γ(0) = −1 and γ(t) = b. The proof for γ(0) = 1 is similar. By (a) and (b) there exists a κ > 0 be such that γ(s) < 1 − κ for all s ∈ [0, t 2 ]. Let ρ ∈ C 1 [0, t] be such that ρ(0) > 0, ρ = 0 on [ t 2 , t] and takes its values in [0, κ], so that γ + ρ attains its values in [−1, 1] for all ∈ [0, 1]. We derive a contradiction by showing the existence of a ∈ (0, 1) such that γ + ρ has a lower cost than γ, i.e., J (γ + ρ) < J (γ) (for I see (13)). We do this by showing that the difference quotient I (γ + ερ) − I (γ) ε = I 0 (γ(0) + ερ(0)) − I 0 (γ(0)) ε + J(γ + ερ) − J(γ) ε ,(55) converges to −∞ as ↓ 0, where J is the path-space cost J(ζ) := t 0 L (ζ(s),ζ(s))ds. As I 0 ∈ C 1,∂ [−1, 1], the first term on the right hand side of (55) converges to −∞. Therefore it is sufficient to show that the second term on the right hand side is bounded from above for small . For θ > 0 we write ψ θ (s) = (γ(s) + θρ(s),γ(s) + θρ(s)). By the mean-value theorem there exists a θ s ∈ (0, ) for all s ∈ [0, t] such that J(γ + ερ) − J(γ) ε = t 0 ∂ x L (ψ θs (s))ρ(s) + ∂ v L (ψ θs (s))ρ(s)ds. The integrand and therefore the integral is bounded for all ≤˜ if {ψ θ (s) : s ∈ [0, t], θ ∈ [0,˜ ]} ⊆ (−1, 1) × R ∪ ({−1} × (v − , ∞)) ∪ ({1} × (−∞, v + )),(56) as the latter set is the domain on which ∂ x L and ∂ v L are continuous, see Lemma 4.8. As (γ, η) satisfies the Hamilton equations, we haveγ(0) = ∂ p H(γ(0), η(0)) > v − (by Assumption 4.1(c)). Whence we can choose˜ small enough such thatγ(0) + θρ(0) > v − for all θ ∈ [0,˜ ], which implies that (56) holds. Now that it is established that γ starts in the interior, we can apply [CS04, Theorem 6.3.3] to obtain that η(0) = I 0 (γ(0)). (a) For all w ∈ (0, ∂ + ) there exists a v ∈ (w, ∂ + ) such that for all b ∈ [−w, w] and any optimal trajectory γ for I t (b) we have γ(s) ∈ [−v, v] for all s ∈ [0, t]. (b) For all m > 0 there exists a v ∈ (0, ∂ + ) such that for all b ∈ (∂ − , −v) ∪ (v, ∂ + ) and any optimal trajectory γ for I t (b) with dual trajectory η η(t) ≥ m if b ∈ (v, ∂ + ), η(t) ≤ −m if b ∈ (∂ − , −v). Proof. (a) follows from Proposition 4.7 together with Proposition 4.13 (for K = R) or together with Proposition 4.14(b) (for K = [−1, 1]). (b) Fix n such that q + n ≥ m and q − n ≤ −m. As I 0 ∈ C 1,∂ (K), there is a v ∈ (0, ∂ + ) be such that sup x∈(∂ − ,−v) I 0 (x) ≤ q − n , inf x∈(v,∂ + ) I 0 (x) ≥ q + n ,(57) Let z n be as in Lemma 4.4. We may and do assume that v > z n . Let b > v and γ the optimal trajectory for I b (b) and η its dual trajectory. We show that η(t) ≥ y + n (in a similar way one proves that b < −v implies η(t) ≤ y − n ). Suppose that γ(0) > v. Then I 0 (γ(0)) > q + n by (57). As γ(0) > 1 − δ > 1 − δ n > y + n (see (47)), (γ, η) starts and, therefore, stays in the quadrant y + n ,q + n , which implies η(t) ≥ q + n . Suppose that γ(0) ≤ v. Because the complement of the region {x : x ≥ v}×(−∞, q + n ] is preserved (by Lemma 4.4), the Hamiltonian trajectory (γ, η) cannot enter this region. Because γ(t) = b ≥ v > z n ≥ y + n , this implies that η(t) ≥ q + n . Properties of the time-evolved rate function To rigorously study the time-dependent rate function u(t, x) := I t (x) and x → I t (x) for both the ±1-space-model and the R-space-model, we establish local semi-concavity and the boundary behaviour. The local semi-concavity implies we can use sub-gradients as in the classical theory and identify that the push-forward of the gradient of the starting rate function under the Hamiltonian flow contains the reachable gradients of I t . Lemma 5.1. Assume Assumption 2.10. Then I t is locally semi-concave on K • for all t ≥ 0. Moreover, (t, x) → I t (x) = u(t, x) is locally semi-concave on (0, ∞) × K • . Proof. Let w ∈ (0, ∂ + ) and v ∈ (w, ∂ + ) be as in Proposition 4.15. Let x, y ∈ K • be such that x < y, [x, y] ⊆ [−w, w] and q := v + y − x < ∂ + . We show that I t is semi-concave on [x, y] (this is sufficient as for all z ∈ K • there exist w, x, y as above with z ∈ [x, y]). Let λ ∈ [0, 1] and ξ be an optimal trajectory for I t (λx + (1 − λ)y). By the choice of v, we have that ξ attains its values in [−v, v]. In addition, consider the trajectories ξ x , ξ y defined by ξ x (s) = ξ(s) + s t (1 − λ) (x − y) , ξ y (s) = ξ(s) + s t λ (y − x) . Note that ξ x (t) = x and ξ y (t) = y, λξ x + (1 − λ)ξ y = ξ and ξ x (0) = ξ(0) = ξ y (0). The trajectories ξ x , ξ y and ξ take their values in [−q, q]. As L is C 2 on K • × R by Lemma 4.8, we find that L is semi-concave on [−q, q] × R. Thus the first statement follows similarly as in the proof of [CS04, Theorem 6.4.3]: λI t (x) + (1 − λ)I t (y) − I t (λx + (1 − λ)y) ≤ t 0 λL (ξ x (s),ξ x (s)) + (1 − λ)L (ξ y (s),ξ y (s)) + L (ξ(s),ξ(s))ds ≤ t 0 C λ(1 − λ) 2 s 2 + 1 t 2 ((1 − λ) 2 + λ 2 )\|y − x\| 2 ds = 2C t 3 + 1 t λ(1 − λ) 2 \|x − y\| 2 . The 'moreover' statement follows in the same lines as the proof of [CS04, Corollary 6.4.4]. Proposition 5.2. Assume assumption 2.10. Let t > 0. (a) Fix b ∈ K • and let γ be an optimizer for I t (b) and let η be the dual trajectory, then η(t) ∈ D + I t (b). (b) For any b ∈ K • the map that associates to (p t , p) ∈ D * u(t, b) the Hamiltonian trajectory (γ, η) with terminal condition γ(t) = b and η(t) = p is a one-to-one correspondence between D * u(t, b) and optimizers of I t (b). (c) We have {(x, p) \| x ∈ K • , p ∈ D * I t (x)} ⊆ G t . For any b ∈ K • and any element p ∈ D * I t (b) the Hamiltonian trajectory (γ, η) with terminal condition γ(t) = b and η(t) = p yields an optimal trajectory γ for I t (b). Proof. (a) and (b) can be proven as [CS04, Theorems 6.4.8 and 6.4.9] using that optimizers are bounded away from the boundary by Proposition 4.15 (a). The proof of (c) uses a variation of the ideas in the proof of [CS04, Theorem 6.4.9]. Let x ∈ K • and p ∈ D * I t (x). By definition, there are (x k , p k ) ∈ K • × R such that (x k , p k ) → (x, p), I t is differentiable at x k and p k = I t (x k ). Let γ k : [0, t] → K be an optimizing trajectory for I t (x k ) and let η k be the dual trajectory. By Propositions 4.13 and 4.14 (γ k , η k ) satisfies Hamilton's equations and η k (0) = I 0 (γ k (0)). [CS04, Theorem 6.4.8] implies that (γ k (t), η k (t)) = (x k , p k ). By continuous dependence on the final conditions, we have (γ k (s), η k (s)) → (γ(s), η(s)) for all s ∈ [0, t] (see [Per01, Theorem 2.3.2]), where (γ, η) solves the Hamilton equations with (γ(t), η(t)) = (x, p). By continuity of ∂ p H, we also obtainγ k (s) →γ(s) for all s ∈ [0, t]. We obtain that indeed (x, p) ∈ G t . By continuity of I 0 , I t and L we find additionally that I t (x) = lim k→∞ I t (x k ) = lim k→∞ I 0 (γ k (0)) + t 0 L (γ k (s),γ k (s))ds = I 0 (γ(0)) + t 0 L (γ(s),γ(s))ds, thus γ is an optimizer for I t (x). Lemma 5.3. Assume Assumption 2.10. Then sup a∈(0,∂ + ) inf b≥a inf D + I t (b) = ∞, inf a∈(∂ − ,0) sup b≤a sup D + I t (b) = −∞. Proof. First note that D + I t (b) = coD * I t (b). By Proposition 5.2 (c) p ∈ D * I t (b) corresponds to a trajectory (γ, η) that solves the Hamilton equations with terminal condition (γ(t), η(t)) = (b, p), the rest follows by Proposition 4.15 (b). Proposition 5.4. Assume assumption 2.10. Then u is a viscosity solution to the Hamilton-Jacobi equation ∂ t u(t, x) + H (x, ∂ x u(t, x)) = 0 (58) on K • × (0, ∞). Proof. The proof follows as in the proof of [CS04, Theorem 6.4.5], using Proposition 4.15 (a) to make sure that optimal trajectories remain in compact sets bounded away from the boundary and Lemma 5.1 to make sure that u is locally semi-concave on K • ×(0, ∞). Topological structure of G t In this section we study the structure of G t . Inspired by the result of Proposition 4.6 we introduce an extension of the Hamiltonian flow to include the points (∂ − , −∞) and (∂ + , ∞). In this way we make sense of Hamilton paths starting at (x, p) for times t > t x,p . In addition, we extend G to include the points at infinity, we can use the properties of connected sets to study the structure of G t in Proposition 6.2. Proposition 6.1. Assume assumption 2.10. Let E := {(t, x, p) ∈ [0, ∞) × K × R : t < t x,p } . We extend the space K×R with two points and write S := K×R∪{(∂ − , −∞)}∪{(∂ + , ∞)}. We equip S with the topology generated by the open subsets of K × R together with the sets • a,b ∪ {(∂ + , ∞)} and • a,b ∪ {(∂ − , −∞)} for a ∈ K • and b ∈ R. (a) The map (x, p) → t x,p is lower semi-continuous and E → K × R, (t, x, p) → (X x,p t , P x,p t ) is continuous. (b) The map Ψ : [0, ∞) × S → S defined by 5 Ψ(t, x, p) :=              (X x,p t , P x,p t ) if (t, x, p) ∈ E, lim s↑tx,p (X x,p s , P x,p s ) if (x, p) ∈ K × R, t ≥ t x,p , (∂ − , −∞) if (x, p) = (∂ − , −∞), (∂ + , ∞) if (x, p) = (∂ + , ∞). is continuous. (b) The continuity on E follows from (a). The continuity of Ψ at (t, ∂ − , −∞) and (t, ∂ + , ∞) follows by the preservation of quadrants, as mentioned in Remark 4.2 and Remark 4.5. Therefore we are left to prove continuity of Ψ at (t, x, p) with t ≥ t x,p . We may restrict to sequential continuity because the topology on S is metrizable by Urysohn's metrization theorem (e.g. [Run05, Theorem 4.1.10]); the topology is second countable and normal (we leave it to the reader to check those properties). Assume (t, x, p) is such that t ≥ t x,p and Ψ(t, x, p) = (∂ + , ∞) (the case Ψ(t, x, p) = (∂ − , −∞) is similar). Suppose that (x n , p n , t n ) → (x, p, t). It is sufficient to show that for all k ∈ N there exists an N such that for all n ≥ N (X xn,pn tn , P xn,pn tn ) ∈ y k ,q k ∪ {(∂ + , ∞)}.(59) Let k ∈ N and s < t x,p be such that (X x,p s , P x,p s ) ∈ y k ,q k .) = (X x,p s , P x,p s ) ∈ y k ,q k .(60) Therefore there exists an N 1 > N 0 such that (X xn,pn s , P xn,pn s ) ∈ y k ,q k for all n ≥ N 1 . Let N > N 1 be such that t n > s for all n ≥ N . As y k ,q k is preserved under the Hamiltonian flow and t n > s for n ≥ N we obtain that (59) holds for all n ≥ N . (b) X x 1 ,p 1 t < X x 2 ,p 2 t for all t ≤ t 0 and P x 1 ,p 1 t < P x 2 ,p 2 t for all t < t 0 and P x 1 ,p 1 t 0 = P x 2 ,p 2 t 0 = q.(61) If (a) holds then we have a contradiction by Lemma 7.1. If (b) holds we first proceed by using variant (i). As in the proof of the lemma above, we find −∂ x H(X x 1 ,p 1 t 0 , p) = P x 1 ,p 1 t 0 ≥Ṗ x 2 ,p 2 t 0 = −∂ x H(X x 2 ,p 2 t 0 , p) which by the assumed strict monotonicity of ∂ x H on A implies that X x 1 ,p 1 t 0 > X x 2 ,p 2 t 0 . This is in contradiction with (b). Next, we use variant (ii). We find p 2 − p 1 = P x 2 ,p 2 0 − P x 1 ,p 1 0 = t 0 0 ∂ x H(X x 2 ,p 2 s , P x 2 ,p 2 s ) − ∂ x H(X x 1 ,p 1 s , P x 1 ,p 1 s )ds. By assumption, the left hand side is strictly greater than 0. For 0 ≤ s ≤ t 0 however, we have X x 1 ,p 1 s < X x 2 ,p 2 s and P x 1 ,p 1 s < P x 2 ,p 2 s , which by (ii) is non-positive. Proof of Theorem 3.5. For both (a) and (b) we use the following for t > 0. By Proposition 6.2(f) it follows that if γ t is strictly increasing, then G t is a graph and thus I t ∈ C 1,∂ (K) by Theorem 2.13. Theorem 3.5(a) follows then immediately from Proposition 7.2 as convexity of a differentiable function implies that its derivative is increasing. That γ t is strictly increasing outside (−a, a) follows by the assumed strict convexity of I 0 and that H preserves order by using Proposition 7.2. We will show that γ t is differentiable and that γ t (x) > 0 for all x ∈ [−a, a] and all t small enough. As [−a, a] × [−b, b] is compact, by Proposition 6.1(a) there exists a t * > 0 such that U := (0, t * ) × (−a, a) × (−b, b) ⊆ E. By [Per01, Theorem 2.5.1] the map Ψ is C 1 on U . As x → (x, I 0 (x)) is C 1 by assumption, the map (t, x) → Ψ(t, x, I 0 (x)) is C 1 on (0, t * ) × [−a, a]. By definition γ t (x) is the first coordinate of Ψ(t, x, I 0 (x)) and therefore (t, x) → γ t (x) is also C 1 on (0, t * ) × [−a, a]. Note that t → inf x∈[−a,a] γ t (x) is continuous and equal to 1 for t = 0. Therefore there exists a 0 < t 0 ≤ t * such that γ t (x) > 0 for all 0 < t < t 0 and x ∈ [−a, a]. Appearance of non-differentiability In this section, we introduce the methods necessary for the proofs of Theorem 3.6. Linearization around stationary points We start by studying linearizations of the Hamiltonian flows around stationary points. In contrast to homeomorphism between flows, see e.g. [Per01], C 1 diffeomorphism are difficult to construct. We will refer to [GHR03] for one such construction. Theorem 8.1. [GHR03,Theorem 3] Let H satisfy Assumption 2.10 and infinitely many times continuously differentiable, assume that x 0 is a stationary point and that m = 0 (m as in (25)). Then H admits a C 1 linearization at (x 0 , 0). Proof of Theorem 3.6 (a). We show that γ t (x 0 ) < 0, which establishes an overhang at x 0 by Proposition 6.2(f). We first establish this for the case that m = 0. The linearized system (24) with (ξ x (0), ζ x (0)) := (x, xI 0 (x 0 )) is solved by ξ x (t) ζ x (t) = exp t m c 0 −m x xI 0 (x 0 ) = x e mt + c 2m (e mt − e −mt )I 0 (x 0 ) e −mt I 0 (x 0 ) . e mt + c 2m (e mt −e −mt )I 0 (x 0 ) < 0 if t > t 0 , whence ∂ x ξ x (t)\| x=x 0 < 0. Denote by Θ the first component of Ψ −1 , the inverse of the linearization (see Definition 3.3). As DΨ(x 0 , 0) = 1l, the identity matrix, by the inverse function theorem we have DΨ −1 (x 0 , 0) = 1l and thus ∂ ξ Θ(x 0 , 0) = 1 and ∂ ζ Θ(x 0 , 0) = 0. Therefore γ t (x 0 ) = ∂ ξ Θ(ξ x 0 (t), ζ x 0 (t))∂ x ξ x (t)\| x=x 0 + ∂ ζ Θ(ξ x 0 (t), ζ x 0 (t))∂ x ζ x (t)\| x=x 0 < 0, In case m = 0, the linearized system (24) with (ξ x (0), ζ x (0)) := (x, xI 0 (x 0 )) is solved by ξ x (t) ζ x (t) = exp t 0 c 0 0 x xI 0 (x 0 ) = x 1 + tcI 0 (x 0 ) I 0 (x 0 ) . As 1 + tcI 0 (x 0 ) < 0 for t > t 0 , again we obtain ∂ x ξ x (t) < 0 so that the result follows as for the case m = 0. Remark 8.2. Note that we actually proved that {y ∈ R : (x 0 , y) ∈ G t } ≥ 3 has at least three elements. Rotating areas in the Hamiltonian flow In this section, we study 'rotating loops' in the Hamiltonian flow, introduced in Definition 8.3, and their connection to the emergence of points of non-differentiability. We give a short overview: • Under the Condition of Theorem 3.6 (b), we find a rotating loop in the Hamiltonian flow that intersects G : solutions of the flow on this loop come back to the same point in some finite time. (Lemma 8.6.) • Consider some fixed rotating loop. A homotopy argument, considering the space punctured space K × R without the interior of this loop, can be used to show that any intersection of the loop with the graph G of the gradient of a rate function I 0 implies that for large t there is an element of G t 'above' the loop. Similarly, one can show that for large t there is an element of G t 'below' the loop. This will lead to the creation of an overhang. (Theorem 8.5.) • First, we characterise rotating loops by their insides. (Lemma 8.4) Definition 8.3. We call a set L ⊂ K × R a loop if there exist a, b ∈ K, a < b and functions g, h ∈ C[a, b] with g(x) < h(x) for x ∈ (a, b), g(a) = h(a) and g(b) = h(b) such that L = g[a, b] ∪ h[a, b]. Let (x, p) ∈ K × R be a non-stationary point, E = H(x, p). If there exists a t 0 > 0 such that (X x,p t 0 , P x,p t 0 ) = (x, p), then L = {(X x,p t , P x,p t ) : t ∈ [0, t 0 ]} is called a rotating loop. E is called the energy of L. Let a be the largest value and b be the smallest value for which L ⊆ [a, b] × R. The set A = {(x, p) ∈ [a, b] × R : H(x, p) < E} is called the inside of L. Note that if H satisfies Assumption 2.10, then a rotating loop is a loop. Then ∂ p H(a, q) = ∂ p H(b, r) = 0 and (a, q) (b, r) h g y z (a, q) (b, r) A yaz A ybz (a) (b)∂ p H(x, h(x)) > 0, ∂ p H(x, g(x)) < 0 for x ∈ (a, b),(62)∂ x H(a, q) = 0 =⇒ ∂ x H(a, q) < 0,(63)∂ x H(b, r) = 0 =⇒ ∂ x H(b, r) > 0.(64) L is a rotating loop if 6 the inside of L is a set A as above with Consequently, if L is a rotating loop, then there exists a t 0 such that G t contains an overhang for all t ≥ t 0 . Proof. We prove (a) only, as the proof of (b) is similar. Suppose that t 1 is as in (a). Write L for the half-line {x 0 } × [h(x 0 ), ∞) and write Θ : [0, ∞) × [∂ − , ∂ + ] → S for the function given by Θ(t, x) = Ψ(t, x, I 0 (x)) (with the convention that I 0 is defined in ∂ − and ∂ + as −∞ and ∞, respectively) where Ψ is as in Proposition 6.1. Note that Θ is continuous by continuity Ψ and because I 0 ∈ C 1,∂ (K). We prove that for all t ≥ t 1 there exists an x ∈ (∂ − , x 1 ] such that Θ(t, x) ∈ L. The idea is that the curve Θ(t, [∂ − , x 1 ]) is pulled through the line L by the rotation of Φ t (x 1 ) over L , and as it is connected to (∂ − , −∞) it will be connected via L for all larger times. We use an argument using homotopy theory to prove this. (1 − s)). Γ is continuous and therefore a path homotopy between the paths γ t : [0, 3] → S \ A for t ∈ [0, ∞) where γ t (s) = Γ(t, s) and A is the inside of L. Whence γ t is path homotopic to γ 0 in the space S \ A and thus to the single point (∂ − , ∞). That a closed path is homotopic to another one in a topological space, basically means that one can continuously transform one path in that space to the other, being homotopic to a point means homotopic to a path that stays at that point. In [Run05, Section 5] one finds the necessary background for homotopy theory. {x 0 } × [h(x 0 ), ∞) Θ(0, [∂ − , x 1 ]) Θ(t, [∂ − , x 1 ]) (a, q) (b, r) Θ(0, x 1 ) Θ(t, x 1 ) Θ([0, t], x 1 ) (∂ − , −∞) We will use the following fact: γ([0, s 1 )) ∪ γ((s 2 , 3]) ⊆ (−∞, x 0 ) × R ∪ {(∂ − , −∞)}. This fact can be proven as follows; for simplicity with (0, 0) instead of (x 0 , p 0 ) and (−1, 0) instead of (∂ − , −∞). Every closed path is homotopic to s → (− cos(2πks), sin(2πks)) for some k ∈ Z (see, e.g. [Run05, Example 5.2.7]). It is straightforward to check that such s 1 and s 2 do not exist in case k = 0. Let p 0 ∈ (g(x 0 ), h(x 0 )). Let t ≥ t 1 . As x 1 < x 0 , s 1 ≥ 1. By the choice of t 1 we have s 1 ∈ [1, 2] and γ t (s 1 ) ∈ {x 0 } × (p 0 , ∞). As the ∂ p H at (x 0 , h(x 0 )) is strictly positive (see (62)), the s 2 as above cannot be in [1, 2). Thus s 2 ∈ [2, 3], which proves that there exists an x ∈ [∂ − , x 1 ] such that Θ(t, x) ∈ {x 0 } × [h(x 0 ), ∞). The following lemma, establishes the existence of a rotating loop L with G ∩ L = ∅ under the assumptions made in Theorem 3.6(b), so that with Theorem 8.5 the statement of Theorem 3.6(b) is proven. Proof. By Lemma 8.4 it is sufficient to show that there exists an E < 0 such that the set A = {(x, p) ∈ (m 1 , m 2 ) × R \| H(x, p) < E} ,(67) is as in Lemma 8.4 such that G ∩ A = ∅ and (65) holds. To find such an E so that A is connected, we consider the function that gives the minimum of H at x, E(x) = inf p∈R H(x, p). We use that a set A as (67) is connected as soon as (x 1 , p 1 ), (x 2 , p 2 ) ∈ A imply that E(x) < E for all x ∈ [x 1 , x 2 ]. Let p(x) = argmin q∈R H(x, q), so that q = p(x) if and only if ∂ p H(x, q) = 0 and E(x) = H(x, p(x)). By Assumption (i) E(m 1 ) = 0 = E(m 2 ) and E < 0 on (m 1 , m 2 ). As H is C 3 , p is C 2 by the implicit function theorem and p (x) = − ∂ x ∂ p H(x, p(x)) ∂ 2 p H(x, p(x)) .(68) For the proof of (d) we apply Theorem 3.6(a). We consider the stationary point x 0 = 0. First, note that m = ∂ x ∂ p H(0, 0) = 2(β − 1) = 0 and c = ∂ 2 p H(0, 0) = 4. As the Hamiltonian is C ∞ , there is a C 1 linearization of the Hamiltonian flow at (0, 0) by Theorem 8.1. Explicit calculation yields I 0 (0) = 0 and I 0 (0) = (1 − α). The condition I 0 (0) < − 2m c ∧ 0 translates into 1 ∨ β < α and (26) into t 1 as in (ii). For (e), first we make the following observations. By Remark 3.8 we see that H(x, p) = 0 if and only if p = 0 or p = f β (x), where f β (x) = arctanh(x) − βx.(71) As β > 1 and arctanh (x) = 1 1−x 2 , the function f β intersects the x-axis at 3 points, m − , 0, m + . As 1 < α < β, then function I 0 , being f α (see (36), f α is as f β in (71)), intersects the x-axis at 3 points too, y − , 0, y + and m − < y − < 0 < y + < m + (see Figure 5). Moreover, the graph of I 0 has a nonempty intersection with B − and B + , where B − = {(x, p) ∈ (m − , 0) × R : H(x, p) < 0}, B + = {(x, p) ∈ (0, m + ) × R : H(x, p) < 0}. First we will show that G t has no overhang at γ t (y − ), at 0 and at γ t (y + ). Let x < y − , then I 0 (x) < 0 and H(x, I 0 (x)) = 0, so that P x,I 0 (x) t < 0 for all t ≥ 0. With Lemma 7.1 this implies that γ t (x) < γ t (y − ) for all t. On the other hand, Φ t (x) ∈ B 1 for all x ∈ (y − , 0) and all t, whence Lemma 7.1 implies γ t (y − ) < γ t (x). So that (by symmetry) we obtain for x 1 , x 2 , x 3 , x 4 with x 1 < y − < x 2 < 0 < x 3 < y + < x 4 and all t ≥ 0 (as long as γ t (x i ) exists) γ t (x 1 ) < γ t (y − ) < γ t (x 2 ) < 0 < γ t (x 3 ) < γ t (y + ) < γ t (x 4 ). So indeed, G t has no overhangs at γ t (y − ), 0 and γ t (y + ). By Proposition 2.15 we obtain that I t is non-differentiable at two points x − ∈ (m − , 0) and x + ∈ (0, m + ), as soon as G t has an overhang in B − and B + . The existence of a t 2 such that for t ≥ t 2 this is the case follows from Theorem 3.6(b), as soon as the conditions (i) and (ii) are satisfied for both (m 1 , m 2 ) = (m − , 0) and (m 1 , m 2 ) = (0, m + ). We show that this is indeed the case. By (31) we see that ∂ p H(x, 0) = 0 if and only if arctanh(x) = βx, whence it is clear that (i) of Theorem 3.6(b) is satisfied. By (33) we see that ∂ p ∂ x H(x, 0) = 0 if and only if 1 − β + βx tanh(βx) = 0. (72) f β has a local maximum or minimum at x if f β (x) = 0, which is the case if and only if 1 − β + βx 2 = 0. At those points f β is not equal to zero. By definition of m ± we have 1 − β + βm ± tanh(βm ± ) = 1 − β + βm 2 ± , from which we conclude that (72) does not hold for x = m ± . Similarly, we have that (72) does not hold at x = 0. This proves condition (ii). Proof of Theorem 3.11. First we make the following observation. Observation Note that the Hamiltonian dynamics for the momentum is autonomous: P = sinh(2P ). Therefore t x 1 ,p = t x 2 ,p < ∞ for all p = 0 and x 1 , x 2 ∈ [−1, 1], and, if 0 < \|p 1 \| < \|p 2 \|, then t x 1 ,p 1 < t x 2 ,p 2 . By Proposition 3.9 H preserves order [−1, 1] × R. (a) Let x 0 be the unique solution to I 0 = 0. Then there exists a δ > 0 and a neighbourhood U of x 0 , such that I 0 is strictly increasing on U and such that I 0 (x) < δ implies x ∈ U . Let t * = t 0,δ . As we saw in our observation this implies that t z,I 0 (z) < t * for z / ∈ U and thus E t as in Proposition 6.2 is a subset of U for t ≥ t * . As H preserves order, this implies that γ t is strictly increasing on E t , i.e., G t is a graph. (b) Fix α > 1. We write g θ (x) = arctanh(x) − αx − θ. There exists a z > 0, such that −z is a local maximum and z is a local minimum of g θ for all θ. Let κ = g 0 (−z) = −g 0 (z) (note that the graph of g κ looks like the graph of I 0 , the dashed blue graph in Figure 3.2). For the rest of the proof we let θ be fixed and such that −θ > κ > 0. (By symmetry we could have also treated θ > κ.) Let t 1 = t −z,g θ (−z) . We show in STEP 1 that there exists an overhang at a time before t 1 , so that we can take t 0 to be be the infimum of all times at which there is an overhang. By the choice of θ, g θ = I 0 has a unique zero, which is the unique global minimiser of I 0 . By (a) there exists a t * such that I t is C 1 and so that there is no overhang for t ≥ t * . We let t 2 be the supremum of all times at which there is an overhang. In STEP 2 we show that I t is non-differentiable for t ∈ (t 0 , t 1 ). STEP 1 By our observation, t z,I 0 (z) < t 1 and thus M := lim t↑t 1 P z,g θ (z) t < ∞. By Lemma 4.4 there exists a u ∈ (0, 1) such that γ t (z) ≤ u for all t ≤ t 1 . As P −z,g θ (−z) t → ∞ as t ↑ t 1 , by Lemma 4.4 again (or Lemma 4.6) we have lim t↑t 1 γ t (−z) = 1. Whence there exists a t < t 1 such that γ t (−z) > u ≥ γ t (z), i.e., there is an overhang. STEP 2 Let y < −z be the unique point such that I 0 (y) = 0, and let w > z be such that g θ (w) = g θ (−z). We show that for all t < t 1 , there is no overhang in (−1, γ t (y)) and in (γ t (w), 1), so that Proposition 2.15 implies that I t is non-differentiable for t ∈ (t 0 , t 1 ). But this follows as g θ is strictly increasing on (−1, −z) and on (z, 1), and g θ < 0 on (−1, y) and g θ > g θ (−z) on (w, 1) by the fact that H preserves order. Remark 9.1. Note that in our proof t 1 is strictly less than t 2 ; Indeed, for t > t 1 and t < t z,g θ (z) the set Φ t (E t ∩ [−1, −z]) connects (−1, −∞) with (1, ∞) and the set Φ t (E t ∩ (−z, 1]) is non-empty as it contains Φ t (z). A The verification of Assumption 4.1 for the main examples In this appendix, we show that Hamiltonians that satisfy Assumption 2.10 (a) in fact satisfy Assumption 2.10 (b). In addition, (e) follows by an explicit computation using Assumption 2.3(c). A.1 Verification for the Curie-Weiss example We are left to verify (b). Pick some some compact set K ⊆ (−1, 1). We consider the function θ K = θ with θ(r) = 1 4 max{r log r, 1}. Property (i) is immediate. We proceed with the proof of (ii). Let M ≥ 0. Note that The latter fraction is indeed bounded. As p → ψ(p) − pe 2p − 1 2 e 2p log(2b) is bounded from below for p ∈ [0, ∞) this implies that there exists a c > 0 such that the inequality in(73) holds for p with \|p\| ≥ p 0 . As (76) implies that p∂ p H(x, p) − H(x, p) ≥ 0 and as θ(\|∂ p H(x, p)\|) is bounded from above for x ∈ K and \|p\| ≤ p 0 , we can choose c such that (73) holds for all p ∈ R. By (75) we have \|∂ x H(x, p)\| + \|p\| ≤ be 2\|p\| + \|p\| for all x ∈ K and p ∈ R. To conclude (iv), by (78) it is sufficient (and not difficult) to see that there exists a c such that be 2\|p\| + \|p\| ≤ c 1 2 e 2\|p\| (2\|p\| + log(2a)) in case \|p\| ≥ p 0 and be 2\|p\| + \|p\| ≤ c in case \|p\| ≤ p 0 (as θ ≥ 1). With this we have ∂ y L (y, 0) = v + (y) v + (y) − v − (y) v − (y) v + (y) − v − (y) . By Assumption 2.3 (a) and (c) it then follows that ∂ y L (y, 0) converges to −∞ at −1 and to ∞ at 1. This shows that (c) is satisfied. For (d) we consider the −1 boundary, the other case follows similarly. By Cauchy's mean-value theorem, there exists y ∈ (−1, x) such that L (x, 0) − L (−1, 0) S(x) − S(−1) = ∂ y L (y, 0) ∂ y S(y) . Whence (using Cauchy's mean-value theorem) lim y→−1 ∂ y L (y, 0) ∂ y S(y) = v − (−1) v + (−1) lim y→−1 −v − (y) − 1 2 1 2 log v − (y) − 1 2 log v + (y) = v − (−1) v + (−1) lim y→−1 v − (y)v − (y) − 3 2 v − (y)v − (y) −1 − v + (y)v + (x) −1 = ∞. • considering the R-space-model that (a) or (b) is satisfied: (a) H is of the form (14), where W satisfies Assumption 2.4. (b) H satisfies Assumptions 4.1 and 4.11 below. • considering the ±1-space-model that (a) or (b) is satisfied: (a) H is of the form (15), where v + , v − satisfy Assumption 2.3. (b) H satisfies Assumptions 4.1 and 4.10 below. Theorem 2.11 (Lemma 5.1 and 5.3). Assume Assumption 2.10. Then I t is locally semi-concave on K • for all t ≥ 0. Moreover, (t, x) → I t (x) is locally semi-concave on (0, ∞) × K • and sup a∈(0,∂ + ) the solution of the Hamilton equations (19) with initial conditions (X x,p 0 , P x,p 0 ) = (x, p) and up to the maximal time of existence t x,p . (See [Per01, Theorem 2.4.1], which by Assumption 4.1(a) can also be applied in the case that K equals [−1, 1].) Definition 3. 3 . 3Assume Assumption 2.10. Let x 0 ∈ K • be a stationary point for the Hamiltonian flow, i.e., ∂ p H(x 0 , 0) = 0. We say that the Hamiltonian flow admits a C 1 linearization at (x 0 , 0) if there exists an open neighbourhoods Figure 1 : 1I 0 of (29), I 0 and I 0 for different values of α and θ. Proposition 3.9. (a) Assume Assumption 2.3 for the ±1-space-model, with H as in (15). In addition, suppose v + (1) < 0 and v − (−1) < 0 on a neighbourhood of −1.Then H preserves order at infinity. and H(x, p) = 0 if and only if either p = 0 or p = arctanh(x)−βx, a Hamilton trajectory in • 0,0 could only leave this set via the point (x, 0) with H(x, 0) = 0, which is a stable point (see Remark 3.8) and therefore cannot be reached. Whence H preserves • 0,0 and similarly • 0,0 . Proposition 7.2 implies that H preserves order on both • 0,0 and • 0,0 , and whence on the union of those sets with {0}. Theorem 3.11 (Loss and recovery). Let K = [−1, 1] and H and I 0 be as in (28) and (29) with β = h = 0. Remark 3. 14 . 14Consider a stationary point x 0 . The condition that ∂ p ∂ x H(x 0 , 0) = 0 in both Theorem 3.6 (a) and (b) is reflected by the exclusion of having b = 0 and β = 1, i.e., ∂ p ∂ x H(0, 0) = 0, in Theorem 3.10 (d). (a) and (b) are there to study the Hamiltonian flow in open subsets of K × R. In particular, these conditions suffice to apply the methods described in [CS04], compare with Conditions (L1)-(L4) on L in [CS04], as long as Hamiltonian trajectories remain inside such open sets.The assumptions (c)-(e) are made to study behaviour of Hamiltonian trajectories at the boundary, or to show that Hamiltonian trajectories stay away from the boundary. Assumption 4. 1 . 1The Hamiltonian H : K × R → R satisfies H(x, 0) = 0 for all x and (a) H is C 2 and ∂ 2 p H(x, p) > 0 for all (x, p) ∈ K × R. If K = [−1, 1], then we additionally assume that there exist an > 0 and a twice continuously differentiable functionH : (−1 − , 1 + ) × R → R such that H equals H on [−1, 1] × R. (b) For every compact set K ⊆ K • , there exists a function θ K : [0, ∞) → [0, ∞), with the properties Remark 4. 2 . 2• By Assumption 4.1 (e) y + n ,q + n and y − n ,q − n are preserved under H. • Assumption 4.1(c) implies that for every compact set K ⊆ K • and c ∈ R that the set (K × R) ∩ H −1 (−∞, c] is compact. Lemma 4. 3 . 3Let K = [−1, 1]. Then Assumption 4.1(a) and (d) imply v − := lim p→−∞ ∂ p H(−1, p) ≥ 0, v + := lim p→∞ ∂ p H(1, p) ≤ 0, a,q ∪ • b,r ) c is a subset of the union of the compact set [−1, 1] × [r, q] with [−1, u] × [0, ∞), [v, 1] × (−∞, 0] and [u, v] × R for any u < a and v > b. Whence to show that (45) is compact, it is by Remark 4.2 sufficient to show that there exist u and v in (−1, 1) such that [−1, u] × [0, ∞) ∩ H −1 (−∞, c] and [v, 1] × (−∞, 0] ∩ H −1 (−∞, c] are compact. There exists an u such that argmin p∈R H(x, p) < 0 for all x ≤ u. Therefore there exists an > 0 such that ∂ p H(x, p) > for all x ∈ [−1, u] and all p ≥ 0 (see also (46)). As H(x, 0) = 0 for all x, this implies that there exists an M > 0 such that for p ≥ M and x ∈ [−1, u] we have H(x, p) > c. This proves that [−1, u]×[0, ∞) ∩H −1 (−∞, c] is compact. The existence of a v can be proved in the same way. ( 48 ) 48and (49) (together with Assumption 4.1(e)) imply that the complements of the sets {x : x ≤ z n } × [q − n , ∞) and {x : x ≥ z n } × (−∞, q + n ] are preserved. See Figure 4.1(b) for a picture. Proof. Fix n. By Assumption 4.1(d) we can choose −z n and z n in K • inf x≤−zn Figure 2 : 2(a) complement of quadrants, (b) push from boundary. ( X x,p t , P x,p t ) takes its values in {(∂ − , −∞), (∂ + , ∞)}.Proof. By [Per01, Theorem 2.4.3] (in case K = [−1, 1], then applied to the Hamilton equations for the extended Hamiltonian as in Assumption 4. Proposition 4. 7 . 7Let H satisfy Assumption 4.1. For all u ∈ (0, ∂ + ) there exists a v ∈ (u, ∂ + ) such that for all t ≥ 0 and all starting points and momenta Lemma 4. 8 . 8Let K = [−1, 1] and let H satisfy Assumption 4.1. Then L (−1 proves that for all v ∈ (v − , ∞) there exists a unique p such thatL(v) = pv − H(−1, p), that lim v→∞ L(v) v = ∞, that H(−1, ·) isthe Legendre transform of L (see the Fenchel-Moreau Theorem [RAS15, Chapter 5]) and that p ∈ D − L(v) if and only if v ∈ D − H(−1, p) (see [RAS15, Theorem 5.22]). With this one proves with the same argument as in the proof of [CS04, Theorem A.2.4] that L is C 1 on Lemma 4. 9 . 9Let K = [−1, 1] and let H satisfy Assumption 4.1. Suppose that (γ, η) : (0, t] → [−1, 1] × R is C 1 and satisfies the Hamilton equations (19). Then (γ(0), η(0)) := lim s↓0 (γ(s), η(s)) exists in [−1, 1] × R and (γ, η) is C 1 on [0, t] and satisfies the Hamilton equations. Proof. Define the time inverted trajectories γ * , η * on [0, t) by γ * (s) := γ(t − s), η * (s) := η(t − s).Thenγ * (s) η * (s) = −∂ p H(γ * (s), η * (s)) ∂ x H(γ * (s), η * (s)) .(52)Because (γ * , η * ) solves the time-inverted Hamilton equations, (52), Assumption 4.1(e) implies that if n ∈ N is such that (γ * (0), η * (0)) is not in the sety + n ,q + n ∪ y − n ,q − n(53)then (γ * (s), η * (s)) is not in this set for all s ∈ [0, t). As H(γ * (s), η * (s)) = H(γ * (0), η * (0) =: c for all s, and the complement of (53) intersected with H −1 ({c}) is compact by Lemma 4.3, by [Per01, Theorem 2.4.3] we find that the maximal interval of existence (γ * , η * ) satisfying (52) is larger than [0, t), which implies that both limits lim s↓0 (γ(s), η(s)) = lim s↑t (γ * (s), η * (s)) and lim s↓0 (γ(s),η(s)) = − lim s↑t (γ * (s),η * (s)) exist. Additionally, we conclude that the trajectory (γ, η) solves the Hamilton equations on the interval [0, t]. Assumption 4.10. Let K = [−1, 1] and H satisfies Assumption 4.1 and there is a function S : [−1, 1] → [0, ∞] that is twice continuously differentiable on (−1, 1) such that (a) S is a Lyapunov function forẋ = ∂ p H(x, 0), i.e. if x(t) solvesẋ = ∂ p H(x, 0) then t → S(x(t)) is decreasing. (b) H(x, S (x) − p) = H(x, p) for all (x, p) ∈ (−1, 1) × R. (c) The map x → L (x, 0) is decreasing on a neighbourhood U −1 of −1 in [ Lemma 4. 12 . 12Let H satisfy Assumption 4.1. If K = R, let H in addition satisfy Assumption 4.11Let I 0 have compact sublevel sets, then the rate function I in (13) has compact sublevel sets. Proposition 4. 14 . 14Let H satisfy Assumption 4.10 in the setting that K = [−1, 1], I 0 ∈ C 1,∂ [−1, 1] and let t > 0 and a, b ∈ [−1, 1]. η) satisfies the Hamilton equations (19) on [0, t) and lim s↑t (γ(s), η(s)) ∈ {(−1, −∞), (1, ∞)}. Proposition 4. 15 . 15Let H satisfy Assumption 4.1 and if K = [−1, 1] additionally assume Assumption 4.10. Let I 0 ∈ C 1,∂ (K) and let t > 0. Theorem 3.5(b). Let a, b > 0, a ∈ K • be such that H preserves order on −a,−b, and on a,b , I 0 ([−a, a]) ⊆ [−b, b] and I 0 is convex on (∂ − , −a] and on [a, ∂ + ). Figure 3 : 3(a) Functions g and h, (b) The arcs A y,a,z and A y,b,z . Lemma 8.4. Let H satisfy Assumption 2.10. Let E ∈ R. Suppose that A is nonempty, relatively compact and a connected component of H −1 (−∞, E) and that ∂A is a connected component of H −1 ({E}). Then ∂A is a loop and with a, b, g, h as in Definition 8.3, the functions g and h are C 1 on (a, b). We write q := g(a) = h(a) and r := g(b) = h(b). ∂ x H(a, q) = 0 0and ∂ x H(b, r) = 0. (65) Proof. Let a and b be the smallest and largest element in the set {x : ∃p, (x, p) ∈ A}, respectively. By strict convexity of p → H(x, p), we have ∂A = {(x, p) ∈ A : H(x, p) = E}. This strict convexity together with the implicit function theorem, implies the existence of g, h as in Definition 8.3 begin C 1 on (a, b) and it implies (62). Let σ : [0, 1] → [∂ − , x 1 ] be a continuous function with σ(0) = ∂ − and σ(1) = x 1 . Define the map Γ : [0, ∞) × [0, 3] as follows, Γ(t, [0, 1]) is the set Θ(0, [∂ − , x 1 ]); Γ(t, [1, 2]) is the set Θ([0, t], x 1 ), i.e., the Hamiltonian path starting at (x 1 , I 0 (x 1 )) up to time t; Γ(t, [2, 3)) is the set Θ(t, [∂ − , x 1 ]) (see also Figure 4); more precisely, for t ∈ [0, ∞) and s ∈ [0, 1] Γ(t, s) := Θ(0, σ(s)), Γ(t, 1 + s) := Θ(ts, x 1 ), Γ(t, 2 + s) := Θ(t, σ Figure 4 : 4Θ(0, [∂ − , x 1 ]), Θ(t, [∂ − , x 1 ]) and Θ([0, t], x 1 ). A closed path γ : [0, 3] → S\{(x 0 , p 0 )} is homotopic to the point (∂ − , −∞) if and only if there exist s 1 , s 2 ∈ [0, 3], s 1 < s 2 such that γ(s 1 ), γ(s 2 ) are either both in {x 0 } × (p 0 , ∞) or both in {x 0 } × (−∞, p 0 ) and s 1 , s 2 are the first and last time such that the path crosses the line {x 0 } × R, respectively, i.e., Lemma 8. 6 . 6Assume Assumption 2.10 and suppose that H is C 3 . Suppose that m 1 , m 2 ∈ K • are two points such that m 1 < m 2 and (i) ∂ p H(m 1 , 0) = 0 = ∂ p H(m 2 , 0) and ∂ p H(x, 0) = 0 for all x ∈ (m 1 , m 2 ), (ii) ∂ x ∂ p H(m 1 , 0) = 0 and ∂ x ∂ p H(m 2 , 0) = 0. If {(x, p) ∈ (m 1 , m 2 ) × R \| H(x, p) < 0} ∩ G = ∅, then there exists a rotating loop L such that G ∩ L = ∅. Figure 5 : 5f β , f α , their zero's and regions B ± . Lemma A. 2 . 2H is as in (15) then for all x ∈ (−1, 1) and v ∈ R there exists a uniquep for which v = ∂ p H(x, p), L (x, v) = p∂ p H(x, p) − H(x, p), ∂ x L (x, v) = −∂ x H(x, p), and ∂ v L (x, v) = p.Proof of Lemma A.1. (a) follows from Assumption 2.3(b). (c) follows directly from Assumption 2.3(a). For (d), note that p(x) := argmin p H(x, p) satisfies ∂ p H(x, p) = 0. By Assumption 2.3(a), we find ∂ p H(−1, p) > 0 and ∂ p H(1, p) < 0 for all p, implying by the continuity of ∂ p H that (d) holds. sup{θ(r + m) : r ∈ [0, M ∨ 2], m ∈ [0, M ]} < ∞.Whence it is sufficient for (ii) to show that θ(r + m)/θ(r) is bounded from above for r ≥ M ∨ 2 and m ∈ [0, M ]. For such m and r we have m ≤ M ≤ r and thusθ(r + m) θ(r) = r + m r log(r + m) log r ≤ 2 log(r + m) log r . Lemma A. 3 . 3Let H satisfy Assumption 2.10(a) for the ±1-space-model, then H satisfies Assumption 4.10. Proof. Assumption 4.1 has been verified in Lemma A.1. We consider S with S(x) = 1 2 log v − (x) v + (x) . Note that lim x↓−1 S (x) = −∞ and lim x↑1 S (x) = ∞ because of Assumption 2.3 (a). The integration constant of S is chosen by choosing the infimum of S equal to 0. As v + , v − are twice-continuously differentiable and positive on the interior, also S is twice continuously differentiable on the interior.We leave the calculations for Assumptions 4.10(a) and (b) for the reader (for (a) one computes that ∂ p H(x, 0)S (x) ≤ 0 for all x ∈ (−1, 1)).By(b), ∂ p H(x, p) = −∂ p H(x, S (x) − p), thus ∂ p H(x,1 2 S (x)) = 0 and so L (x, 0) = H x, 1 2 S (x) . an extensions to an open set V ⊆ R that contains [−1, 1] that are two times continuously differentiable, satisfies the Hamilton equations (19) on the interval [0, t]. By (a) we have lim inf n→∞ t xn,pn ≥ t x,p . Whence there exists a N 0 such that t xn,pn > s for all n ≥ N 0 . As s < t xn,pn and s < t x,p we have by (a)lim n→∞ (X xn,pn s , P xn,pn s Lemma A.1. Let H satisfy Assumption 2.10(a) for the ±1-space-model, then H satisfies Assumption 4.1. For convenience in the proof, we recall part of [CS04, Corollary A.2.7]. In case K = R, the conditional measure on the left hand side of (8) has to be understood in terms of weakly continuous regular conditional probabilities as is done in[HRZ15]. D K ([0, ∞)) is the Skorohod space of càdlàg paths [0, ∞) → K, see also [EK86, Section 3.5]. See Theorem 8.1. As β > 1 such m− and m+ exist. Note that lim s↑tx,p (X x,p s , P x,p s ) is welldefined by Lemma 4.6. Actually "if and only if"; as we don't use this we leave out the proof. WvZ is supported by the German Science Foundation (DFG) via the Forschergruppe FOR2402 "Rough paths, stochastic partial differential equations and related topics".Proposition 6.2. Assume Assumption 2.10. Define by E t := {x ∈ K : t x,I 0 (x) > t} and Φ t : E t → K × R be defined by Φ t (x) := X x,I 0 (x) t , Px,I 0 (x) t . Note that G t = Φ t (E t ).(a) E t is open for all t, Φ t is continuous and injective on E t .(b) Let t > 0 and suppose that (a, b) ⊆ E t , a, b / ∈ E t . Then Φ t (a, b) is path connected and both lim z↓a Φ t (z) and lim z↑b Φ t (z) are in {(∂ − , −∞), (∂ + , ∞)}.(c) If (a, b), (c, d) are consecutive maximally (as in (b)) connected components in E t , then lim z↑b Φ t (z) = lim z↓c Φ t (z).(d) LetThenProof. Note that I 0 is continuous and that Φ t (x) = Ψ(t, x, I 0 (x)) for x ∈ E t , with Ψ as in Proposition 6.2. By this one deduces (a), (b) and (c). (d) follows from preservation of quadrants and the fact that I 0 ∈ C 1,∂ (K). (e) then follows as K → S given by x → Ψ(t, x, I 0 (x)) is continuous and connects (∂ − , −∞) with (∂ + , ∞) by (d). (f) follows from the previous. Remark 6.3. By Proposition 6.2(e), we see that if E t has at least two disconnected components, then G t has an overhang at an x 0 . But in this situation, generally one cannot find x 1 , x 2 at which G t has no overhang, with x 1 < x 0 < x 2 , so that we cannot apply Proposition 2.15 to conclude that I t is not differentiable.Dynamics that preserves orderIn this section, we give the proof of Theorem 3.5 and give sufficient conditions on H for it to preserve order (in Proposition 7.2, see Definition 3.2(c)). First we prove in Lemma 7.1 that if the order of the momentum is preserved along some time interval, that the order of the position is also preserved.Proof. Assume the contrary, that there is someThen H preserves order on A.Proof. Assume the contrary, that there is some. We assume that t 0 is the smallest of such times s. Then we have exactly one of the following two situations as the Hamiltonian trajectories do not meet:We prove (63), the proof of (64) being similar. Suppose that ∂ x H(a, q) = 0. By the implicit function theorem, ∂A can be described bya C 1 function around (a, q), i.e., there exists a δ > 0 and a C 1 function j :Note that j (p) = 0 if and only if p = q. Therefore for p ∈ (q, q+δ) we have j (q) > 0 and thus by (66) and (62) ∂ x H(j(p), p) < 0. Then ∂ x H(a, q) < 0 follows by taking a limit.We will now show that ∂ x H(a, q) = 0 implies that one has a rotation around a in the following sense. Let y, z ∈ (a, b) and A y,a,z be the arc from y to z via a (seeFigure 3We show that every point in the arc passes thought the point (z, h(z)) in a finite and bounded time: We show that there exists a t a such that for all (x, p) ∈ A y,a,z there exists a t 0 < t a such thatLet q 1 , q 2 ∈ (q − δ, q + δ) be such that q 1 < q < q 2 . We may assume that x 1 := j(q 1 ) < y and x 2 := j(q 2 ) < z. There exists an > 0 such thatTherefore there exist t 1 , t 2 , t 3 such that for all x ∈ [x 1 , y] there exists a t 0 ≤ t 1 such that Xand for all p ∈ [q 1 , q 2 ] there exists a t 0 ≤ t 3 such that P j(p),p t < q 2 for t < t 0 , P j(p),p t 0 = q 2 . Now by taking t a = t 1 + t 2 + t 3 , we leave it to the reader to check that the claim follows.Similarly, one shows that ∂ x H(b, r) = 0 implies rotation around b (along arcs A y,b,z , seeFigure 3(b)). Therefore it follows that L = ∂A is a rotating loop if ∂ x H(a, q) = 0 and ∂ x H(b, r) = 0.Theorem 8.5. Let H satisfy Assumption 2.10. Suppose that L is a loop, G ∩ L = ∅, and that (x 1 , p 1 ) and (x 2 , p 2 ) withFor (iv), by Lemma A.2, it suffices to show the existence of a constant c such thatWe will consider the following computations and estimationsSet ψ(p) = 2pe 2p − e 2p + 1. Then ψ ≥ 0, andLet p 0 > 0 be such thatSo that for all x ∈ K (e.g., vNote that (78) also implies \|∂ p H(x, p)\| ≥ 2 for \|p\| ≥ p 0 and thus that θ(r) = r log r for r = \|∂ p H(x, p)\|.Using(76)and(77)we obtain the following lower bounds For (iii), by Lemma A.2, it suffices to show the existence of a constant c such that p∂ p H(x, p) − H(x, p) ≥ θ(\|∂ p H(x, p)\|) − c for all x ∈ K. p ∈ RFor (iii), by Lemma A.2, it suffices to show the existence of a constant c such that p∂ p H(x, p) − H(x, p) ≥ θ(\|∂ p H(x, p)\|) − c for all x ∈ K, p ∈ R. . A , A Dynamical moderate deviations for the Curie-Weiss model. Francesca Collet, Richard C Kraaij, Stochastic Processes and their Applications. 127Francesca Collet and Richard C. Kraaij. "Dynamical moderate deviations for the Curie-Weiss model". In: Stochastic Processes and their Applications 127.9 (2017), pp. 2900 -2925. Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. 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[ "Lebedev Physical Institute, Pushchino Radio Astronomy Observatory", "Lebedev Physical Institute, Pushchino Radio Astronomy Observatory" ]	[ "B A Nizamov ", "M S Pshirkov ", "\nFaculty of Physics\nLomonosov Moscow State University\nLeninskie Gory119991MoscowRussia\n", "\nSternberg Astronomical Institute\nLomonosov Moscow State University\nUniversitetsky prospekt 13119992MoscowRussia\n", "\nSternberg Astronomical Institute, Lomonosov Moscow State University\nUniversitetsky prospekt 13119992MoscowRussia\n" ]	[ "Faculty of Physics\nLomonosov Moscow State University\nLeninskie Gory119991MoscowRussia", "Sternberg Astronomical Institute\nLomonosov Moscow State University\nUniversitetsky prospekt 13119992MoscowRussia", "Sternberg Astronomical Institute, Lomonosov Moscow State University\nUniversitetsky prospekt 13119992MoscowRussia" ]	[ "60th October Anniversary st. 7a, 117312" ]	It is well known that if the ultra-high energy cosmic rays (UHECRs, E > 10 20 eV) were protons then their acceleration sites should possess some extreme properties, including gigantic luminosity. As no stationary sources with such properties are known in the local (D < 200 Mpc) neighborhood of the Milky Way, it is highly likely that the UHECR acceleration takes place in some transient flares. In this paper we investigate scenario where the UHECRs are produced in strong AGN flares. Using more than 7 years of the Fermi-LAT data and gamma-ray luminosity as a proxy for the bolometric luminosity, we select candidate flares and, using correlation between jet kinetic luminosity and its bolometric luminosity, obtain local kinetic emissivity of giant AGN flares: L ∼ 1.1×10 45 erg Mpc −3 yr −1 . This value is about an order of magnitude larger than the emissivity in the UHECRs, thus making this scenario feasible, if the UHECR escape spectrum is rather hard and/or narrow. This shape of spectrum is predicted in a number of present models of strong relativistic collisionless shocks. Also the scenario of acceleration in AGN flares can accommodate constraints coming from the observed arrival distribution of UHECRs.	10.1088/1475-7516/2020/03/060	[ "https://arxiv.org/pdf/1804.01064v3.pdf" ]	54,217,912	1804.01064	4b7782a07a2d5bf2e123e6e89ae6efb2c48059c6	Lebedev Physical Institute, Pushchino Radio Astronomy Observatory 142290 B A Nizamov M S Pshirkov Faculty of Physics Lomonosov Moscow State University Leninskie Gory119991MoscowRussia Sternberg Astronomical Institute Lomonosov Moscow State University Universitetsky prospekt 13119992MoscowRussia Sternberg Astronomical Institute, Lomonosov Moscow State University Universitetsky prospekt 13119992MoscowRussia Lebedev Physical Institute, Pushchino Radio Astronomy Observatory 60th October Anniversary st. 7a, 117312 Moscow, Russia; Pushchino, Russia142290AGNUltra-high energy cosmic raysGamma rays It is well known that if the ultra-high energy cosmic rays (UHECRs, E > 10 20 eV) were protons then their acceleration sites should possess some extreme properties, including gigantic luminosity. As no stationary sources with such properties are known in the local (D < 200 Mpc) neighborhood of the Milky Way, it is highly likely that the UHECR acceleration takes place in some transient flares. In this paper we investigate scenario where the UHECRs are produced in strong AGN flares. Using more than 7 years of the Fermi-LAT data and gamma-ray luminosity as a proxy for the bolometric luminosity, we select candidate flares and, using correlation between jet kinetic luminosity and its bolometric luminosity, obtain local kinetic emissivity of giant AGN flares: L ∼ 1.1×10 45 erg Mpc −3 yr −1 . This value is about an order of magnitude larger than the emissivity in the UHECRs, thus making this scenario feasible, if the UHECR escape spectrum is rather hard and/or narrow. This shape of spectrum is predicted in a number of present models of strong relativistic collisionless shocks. Also the scenario of acceleration in AGN flares can accommodate constraints coming from the observed arrival distribution of UHECRs. I. INTRODUCTION Despite huge efforts the problem of ultra-high energy cosmic rays (UHECR, E > 10 18 eV) origin is far from being resolved -we do not know where these particles are accelerated [1]. Pure common sense suggests that the acceleration to such extreme energies takes place in regions with some extreme conditions, and it can be demonstrated more rigorously using so-called "Hillas plot" [2]: acceleration of energetic particles requires nontrivial combination of source parameters, primarily magnetic field strength B and acceleration region size L and that severely decreases the number of potential sources (e.g., [3]). Active galactic nuclei (AGNs) are considered to be one of the most plausible candidates -it is very tempting to tap into a huge reservoir of energy coming from accretion of matter onto supermassive black hole. However, this model faces certain difficulties -a configuration needed for an effective acceleration is a very efficient photon source as well. It can be estimated that sources that can accelerate protons up to the highest energies exceeding 10 20 eV will have a bolometric luminosity larger than 10 47 erg s −1 [4,5]. On the other hand, production of observed UHECR is a local phenomenon, * [email protected] † [email protected] because due to an interaction with background photons the horizon of propagation of 10 20 eV protons is smaller than 150 Mpc [6,7]. The existence of Greisen-Zatsepin-Kuzmin (GZK) cut-off in the UHECR energy spectrum was discovered by the HiRes experiment [8] and lately confirmed with a very high statistical significance by the Auger and the Telescope Array observatories [9,10]. The problem is rather self-evident -there are no steady sources with required luminosity in the local volume. It was suggested that this difficulty can be solved by dropping the 'steadiness' condition -UHECRs can be accelerated in flares of AGNs. Flares with isotropic luminosity > 10 50 erg s −1 were observed [11], so it is theoretically possible to accelerate protons up to energies exceeding 10 21 erg. This model can be tested observationally. In order to do that we have used γ-ray observations of the Fermi LAT in the high energy (HE, E > 100 MeV) range. Fermi LAT observes the whole sky every three hours since August 2008, providing almost uniform coverage with a high temporal resolution. We have focussed at the 'local sources' (z < 0.3) because we wanted to avoid complications that arise due to a redshift evolution but still retain a decent statistics, with a probed volume that is more than 1000 times larger than a GZK volume. We have selected all flares with a luminosity above threshold corresponding to acceleration of UHECR protons to energies E > 10 20 eV and calculated total fluence of these flares. That allowed us to obtain local emissiv-ity in the HE γ-rays, estimate the bolometric and kinetic emissivities, and finally obtain the ratio of UHECR to kinetic emissivity. The paper is organized as follows: in Section II we describe the data and method of data analysis, Section III contains our results and discussion, and we draw conclusions in Section IV. II. DATA AND METHOD The goal of the present work is to investigate whether the AGNs can be possible sources of the UHECRs with E > 10 20 eV even in the case of the lightest (protonic) composition. Only sources with some extreme properties can accelerate protons to these energies [2,3] and the closest stationary source of the kind resides far out the GZK volume V GZK , which we define as a sphere with a radius R GZK = 150 Mpc -a mean attenuation free path of a particle with the initial energy of 10 20 eV. However, strong flares of AGNs can possibly accelerate protons to the very highest energies [4,5]. Stringent requirements on the size of the acceleration region and strength of the magnetic field there naturally translate into the lower limit on the 'magnetic luminosity' L B (e.g. [4]): L B ≥ 10 45 Γ 2 E 2 20 erg s −1 ,(1) which is adopted in this paper as a threshold value; E 20 is the UHECR energy in units of 10 20 eV. Such extreme flares have never been observed in V GZK (see below for the method of L B estimation). Nevertheless it is possible to calculate their local energetics using much larger test volume V 0 : we have selected a sphere with a radius corresponding to z = 0.3, V 0 > 10 3 V GZK . This volume is large enough to avoid significant statistical fluctuations and still it is appropriate to neglect effects of cosmological evolution at z < 0.3. Firstly, we selected all flaring sources satisfying Eq. (1), that potentially could be the sites of UHECR acceleration. It is impossible to directly measure L B , so we used bolometric luminosity L bol as a proxy. That implies that energy density of the magnetic field and electrons are close, B ∼ e and AGN jets radiate efficiently. The whole approach is made feasible by the fact that the bolometric luminosity can be rather reliably estimated using only the high-energy luminosity L γ , 0.1 GeV < E γ < 100 GeV [12]: L bol ∼ 2L γ .(2) The spectral-energy distribution of the blazars -and all selected flaring sources belong to this class -consists of two peaks. The first one lies around optical frequency range, and the other one is usually in the HE region; the luminosities in two peaks are comparable. This justifies the adoption of the factor 2 in the Eq. (2). There can be a slight overestimation of this coefficient due to the fact that for the sources being in the high state the second peak becomes more pronounced. For almost 10 years Fermi LAT has been continuously observing the celestial sphere at energies > 100 MeV. That allowed us to compile the full census of the very bright flares in the local Universe. We have made use of the second catalog of flaring gamma-ray sources (2FAV) based on Fermi All-Sky Variability Analysis (FAVA) [13]. This catalog is a collection of gamma-ray sources which show significantly higher (or lower) photon flux in a given time window as compared to what could be expected from the average flux from the source direction. The catalog is based on the Fermi observations during the first 7.4 years of the mission. Among the AGNs in 2FAV we selected those within the test volume V 0 . There are controversial determinations of z for the source PMN J1802-3940: there are a number of works with low estimation, z = 0.296 while in several others the source is placed at much higher redshift, z = 1.32. For the sake of robustness, we included the source in our list with the lower value of z. Moreover, it is possible that some AGNs could be in high state for a time span longer than 7.4 years and therefore enter the catalog as steady sources. In order not to miss such objects, we checked 3FGL catalog for superluminous sources within z = 0.3. For the sources with known redshift we approximated the isotropic equivalent bolometric luminosity with (2) where L γ ∼ 4πd 2 F 100 , d is the distance to the source and F 100 is the energy flux above 100 MeV taken from 3FGL. Only one source with the known z ≤ 0.3, S5 0716+71, exceeded the threshold (1). This source is also in 2FAV catalog and we included it in our analysis. The final list of selected AGNs consists of thirty nine sources and is shown in Tab. I. In this table N flare is the number of flares detected in a given source 1 . There are different ways to estimate the gamma flux with their own advantages and drawbacks. First, one can use aperture photometry (AP) light curves which are available, e.g., on the SSDC website 2 . This simple approach does not distinguish between source and background photons, therefore the flux is inevitably overestimated, especially at low galactic latitudes. Second, one can obtain maximum likelihood (ML) flux estimates for each flare using a Fermi Science Tools routine gtlike. This approach is the most rigorous, but the total length of the time intervals to be analyzed and their number makes such task very challenging. In this work, we use the first approach, i.e. we obtain the flux from AP data obtained from SSDC and use it to estimate the total kinetic power. There is a strong correlation between the flux in the high energy range, 0.1 -100 GeV and bolometric luminosity. At the same time it is desirable to use as higher energies as possible in order to maximally avoid contamination by the background photons because the point-spread function of Fermi LAT quickly deteriorates with decreasing energies. Our strategy is to calculate the AP flux in the range 1 -100 GeV within 2 degrees from the source and extrapolate it down to 0.1 GeV in the assumption of a flat spectrum. We check the correctness of such procedure in two steps. First, we compare the fluxes in the range 1 -100 GeV obtained by AP and ML. It appears that AP is mostly consistent with ML within the statistical uncertainties (see Appendix A for details). Second, we estimate what error can be introduced to our estimations by the flux extrapolation to the range 0.1 -100 GeV in the assumption of a flat source spectrum and find that it underestimates the flux by up to 40% for typical blazar spectra (see Appendix B). The threshold luminosity given by Eq. (1) depends on the bulk Lorentz factor of the jet. We used Lorentz factor Γ = 10 as a robust benchmark value [14][15][16] and considered acceleration to energies higher than 10 20 eV. These parameters set the threshold (1) at the level of 10 47 erg/s. We have explicitly checked that the acceleration in this regime was not limited by the synchrotron losses. Due to a relatively long duration of flares (> 10 5 s) constraints on the minimal size of the acceleration region are greatly relaxed, which in turn allows to decrease magnitude of needed magnetic field and, therefore, importance of synchrotron losses, see e.g. eqs. (1) and (2) in [4]. After that we selected the time intervals where the estimated luminosity exceeds the threshold. In other words, we choose the time intervals when a given AGN can accelerate UHECRs. As a proxy of the bolometric luminosity we use the radiative energy flux in the range 0.1 -100 GeV multiplied by a factor of 2 (see (2)). Now it is possible to estimate total kinetic energy of these flares: within the chosen intervals we convert gamma-ray luminosity to the jet kinetic power as proposed by [12]: log P jet = A log L γ + B.(3) We assume that, due to the beaming of the photons, we can not observe all blazars, but only a fraction of order Ω b /2π ∼ 1/2Γ 2 where Ω b is the surface angle of each of the two jets. Therefore in order to obtain the estimate of the total kinetic energy within the test volume we multiply P jet by 2Γ 2 . The kinetic energy thus obtained is divided by the time of observation (7.4 years) and by the volume V 0 which gives the total kinetic emissivity. The value obtained is 1.1 × 10 45 erg Mpc −3 yr −1 . This should be compared with the emissivity in UHE-CRs. It was shown that the emissivity required to reproduce the UHECR data above 10 18 eV should be of the order 10 45 − 10 46 erg Mpc −3 yr −1 [17] which corresponds to ∼ 10 43 − 10 44 erg Mpc −3 yr −1 above 10 20 eV. Our own calculation which follows the approach of [18] using the photo-pion production cross-section from [19] gave the values of 7.8 × 10 43 erg Mpc −3 yr −1 and 1.5 × 10 43 erg Mpc −3 yr −1 (see Appendix C). The first value corresponds to the UHECR intensity as observed by Telescope Array collaboration [10], and the second value corresponds to the intensity reported by the Pierre Auger collaboration [20]. Thus, the ratio of the UHECR emissivity to the AGNs' jet kinetic power, according to our estimations, varies from 1.4 × 10 −2 in case of the lower UHECR flux of Pierre Auger to 7.5 × 10 −2 in case of the higher UHECR flux of Telescope Array. In Appendix D we show how this estimation is affected by the uncertainty of the threshold (1). III. DISCUSSION In the previous section we have calculated emissivity in UHECRs L UHECR = 7.8 × 10 43 erg Mpc −3 yr −1 or 1.5 × 10 43 erg Mpc −3 yr −1 from the observed flux (reported by Telescope Array or Pierre Auger) of these particles and compared it with a kinetic emissivity in the strong AGN flares which satisfy Eq. (1): L kin = 1.1×10 45 erg Mpc −3 yr −1 . The latter one was calculated taking into account the fact that we can observe only small fraction of all flaring sources and total observed kinetic emissivity should be multiplied by the number of unseen sources, n = 2π/Ω b ∼ 2Γ 2 . The viability of the scenario crucially depends on the degree of UHECR beaming, Ω UHECR . If the accelerated UHECRs remain strongly beamed with Ω UHECR ∼ Ω b , the UHECR emissivity L UHECR will be corrected for the beaming factor as well. As after this correction L UHECR will be much larger than L kin it would clearly make the model unfeasible. The UHECRs can be effectively isotropised during their propagation towards the observer, increasing Ω UHECR up to value of order unity. The isotropisation can take place either in the immediate vicinity of the flaring region, which in turn shall be very close to the AGN engine, or in the magnetized intracluster medium. Large values of Ω UHECR mean that the observed flux of cosmic rays is generated by large number of flares with only small fraction of them pointing at us. Even then, the luminosity in the UHECRs with E > 10 20 eV amounts to a sizable fraction of the full kinetic luminosity of suprathreshold AGN flares in the local Universe, 1.4 × 10 −2 or 7.5 × 10 −2 depending on the UHECR spectrum selected, either PA or TA. Nevertheless, from the point of view of pure energetics it is not impossible that these flares can be the primary sources of the CRs of the highest energies. More than that, relatively large value of the ratio of UHECR to the full kinetic luminosity allows to put stringent constraints on the properties of the spectrum of CRs produced in the flares. We demanded that the total amount of energy in these CRs will not exceed one half of the kinetic energy [21]. The spectrum of the UHECRs was described as a simple powerlaw, ∝ E −α , E min < E < E max = 10 21 eV bounded from above and below. The results in form of E min (α) curves are presented in the Fig. 1. We have also checked that the results are not considerably changed with an increased value of E max = 10 22 eV. It can be seen that soft extended spectra are excluded with a high degree of confidence, and the spectrum of produced UHECR must be sufficiently hard and/or narrow. A lot of models predict the very same shape for the spectrum of particles escaping from relativistic collisionless shocks (e.g., [22][23][24]) or accelerated immediately in BH magnetospheres [25]. Additional constraints on the spectral shape can be obtained using multiwavelength and multimessenger observations -large regions of parameter space can be excluded depending on realized scenario of the cosmic source evolution [26]. This question with detailed treatment of the redshift dependence of AGN flares energetics will be studied elsewhere. Small number of observed events with E > 10 20 eV does not allow to perform meaningful large-scale anisotropy studies, it is still impossible to reliably distin- guish between isotropic or some anisotropic distributions. Nevertheless, it is safe to claim that at any given moment the number of active sources in the V GZK must be larger or equal than one [27]. We have observed N = 13 potential sources in t = 7.4 years inside V 0 = 10 3 V GZK volume which gives the following estimate for the rate of extreme UHECR producing flares in the GZK volume: R ∼ N t × V GZK V 0 Γ 2 ∼ 0.2 yr −1(4) Number of active UHECR sources can be readily estimated as N = Rt UHECR ,(5) where t UHECR is the duration of observed UHECR signal. At first glance, given that the typical duration of gamma flares t γ are of order of weeks 3 , this low rate seems to be at variance with the observations. However, being charged particles, UHECRs are subject to considerable deflection in the magnetic fields; when propagating short pulses are broadened by scattering: τ ∼ dθ 2 s /2,(6) where d is the propagation path length, θ s is the characteristic angle of scattering. The observed duration t UHECR = τ can be much longer than t γ . There are two closely related causes of scattering: first, there is a part due to the deflection in the random magnetic fields; second, there is considerable scattering due to a finite energy width of the pulse -cosmic rays at different energies are deflected by different angles. There are several regions where scattering can possibly take place. At the very least, a pulse of UHECRs will be deflected in the Galactic magnetic fields. At these energies, the magnitude of scattering due to the random parts of the galactic magnetic field will be several tenth of degree for out of the Galactic plane directions [28], corresponding length is of order of kpc. Also the signal will be spread by several degrees in the regular galactic magnetic fields [29]. That translates to characteristic time scales τ around several decades. Also the UHECRs will be scattered during inevitable isotropization -the corresponding timescales are highly uncertain, they could be very large if the UHECRs were scattered in the magnetic fields of clusters, or, on the other hand, they could be insignificantly small if the primary site of the isotropization was very close to the source. Another sites of possible scattering include the local filament of the Large scale structure [30] or the extragalactic voids [31]. In the former case possible time delays could be of order 10 5 years, when in the latter case they could exceed 10 7 years. That means that at the highest energies we will simultaneously observe at least O(10) sources and probably many more. There are several underlying assumptions that are important to our approach and should be emphasized. First, we assume that the jets are radiatively efficient and magnetic field energy is in a rough equipartition with energy of relativistic electrons. Only that premise allows us to estimate magnetic luminosity from the observed bolometric luminosity, and in case of radiatively inefficient jet and/or large deviation of B / e ratio from unity our estimate will be inaccurate. Still, our algorithm for search for extreme flares is likely to be on the conservative side, as one would expect that B / e ratio rather deviates towards larger values, meaning that we underestimate number of potential sources. Also some underestimation of real L kin can be due to a number of blazars without assigned redshift. There are 65 associated sources without assigned redshift in the 2FAV catalog, 57 of them would satisfy Eq. 1 if they were situated at z = 0.3. In the highly unlikely scenario that all sources without estimated redshifts reside so close inside V 0 volume it will shift upwards our estimate of L kin by factor of 3 and, accordingly, the estimate of energy available for UHECR acceleration, see Appendix D. We have chosen the lightest, protonic, composition for the UHECRs as a limiting scenario. From the theoretical point of view, acceleration of protons to the highest energies is very difficult, i.e. if it is possible for some class of sources to accelerate them, then, a fortiori, these sources can potentially produce UHECRs nuclei with larger atomic mass A. From the observational point of view, the composition of the UHECRs at the highest energies is still far from being certain: while the results of the Telescope Array experiment favour lighter one [32], the ones of Pierre Auger Observatory indicate that there is progressive increase in A at higher energies [33], although the detailed analysis shows that there is a good agreement between distributions in shower depth maximum within the systematic uncertainties [34]. Still, even the PAO results are fully consistent with existence of large fraction of protons even at the energies E > 10 19 eV. More than that, due to a very low statistics 4 there is no information about UHECR composition at the energies E > 10 20 eV, so it is not inconceivable that we will eventually observe decrease in A. Finally, we have tried to evaluate how our results depended on the exact value of threshold (Eq. (1)) and repeated our calculations for two bracketing cases, i.e. L B ≥ 3 × 10 46 erg s −1 and L B ≥ 3 × 10 47 erg s −1 , see Appendix D for details. IV. CONCLUSIONS The full census of strong local gamma-ray flares at redshifts z < 0.3 from the Fermi-LAT data was used in order to find ones that can possibly accelerate protons to the highest energies E > 10 20 eV and estimate maximal disposable amount of kinetic energy that can be potentially used for UHECRs acceleration. The estimated kinetic emissivity is more than one order of magnitude higher than the UHECR emissivity obtained from the observations, that makes the scenario feasible from the point of view of total energetics, if the escape spectrum of cosmic rays is not too soft. Also, the number of potential sources in the Greisen-Zatsepin-Kuz'min volume is high enough, so no markedly anisotropic distribution of UHECRs is expected which is perfectly in line with the current observations. Thus, the giant flares of the AGNs similar to ones observed with the Fermi LAT experiment can be primary sources of the UHECRs with energies E > 10 20 eV. F = F γ − 1 γ − 2 E −γ+2 1 − E −γ+2 max E −γ+1 2 − E −γ+1 max (B1) if γ = 2 and F = F E −1 2 − E −1 max log E max E 1 (B2) if γ = 2. Now we define a function f (γ) ≡ F(2)/F(γ) which is equal to the ratio of the energy flux above 100 MeV in the assumption of a flat spectrum to the "correct" one. Fig. 2 shows the function f (γ) for a range of typical values of γ. One can see that in the wide range of typical values of γ from 1.5 to 2.6 the inferred error does not exceed 40%. Here we present the calculation of the UHECR emissivity in the local volume needed to provide the observed UHECR intensity above 10 20 eV. Let N be the particle distribution function in space and energy and E is the particle energy. In general, N depends on the three space coordinates so that N (r, E)dV dE is the number of the particles inside the volume V . Consider the propagation of the particles along one coordinate s. The kinetic (transport) equation in one dimension is essentially the continuity equation in the space (s, E) and can be written in the form ∂N ∂t = − ∂ ∂s (N v) − ∂ ∂E N dE dt + S (C1) where t is time, v is the particle velocity, dE/dt = vdE/ds is the energy loss rate, or the velocity in the energy space, S represents the source term. For our simple estimation we will assume that the medium is uniform and isotropic on the scale of 100 Mpc which justifies the use of one-dimensional approach and implies ∂/∂s ≡ 0. We also assume stationarity on the scales of ∼300 Myr which leads to following relation: ∂/∂t ≡ 0. Moreover, at such relatively small scale we neglect the adiabatic losses. Losses due to pair production contribute very little at the energies above 10 20 eV and we do not take them into account. The different contributions to energy losses can be compared, e.g., using Fig. 1 of [36]. These assumptions leave us with S = ∂ ∂E N v dE ds . (C2) The energy losses are solely due to photo-pion production, therefore dE/ds = −E/λ attn (C3) where λ attn is the mean attenuation free path of a CR which can be obtained from Eq. 10 of [18] via multiplication by the CR velocity c and changing the order of integration: λ attn = 2γ 2 3 π 2 c 3 kT × × ∞ th d σ( )K p ( )[− log(1 − e − /2γkT )] −1 . (C4) Here T is the temperature of CMB, γ is the Lorentzfactor of the UHECR, is the CMB photon energy in the UHECR rest frame, th is the pion production threshold energy, σ( ) is the photo-pion production cross-section and K p ( ) is the interaction inelasticity, or the mean fraction of the energy lost by an UHECR in a photo-pion production event. Equation (C3) is equivalent to Eq. 1 of [36]. For ultrarelativistic particles the density and intensity are related via I = c 4π N,(C5) hence, we obtain S = −4π ∂ ∂E IE λ attn (C6) which in principle expresses the spectrum of the UHECR sources given the observed spectrum and the photo-pion production cross-section. Now the total energetic emissivity in UHECRs can be obtained via integration of the last equation over energies with the weight E: where E min = 10 20 eV and E max = 10 21 eV is the assumed upper energy limit of UHECR acceleration. The spectrum of cosmic rays above 10 20 eV as observed by the Telescope Array collaboration can be represented in the form I(E) = A E, eV 10 20 −γ(C8) where A = 6.25 × 10 −29 erg −1 cm −2 s −1 sr −1 and γ = 4.6 [10]. The spectrum in this energy range provided by the Pierre Auger collaboration reads I(E) = I 0 E E a −γ2 1 + E a E s ∆γ 1 + E E s ∆γ −1 (C9) where I 0 = 3.30 × 10 −19 eV −1 km −2 sr −1 yr −1 , E a = 4.8 EeV, E s = 42 EeV, γ 2 = 2.60 and ∆γ = 3.14 [20]. Substituting these expressions to (C7) we obtain for TA and PA data 7.8 × 10 43 and 1.5 × 10 43 erg Mpc −3 yr −1 respectively. Appendix D: Uncertainty of threshold luminosity and unknown redshifts The threshold (1) is not a well-defined quantity and its real value can deviate to a certain extent from the adopted value. We estimated the total kinetic luminosity as described in Section II with the threshold luminosities of 3 × 10 46 erg/s and 3 × 10 47 erg/s. The values obtained are 2.4 × 10 45 erg Mpc −3 yr −1 and 1.8 × 10 44 erg Mpc −3 yr −1 . The latter value is very close to the inferred UHECR emissivity which can make this scenario problematic. Next, we note that in 2FAV catalog there are 65 sources with unknown redshifts. If we place them at the boundary of the test volume (z = 0.3), then 57 of them exceed the threshold (1) at some times. Then we can calculate the 'hypothetical' contribution of these sources to the overall kinetic luminosity. This contribution is equal to 3 × 10 45 erg Mpc −3 yr −1 (we excluded the sources within 5 • from the galactic plane). This is more than twice higher than the value given in Section II, however we stress that we considered a highly unlikely case when all the sources with unknown z reside at z = 0.3 thus making the largest possible contribution to the total kinetic emissivity. Finally we note that in the 3FGL catalog there are also a number of sources with the unknown redshift. Analogously, we 'put' them at the distance of z = 0.3 and estimate their isotropic equivalent luminosity as 4πd 2 F 100 (see II). We find that in such a case no source with unknown z exceeds the threshold (1). FIG. 1 . 1Constraints on the spectrum of the UHECRs produced in extreme AGN flares. The spectrum is described by a power law: dN/dE ∼ E α , Emin(α) < E < Emax = 10 21 eV. The upper/lower curve corresponds to the TA/PAO energy spectrum parametrization. UHECRs. In order to check the accuracy of AP in the energy range 1 -100 GeV, we selected a number of flares (one per each of these 13 sources) and evaluated the fluxes with AP and ML. These sources are located at various galactic latitudes and are of different brightness. The flares analyzed are given in Tab. II. One can see that AP is mainly consistent with ML within the statistical uncertainties, except for the sources strongly contaminated by the Milky Way.Appendix B: Systematic uncertainties from the a priori unknown spectral shape Let us show how the uncertainty in the spectral index affects the estimate of the energy flux in the range 0.1 -100 GeV given the photon flux in the range 1 -100 GeV which is measured with the aperture photometry. Let the flux of photons be N (E)dE = AE −γ dE, E 1 = 100 MeV, E 2 = 1 GeV, E max = 100 GeV. Then the photon flux in the range 1 -100 GeV equals F = A(E −γ+1 2 − E −γ+1 max )/(γ −1) and the energy flux in the range 100 MeV -100 GeV is TABLE I . IThe list of the analyzed AGNs. z, l, b are the redshift and galactic longitude and latitude, N flare is the number of flares, Lmax is the maximum isotropic equivalent bolometric luminocity in the observation period, in units of 10 47 erg s −1 .Name z l b N flare Lmax PKS 0056-572 0.02 300.9 -60.1 0 0.00170 PKS 0131-522 0.02 288.3 -63.9 0 0.00670 Mkn 421 0.03 179.8 65.0 0 0.0892 3C 120 0.03 190.4 -27.4 0 0.0218 Mkn 501 0.03 63.6 38.9 0 0.0299 1ES 1959+650 0.05 98.0 17.7 0 0.0724 SBS 1646+499 0.05 76.6 40.1 0 0.0278 3C 111 0.05 161.7 -8.8 0 0.0412 AP Librae 0.05 340.7 27.6 0 0.0471 5BZBJ1728+5013 0.06 77.1 33.5 0 0.0461 PKS 0521-36 0.06 240.6 -32.7 0 0.129 1H 0323+342 0.06 155.7 -18.8 0 0.900 PKS 1441+25 0.06 34.6 64.7 0 0.288 BL Lacertae 0.07 92.6 -10.4 0 0.278 TXS 0518+211 0.11 183.6 -8.7 0 0.681 PKS 2155-304 0.12 17.7 -52.2 0 0.936 GB6 J1542+6129 0.12 95.4 45.4 0 0.217 1ES 1215+303 0.13 188.9 82.1 0 0.787 ON 246 0.14 232.8 84.9 2 1.20 PKS 1717+177 0.14 39.5 28.1 0 0.588 1ES 0806+524 0.14 166.2 32.9 0 0.402 OQ 530 0.15 98.3 58.3 0 0.360 3C 273 0.16 290.0 64.4 3 2.78 PKS 0829+046 0.17 220.7 24.3 0 0.884 PKS 0736+01 0.19 217.0 11.4 5 2.71 MG1 J021114+1051 0.20 152.6 -47.4 2 1.43 B2 2107+35A 0.20 80.3 -8.4 0 0.564 OX 169 0.21 72.1 -26.1 1 1.13 1H 1013+498 0.21 165.5 52.7 2 2.00 B2 2023+33 0.22 73.1 -2.4 36 3.22 PMN J0017-0512 0.23 101.2 -66.6 1 1.39 5BZQJ0422-0643 0.24 200.8 -36.1 0 0.725 PMN J1903-6749 0.26 327.7 -26.1 1 1.62 PKS 0301-243 0.26 214.6 -60.2 10 7.42 S2 0109+22 0.27 129.1 -39.9 31 2.92 GB6 J0937+5008 0.28 167.4 46.7 0 0.934 PMN J1802-3940 0.30 352.5 -8.4 61 5.06 NVSS J223708-392137 0.30 0.59 -59.6 2 1.55 S5 0716+71 0.30 144.0 28.0 42 8.73 TABLE II . IIAP and ML fluxes for a number of flares. l, b, z are galactic longitude, latitude, and redshift respectively, start and end times are given in MJD, AP and ML are the fluxes obtained with Aperture photometry and maximum likelihood respectively, in units of 10 −8 cm −2 s −1 , σAP and σML are the errors of these quantities.Source l b z Start End AP σAP ML σML (AP-ML)/ML ON 246 232.8 84.9 0.14 57177.5 57181.9 9.60 1.35 10.71 1.68 −0.10 3C 273 290.0 64.4 0.16 55061.2 55068.6 9.15 1.44 7.07 1.16 0.29 PKS 0736+01 217.0 11.4 0.19 56974.0 56988.0 5.66 1.12 5.10 0.79 0.11 MG1 J021114+1051 152.6 -47.4 0.20 55576.7 55586.5 5.30 0.95 3.75 0.70 0.41 OX 169 72.1 -26.1 0.21 54936.4 54939.5 3.72 0.89 3.91 1.72 −0.05 1H 1013+498 165.5 52.7 0.21 56243.8 56249.2 3.58 0.72 2.78 0.69 0.29 B2 2023+33 73.1 -2.4 0.22 55029.2 55043.2 7.16 1.15 3.23 0.66 1.22 PMN J0017-0512 101.2 -66.6 0.23 56434.0 56439.5 3.86 0.95 2.69 1.00 0.43 PMN J1903-6749 327.7 -26.1 0.26 56964.0 56971.2 3.47 0.65 2.57 0.57 0.35 PKS 0301-243 214.6 -60.2 0.26 55309.5 55341.1 6.92 1.05 5.14 0.41 0.35 S2 0109+22 129.1 -39.9 0.27 56117.7 56125.8 2.76 0.68 1.65 0.50 0.67 PMN J1802-3940 352.5 -8.4 0.30 54934.6 55019.6 2.45 0.20 1.24 0.15 0.98 S5 0716+71 144.0 28.0 0.30 55611.4 55637.9 5.63 0.86 3.73 0.37 0.51 Emax Emin EmaxESdE = −4πEmax Emin E ∂ ∂E IE λ attn dE = = 4πE 2 I λ attn Emin Emax + 4π Emax Emin IE λ attn dE (C7) We call a 'flare' each time interval when a source exceeds the threshold (1). 2 http://www.asdc.asi.it/ Also one can not exclude the possibility that individual flares can emerge during much longer interval when the source is in a state of increased activity. 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[ "Occurrence of exact R 2 inflation in non-local UV-complete gravity", "Occurrence of exact R 2 inflation in non-local UV-complete gravity" ]	[ "Alexey S Koshelev [email protected] \nDepartamento de Física and Centro de Matemática e Aplicações (CMA-UBI)\nUniversidade da Beira Interior\n6200CovilhãPortugal\n\nThe International Solvay Institutes\nTheoretische Natuurkunde\nVrije Universiteit Brussel\nPleinlaan 2B-1050BrusselsBelgium\n", "Leonardo Modesto [email protected] \nDepartment of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina\n\nDepartment of Physics & Center for Field Theory and Particle Physics\nLandau Institute for Theoretical Physics RAS\nFudan University\n200433, 119334Shanghai, MoscowChina, Russian Federation\n", "Les Law \nDepartment of Physics & Center for Field Theory and Particle Physics\nLandau Institute for Theoretical Physics RAS\nFudan University\n200433, 119334Shanghai, MoscowChina, Russian Federation\n", "Rachwa L [email protected] ", "Alexei A Starobinsky \nKazan Federal University\n420008Kazan\n\nRepublic of Tatarstan\nRussian Federation\n" ]	[ "Departamento de Física and Centro de Matemática e Aplicações (CMA-UBI)\nUniversidade da Beira Interior\n6200CovilhãPortugal", "The International Solvay Institutes\nTheoretische Natuurkunde\nVrije Universiteit Brussel\nPleinlaan 2B-1050BrusselsBelgium", "Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina", "Department of Physics & Center for Field Theory and Particle Physics\nLandau Institute for Theoretical Physics RAS\nFudan University\n200433, 119334Shanghai, MoscowChina, Russian Federation", "Department of Physics & Center for Field Theory and Particle Physics\nLandau Institute for Theoretical Physics RAS\nFudan University\n200433, 119334Shanghai, MoscowChina, Russian Federation", "Kazan Federal University\n420008Kazan", "Republic of Tatarstan\nRussian Federation" ]	[]	The R + R 2 , shortly named "R 2 " ("Starobinsky") inflationary model, represents a fully consistent example of a one-parameter inflationary scenario. This model has a "graceful exit" from inflation and provides a mechanism for subsequent creation and final thermalization of the standard matter. Moreover, it produces a very good fit of the observed spectrum of primordial perturbations. In the present paper we show explicitly that the R 2 inflationary spacetime is an exact solution of a range of weakly non-local (quasi-polynomial) gravitational theories, which provide an ultraviolet completion of the R 2 theory. These theories are ghost-free, super-renormalizable or finite at quantum level, and perturbatively unitary. Their spectrum consists of the graviton and the scalaron that is responsible for driving the inflation. Notably, any further extension of the spectrum leads to propagating ghost degrees of freedom. We are aimed at presenting a detailed construction of such theories in the so called Weyl basis. Further, we give a special account to the cosmological implications of this theory by considering perturbations during inflation. The highlight of the non-local model is the prediction of a modified, in comparison to a local R 2 model, value for the ratio of tensor and scalar power spectra r, depending on the parameters of the theory. The relevant parameters are under control to be successfully confronted with existing observational data. Furthermore, the modified r can surely meet future observational constraints.	10.1007/jhep11(2016)067	[ "https://arxiv.org/pdf/1604.03127v2.pdf" ]	59,457,195	1604.03127	f917a27a147353d354c537c9a7359b12cf1fe3c7	Occurrence of exact R 2 inflation in non-local UV-complete gravity 24 Nov 2016 Alexey S Koshelev [email protected] Departamento de Física and Centro de Matemática e Aplicações (CMA-UBI) Universidade da Beira Interior 6200CovilhãPortugal The International Solvay Institutes Theoretische Natuurkunde Vrije Universiteit Brussel Pleinlaan 2B-1050BrusselsBelgium Leonardo Modesto [email protected] Department of Physics Southern University of Science and Technology 518055ShenzhenChina Department of Physics & Center for Field Theory and Particle Physics Landau Institute for Theoretical Physics RAS Fudan University 200433, 119334Shanghai, MoscowChina, Russian Federation Les Law Department of Physics & Center for Field Theory and Particle Physics Landau Institute for Theoretical Physics RAS Fudan University 200433, 119334Shanghai, MoscowChina, Russian Federation Rachwa L [email protected] Alexei A Starobinsky Kazan Federal University 420008Kazan Republic of Tatarstan Russian Federation Occurrence of exact R 2 inflation in non-local UV-complete gravity 24 Nov 2016Prepared for submission to JHEPModels of Quantum Gravity, Cosmology of Theories beyond the SM The R + R 2 , shortly named "R 2 " ("Starobinsky") inflationary model, represents a fully consistent example of a one-parameter inflationary scenario. This model has a "graceful exit" from inflation and provides a mechanism for subsequent creation and final thermalization of the standard matter. Moreover, it produces a very good fit of the observed spectrum of primordial perturbations. In the present paper we show explicitly that the R 2 inflationary spacetime is an exact solution of a range of weakly non-local (quasi-polynomial) gravitational theories, which provide an ultraviolet completion of the R 2 theory. These theories are ghost-free, super-renormalizable or finite at quantum level, and perturbatively unitary. Their spectrum consists of the graviton and the scalaron that is responsible for driving the inflation. Notably, any further extension of the spectrum leads to propagating ghost degrees of freedom. We are aimed at presenting a detailed construction of such theories in the so called Weyl basis. Further, we give a special account to the cosmological implications of this theory by considering perturbations during inflation. The highlight of the non-local model is the prediction of a modified, in comparison to a local R 2 model, value for the ratio of tensor and scalar power spectra r, depending on the parameters of the theory. The relevant parameters are under control to be successfully confronted with existing observational data. Furthermore, the modified r can surely meet future observational constraints. Introduction The inflationary spacetime produced by a local R + R 2 gravity where R is the Ricci scalar ("Starobinsky" inflation, which we will refer to as "R 2 " inflation 1 ) [1][2][3] stays as a very successful model already for several decades. Theory-wise this model is the simplest extension to Einstein's general relativity (GR) with just one extra parameter (being the large dimensionless coefficient in front of the R 2 term in the action). The minimal number of parameters is a very elegant property of this model. Physics-wise this model features a "graceful exit" from inflation and provides a natural mechanism for the subsequent creation and final thermalization of the standard matter. Importantly, it produces a very good fit of the observed spectrum of primordial scalar perturbations and related parameters, such as the ratio of tensor to scalar power spectra r [4][5][6]. Being good at the classical level, R 2 gravity by itself does not admit an ultravioletcomplete (UV-complete) quantum treatment since it is not renormalizable. Stelle has shown [7,8] that adding the Weyl tensor squared term to the action makes it renormalizable, but then a tensor ghost appears. However, if the dimensionless coefficient in front of this term is of order of a unit and is much smaller than the one in front of the R 2 term, this ghost appears at very large values of the spacetime Riemann curvature close to the Planck scale and thus is much exceeding the curvature during the observable part of inflation which is 10 orders of magnitude less. Thus, we may use the R 2 model as an effective theory for description of inflation and post-inflationary evolution of the Universe. Anyway, if we believe that R + R 2 theory is the correct one in the inflationary era, and, at the same time, we assume renormalizability to be a guiding principle (so successful for all the other fundamental interactions), then we are forced to look for a completion of the local R 2 gravity in the UV regime [9][10][11][12]. Moreover, working in the quantum field theory framework we require the theory to be consistent at any perturbative loop order. In this paper we show explicitly that the R 2 inflationary metric is an exact solution to the equations of motion (EOM) for a large class of weakly non-local (quasi-polynomial) theories of gravity [13][14][15][16][17]. Actions of such theories are characterized by a polynomial structure in Riemannian curvatures with insertions of fully covariant differential operators with infinite number of derivatives. An example operator can be written as F( ) with being the covariant d'Alembertian operator and F(z) an entire function. The crucial property assumed in constructions of this class of actions is that all such non-local operators are analytic in their arguments. That is these theories are non-local and hereafter we refer to them as to "non-local" ones. 2 The consideration of non-local gravity theories in its current form was initiated by Tomboulis [13] in an attempt to find a UV-complete gravitational theory. This line of research propagated to the construction of a quasi-polynomial theory [17], that realizes a unitary completion of GR. There exists a subclass of theories, such that at quantum level they are super-renormalizable or finite and moreover ghost-like perturbative degrees of freedom do not show up [11,[19][20][21][22]. At the level of classical solutions of these theories the Newtonian potential turns out to be a regular at short distances (high energies) with a universal constant limit at zero distance [7,[23][24][25][26][27][28][29][30][31][32]. The spectrum of these theories consists solely of the graviton and the scalaron -the quantum scalar particle corresponding to the gravitational scalar degree of freedom arising in f (R) garvity with d 2 f dR 2 ≡ 0. It was shown in [17] that any extension of the spectrum beyond this leads to a propagation of ghosts. Moreover as opposite to GR or the local quadratic gravity in which understanding of stability has taken a long-time effort [33][34][35][36][37], the Minkowski vacuum is naturally stable in non-local gravity [24,38,39]. Account of non-local gravity effects, namely, backreaction of one-loop corrections of quantized matter fields (both massless and massive) on the evolution of a FLRW background and metric perturbations, had been already a crucial part of the model in [1] since it produced decay of scalarons into pairs of particles and antiparticles of all quantum matter fields after the end of inflation, thus, providing the final transition of the Universe to the radiation dominated stage. 3 After that cosmological considerations of non-local gravity were re-initiated in [41], where non-local scalar field Lagrangian coming from string field theory (SFT) [42][43][44] was minimally coupled to GR in an attempt to describe Dark Energy. Notice that SFT natively produces the non-local structures essentially similar to those suggested ad hoc in [13]. Moreover, SFT by its construction is unitary and UVcomplete. The first explicit gravitational action that includes non-local terms involving the scalar curvature and an exact analytic solution for a non-singular bounce in this model were described in [16]. Further cosmologically oriented explorations of this gravitational action can be found in e.g. [45,46]. 4 It was shown in [51] that the most general quadratic in curvature non-local gravity action may contain R 2 , R 2 µν and R 2 µναβ terms sandwiched by non-local differential operators. As long as we are focusing on inflation, which is a nearly de Sitter (dS) expansion phase, curvature squared terms give the essence of the underlying physics. A rigorous proof of this fact and a perturbative treatment of such theories around maximally symmetric spacetimes are accumulated in [52]. In [53] for the first time the R 2 inflation was considered in a framework of non-local gravity of the type R + RF( )R. It was shown that predictions for power spectra and their ratio for tensor and scalar perturbations remain as in a local R + R 2 model. This can be understood as follows. The scalar curvature squared piece does not generate tensor modes thus leaving them as they were in GR. Focusing on the nearly dS phase effectively eliminates the non-locality as all background quantities are covariantly constant. In the present paper we perform an analysis similar to the one in [53] in a presence of the non-local Weyl tensor squared term in the action. We thoroughly study classical perturbations and their quantization. As the net result we deduce power spectra for tensors and scalars and compute their ratio. It is expected that this time there will be extra tensor modes contributions generated from new terms with tensorial structure in the action and that the ratio r will be modified. This is the novel result of the paper, which gives the theory the predictive power as now it can be checked with existing or future observational data about cosmological perturbations. The paper is organized as follows. In the second section we introduce a generic class of weakly non-local theories, we evaluate the propagator, and we shortly remind the power counting analysis of super-renormalizability in these theories. In particular, we construct a theory in the "Weyl basis", which is slightly different from the one proposed in [17]. The Weyl basis proves useful for considering Friedmann-Lemaître-Robertson-Walker (FLRW) solutions. In the third section we explicitly show that the R 2 inflationary metric is an exact solution of the constructed weakly non-local theory. In the fourth section we compute the second order variation of the action around the nearly dS phase of inflation and derive cosmological parameters. In the fifth section we conclude by summarizing the results and promoting open questions. Appendices contain notations used throughout the paper as well as additional derivations and verifications of some computations. General multidimensional non-local gravity The Lagrangian density of the most general D-dimensional theory weakly non-local (quasipolynomial) and quadratic in curvature reads [13,17,19,20,24,[54][55][56][57][58][59], L g = 1 2 κ −2 D \|g\| [R + R γ 0 ( )R + Ric γ 2 ( )Ric + Riem γ 4 ( )Riem + V ] . (2.1) We refer the reader to appendix A for notations and conventions used in this paper. Here we notice that = g µν ∇ µ ∇ ν is the covariant d'Alembertian operator while operators γ i ( ) are termed form-factors. These form-factors are analytic functions of their argument. Physically speaking the proper argument should be written as z ≡ M ≡ /M 2 , where M is an invariant fundamental mass scale in our theory. Since the main problem of the quantum Einstein gravity (which remains in the local R + R 2 quantum gravity, too) is the appearance of the massive tensor ghost with mass ∼ κ D −1 at the one-loop level [8], any UV-complete non-local generalization of gravity has to begin below this scale. Thus, we assume that M M P ≡ κ −1 D . On the other hand, observational data on the amplitude and slope of the primordial power spectrum of scalar (density) perturbations [4][5][6], if we want to describe them using the R 2 inflationary model, require the dimensionless coefficient in front of the R 2 term in the action to be very large and equal to ≈ 5 × 10 8 [3]. As a result, the rest mass M of the scalaron should be M = 1.3 × 10 −5 55 N 0 M P = 3.2 × 10 13 55 N 0 GeV , where N 0 is the number of e-folds from the end of inflation corresponding to the pivot point k = k 0 = 0.05 Mpc −1 in the measurements of the scalar power spectrum (here k is the spatial momentum). Moreover, the same data show that the terms higher order in R, namely R n with n > 2, should be strongly suppressed during the observable part of inflation which occurs in the range R = (4 − 240)M 2 (see Sec. 3.2 below) [60], so that gravity has approximately scale-free behaviour in some range of Ricci curvatures exceeding M 2 . Thus, the cosmological data exclude the "naive" conjecture M ∼ M , and we have to assume the following hierarchy of scales: M ≪ M M P . (2.2) This phenomenological hierarchy suggests that the microscopical origins of inflation and non-local UV-completion of gravity are not the same. It is important to emphasize that this hierarchy between Ricci scalar and Weyl type corrections to general relativity is not destroyed by quantum gravitational corrections, at least in the first order, as our calculation of the generated graviton spectrum (finite and small) presented below shows. We set M to unit hereafter unless stated otherwise. We can rewrite the theory making use of a more compact notation introducing a tensorial form-factor, namely L g = 1 2 κ −2 D \|g\| (R + Riem γ( )Riem + V ) ≡ ≡ 1 2 κ −2 D \|g\| R + R µνρσ γ( ) µνρσ αβγδ R αβγδ + V , (2.3) where the operator in between the Riemann tensors is γ( ) µνρσ,αβγδ ≡ g µρ g αγ g νσ g βδ γ 0 ( ) + g µρ g αγ g νβ g σδ γ 2 ( ) + g µα g νβ g ργ g σδ γ 4 ( ) . The theory consists of an Einstein-Hilbert (EH) term, a kinetic weakly non-local operator quadratic in the curvature, and a local potential-like term V made of the following three sets of operators, V = N+2 j=3 j k=3 i c (j) k,i ∇ 2(j−k) R k i + γ+N+1 j=N+3 j k=3 i d (j) k,i ∇ 2(j−k) R k i + γ+N+2 k=3 i s k,i ∇ 2(γ+N+2−k) R k i , where the operators in the third set are called "killers", because they are crucial in making the theory finite in any dimension. The coefficients c (j) k,i , d (j) k,i , s k,i are running or non RGrunning coupling constants, while the tensorial structures of terms have been neglected. With symbol R we generally denote one of the above curvature tensors. The integer parameter γ will be defined shortly. The capital N is defined to be the following function of the spacetime dimension D: 2N + 4 = D. The form-factors γ i ( ) are defined in terms of exponentials of entire functions H ℓ ( ) (ℓ = 0, 2), namely γ 0 ( ) = − (D − 2)(e H 0 ( ) − 1) + D(e H 2 ( ) − 1) 4(D − 1) + γ 4 ( ) , (2.4) γ 2 ( ) = e H 2 ( ) − 1 − 4γ 4 ( ) ,(2.5) while γ 4 ( ) stays arbitrary. It is only constrained by renormalizability of the theory to have the same asymptotic polynomial UV behaviour as the other two form-factors γ ℓ ( ) (ℓ = 0, 2). The minimal choice compatible with unitarity and super-renormalizability corresponds to γ 4 ( ) = 0. As a matter of fact we can also add other operators quadratic in the curvature and equivalent to the above operators up to interaction vertices. These operators correspond to a different ordering in introducing the form-factors in between of the Riemann, Ricci, and scalar curvatures. We name these operators "terminators" to distinguish them from the "killer operators" present in the potential-like term V . Such non-local operators can be crucial in making the theory finite [61], if we do not introduce any local (or non-local) potential-like terms V higher than quadratic in the curvature. The non-local terminators, if expressed through entire functions do not affect the unitarity. Some examples of terminators are: R∇ α e H 3 ( ) − 1 2 ∇ α R , R µν ∇ α ∇ β e H 4 ( ) − 1 3 ∇ α ∇ β R µν , R µνρα e H 5 ( ) − 1 2 ∇ β ∇ α R µνρβ , . . . Finally, the entire functions V −1 ℓ (z) ≡ exp(H ℓ (z)) (ℓ = 0, 2) in the action satisfy the following general conditions [13]: (i). V −1 ℓ (z) is real and positive on the real axis and it has no zeros on the whole complex plane \|z\| < +∞. This requirement implies that there are no gauge-invariant poles other than the physical pole of transverse massless graviton; (ii). \|V −1 ℓ (z)\| has the same asymptotic behaviour along the real axis at ±∞; (iii). there exists 0 < ϕ < π/2, such that asymptotically \|V −1 ℓ (z)\| → \|z\| γ+N+1 , when \|z\| → +∞ with the integer parameter γ satisfying: (a) γ D even 2 , (b) γ D odd − 1 2 , respectively in even and odd dimension for the complex values of z in the conical regions defined by: −ϕ < argz < +ϕ , π − ϕ < argz < π + ϕ . The last condition is necessary to achieve the maximum convergence of the theory in the UV regime. The necessary asymptotic behaviour is imposed not only on the real axis, but also on the conical regions, that surround it. In Euclidean spacetime, the condition (ii) is not strictly necessary, if (iii) applies. An example of an exponential form-factor exp H ℓ (z) compatible with (i)-(iii), and the guiding principles of quantum field theory (locality of counterterms) is [13]: e H ℓ (z) = e a 2 [Γ(0,p(z) 2 )+γE+log(p(z) 2 )] = = e a γ E 2 p(z) 2a 1 + a e −p(z) 2 2 p(z) 2 1 + O 1 p(z) 2 + O e −2p(z) 2 , (2.6) where the last equality is correct only on the real axis. In the formula above a is a positive integer, γ E ≈ 0.577216 is the Euler-Mascheroni constant and Γ(0, z) = +∞ z dt e −t /t is the incomplete gamma function with its first argument vanishing (notice that it follows from the theory of special functions that Γ(0, z) = Ei(1, z)). The polynomial p(z) of degree γ + N + 1 is such that p(0) = 0, which gives the correct low energy limit of our theory (coinciding with GR). This entire function has asymptotic polynomial behaviour z a(γ+N+1) in a conical region around the real axis with angular opening ϕ = π/(4(γ + N + 1)). For γ = 0 and N = 0 we have the maximal conical region characterized by the opening angle ϕ = π/4. Propagator and unitarity Splitting the spacetime metric into the flat Minkowski background and the fluctuation h µν defined by g µν = η µν + κ D h µν , we can expand the action (2.1) to the second order in h µν . The result of this expansion together with the usual harmonic gauge fixing term reads [11] L quad + L GF = 1 2 h µν O µν,ρσ h ρσ , where the operator O is made out of two terms, one coming from the quadratic part of (2.1) and the other from the following gauge-fixing term, L GF = ξ −1 ∂ ν h µν w( )∂ ρ h ρµ , and w( ) is a weight functional [8,9]. The d'Alembertian operator in L quad and the gauge fixing term must be conceived on the flat spacetime. Inverting the operator O [11] and making use of the form-factors (2.4) and (2.5), we find the two-point function in the harmonic gauge (∂ µ h µν = 0) and in momentum space, O −1 = − ξ(2P (1) +P (0) ) 2p 2 w(−p 2 ) − P (2) p 2 e H 2 (−p 2 ) − P (0) (D − 2) p 2 e H 0 (−p 2 ) . (2.7) Here p α is the 4-momentum and p 2 = p α p α . Above we omitted the tensorial indices for the propagator O −1 and the projectors {P (0) , P (2) , P (1) ,P (0) } defined by [11,62] P (2) µν,ρσ (p) = 1 2 (θ µρ θ νσ + θ µσ θ νρ ) − 1 D − 1 θ µν θ ρσ , P (1) µν,ρσ (p) = 1 2 (θ µρ ω νσ + θ µσ ω νρ + θ νρ ω µσ + θ νσ ω µρ ) , P (0) µν,ρσ (p) = 1 D − 1 θ µν θ ρσ ,P (0) µν,ρσ (p) = ω µν ω ρσ , θ µν = η µν − p µ p ν p 2 , ω µν = p µ p ν p 2 . We also have replaced → −p 2 in the quadratic action, so going to the momentum space. The propagator (2.7) is the most general one compatible with unitarity describing a spectrum without any other degree of freedom besides the graviton field. In relation to unitarity issue, the optical theorem here is trivially satisfied, namely 2 Im T (p) µν O −1 µν,ρσ T (p) ρσ = 2π Res T (p) µν O −1 µν,ρσ T (p) ρσ p 2 =0 > 0 ,(2.8) where T (p) µν is the most general conserved energy tensor written in Fourier space [11]. The most general theory compatible with optical theorem at tree-level will contain also a scalar particle, the scalaron. Further extensions of the physical spectrum will inevitably introduce real ghost [17]. We obtain this most general theory provided we make the following replacement, e H 0 ( ) → e H 0 ( ) 1 − M 2 ,(2.9) on the level of definition of the form-factor γ 0 ( ) in (2.4). In this case the gauge-invariant part of the propagator for all modes contained in the fluctuation field h µν is slightly different, namely O −1 = −   P (2) p 2 e H 2 (−p 2 ) − P (0) (D − 2) p 2 e H 0 (−p 2 ) 1 + p 2 M 2   . (2.10) In this kind of theories we really need the asymptotically polynomial behaviour of the form-factors. Moreover to have renormalizability the degree of the asymptotic polynomial appearing in the entire function e H 0 ( ) with the scalar projector P (0) must be smaller by one than the corresponding one in e H 2 ( ) , that is it must be given by γ + N. We see that M is the mass of the scalaron. The version of the optical theorem satisfied here is slightly modified [17] and reads 2 Im T (p) µν O −1 µν,ρσ T (p) ρσ = 2π Res T (p) µν O −1 µν,ρσ T (p) ρσ p 2 =−M 2 > 0 . (2.11) This is the theory that we will analyze deeply in the course of this paper. Simplified power counting We now review the power counting analysis of the quantum divergences. In the UV regime, the above propagator (2.7) in momentum space schematically scales as O −1 (p) ∼ 1 p 2γ+D . (2.12) The vertices can be collected in different sets, that may or not involve the entire functions exp(H ℓ (z)). However, to find a bound on the quantum divergences it is sufficient to concentrate on the leading operators in the UV regime. These operators scale as the propagator giving the following upper bound on the superficial degree of divergence d of any graph G [19], d(G) = DL + (V − I)(2γ + D) (2.13) in a spacetime of even or odd dimensionality D. We simplify this to d(G) = D − 2γ(L − 1) . (2.14) In (2.14), we used the topological relation between the number of vertices V , internal lines I and the number of loops L: I = V + L − 1. Thus, if γ > D/2, only 1-loop divergences survive in the theory. Therefore, the theory is super-renormalizable [54,63,64] and only a finite number of operators of mass dimension up to [mass] D has to be included in the action in an even dimension to secure renormalizability. The conclusions obtained here are valid not only for theories with a graviton in the spectrum, but also for theories, where we have an additional scalar particle. Theory in Weyl basis In this section we consider a different gravitational action written in the Weyl basis, which is equivalent to the previous one (2.1) for everything, which concerns unitarity (the propagator is given again by (2.7)) and super-renormalizability or finiteness. The Lagrangian density reads, L C = 1 2 κ −2 D \|g\| R + Cγ C ( )C + Rγ S ( )R + Riem γ R ( )Riem + V g (C) . (2.15) Here the form-factors are found to be, γ C ( ) = − D − 2 4 γ 2 ( ) , γ S ( ) = γ 0 ( ) + 1 2(D − 1) γ 2 ( ) , γ R ( ) = γ 4 ( ) + D − 2 4 γ 2 ( ) , where all the form-factors γ ℓ ( ) (ℓ = 0, 2, 4) are defined in (2.4) and (2.5). Expressing γ C ( ), γ S ( ) and γ R ( ) via entire functions H 0 ( ), H 2 ( ) and γ 4 ( ) we find γ C = (D − 2) −e H 2 ( ) + 4γ 4 ( ) + 1 4 , (2.16) γ S = (2 − D) e H 0 ( ) + e H 2 ( ) − 2 + 4γ 4 ( )(D − 3) 4(D − 1) , (2.17) γ R = (D − 2) e H 2 ( ) − 1 − 4γ 4 ( )(D − 3) 4 . (2.18) For the sake of simplicity we can assume γ R ( ) = 0 , then the theory (2.15) reduces to L C = 1 2 κ −2 D \|g\| R + Cγ C ( )C + Rγ S ( )R + V g (C) , (2.19) γ C = D − 2 4(D − 3) e H 2 ( ) − 1 , γ S = − D − 2 4(D − 1) e H 0 ( ) − 1 . If a potential-like term V g (C), that we can always built up with only Weyl tensor, is included in the action, this theory likely turns out to be UV-finite at the quantum level, because all remaining divergences at one loop level can be consistently cancelled. In D = 4 it is enough to include in V a term made out of two Weyl killer operators to end up with a finite quantum gravitational theory at any perturbative order in the loop expansion. For example we can choose the following two operators, V g (C) = s (1) C C µνρσ C µνρσ γ−2 C αβγδ C αβγδ + s (2) C C µνρσ C αβγδ γ−2 C αβγδ C µνρσ . (2.20) Since the beta-functions can only be linear in the front coefficients s (1) C and s (2) C , we can always find a solution for β R 2 = 0 and β Ric 2 = 0 at any energy scale. (Actually the betafunctions here do not depend on any scale). We proved the multi-loop Feynman diagrams to be finite and, therefore, the beta functions are actually one-loop exact (see [20] for more details). In the Weyl basis presented here the FLRW metric for conformal matter (T matter ≡ T matter µ µ = 0) solves exactly the non-local EOM. The Big-Bang singularity may show up in some solutions of our finite theory of quantum gravity [65]. In this paper we are mostly interested in a particular solution, namely the R 2 selfinflating early Universe. Since we are looking for FLRW solutions the Weyl square or quartic terms do not give contribution to the background EOM, because Weyl tensor on FLRW background vanishes. Then we can concentrate on the following effective non-local f ( , R) theory, L eff = 1 2 κ −2 D \|g\| R + Rγ S ( )R with γ S ( ) = − D − 2 4(D − 1) e H 0 ( ) − 1 . (2.21) To have the scalar degree of freedom in the spectrum of perturbations we perform the substitution given by (2.9). This theory is still unitary (as explained above) and superrenormalizable (eventually finite after adding killers), but the spectrum now consists of an extra degree of freedom: the R 2 scalaron. Moreover, if we expand the action in inverse powers of the scale M and use the asymptotically polynomial form of the form-factor given in (2.6), then the leading operators reconstruct exactly the effective R 2 theory, L R 2 = 1 2 κ −2 D \|g\| R + R 2 6M 2 + O R R M 2 M 2 . (2.22) Clearly, for M < M the R 2 theory is a good approximation of the non-local theory. Therefore, we expect all the features of the R 2 inflation to be perfectly reproduced by our theory. R 2 inflationary spacetime in non-local gravity The goal in this section is to study the inflationary scenarios in the framework of action (2.19). Such scenarios are described by the FLRW metric, which is conformally flat (moreover for the sake of simplicity we here focus on spatially flat solutions only), hence the Weyl tensor is identically zero on the background. This implies that the Weyl-dependent terms in the action (2.19) do not contribute to the background EOM at all. However, the Cγ C C term contributes to the quadratic variation of the action. This variation is crucial and will be analyzed later on in order to study the perturbations. The V g (C) term does not influence the subsequent analysis entirely. Therefore the action in question is S = d 4 x \|g\| M 2 P 2 R + λ 2 RF( )R + λ 2 C µνρσ F C ( )C µνρσ − Λ cc . (3.1) Here we adopt the notations for non-local operators utilized in [53] in order to make use of and comparison with the previous results simpler. The identification with the present notations is rather straightforward, namely γ S = λ M 2 P F( ) , γ C = λ M 2 P F C ( ) . (3.2) λ is a dimensionless constant which is convenient to control the scale of the R 2 modification. Also the cosmological constant is (re)-introduced explicitly for generality and the term with it in the action is denoted by −Λ cc . The EOM for the latter action were found already in [51] and read as follows, E µ ν ≡ −(M 2 P + 2λF( )R)G µ ν − Λ cc δ µ ν − 1 2 λδ µ ν RF( )R + 2λ(∇ µ ∂ ν − δ µ ν )F( )R + λL µ ν − λ 2 δ µ ν L σ σ +L + 2λ (R αβ + 2∇ α ∇ β ) F C ( )C αβµ ν + O(C 2 ) = 0 . (3.3) Here G µ ν is the Einstein tensor and we have defined L µ ν = ∞ n=1 f n n−1 l=0 ∂ µ R (l) ∂ ν R (n−l−1) ,L = ∞ n=1 f n n−1 l=0 R (l) R (n−l) , where f n are coefficients of the Taylor expansion of the function F(z) = n f n z n and R (l) ≡ l R. We recall that function F(z) is analytic. Notice that the mentioned above second variation of the action corresponds to the linear variation of the EOM. Since the Weyl tensor is zero, any term more than linear in C in EOM will vanish at the perturbative level. At the background level all (even linear) terms containing the Weyl tensor vanish. Therefore, all the killer operators cubic or quartic in C do not take part in the analysis of linear perturbations. From (3.3) the trace equation is derived E = M 2 P R − 4Λ cc + 6λ F( )R + λ(L + 2L) + O(C 2 ) = 0 . (3.4) It does not contain the linear Weyl tensor contribution as the Weyl tensor is absolutely traceless. The simplifying ansatz proposed in [16] R = r 1 R + r 2 (3.5) really simplifies the EOM considerably. Indeed, it turns out that n R = r n 1 (R + r 2 /r 1 ) for n > 0 , F( )R = F 1 R + F 2 , F 1 = F(r 1 ) , F 2 = r 2 r 1 (F 1 − f 0 ) . Upon substitution of these latter relations into the EOM (3.3), some further algebra brings to the result obtained in [16], namely, a solution of the ansatz (3.5) is also a solution of the full non-local EOM (3.3) provided the following algebraic conditions are satisfied, F (1) (r 1 ) = 0 , F 2 = − M 2 P 2λ + 3r 1 F 1 , 4r 1 Λ cc = −r 2 M 2 P . (3.6) Here F (1) (r 1 ) is the first derivative with respect to an argument. Technically, the above conditions can be re-written in a number of equivalent forms using the definitions of F 1 and F 2 , but one has to be careful about possible divisions by "zero" that may arise for certain values of parameters. We additionally stress here that presence of a cosmological term forces to have a nonzero r 2 while zero cosmological term forces r 2 = 0. Moreover, a reverse statement is also true. Given a solution to (3.5) such that r 2 = 0 we must have a non-zero cosmological term in the theory. Zero r 2 forces that no cosmological term is present. Moreover, we emphasize that, of course, the trace equation (3.4) does not exhaust the whole system of gravitational field equations and, for a FLRW background, we have to consider the (00)-component of Einstein equations, too. Then, accounting the Bianchi identities, in addition to the terms following from the variation of action (3.1), one can add the term ∝ a −4 , i.e. energy density of "dark radiation", to the (00)-component of equations of motion. For non-local models this aspect was discussed in details in [25,66]. Note that this dark radiation may even have negative energy density. Explicitly, assuming that the solution satisfies the ansatz (3.5) and using relations (3.6) one gets the following Einstein equations from (3.3) 2λF 1 −(R + 3r 1 )G µ ν + ∇ µ ∂ ν R − δ µ ν R 2 4 + r 1 R + r 2 4 + T r µ ν = 0 , (3.7) where T r is the stress-energy tensor of the additional radiation source. The (00)-component with both lower indices becomes 2λF 1 −(R + 3r 1 )3H 2 +R + R 2 4 + r 1 R + r 2 4 + ρ r = 0 , (3.8) while some extra algebra brings this expression to a very neat form 2λF 1 3 2 RḢ − 3HṘ − 9r 1 H 2 − 3 4 r 2 + ρ r = 0 . (3.9) However, and this was one of the main results of [1], the existence of inflation is incompatible with the radiation term being significant, so it should be very small both during and after inflation. It may be important before the beginning of inflation though. Nevertheless, then one has to consider a more general case with a non-zero spatial curvature, since the latter generates a similar term (in addition to the radiation one) with the large coefficient M 2 P /M 2 in the (00)-component of equations in the case of R 2 gravity. Thus, it is not consistent to add dark radiation to the model involved while still neglecting the spatial curvature. Hence, we leave this for a future work. Relation with local R 2 model Consider a local R 2 model of the form S = d 4 x \|g\| M 2 P 2 R + λ 2 Rf 0 R −Λ cc . (3.10) The parameters are here designated by the hats, and (3.10) is technically (3.1) where the non-local operators reduced to a constant term. However, in contrary to the situation with higher powers of the operator we can derive (3.5) as one of the equations of motion for (3.10) (indeed, the trace). Moreover, conditions (3.6) are simplified tô M 2 P 2λ = 3r 1f0 , 4r 1Λcc = −r 2M 2 P . (3.11) Indeed, the non-local operator in between the Ricci scalars is just a constant and its derivative is always zero, moreover, F 2 is identically zero here thanks to the same argument. Comparing (3.6) and (3.11) we arrive to the conclusion [67] that any (conformally flat) solution of a local R 2 model (3.10) is a solution of the EOM coming from (3.1) with the following identification of parameters, M 2 P = M 2 P + 2λF 2 ,Λ cc = Λ ccM 2 P M 2 P . (3.12) Technically, M 2 P orM 2 P , or both can turn out to be negative. Their identification requires F 2 = 0, provided we want the same Planck masses in both theories. Additionally a radiation source may be needed in either local or non-local model to support a particular solution to equations of motion. Density of radiation in either case can be computed using (3.9) while the sign of the radiation energy density is "solution dependent". We now understand that even though the non-local model with the ansatz (3.5) looks pretty much the same as a local R 2 theory, they are different in a number of parameters left. Therefore, we still have more freedom of adjustment in the non-local model even upon the ansatz (3.5) imposition. R 2 inflation The R 2 inflationary model is given by a local R 2 gravitational theory of the type (3.10) without a cosmological term at all, i.e.Λ cc = 0. Relations (3.6) imply that r 2 = 0 and the ansatz reduces to R = r 1 R , (3.13) which is exactly the trace equation in R 2 gravity. An analytic solution to the above equation cannot be obtained in full. However, the vital piece of information for us is that R 2 inflation is indeed a solution of a local R 2 gravity model. As perhaps the most crucial consequence supported by the previous subsection is that the same solution is a solution in our non-local model upon adjustment of the parameters dictated by (3.6). In this paper we are mostly aimed at studying the nearly dS phase of R 2 inflation. This regime is given by the following expressions: a(t) ≈ a 0 (t s − t) −1/6 e −r 1 (ts−t) 2 /12 , (3.14) H =ȧ a = r 1 (t s − t) 6 + 1 6(t s − t) + ... , (3.15) R = 6(Ḣ + 2H 2 ) = r 2 1 (t s − t) 2 3 − r 1 3 + 4 3(t s − t) 2 + ... ,(3.16) where t s corresponds to the end of inflation when \|Ḣ\| ∼ H 2 , dot means the derivative with respect to the cosmic time t, and all formulae are valid for √ r 1 (t s − t) ≫ 1. Note that we must have r 1 > 0 for inflation to be metastable. In this case, the Universe undergoes inflation when t < t s , and for t > t s , it has a graceful exit to the subsequent power-law expansion stage with a(t) ∝ t 2/3 modulated by small oscillations. 5 In principle, t s − t can be arbitrary large in these formulae, so the local R 2 inflation never approaches the exact dS phase with a constant R. Note that to obtain the power-law multiplier in. (3.14) and the last terms in the other two equations, one has to go beyond the leading order in the slow-roll approximation, see the redivation in the Appendix B. For this reason, the power-law exponent is different from that in Eq. (7.26) in [68] where this correction was not taken into account. In order to see what happens to a possible radiation source we analyze equation (3.9) accounting that r 2 = 0 as well as identifying r 1 = M 2 and λF 1 = M 2 P /(6M 2 ) where M is the scalaron mass. All this together yields M 2 P 3M 2 3 2 RḢ − 3HṘ − 9M 2 H 2 + ρ r = 0 . (3.17) One can see that a simple substitution of expressions (3.15) and (3.16) up to the leading order (i.e. keeping only the first terms) yields ρ r = 0 which is in the total agreement with the physical arguments presented above right after equation (3.9). Non-local R 2 inflation To embed R 2 inflation as it is in a local model in our non-local framework we essentially should satisfy relations (3.6) when we already assume Λ cc = r 2 = 0. We therefore focus on parameters and non-local operators. Combining (2.21) and (2.9) we arrive to the following wishful form-factor in D = 4, λ M 2 P F( ) = − 1 6 e H 0 ( ) 1 − M 2 − 1 . (3.18) Evaluating the second relation in (3.6) one, after canceling all manifestly non-zero factors on opposite sides, gets: r 2 r 1 (1 − e −H 0 (r 1 ) ) 1 − r 1 M 2 = 3r 1 1 − r 1 M 2 . (3.19) In the case r 1 = M 2 one can cancel the corresponding factors, but as a consequence one gets with necessity r 2 = 0. The latter implies a non-zero cosmological constant due to the last relation in (3.6) and contradicts our initial assumptions. Notice that this outcome appears irrespectively of the particular solution solely as the result of our choice of the form-factor. The absence of a cosmological constant term, given the form-factor (3.18), is only compatible with r 1 = M 2 . Notice that this value for the parameter r 1 is exactly like in a local R 2 gravity theory. The remaining question is to maintain the first relation in (3.6). Starting from (3.18) and assuming r 1 = M 2 we arrive at H 0 (r 1 ) = 0 . (3.20) Even though the condition looks extremely simple it is incompatible with a proposed above form for H 0 (z) given by (2.6). Indeed, such form produces H 0 (z) positive for any non-zero argument and H 0 (0) = 0. In our case r 1 should not be trivial and therefore we are set to slightly modify the form of entire function H 0 (z). A natural and very simple form-factor compatible with H 0 (r 1 ) = 0 reads as follows, 6 H 0 ( ) = a 2 Γ 0, p γ ( ) 2 + γ E + log p γ ( ) 2 , p γ ( ) = γ−1 ( − M 2 ) . (3.21) We here remind that in D = 4 we achieve super-renormalizability with only one-loop divergences for the minimal choice γ = 3, therefore for this particular value of γ we get: p 3 ( ) = 2 ( − M 2 ) =⇒ e H 0 (p 3 ( )) =0 = e H 0 (p 3 ( )) =M 2 = 1 . (3.22) Analyzing a possible radiation source we look at equation (3.9) accounting that r 2 = 0. Moreover, the above construction of the operator function F( ) (see (3.18)) and especially conditions r 1 = M 2 and (3.20) imply that λF 1 ≡ λF(r 1 ) = M 2 P /(6M 2 ). This is exactly the coupling in front of the R 2 term in a local inflationary model. Combining all of this together one yields M 2 P 3M 2 3 2 RḢ − 3HṘ − 9M 2 H 2 + ρ r = 0 ,(3.23) which is an equation identical to the local model case (3.17). This means that for a dense open subset of all solutions of the local R 2 model having a sufficiently long slow-roll inflationary stage (3.14-3.16) required by observational data, the radiation term ρ r becomes unimportant once inflation begins (and even after its end) in our non-local model as well. Finite quantum gravity with three propagating degrees of freedom Let us here expand about finiteness of the theory for what concerns the Newton and cosmological constant couplings. In section (2.3) we reviewed a special class of finite theories with monomial UV behaviour of the form factors [20]. However, for the UV completion of the local R 2 model the form factor F is polynomial, and not monomial, in the UV regime. In particular, the next to the leading order term in the UV behaviour in the large momenta limit is forced to depend on the scalaron mass M . Or in other words the effective mass of the scalaron is determined from the next to leading higher derivative term in the UV regime. We would like to remind that in this theory we have two propagating degrees of freedom in the metric fluctuations and one in the scalaron field. Focusing on the structure of the theory in UV we can have non-zero beta functions for the Newton constant (β G N ) and the cosmological constant (βλ). Moreover if we treat the scalaron mass as the fundamental coupling the integration of the RG equations for the two couplings in front of the two quadratic in curvature operators is non-trivial. Namely, the beta functions depend on the coupling itself, which is actually the mass of the scalaron. We can view the system of beta functions from the other perspective too and it still can be integrated quite easily. The mass of the scalaron may be considered not as a fundamental coupling, it is instead read from ω γ−1 or ω γ−2 . The last parameters do not run under change of energy scale. The theory is like any higher derivative theory, where beta functions are plain. Only specific relation between them define the scalaron mass M . The only issue is how to have non-vanishing mass of the scalaron but at the same time vanishing of the beta functions for G N and cosmological constant. It is obvious since divergences are local that divergences (and hence beta functions) of the theory do not depend on the curvature of background and we easily read them as from the theory on flat spacetime background (in the short distance limit any non-singular spacetime is effectively flat). Below we show how to obtain this for a specific choice of the form factor. There is a simple way to make harmless the UV non-monomial contribution resulting in the limit → +∞ of the form factor F. We can just replace the polynomial (3.21) with the following one, p γ ( ) = M 2 M 2 1 M 2γ γ−5 ( 2 + M 2 + M 4 ) 2 ( − M 2 ) , γ > 5 . (4.1) The asymptotic behaviour of F now reads F( ) → M 2 P λ M 2 M 2γ+2 γ−5 ( 2 + M 2 + M 4 ) 2 ( − M 2 ) ( − M 2 ) 6M 2 = M 2 P λ 1 6M 2γ+2 γ−6 ( 3 − M 6 ) 2 = M 2 P λ 1 6M 2γ+2 ( γ − 2 γ−3 M 6 + γ−6 M 12 ). (4.2) Following the notation of reference [10], now the coefficients ω γ−1 and ω γ−2 are identically zero, which means there are no operators quadratic in gravitational curvature with in between neither γ−1 nor γ−2 . Then the beta functions for G N and the cosmological constant are identically zero. Moreover, the coefficient ω γ (we again refer to the paper [10]) is independent on the scalaron mass M . Therefore, the beta functions for R 2 and C 2 can only depend on the scale M and the coefficients s C , the latter two can be selected to make zero the beta functions for the operators quadratic in the curvature. It is crucial that the coefficients ω γ , s C and s C do not get any divergent renormalization, or, which is the same, that M is not renormalized. Indeed, the theory is super-renormalizable and such coefficients appear in front of higher derivative operators. Note that the polynomial (4.1) takes zero value in = 0 and in = M 2 as it is required by the classical solution (see the previous section). If the theory is only super-renormalizable we expect perturbative logarithmic corrections to the operators R 2 and Ric 2 (for the sake of simplicity we here consider the case β G N = βλ = 0), namely in the UV R log − µ 2 R and Ric log − µ 2 Ric , (4.3) where µ is a general renormalization scale. We can write the one-loop dressed propagator in momentum space as follows [17], O −1 = − P (2) e −H 2 (p 2 ) p 2 Π 2 (p 2 ) + P (0) e −H 0 (p 2 ) (D − 2)p 2 Π 0 (p 2 ) (4.4) = − P (2) e −H 2 (p 2 ) p 2 1 + e −H 2 p 2 c 0 log p 2 µ 2 + P (0) e −H 0 (p 2 ) (D − 2)p 2 1 + p 2 M 2 + e −H 0 (p 2 ) p 2c 0 log p 2 µ 2 . The constant c 0 andc 0 have both inverse mass square dimension and are related to the beta functions for the counterterms quadratic in the curvature. The dressed propagator (4.4) develops an infinite number of complex conjugate poles in both sectors: the spin two and the spin zero. Therefore, the scalaron pole survives at quantum level iffc 0 = 0, which means that at least the beta function for the counterterm R 2 must vanish. This is consistent with the finiteness of the theory at quantum level. Now we apply the most popular subtraction point at p 2 = −M 2 , where the scalaron is on the mass shell. This means we want the finite part of Π 0 (p) to vanish when p 2 = −M 2 . Therefore, the finite part is uniquely specified by defining Π 0,R (p 2 ) = Π 0 (p 2 ) − Π 0 (−M 2 ). (4.5) Finally, the one-loop propagator reads − P (2) e −H 2 (p 2 ) p 2 1 + c 0 e −H 2 (p 2 ) p 2 log − p 2 M 2 + P (0) e −H 0 (p 2 ) (D − 2)p 2 1 + p 2 M 2 +c 0 e −H 0 (p 2 ) p 2 log − p 2 M 2 . This is of course just a particular identification of the renormalization scale µ that consistently preserves the mass spectrum of the theory without introducing mixing between M and M . Notice that at perturbative level there is no hierarchy problem for the finite theory because there are no divergences forcing us to renormalize M 2 P = 1/8πG N and/or the scalaron mass M . Both the scales can coexist in a finite theory of gravity without perturbative mixing (see the discussion above). Moreover, the finite contributions in the ultraviolet regime look like: R(1/ n )R or C(1/ n )C (n > 0), and they can not significantly move the poles. It deserves to be noticed that both the finite and the super-renormalizable theory with polynomial behaviour (4.1) are perturbatively consistent with zero value (or any small value) of the cosmological constant at quantum level. In addition to these theoretical arguments, phenomenology of inflation, namely the specific form of deviation of the power spectrum of primordial scalar (density) perturbations measured in [4][5][6] from the exactly scale-invariant (or, Harrison-Zeldovich) one, suggests that gravity is approximately scale-free in the range of Ricci curvatures between M 2 and M 2 . So, in the local limit, terms containing higher order powers of the Riemann curvature are somehow suppressed relative to quadratic ones for a large range of curvatures. That is why it is interesting and important to study in the first approximation what happens if they are absent at all. Moreover, in our theory the higher in curvature Weyl killer operators in (2.20) do not influence the two-point function around the de Sitter space at the classical level because they are constructed out of the Weyl tensor, which vanishes on the background. At quantum loop-level we expect such operators to give perturbative contributions only to finite terms in the UV-finite complete theory. These last contributions will be perturbatively suppressed in the coupling constant. Cosmological perturbations during inflation A vast amount of a related work in non-local gravities was already done in [52,69] and [53]. In the paper [52] the second variation of the most general action including the Weyl tensor was derived using the spin decomposition of the gravitational fluctuation that we will review and use in section 5.2. However, in section 5.1 we present a general self-adjoint second order variation of the nonlocal Weyl squared term without using any decomposition of the fluctuations (see formula (5.8)). The general variation (5.8) is essential in the evaluation of the beta functions for the G N coupling, but it is also useful to obtain the combination of beta functions for the couplings in front of the R 2 and Ric 2 terms. Moreover, we also need (5.8) to compute the exact graviton propagator around any maximally symmetric background (which is either of dS, anti-dS or Minkowski) 7 . Here we breakdown the essence of the previous results complementing it with new developments related to the Weyl tensor term. Moreover, we gain certain simplifications considering a model without an explicit cosmological term. To find out the parameters of inflation such as power spectra, spectral tilts and the tensor to scalar power spectra ratio we have to quantize perturbations in the inflationary background. To accomplish this task we need the second order variation of the action around the inflationary (in case of R 2 inflation in fact a nearly dS) background. However, to perform certain mathematical manipulations we have to have the linear variation of equations of motion. Since the latter is even simpler to derive we will elaborate on both: linear variation of the EOM and the quadratic variation of the action. We start by introducing the following basic notation, g µν =ḡ µν + h µν . (5.1) Hereafter the bar is used to designate the background values. We shall proceed according to the following plan: • First, we will derive the second variation of the action around a nearly dS background in terms of covariant metric perturbation h µν . • Second, we will analyze the spin-2 mode, i.e. transverse and traceless part of h µν . This will lead to an important constraint on the form-factor F C . • Third, we will introduce the canonical, in cosmology, ADM formalism and we will perform the study of tensor perturbations around the dS background. • Fourth, we will derive the linearized equations for scalar perturbations and quantize the scalar perturbations around a nearly dS background. In the last subsection we will compute the ratio of power spectra of tensor and scalar perturbations at the end of inflation in our non-local theory. Second order variation of action (3.1) in terms of h µν Considering Einstein-Hilbert and cosmological (just for the generality) terms in (3.1) one can derive the following second order variation δ 2 S EH+Λcc = d 4 x \|ḡ\| M 2 P 2 δ EH − 2 M 2 P Λ cc δ g ,(5.2) where we have introduced the following definitions, δ EH = 1 4 h µν¯ h µν − 1 4 h¯ h + 1 2 h∇ µ∇ρ h µρ + 1 2∇ µ h µρ∇ ν h ν ρ + (hh µν − 2h µ σ h σν ) 1 8ḡ µνR − 1 2R µν − 1 2R σν h σ ρ h νρ + 1 2R σ ρνµ h µ σ h νρ , (5.3) δ g = h 2 8 − h 2 µν 4 . (5.4) One can verify this result against, e.g. [71]. This is a generic fully covariant variation around any background. One important remark must be made here. Below we are greatly focused on considering the nearly dS phase of the evolution of the R 2 inflation. It is crucial that this phase happens due to the presence of the R 2 term and not due to a cosmological term, since the latter is absent at all in our model of inflation. This implies that we do not have the usual in GR relation between Λ cc and the background scalar curvatureR. We nevertheless are allowed to consider the Riemann tensor as having a nearly dS form. Next we vary the term quadratic in the scalar curvature. Doing this around a nearly dS spacetime 8 we can show that the terms related to the variation of d'Alembertians inside the function F cancel out. The fact thatR is almost constant in this approximation helps a lot. We thus obtain the following variation (this can be checked against [52,53]), δ 2 S R 2 = d 4 x \|ḡ\| λ 2 (2δ EH −Rδ g )f 0R + δRF(¯ )δR ,(5.5) where the variation of the scalar curvature reads, δR = (−R µν + ∇ µ ∇ ν − g µν¯ )h µν . (5.6) Note that the expression for δR is valid for any background. The last piece we need is the second variation of the Weyl-dependent term. It turns out that this term is quite simple as long as we consider conformally flat backgrounds (like FLRW in general and the R 2 inflationary solution in particular). Indeed, the Weyl tensor is zero on such backgrounds and therefore the only non-trivial combination which persists at the second order in the gravitational fluctuation reads as follows, δ 2 S C 2 = d 4 x \|ḡ\| λ 2 (δC µναβ )F C (¯ )(δC µναβ ) . (5.7) We stress that this formula is valid around any conformally flat background. In the next sections we will present this formula for scalar perturbations in the ADM formalism and for an arbitrary scale factor. However, its expression in terms of h µν for the moment could be derived only around a nearly dS background (strictly speaking around a nearly maximally symmetric background). The corresponding expression reads as follows, 1 2 δ 2 d 4 x \|g\|C µνρσ F C ¯ C µνρσ = = d 4 x \|ḡ\| h µν R 2 36 −R 4¯ + 1 2¯ 2 F C ¯ +R 3 h µν − h R 2 144 −R 16¯ + 1 8¯ 2 F C ¯ +R 3 + R 48¯ + 1 24¯ 2 F C ¯ +R h − h µν∇ µ R 12 +¯ F C ¯ + 3 4R ∇ ρ h ν ρ + 1 3 h µν∇ µ∇ν F C ¯ +R ∇ ρ∇σ h ρσ + h R 6 + 1 3¯ F C ¯ +R ∇ µ∇ν h µν . (5.8) Covariant spin-2 excitations and restrictions on form-factor F C ( ) The covariant spin-2 excitation of the generic metric variation h µν is the transverse and traceless part which we denote as h ⊥ µν . Since it enjoys the conditions∇ µ h ⊥ µν =ḡ µν h ⊥ µν = 0, its substitution to the general second variation is rather simple (moreover, once can use the results of [52] in order to check the subsequent expression). As long as we use formula (5.8) the subsequent analysis in this subsection is valid only around a nearly dS background. Performing some algebra one gets δ 2 S ⊥ = d 4 x \|ḡ\| M 2 P 2 1 4 h ⊥ µν ¯ − 2R 3 h ⊥ µν + 2 M 2 P Λ cc 1 4 h ⊥ µν 2 + λ 2 2 1 4 h ⊥ µν ¯ − 2R 3 h ⊥ µν +R 1 4 h ⊥ µν 2 f 0R + λ 2 2 1 4 h ⊥ µν ¯ −R 6 ¯ −R 3 F C ¯ +R 3 h ⊥ µν . (5.9) Here we keep some factors not cancelled and some similar terms not grouped in order to make the computation track transparent. We recall that the relations (3.6) must hold for the solution. F 2 = 0 , f 0 = F 1 , 2f 0R + M 2 P λ = 2F 1 (R + 3r 1 ) . Moreover, in the nearly dS phase one has r 1 ≪R . (5.10) The latter comparison follows from (3.15) and the discussion thereafter. Under all these approximations we can neglect the first line in formula (5.9) while the rest neatly combines in the following variation, δ 2 S ⊥,inf = d 4 x \|ḡ\| λ 4 h ⊥ µν ¯ −R 6 F 1R + ¯ −R 3 F C ¯ +R 3 h ⊥ µν , (5.11) where "inf" stands for inflation. Here we easily distinguish the standard dS propagator factor¯ −R/6 for the spin-2 mode, but on top of this we have a new non-local operator in the curly brackets. The concept of non-local theories stays that the quadratic form of the d'Alembertian operator defines as many degrees of freedom as many roots this form as the function of¯ has. The factor¯ −R/6 gives the standard pole of the propagator. If we require that extra poles do not appear (in fact an extra pole will necessarily be a ghost), then the following function has no roots, P(¯ ) = 1 + 1 F 1R ¯ −R 3 F C ¯ +R 3 . (5.12) Mathematically this implies that P(¯ ) is the exponential of an entire function of¯ , namely P(¯ ) = e −2ω(¯ ) , (5.13) where ω(¯ ) is some entire function. We emphasize that the above derived restriction on the form-factor F C must be treated as a general requirement and should propagate to any regime in the evolution of the background because we want the theory to be always healthy. However, the value of scalar curvature used to obtain the restriction is the one during the nearly dS inflationary phase. An example of form-factor F C compatible with (5.13) reads F C ( ) = F 1R e H 2 ( − 2 3R ) − 1 − 2 3R . (5.14) Indeed, (5.14) generates the desired behaviour for the operator P, namely P( ) = e H 2 ( − 2 3R ) . (5.15) Redefining the fieldĥ ⊥ µν = e −ω(¯ ) h ⊥ µν we finally get: δ 2 S ⊥,inf = d 4 x \|ḡ\| λ 4 F 1Rĥ ⊥ µν ¯ −R 6 ĥ ⊥ µν . (5.16) One can check that the action of the¯ operator does not spoil the properties of the original field to be transverse and traceless. Tensor perturbations Let us recall that the most standard formalism for studying the cosmological perturbations is the ADM decomposition of the spacetime. In what follows we proceed in the canonical way classifying the perturbations with respect to the representation of the 3-dimensional symmetry group. That is this section is about standard tensorial perturbations. In contrast to this, we used the covariant approach in the previous subsection mainly in order to derive the constraint (5.13). This constraint will manifest its usefulness in the analysis of scalar perturbations. We remind that the scalars, vectors, and tensors do not mix at the linear order and can be studied independently. Moreover, we omit vectors and concentrate on cosmologically important scalars and tensors only starting with the tensors. The line element for tensor perturbations reads ds 2 = a(τ ) 2 −dτ 2 + (δ ij + 2h ij )dx i dx j , (5.17) While all the notations are collected in the appendix A, we just recall that τ is the conformal time. The three-dimensional metric is just the delta symbol as we already stated our wish to consider only spatially flat solutions in this paper. Tensor h ij is transverse and traceless and, moreover, it is gauge invariant. One can check that h ij forms a subset of the fourdimensional tensor h ⊥ µν considered in the previous subsection. This very nice fact saves us from a lot of computations. Indeed, the last formula of the previous subsection is ready for the quantization of the tensor perturbations. Furthermore, reviewing the results of [53] we see that our formula (5.16) is almost exactly equation (4.23) in [53] modulo a normalization of the tensor field. Going through the results of [53] we write down the power spectrum for the tensor modes as \|δ h \| 2 = 1 2π 2 λF 1R k 2 a 2 e 2ω(R/6) , (5.18) in the sub-Hubble regime and \|δ h \| 2 = H 2 2π 2 λF 1R e 2ω(R/6) , (5.19) in the super-Hubble regime. The additional exponential factor in the power spectrum of the tensor modes is a very important difference. It will appear crucial later when we will discuss the tensor to scalar ratio r. Also here k denotes the length of the comoving spatial momentum, which in turn originates from the spatial Fourier transform. The latter has the form of the plane wave e i k x . We follow the standard convention that indices of k i are raised and lowered by the 3-dimensional spatial metric that is δ ij in our consideration. Scalar perturbations We start by noting that for sure using the spatially Fourier transformed quantities is very handy. We therefore proceed working with all fields being spatially Fourier transformed. Such fields are function of τ and the comoving spatial momentum k, such as ϕ(τ, k). We recite here the following crucial notation from (A.18): k ϕ(τ, k) = − 1 a 2 (∂ 2 0 + 2H∂ 0 + k 2 )ϕ(τ, k) . (5.20) Note that we use ∂ 0 ≡ ∂/∂τ . Linear variation of EOM for scalar perturbations We here use standard notations. In particular the perturbed line element reads ds 2 = a(τ ) 2 −(1 + 2φ)dτ 2 − 2∂ i βdτ dx i + ((1 − 2ψ)δ ij + 2∂ i ∂ j γ)dx i dx j . (5.21) We immediately move to gauge-invariant variables putting aside all possible gauge-fixing issues. For the scalar perturbations these variables are known as Bardeen potentials [68,72] defined as follows, Φ = φ − 1 a (aϑ) ′ = φ −χ , Ψ = ψ + Hϑ = ψ + Hχ , (5.22) where χ = aβ + a 2γ , ϑ = β + γ ′ , H(τ ) = a ′ /a, and prime hereafter denotes the derivative with respect to the conformal time τ . One more useful gauge-invariant quantity is defined by where ζ = δ R + (¯ k − r 1 )δR GI , δR GI = δR −R ′ (β + γ ′ ) = = 2(R + 3¯ k )Ψ − 2R(Φ + Ψ) − 6 a ′ a 3 (Φ ′ + Ψ ′ ) + 2 k 2 a 2 (Φ + Ψ) .P ≡ ∂ µR ∂ µ − 2 a 2 R ′′ + 2HR ′ F ¯ k − F 1 ¯ − r 1 2 + 3F ¯ k + (R + 3r 1 ) F ¯ k − F 1 k − r 1 , δ = 1 a 2 2Φ ∂ 2 0 + 2H∂ 0 + Φ ′ + 3Ψ ′ ∂ 0 . We notice that ζ actually is the variation of the ansatz relation (3.5), and in the local R 2 gravity theory ζ = 0. The fractions with denominators containing operator¯ k are still analytic functions. This can be shown by using the Taylor series expansion of F(¯ k ) in the numerators near the point r 1 and the required condition F (1) (r 1 ) = 0. Equation (5.25) is exactly the same as in [69] because the additional term containing the Weyl tensor enters the trace equation only as a O(C 2 ) term and thus does not appear in perturbations around a conformally flat background. Even though the expression for δE is manifestly homogeneous with respect to ζ it may be convenient to rewrite it in the following different way, δE = 2λ a 2 R ′ Ξ ′ + 2 R ′′ + 2HR ′ Ξ − a 2 (R + 3r 1 )(Υ − F 1 δR GI ) − 3a 2 F(¯ k )ζ = 0 , (5.25) where Υ = F(¯ k ) − F 1 k − r 1 ζ + F 1 δR GI and Ξ = F(¯ k ) − F 1 (¯ k − r 1 ) 2 ζ . The variation of the ( i j )-equation with i = j in the system (3.3) yields δE i j = −2λ k i k j a 2 F 1 (R + 3r 1 )(Φ − Ψ) + Υ + 2λc i j = 0 . (5.26) Here we have introduced the following notation, c µ ν = (R α β + 2∇ α k∇kβ )F C (¯ Ck )δ (s) C βµ να . (5.27) The whole expression above is explained and evaluated explicitly in the appendix C. Moreover, its explicit form follows below after all necessary equations are given. Equations (5.25) and (5.26) are in principle sufficient to study the classical dynamics of the perturbations because they provide a coupled system of two equations for the two Bardeen potentials. However, one must understand that all equations are in the game, and we may expect that certain constraints arise because we have more than two equations for two functions. These constraints must be accounted. This actually happens in the pure GR. Furthermore, at least in GR those constraints provide a considerable simplification and they help to write the second variation of the action that we will derive later. The variation of the ( 0 i )-equation in the system (3.3) yields δE 0 i = 2λ ik i a 2 2F 1 (R + 3r 1 )(Ψ ′ + HΦ) − (Υ ′ − HΥ) + F 1R ′ Φ − 1 2R ′ Ξ + 2λc 0 i = 0 . (5.28) Finally the variation of the ( 0 0 )-equation in (3.3) (perhaps, the most tedious derivation in this section) yields δE 0 0 = 2λ a 2 − 2F 1 (R + 3r 1 )(3HΨ ′ + 3H 2 Φ + k 2 Ψ) + 3HΥ ′ − 3H ′ Υ + k 2 Υ − 3F 1R ′ (Ψ ′ + 2HΦ) −R ′ 2 Ξ ′ + 1 2 R ′′ + 2HR ′ Ξ + 2λc 0 0 = 0 . (5.29) For completeness we rewrite here from the appendix C, equation (C.5), the following components of c µ ν , c i j = k i k j a 2 Θ ′ + 2HΘ + (H ′ − H 2 )Ω + k 2 3 Ω − a 2Ḣ Ω where i = j , c 0 i = − 2 3 ik i k 2 a 2 Θ , c 0 0 = 2 3 k 4 a 2 Ω , Θ = Ω ′ + 2HΩ , Ω = F C (¯ k + 6H 2 ) Φ + Ψ a 2 . We emphasize that in the definition of Ω the d'Alembertian operator is the one given in (5.20). 9 We performed a verification of the equations (5.25), (5.26), (5.28), (5.29) by evaluating the Bianchi identities in appendix D. One more equation, namely the ( i i ) component with no sum over i is needed for this verification and it is presented in the appendix D as well. Such an equation is not necessary for the main computations here although may be useful elsewhere. de Sitter limit of linearized equations for scalar perturbations The dS limit is what actually must be analyzed to get the inflationary observables. Moreover, assuming a nearly dS background we greatly simplify the perturbation equations. Equation (5.26) multiplied by k 2 reduces to − k 2 F 1 (R + 3r 1 )(Φ − Ψ) − k 2 Υ + k 2 Θ ′ + 2HΘ + k 2 3 Ω = 0 ,(5.6HF 1 (R + 3r 1 )(Ψ ′ + HΦ) − 3H(Υ ′ − HΥ) − 2Hk 2 Θ = 0 , (5.31) while equation (5.29) reduces to − 2F 1 (R + 3r 1 )(3HΨ ′ + 3H 2 Φ + k 2 Ψ) + 3HΥ ′ − 3H ′ Υ + k 2 Υ + 2 3 k 4 Ω = 0 . (5.32) Summing up all three equations above, using that in dS limit H ′ = H 2 , expanding Θ in terms of Ω, using the explicit representation for Ω in terms of Bardeen potentials and cancelling common k 2 factor we end up with [F 1 (R + 3r 1 ) + (¯ k − 2H 2 )F C (¯ k + 6H 2 )] Φ + Ψ a 2 = 0 . (5.33) From here it is obvious that, whether there is no Weyl tensor contribution in the original action (that means F C ( ) = 0), the following very neat condition arises: Φ + Ψ = 0. At this moment we observe the very crucial fact. The non-local operator in the latter equation (5.33) is nothing but the operator P(¯ ) defined in (5.12) provided we identify the operators¯ k + 2H 2 and¯ , take the approximation r 1 ≪R, and factor out the constant multiplier F 1 r 1 . This observation immediately guarantees that in a general situation even with the Weyl tensor term in the model the only solution to equation (5.33) reads Φ + Ψ = 0 . (5.34) This in turn has the following dramatic consequences: c µ ν = Ω = Θ = 0 ,(5.35) meaning that the Weyl term in the action does not influence the scalar perturbations during the nearly dS expansion at all. Furthermore, in the dS limit we have 36) and the perturbation of the trace equation (5.25) simplifies to ζ = (¯ k − r 1 )δR GI , Υ = F(¯ k )δR GI ,(5.F(¯ k )(R + 3¯ k )δR GI = (R + 3¯ k )Υ = F 1 (R + 3r 1 )δR GI . (5.37) Explicitly replacing δR GI in the RHS of the latter equation yields Υ = 2F 1 (R + 3r 1 )Ψ . (5.38) Notice that our results for the constraints coincide with those obtained in [53] (as it should be as the presence of Weyl-dependent term in the action was just proven not to influence the scalar perturbations in the nearly dS phase). Quadratic variation of action for scalar perturbations Computing the complete quadratic variation around an arbitrary solution in our nonlocal model seems a very hard task. The main and only obstacle is the RF( )R term. The variation of the Weyl tensor term is generic for any conformally flat background. Even though we consider already special backgrounds satisfying a simplifying ansatz (3.5) (which made it possible to compute the complete linearized equations of motion) we leave the problem of computing the complete second order action variation for the future work. For the present paper we concentrate on questions only requiring to compute the second order variation around a nearly dS background. We start with the variation of the Einstein-Hilbert action that we already prepared in (5.1) and we consider the scalar perturbations using the following line element, ds 2 = a(τ ) 2 −(1 + 2Φ)dτ 2 + (1 − 2Ψ)δ ij dx i dx j ,(5.39) The substitution of this simple line element in (5.4) is considerably long, but he result upon omitting total derivative terms, reads δ (s) EH = 1 a 2 −6Ψ ′ 2 + 6k 2 Ψ 2 − 12HΨ ′ (Φ + Ψ) − 9H 2 (Φ + Ψ) 2 − 4k 2 Ψ(Φ + Ψ) , δ (s) g = − 1 2 Φ 2 + 6ΦΨ − 3Ψ 2 . (5.40) Here the superscript (s) stands for scalars. This result is valid for any scale factor and one can check that it coincides with the result presented in [68]. Next we make the variation of the operator quadratic in the scalar curvature. Doing this around dS spacetime we write δ 2 S (s) R 2 = dτ d k \|ḡ\| λ 2 (2δ (s) EH −Rδ (s) g )f 0R + δR GI F(¯ k )δR GI ,(5.41) which differs from (5.5) in using δ (s) instead of δ and by the use of δR GI in place of a generic δR. The last piece we need is the variation of the operator quadratic in the Weyl tensor, which can be immediately written using the results in appendix C, namely δ 2 S (s) C 2 = dτ d k \|ḡ\| λ 2 4k 4 3 Φ + Ψ a 2 F C (¯ k + 6H 2 ) Φ + Ψ a 2 ,(5.42) where the factor 4k 4 /3 comes from the full contraction of two K-tensors also defined in appendix C. This particular result is valid for a general scale factor. Furthermore, thanks to the constraint (5.34) this contribution vanishes around the nearly dS inflationary phase. Now technically we have all pieces of the second variation for the non-local action around a nearly dS background. Quantization of scalar perturbations We must derive an effective action for a canonical variable that in turn has to be cooked up in order to make use of the quantization procedure described in [68]. Generically it is a tedious task, but in this paper we consider only the nearly dS phase of inflation and, therefore, we quantize the perturbations in a nearly dS background. Moreover, we have proven that the Weyl tensor contribution for the scalar perturbations goes away and we just come to the configuration considered in [53]. Therefore, we do not have to do anything rather than list the results obtained previously. Those results absolutely support the statement that the non-local modification of R 2 theory retains the same value for the scalar power spectrum as a local R 2 theory as long as the nearly dS phase of evolution is considered. In particular the following quantity was analyzed, R = Ψ + Ḣ R δR GI ,(5.43) which during the nearly dS phase can be approximated by R ≈ H 2 H Ψ . (5.44) Moreover, this quantity enjoys the conservation in the slow-roll inflation in the super-Hubble limit. Then the power spectrum during the crossing of the Hubble radius is: \|δ R (τ, k)\| 2 ≈ H 6 k=Ha 16π 2Ḣ 2 k=Ha 1 3λF 1R ,(5.45) which is manifestly a scale invariant expression. Tensor to scalar ratio r This ratio is simply defined as the ratio of power spectra of tensor and scalar perturbations. At the moment of the Hubble radius crossing (when (5.18) and (5.19) coincide) we get r = 2\|δ h \| 2 \|δ R \| 2 = 48Ḣ 2 H 4 e 2ω(R/6) ,(5.46) where the factor 2 accounts for two polarizations of the tensor modes. Next using the connection between the slow-roll parameter ǫ 1 = −(Ḣ/H 2 )\| k=Ha and the number of efoldings N , namely N = t f t i Hdt = 1 2ǫ 1 , we arrive at r = 48ǫ 2 1 e 2ω(R/6) = 12 N 2 e 2ω(R/6) , (5.47) which is the familiar result (see [73]) modulo the exponential factor. Conclusions In this paper we have presented a class of weakly non-local gravitational theories in the Weyl basis. These theories provide a super-renormalizable or finite UV completion of the Einstein-Hilbert theory as well as the local R 2 theory. The theory presented here is very general and all the freedom is encoded in the choice of two form-factors (entire functions). Here we selected two of them compatible with stability (no tachyons) and unitarity (no ghosts). However, we did not investigate in this paper the whole landscape of theories likely compatible with general principles. This sounds like a very important question to determine possible classes of form-factors for which the resulting theory is healthy, i.e. unitary and UV-complete. Next, we explicitly showed that any solution of a local R 2 gravity is a solution in our non-local framework. This is including a situation without a cosmological constant. To embed a solution of a local R 2 gravity in our non-local model one has to fulfill relations (3.6). In this case a given solution of a local R 2 gravity is an exact solution in our nonlocal theory. Perturbations, however, must and do differ in local and non-local theories. It is an outstanding mathematical problem to construct explicitly more general analytic solutions beyond this embedding. While some examples are presented in [74], in general this question is far from being answered in full. The idea of adopting solutions of a local R 2 gravity applies in particular to the R 2 inflationary scenario for which all parameter relations can be easily satisfied. This is a very important solution. Indeed, for the moment the accuracy of observational data related to inflation has been increased considerably. It is therefore a test bed for our model. Most of the interesting observables are related to perturbations around the nearly dS expansion during inflation. We therefore have thoroughly elaborated on linearized equations of motion, quadratic variation of action and quantization of perturbations. It turns out that only the term RF( )R is not yet tamed in full. We can derive its quadratic variation only around a nearly dS background. We hope to extend this to any solution satisfying ansatz (3.5) in our future works. Nevertheless, we managed to deduce all the linearized equations of motion around any background satisfying (3.5), and not only in a dS limit. As the result we found the full quadratic variation of the action around a nearly dS background and quantized scalar and tensor perturbations. A very interesting and useful formula (5.8) for the second variation of a quadratic Weyl tensor term in the action was obtained in passing. Upon analysis of quantized perturbations we came to a modified ratio of power spectra of tensors and scalars r given in (5.47). While other essential parameters, spectral tilts for example, retain their values like in a local R 2 gravity the modified r deserves more discussion. Namely r R 2 = 12 N 2 vs. r non−local = 12 N 2 e 2ω R 6M ,(6.1) where we have restored the scale of our theory M. The change is always a positive factor originating from the structure of the non-local operator F C ( ), yet this factor can be smaller or grater than 1. Explicit expression for ω(z) comes from (5.12, 5.13).R is the scalar curvature during the nearly dS inflationary phase. The function ω(z) cannot be arbitrarily normalized, but is rather determined from many conditions imposed from various considerations like absence of ghosts in Minkowski background, renormalizability, etc. In other words even though we have freedom to adjust the form-factors, once given, we must accept their respective behaviour. This means that apparently the latter formula can be understood as a one more constraint. More and more accurate measurements of r will give information on how to fix two things: the scale M and the first Taylor coefficients of the form-factors γ i (z) around z =R/(6M). It is currently a work in progress to obtain constraints on M for model form-factors γ i (z) based on the cosmological data. Existing data are not enough to say somewhat definitive about this new extra multiplier though. Indeed, it is important to have N ∼ 55 e-foldings. This gives r R 2 ≈ 0.004 while the most strict upper bound from PLANCK 2015, BICEP2 and Keck Array data [4][5][6] is r < 0.07. It is however very important that given that there come incredible improvements in accuracy of the measurements our modified formula r non−local is ready to explain values of r different from a pure local R 2 inflation. We also notice that a modification of r established in this paper mathematically resembles the effect of r modification in the so called α-attractor inflationary models [75]. Nevertheless, physically the effects are different. In our present consideration we observe a modified r due to the inclusion of tensorial structures which generate a modified tensor power spectrum. The scalar power spectrum in our setup is preserved at the level of a pure R 2 inflation. This automatically preserves the value and scale dependence of n s , the spectral tilt for the scalar power spectrum. For a narrow subclass of inflaton potentials considered in [75], namely for the potential V (φ) ∝ tanh 2 (αφ), it is possible to keep the same scale dependence n s (k), too. However, this potential is modified compared to the one in the Einstein frame representation of the R 2 model. It is worth to mention also that our results regarding changes to the tensor perturbations during inflation, in particular a modified value of r, seem to have no local counterpart. Given a local curvature squared gravity with R 2 and C 2 µναβ terms we readily observe 2 poles in the spin-2 propagator (this is read from (5.11) with F C ( ) = const). The second pole will be either ghost or tachyon creating instabilities. Even if someone is ready to live with this, corresponding changes to parameters like r will go way beyond just a simple multiplier. A model of this kind was analyzed in [55]. The same should apply to higher but finite number of derivatives as a finite number of poles will be generated. In the non-local gravity however, infinite number of derivatives can be combined to an operator with no poles at all. There are many open questions of the cosmological origin around obtained here results. One may want to analyze perturbations beyond dS phase. This requires an anticipated quadratic variation of the non-local gravity action around any background (at least satisfying ansatz (3.5)). A much more involved problem is a generalization of our considerations to the full inflationary model considered in [1,2] which uses the local R 2 model to get inflation and the graceful exit from it to a matter-dominated stage driven by scalarons at rest, but needs inclusion of one-loop quantum gravitational corrections to obtain decay of scalarons into pairs of particles and anti-particles of all kinds of matter (including the known ones) with their subsequent thermalization leading finally to the hot radiation-dominated stage (the standard hot Big Bang). Another interesting question is a re-computation of the multipole curve in the non-local theory. While the curve itself finds strong observational support, we can use it to constrain further non-local form-factors. As a more ambitious and a very intriguing question we see the development of a model, which could unify bounce and inflation in the non-local gravity framework. We recall that from the point of view of the non-local gravity theories presented in this paper the question of building a model is translated into finding an appropriate form-factors, which allow a given solution to EOM to exist. Enriched, compared to a local f (R) gravity, parameter space in principal allows a co-existence of bounce and inflation. Early studies of this question in a local R 2 gravity, where such bounce is possible in closed FLRW backgrounds with a positive spatial curvature, were already done in [1]. Bounce in non-local gravities was studied in, e.g. [16,51,74,76,77] and many interesting and important results are already uncovered. Nevertheless, a complete scenario joining bounce and inflation in one is still missing. Returning to the present paper we conclude that for the moment we have provided a class of gravity theories, which accommodate at the same time unitarity, stability, asymptotic polynomial behaviour of the form-factors, and inflation. We can successfully confront inflationary parameters being derived in our model with their respective observed values. A Notations We use κ D for the gravitational coupling and M P for the Planck mass. D is the dimension. Unless specified explicitly we work in D = 4. In this case we have 1 2κ 2 4 = M 2 P 2 = 1 16πG N , (A.1) where G N is the Newtonian constant. The metric signature is: g µν = (−, +, +, +, . . . ), g µν g µν = D . (A.2) The 4-dimensional indices are labelled by small Greek letters. The metric-compatible connection (Christoffel symbols) reads Γ ρ µν = 1 2 g ρσ (∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) . (A.3) The covariant derivative is denoted by ∇ µ and acts as follows, ∇ µ F .α. .β. = ∂ µ F .α. .β. + Γ α µχ F .χ. .β. − Γ χ µβ F .α. .χ. . (A.4) It follows that ∇ ρ g µν ≡ 0. The Riemann tensor, curvatures and the Einstein tensor are defined as R σ µνρ = ∂ ν Γ σ µρ − ∂ ρ Γ σ µν + Γ σ χν Γ χ µρ − Γ σ χρ Γ χ µν , R µρ = R σ µσρ , R = R µ µ , (A.5) G µν = R µν − 1 2 Rg µν . (A.6) The commutator of covariant derivatives and the d'Alembertian (box) operators are [∇ µ , ∇ ν ]A ρ = R χ ρνµ A χ , = g µν ∇ µ ∇ ν . (A.7) The Weyl tensor follows from the Ricci decomposition, namely C µ ανβ = R µ ανβ − 1 D − 2 (δ µ ν R αβ − δ µ β R αν + g αβ R µ ν − g αν R µ β ) + R (D − 2)(D − 1) (δ µ ν g αβ − δ µ β g αν ) . (A.8) The Weyl tensor has all the symmetry properties of the Riemann tensor and it is absolutely traceless, i.e. C µ αµβ = 0 . (A.9) Moreover it is invariant under Weyl rescalings, i.e. C µ αβγ = C µ αβγ forĝ µν = Ω 2 (x)g µν . (A.10) The latter implies that the Weyl tensor is zero for conformally flat manifolds (i.e. those manifolds, where the metric can be brought to the form ds 2 = a(x) 2 η µν dx µ dx ν with η µν being the Minkowski metric with the same signature). When the index structure is not crucial we use R for R , Ric for R µν , Riem for R µναβ , C for C µναβ , for brevity. The bar is used to designate background quantity upon perturbations such that ϕ =φ + δϕ . The FLRW Universe is described by the following metric, ds 2 = −dt 2 + a(t) 2 dr 2 1 − Kr 2 + r 2 dΩ 2 . (A.11) Here t is the cosmic time and a(t) is the scale factor. K = ±1, 0 designates open, closed and spatially flat configurations. In the present paper we focus on the spatially flat case with K = 0. Then we can write the FLRW metric as ds 2 = −dt 2 + a(t) 2 δ ij dx i dx j . (A.12) The Hubble function is defined as H =ȧ/a with dot denoting the derivative with respect to t. Equivalently one can rewrite (A.12) as ds 2 = a(τ ) 2 −dτ 2 + δ ij dx i dx j . (A.13) Here τ is the conformal time and the relation between it and the cosmic time is adτ = dt. Hence, the FLRW Universe is conformally flat and the Weyl tensor in it is identically zero. The background quantities in the latter metric are Γ µ 0ν = Hδ µ ν , Γ 0 µν = δ µν H , H = a ′ /a , (A.14) R = 6 a 2 (H ′ + H 2 ) , R µν = −3H ′ 0 0 (H ′ + 2H 2 )δ ij , (A.15) R 0 i0j = H ′ δ ij , R i 0j0 = −H ′ δ i j , R i jkm = H 2 (δ i k δ jm − δ i m δ kj ) . (A.16) We use the index "0" for the tau component of any tensor (the cosmic time is used less often and wherever needed we designate it with the index t). Latin small letters from the middle of the alphabet are used for the spatial indices, while with ′ we denote the derivative with respect to the conformal time τ . We finally introduce the following useful relations, ∂ 0 = a∂ t , H = H/a , S = − 1 a 2 (∂ 2 0 + 2H∂ 0 − δ ij ∂ i ∂ j ) , (A.17) R = 6Ḣ + 12H 2 , where S is the d'Alembertian operator acting on scalars. Perturbation functions are not space-homogeneous. It is convenient to perform a spatial Fourier transform which for some function ϕ(τ, x) in the case K = 0 reads ϕ(τ, x) = ϕ(τ, k)e i k x d k . Using the above Fourier representation one readily observes that S ϕ(τ, x) = − 1 a 2 (∂ 2 0 + 2H∂ 0 + k 2 )ϕ(τ, k)d k , where the bar means that the scalar d'Alembertian operator is not perturbed. The following notation is extremely useful: k ϕ(τ, k) = − 1 a 2 (∂ 2 0 + 2H∂ 0 + k 2 )ϕ(τ, k) . (A.18) In the dS limit we have: B Derivation of equations (3.14), (3.15) and (3.16) For the Lagrangian density (2.22), the (00)-component of the gravitational field equations for an isotropic homogeneous spatially flat universe, a spatially flat FLRW background, is the following third order differential equation with respect to the scale factor a(t): H ≈ const ,HṘ M 2 − 3(Ḣ + H 2 ) 1 + R 3M 2 + 1 2 R + R 2 6M 2 = κ 2 4 ρ , (B.1) Further, we consider the stage of expansion H > 0. In the absence of any kind of matter (ρ = 0), it has two symmetries: invariance under time translation and scale factor dilatation. Thus, it can be reduced to the first order equation [1]: dy dx = − M 2 12x 1/3 y − 1 , x = H 3/2 , y =Ḣ 2H 1/2 , dt = dx 3x 2/3 y . (B.2) The inflationary regime \|Ḣ\| ≪ H 2 corresponds to y < 0, \|y\| ≪ x, \|dy/dx\| ≪ 1. Thus, in the leading order of the slow-roll approximation: y = y 0 = − M 2 12x 1/3 , x ≫ M 3/2 , t s − t = 6x 2/3 M 2 = 6H M 2 . (B.3) Let us calculate the next order correction: y = y 0 + y 1 . Then where δ (s) means that the variation is evaluated on scalar perturbations. K βµ να is a constant tensor whose components depend on k i only. Moreover it is absolutely traceless and retains all the symmetry properties of the Weyl tensor. Its explicit form is K 0i 0j = − 1 6 k 2 δ i j + 1 2 k i k j , K ki mj = 1 3 k 2 (δ k m δ i j − δ k j δ i m ) − 1 2 δ k m k i k j − 1 2 δ i j k k k m + 1 2 δ k j k i k m + 1 2 δ i m k k k j . Few comments are in order. First, variation of the Weyl tensor is manifestly gauge-invariant in our case, because this tensor is zero on the background. Second, again because the Weyl tensor is zero on the background its variation must retain the property to be trace-free. 10 Third, the appearance of a factor Φ + Ψ is easy to understand as the case Φ + Ψ = 0 corresponds to a specific form of the metric perturbation such that h µν = 2Φg µν . One easily sees here a conformal rescaling of the metric. The Weyl tensor in turn is known to transform covariantly (rescaling by the conformal function) under conformal transformations. It is even conformally invariant provided that one of its indices is up and others are down. The latter implies that for Φ + Ψ = 0 combined with the fact that the Weyl tensor is zero on the background the variation of the Weyl tensor must be zero as it is obvious from equation (C.1). The next step is to computē Ck δ (s) C βµ να = (¯ k + 6H 2 ) Φ + Ψ a 2 K βµ να . (C.2) We pay attention to the fact that¯ Ck is a new operator. Bar as before indicates that it is not perturbed, the subscript C reminds that it is the d'Alembertian operator with all accompanying connection terms due to the rank of the tensor on the right, the subscript k means that the tensor on the right is Fourier transformed with respect to the spatial coordinates. The essential result here is that on the right hand side box acts on a scalar function only. This results through a recursion relation in F C (¯ Ck )δ (s) C βµ να = F C (¯ k + 6H 2 ) Φ + Ψ a 2 K βµ να . (C.3) To finalize the evaluation of (5.27) we have to compute c µ ν = (R α β + 2∇ α k∇kβ )Ω(τ )K βµ να where Ω(τ ) = F C (¯ k + 6H 2 ) Φ + Ψ a 2 . (C.4) The subscript k in ∇ technically means that wherever encountered, we replace ∂ i → ik i . Some long, careful and rather tough computation gives c i j = k i k j a 2 Ω ′′ + 4HΩ ′ + (3H ′ + 3H 2 )Ω + k 2 3 Ω − a 2Ḣ Ω where i = j , c 0 i = − 2 3 ik i k 2 a 2 (Ω ′ + 2HΩ) , c 0 0 = 2 3 k 4 a 2 Ω , c i i = − 1 3a 2 (k 2 − 3k i k i )(Ω ′′ + 4HΩ ′ + (3H ′ + 3H 2 + a 2Ḣ )Ω) + k 2 (k 2 − k i k i )Ω , (C.5) with no sum over i in the last expression. First we stress that the above result is for a generic scale factor a(τ ). As a verification of the above formulae one can check that the trace c µ µ vanishes. Also we notice that the terms proportional toḢ originate from the Ricci tensor term. Hence, this contributions cancel in the dS limit when H is almost a constant. This is very much correct, since the Ricci tensor becomes proportional to the metric tensor and being contracted with the K-tensor results in the zero trace. We verify the equations for perturbations by evaluating the Bianchi identity. The identity dictates that ∇ µ G µ ν ≡ 0 . (D.1) This implies that in any generally covariant gravity modification EOM should form a rank-2 tensor obeying Bianchi identity. Therefore, for system (3.3) we can write ∇ µ E µ ν = 0 . (D.2) This fact at the background level was explicitly verified in [25,51]. At the perturbed level up to the linear order we must have (δ∇ µ )Ē µ ν +∇ µ (δE µ ν ) = 0 . (D.3) However, to simplify the task, we will assume that we do the computation around a given background, meaning thatĒ µ ν = 0 even though the identical cancellation should hold in general. This simplification is valid since we were already using the ansatz (3.5) and relation among parameters (3.6) in derivation of the perturbation equations. Having said this we must show that∇ µ (δE µ ν ) = 0 , (D.4) for all ν. Moreover, since this is an identity it must hold for any values of constant parameters in δE µ ν . This in turn implies that it must hold independently for the c µ ν tensor. We thus start by showing that∇ µ c µ ν = 0 . (D.5) Accounting (C.5) and performing a lengthy algebra one can demonstrate explicitly that (D.5) holds. This in the meantime verifies (C.5) itself. The next step is to verify that ∇ µ (δE µ ν (F C = 0)) = 0 . (D.6) For this aim we need an explicit form of δE i i , where no sum is taken and F C = 0. This is given as δE i i (F C = 0) = = − 2λ a 2 F 1 (R + 3r 1 ) (Ψ − Φ)(k 2 − k i k i ) + 2(Ψ ′′ + 2HΨ ′ ) + 2(HΦ ′ + (H 2 + 2H ′ )Φ) − 2λ a 2 HΥ ′ + 2λ a 2 (2H ′ + H 2 − k i k i ) − λ(R + 2r 1 ) Υ + λ a 2 (R ′ Ξ ′ + (R ′′ + 2HR ′ )Ξ) + 2λ a 2 F 1R ′ (Ψ ′ + 2HΦ) − 2λF(¯ k )ζ . (D.7) As a first check for the latter equation itself one can prove that performing the trace δE µ µ one indeed gets (5.25). Then after indeed a tedious and long computation one proves that (D.6) is true. order variation of action (3.1) in terms of h µν 19 5.2 Covariant spin-2 excitations and restrictions on form-factor F C Acknowledgments AK is supported by the FCT Portugal fellowship SFRH/BPD/105212/2014 and in part by FCT Portugal grant UID/MAT/00212/2013 and by RFBR grant 14-01-00707. AS was supported by the RSF grant 16-12-10401. order correction to y is proportional to x −3 . It follows from this thaṫ zero-order result for H is used to obtain the last term. Integration of (B.6) leads to (3.14), (3.15), (3.16) with r 1 = M 2 . a ≈ a 0 e Ht , (A.19)R σ µνρ ≈ R 12 (δ σ ν g µρ − δ σ ρ g µν ) , R µ ν ≈ R 4 δ µ ν , R ≈ 12H 2 ≈ const. (A.20) In the conformal time we observe that a ≈ − 1 Hτ , H ≈ − 1 τ and H ′ ≈ H 2 ≈ 1 τ 2 . (A.21) In the course of this paper terms "R 2 inflationary model (spacetime, metric, etc.)" or just "R 2 model (inflation, gravity, etc.)" refer to the local R + R 2 model and its solutions. Note also that this is a simplified form of the model studied in[1] in the limit M ≪ H using notations of that paper.2 Analyticity of functions of differential operators implies that theories based on the pure inverse d'Alembertian[18] are strictly separated from our considerations. In[1], the final results of corresponding calculations were presented only. The detailed calculations of particle creation using the formalism of α, β-coefficients of the Bogolyubov transformation can be found in[2]. Note also that decay of scalarons into pairs of gravitons appears to be strongly suppressed[40] that it is very important for the viability of the model, too.4 We also point out related studies addressing aspects of non-local gravity theories while having inverse powers of the d'Alembertian as well[47][48][49][50]. It was shown in[53] that the scale factor in(3.14) gives an exact solution to the full system of nonlocal (or local as well) equations of motion provided we introduce a small negative cosmological term. The presence of a cosmological term in a local R 2 model describing inflation is questionable as this introduces an unnecessary extra parameter. Notice that even if Λcc = 0 and consequently r2 = 0 an assumption r1 = M 2 leads to an impossible (in real numbers) relation e −H 0 (r 1 ) = − 5 4 < 0. So, we again must require(3.20) and therefore implement changes to the form-factor H0. After the present paper was submitted to arxiv and to the journal, the paper[70] co-authored by one of the present authors appeared in which the detailed computation of the second order variation of non-local quadratic in curvature actions around maximally symmetric background using the spin decomposition of the metric fluctuation was presented. In fact this is the only term that for the moment we can analyze around a nearly dS background only. Notice that the present equations (5.28), (5.29) differ from analogous equations presented in[69] in what they must coincide (i.e. without c µ ν ) by terms which were irrelevant for the analysis in[69]. We assume that it was a misprint type omission in that paper. Our formula for the Weyl tensor variation differs from the one presented in[78]. Namely[78] claims that only (0i0j) components are non-trivial (for which our results coincide). 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[ "Vote for Me! Election Control via Social Influence in Arbitrary Scoring Rule Voting Systems", "Vote for Me! Election Control via Social Influence in Arbitrary Scoring Rule Voting Systems" ]	[ "Federico Corò [email protected] \nGSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly\n", "Emilio Cruciani [email protected] \nGSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly\n", "Gianlorenzo D ' Angelo [email protected] \nGSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly\n", "Stefano Ponziani [email protected] \nGSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly\n" ]	[ "GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly", "GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly", "GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly", "GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7\n67100L'AquilaItaly" ]	[]	Online social networks are used to diffuse opinions and ideas among users, enabling a faster communication and a wider audience. The way in which opinions are conditioned by social interactions is usually called social influence. Social influence is extensively used during political campaigns to advertise and support candidates.Herein we consider the problem of exploiting social influence in a network of voters in order to change their opinion about a target candidate with the aim of increasing his chance to win/lose the election in a wide range of voting systems.We introduce the Linear Threshold Ranking, a natural and powerful extension of the well-established Linear Threshold Model, which describes the change of opinions taking into account the amount of exercised influence. We are able to maximize the score of a target candidate up to a factor of 1 − 1/e by showing submodularity. We exploit such property to provide a 1 3 (1 − 1/e)-approximation algorithm for the constructive election control problem. Similarly, we get a 1 2 (1 − 1/e)-approximation ratio in the destructive scenario. The algorithm can be used in arbitrary scoring rule voting systems, including plurality rule and borda count. Finally, we perform an experimental study on real-world networks, measuring Probability of Victory (PoV) and Margin of Victory (MoV) of the target candidate, to validate the model and to test the capability of the algorithm.	null	[ "https://arxiv.org/pdf/1902.07454v1.pdf" ]	67,771,623	1902.07454	070143f5e8521d23ed723aa2a5fe898c2c7f5a17	Vote for Me! Election Control via Social Influence in Arbitrary Scoring Rule Voting Systems 20 Feb 2019 Federico Corò [email protected] GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7 67100L'AquilaItaly Emilio Cruciani [email protected] GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7 67100L'AquilaItaly Gianlorenzo D ' Angelo [email protected] GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7 67100L'AquilaItaly Stefano Ponziani [email protected] GSSI -Gran Sasso Science Institute Viale Francesco Crispi 7 67100L'AquilaItaly Vote for Me! Election Control via Social Influence in Arbitrary Scoring Rule Voting Systems 20 Feb 20191 Online social networks are used to diffuse opinions and ideas among users, enabling a faster communication and a wider audience. The way in which opinions are conditioned by social interactions is usually called social influence. Social influence is extensively used during political campaigns to advertise and support candidates.Herein we consider the problem of exploiting social influence in a network of voters in order to change their opinion about a target candidate with the aim of increasing his chance to win/lose the election in a wide range of voting systems.We introduce the Linear Threshold Ranking, a natural and powerful extension of the well-established Linear Threshold Model, which describes the change of opinions taking into account the amount of exercised influence. We are able to maximize the score of a target candidate up to a factor of 1 − 1/e by showing submodularity. We exploit such property to provide a 1 3 (1 − 1/e)-approximation algorithm for the constructive election control problem. Similarly, we get a 1 2 (1 − 1/e)-approximation ratio in the destructive scenario. The algorithm can be used in arbitrary scoring rule voting systems, including plurality rule and borda count. Finally, we perform an experimental study on real-world networks, measuring Probability of Victory (PoV) and Margin of Victory (MoV) of the target candidate, to validate the model and to test the capability of the algorithm. Introduction As humans, we usually have a specific personal opinion on certain topics, such as lifestyle or consumer products. These opinions, normally formed on personal life experience and information, can be conditioned by the interaction with our friends leading to a change in our original opinion on a particular topic if a large part of our friends holds a different opinion. In the last decades, this phenomenon of opinion diffusion has been intensely investigated from many different perspectives, from sociology to economics. In recent years, there has been a growing interest on the relationship between social networks and political campaigning. Political campaigns nowadays use online social networks to lead elections in their favor; for example, they can target specific voters with fake news [AG17]. A real-life example of political intervention in this context occurred in the US Congressional elections in 2010, where a set of users were encouraged to vote with a message on Facebook. These messages directly influenced the real-world voting behavior of millions of people [BFJ + 12]. Another example is that of French elections in 2017, where automated accounts in social networks spread a considerable portion of political content, mostly fake news, trying to influence their outcome [Fer17]. There exist an extensive literature on manipulating elections without considering the underlying social network structure of the voters, e.g., swap bribery [EFS09], shift bribery [BFNT16]; we point the reader to a recent survey [FRM16]. The study of opinion diffusion modeled as a majority dynamics has attracted much attention in recent literature [ACF + 15, BEEG16,BGP17]. In these models each agent has an initial preference list and at each time step a subset of agents updates their opinions, i.e., their preference lists, according to some majority-based rule that depends on the opinions of their neighbors in the network. Nevertheless, there are only few studies on the opinion diffusion on social networks. The Independent Cascade Model [KKT03] has been considered as diffusion process to guarantee that a target candidate wins/loses [BTT92,HHR07]. The constructive (destructive) election control problem consists in targeting a specific candidate to change voters' opinions about him with the aim of maximizing (minimizing) his margin and probability of victory [WV18b]. A variant of the Linear Threshold Model [KKT03] with weights on the vertices has been considered on a graph in which each node is a cluster of voters with a specific list of candidates and there is an edge between two nodes if they differ by the ordering of a single pair of adjacent candidates [FGKT18]. Moreover, it has been studied how to manipulate the network in order to have control on the majority opinion, e.g., bribing or adding/deleting edges, on a simple Linear Threshold Model where each node holds a binary opinion, each edge has the same fixed weight, and all vertices have a threshold fixed to 1/2 [BE17]. In this work we focus on a variant of the election control through social influence problem defined in [WV18b]: Given a social network of people willing to vote, we want to select a fixed-size subset of voters such that their influence on the others will change the election outcome, maximizing the chances of a target candidate to win or lose, in the specific scenario in which only the opinions about a target candidate can be changed. Differently from previous work [WV18b] we consider more general voting systems and a different diffusion model, that takes into account the degree of influence that voters exercise on the others and is able to describe the scenario in which a high influence on someone can radically change its opinion. In this setting we prove the nontrivial fact that any scoring function is monotone and submodular with respect to the initial set of active nodes. Moreover we exploit this fact to prove a constant factor approximation of the election control problem in our model. Original Contribution • We introduce the Linear Threshold Ranking, a natural and powerful extension of the Linear Threshold Model for the election scenario that takes into account the degree of influence of the voters on each other. • We prove that, in such model, we can achieve a (1 − 1/e)-approximation to the problem of maximizing the score of a target candidate by proving submodularity in the general case of the scoring rule for arbitrary scoring function (including popular voting systems, e.g., plurality rule or borda count), with any number of candidates. • We exploit the (1 − 1/e)-approximation algorithm that maximizes the score to achieve an extra approximation factor of 1 3 to the problem of maximizing the Margin of Victory of a target candidate in arbitrary scoring rule voting systems with any number of candidates. • We give a simple reduction that maps destructive control problems to constructive control ones and allows us to achieve a 1 2 (1 − 1/e)-approximation to the destructive control problem. • We perform simulations of our model on four heterogeneous real-world networks and test the capability of our algorithm. Background In this section we present some notions and concepts about voting systems and influence maximization on social networks that will be used in the design and analysis of the algorithm. Moreover we formally introduce the problem and the notation used to analyze it. Voting Systems Voting systems are sets of rules that regulate all aspects of elections and that determine their outcome. In particular a voting system decides candidates and voters eligibility, other than fixing the rules for determining the winner of the elections. Social choice theory formally defines and analyzes voting systems, studying how the combination of individual opinions or preferences reaches a collective decision; computational social choice, instead, studies the computational complexity of outcomes of voting rules and can serve as a barrier against strategic manipulation in elections [CELM07, FP10, BCE + 16, End17]. We focus on two single-winner voting systems: • Plurality rule: Each voter can only express a single preference among the candidates and that with the plurality of the votes wins, i.e., it is sufficient to have the highest number of votes and there is no need of an absolute majority (50%+1 of votes). • Scoring rule: Each voter expresses his preference as a ranking; each candidate is then assigned a score, computed as a function of the positions he was ranked among the voters. The former is arguably the simplest scenario and is one of the most commonly used for national legislatures and presidential elections. The latter is a very general definition, but can include several popular election methods by choosing an adequate scoring function: • if the scoring function assigns 1 point to the first candidate and 0 to all the others this is equivalent to the plurality rule; • if the scoring function assigns 1 point to the first t candidates and 0 to the others then it is equivalent to the t-approval, where each voter approves t candidates; • if the scoring function assigns 1 point to the first m − t candidates and 0 to the remaining t, where m is the total number of candidates, then it is equivalent to the t-veto or anti-plurality rule; • if the scoring function assigns m − l points to the candidate in position l then it is equivalent to the borda count, in which each voter ranks the candidates and each candidate gets a score equal to the number of candidates ranked lower in each list. Influence Maximization The influence maximization problem studies a social network represented as a graph and has the goal of finding the B-sized set of influential nodes that can maximize the spread of information [KKT15]. In general, all existing diffusion models can be categorized into three classes: cascade models, threshold models, and epidemic models. The most popular for studying social influence problems are the Independent Cascade Model (ICM) and the Linear Threshold Model (LTM). These models are graph-based, namely they assume an underlying directed graph where nodes represent agents and edges represent connections between them. Each node can be either active, that is it spreads the information, or inactive. With some probability, active nodes diffuse the information to their neighbors. The ICM model requires a diffusion probability to be associated with each edge, whereas LTM requires an influence degree to be defined on each edge and an influence threshold on each node. For both models, the diffusion process proceeds iteratively in a synchronous way along a discrete time-axis, starting from an initial set of nodes, usually called seeds. In this work we focus on LTM [KKT03]. Given a graph G = (V, E), in LTM each node v ∈ V has a threshold t v ∈ [0, 1] sampled uniformly at random and independently from the other nodes and each edge (u, v) ∈ E has a weight b uv ∈ [0, 1] with the constraint that, for each v ∈ V , the sum of the weights of the incoming edges of v is less or equal to 1, i.e., (u,v)∈E b uv 1. Let A 0 ⊆ V be the set of active nodes at the beginning of the process. More in general, let A t ⊆ V be the set of nodes active at time t. In LTM an inactive node v becomes active if the sum of the weights of the edges coming from nodes that are active at the previous round is greater than or equal to its threshold t v , i.e., v ∈ A t if and only if v ∈ A t−1 or u∈A t−1 :(u,v)∈E b uv t v . When a node is active, it influences its neighbors and increases the chance of making them change their preference list. The process has quiesced at the first timet such that the set of active nodes would not change in the next round, i.e., timet is such that At = At +1 . We define the eventual set of active nodes as A := At. The distribution of the set of active nodes in the graph starting with A 0 under the LTM process is equivalent to the distribution reachable from the same set A 0 in the set of random graphs called live-edge graphs (Theorem 1). A live-edge graph is built as follows: Given an influence graph G = (V, E), for every node v ∈ V select at most one of its incoming edges with probability proportional to the weight of that edge, i.e., edge (u, v) is selected with probability b uv , and no edge is selected with probability 1 − u∈Nv b uv . Let us denote by G the set of all possible live-edge graphs that can be generated from G. Theorem 1 (Kempe, Kleinberg, and Tardos [KKT15]). Given a graph G = (V, E) and an initial set of nodes A 0 ⊆ V , the distribution of the sets of active nodes in G after LTM has quiesced starting from A 0 is equal to the distribution of the sets of nodes that are reachable from A 0 in the set of live-edge graphs G. Moreover, under the live-edge model, the problem of selecting the initial set of nodes in order to maximize the diffusion is submodular [KKT15]. Therefore, exploiting a classical result [NWF78], the influence maximization problem can be approximated to a constant factor of 1 − 1/e using a simple greedy hill-climbing approach that starts with an empty solution and, for B iterations, selects a single node that gives the maximal marginal gain on the objective function with respect the solution computed so far. This algorithm guarantees the best approximation, but is still very computational expensive: Evaluating the expected number of active nodes is #P -hard [CYZ10]. There exists a simulation-based approach in which a Monte-Carlo simulation is performed to evaluate the influence spread of a given seed set A [KKT15]. The standard Chernoff-Hoeffding bounds imply 1 ± approximation to the expected number of active nodes by simulating a polynomial number of times the diffusion process. Linear Threshold Ranking and Election Control We consider the scenario in which a set of candidates are running for the elections and a social network of voters will decide the winner. In particular we focus on the simple plurality rule and on the more general case of the scoring rule. Some attacker could be interested in changing the outcome of the elections by targeting a subset of voters with advertisement or (possibly fake) news about one specific candidate. Such voters, with some probability, can influence their friends by sharing the news. Suppose the attacker has a budget that can use to target some voters and that they will start a diffusion process that changes opinions in the network. Is it possible for the attacker to select a set of voters in order to have constructive/destructive control over a target candidate, i.e., to change the voters' opinions on this candidate in order to maximizes his chances to win/lose the elections? More formally, let G = (V, E) be a directed graph representing the underlying social network of people willing to vote. For each node v ∈ V we call N v the sets of incoming neighbors of v. Let C = {c 1 , . . . , c m } be a set of m candidates nominated for the elections; we refer to our target candidate, i.e., the one that we want to make win/lose the elections, as c . Each v ∈ V has a permutation π v of C, i.e., its list of preferences for the elections; we denote the position of candidate c i in the preference list of node v as π v (c i ). We consider the LTM process starting from an initial set of active nodes A 0 ⊆ V . Recall that, according to LTM, each node v ∈ V has a threshold t v , each edge (u, v) ∈ E has a weight b uv , and that A ⊆ V is the set of active nodes at the end of the process. Let B ∈ N be an initial budget that can be used to select the nodes in A 0 , i.e., the set of active nodes from which the LTM process starts. In particular, the budget constrains the size of A 0 , namely \|A 0 \| B. After the LTM process has quiesced, the position of c in the preference list of each node changes according to a function of its incoming active neighbors. The threshold t v of each node v ∈ V models its strength in retaining its original opinion about candidate c : The higher is the threshold t v the lower is the probability that v is influenced by its neighbors. Moreover the weight on an edge b uv measures the influence that node u has on node v. Taking into account the role of such parameters, we define the number of positions that c goes up in π v as π ↑ v (c ) := min   π v (c ) − 1,     α(π v (c )) t v u∈A, (u,v)∈E b uv       , where α : {1, . . . , m} → [0, 1] is a function that depends on the position of c in π v and models the rate at which c shifts up. Note that α can be set arbitrarily to model different scenarios, e.g., shifting up of one position from the bottom of the list could be easier than going from the second position to the first with a suitable choice of α. As for π ↑ v (c ), it can be any integer value in {0, . . . , π v (c ) − 1}: The floor function guarantees a positive integer value; the minimum between such value and π v (c ) − 1 guarantees that final position of c is at least 1, since the floor function could output too high values when the threshold is small w.r.t. the neighbors' influence. We call this process the Linear Threshold Ranking (LT R). After the modification of the lists at the end of LT R, the candidates might have a new position in the preference list of each node v ∈ V ; we denote such new preference list asπ. In particular, the new position of candidate c will bẽ π v (c ) := π v (c ) − π ↑ v (c ) ; the candidates that are overtaken by c will shift one position down. In the problem of election control we want to maximize the chances of the target candidate to win the elections under LT R. To achieve that, we maximize its expected Margin of Victory (MoV) w.r.t. the most voted opponent, akin to that defined in [WV18b]. 1 Let us consider the general case of the scoring rule, where a nonincreasing scoring function f : {1, . . . , m} → N assigns a score to each position. Let c andc respectively be the candidates, different from c , with the highest score before and after LT R. Let µ(∅) := v∈V f (π v (c)) − f (π v (c )) (1) µ(A 0 ) := v∈V f (π v (c)) − f (π v (c ))(2) be the margin (i.e., difference in score) between the most voted opponent and c before and after LT R (Equations (1) and (2)). Thus, the election control problem is formalized as that of finding a set of nodes A 0 such that max A 0 E [MoV(A 0 )] := E [µ(∅) − µ(A 0 )] s.t. \|A 0 \| B, namely to find an initial set of seed nodes of at most size B that maximizes the expected MoV, i.e., change in margin. 2 Algorithm 1 Greedy Require: Social graph G = (V, E); Budget B; Score function F 1: A 0 = ∅ 2: while \|A 0 \| B do 3: v = arg max w∈V \A 0 F (A 0 ∪ {w}) − F (A 0 ) 4: A 0 = A 0 ∪ {v} 5: return A 0 To solve the problem we focus on the score of the target candidate. Let us define F (∅) := v∈V f (π v (c )) (3) F (A 0 ) := E v∈V f (π v (c ))(4) as the total expected score obtained by candidate c before and after LT R (Equations (3) and (4)). In Sections 4 and 5 we prove that the score of the target candidate is a monotone submodular function w.r.t. the initial set of seed nodes A 0 in both the plurality and the scoring rule; this allows us to get a (1 − 1/e)-approximation of the maximum score through the use of Greedy (Algorithm 1). Note that maximizing the score of the target candidate is a NP -hard problem: Consider the case in which there are only two candidates, α(1) = α(2) = 1, all nodes have c as second preference, and the scoring function is that of the plurality rule; maximizing the score is equal to maximizing the number of active nodes in LTM because when a node becomes active the target candidate shifts of at least one position up (in this case, in first position); thus the two problems are equivalent. Since influence maximization in LTM is NP -hard, then also maximizing the score in LT R is NP -hard because it is a generalization of LTM. Moreover, in this instance, the maximum value of MoV is equal to twice the maximum score; then the problem of maximizing MoV is also NP -hard. Although maximizing the score is not equivalent to maximizing MoV, in Section 6 we show that it gives a constant factor approximation to MoV. Finally, in Section 7, we consider the problem of destructive control, in which we want the target candidate to lose the elections. We prove a constant factor approximation to MoV also in this case by exploiting a simple reduction that maps it to the constructive case. Maximizing the Score: Plurality Rule As a warm-up, in this section we focus on the plurality rule. We give an algorithm to select an initial set of seed nodes to maximize the expected number of nodes that will change their opinion and have c as first preference at the end of LT R. Given a set of initially active nodes A 0 , let A be the set of nodes that are active at the end of the process. An active node v with π v (c ) > 1 will have c as first preference if π ↑ v (c ) = π v (c ) − 1, that is if and only if α(π v (c )) t v u∈A∩Nv b uv π v (c ) − 1 or, equivalently, t v α(π v (c )) π v (c ) − 1 u∈A∩Nv b uv . As for the influence maximization problem, we define an alternative random process based on live-edge graphs. One possibility could be the following: For each live-edge graph evaluate which active nodes satisfy the above formula; however, in the live-edge graph process, we don't know the value of t v since they are sampled uniformly at random at the beginning of LTM. To overcome this limitation we introduce a new process, Live-edge Coin Flip (LCF ). Definition 1. (Live-edge Coin Flip process) 1. Each node v ∈ V selects at most one of its incoming edges with probability proportional to the weight of that edge, i.e., edge (u, v) is selected with probability b uv , and no edge is selected with probability 1 − u∈Nv b uv . 2. Each node v with π v (c ) > 1 that is reachable from A 0 in the live-edge graph flips a biased coin and changes its list according to the outcome. This is equivalent of picking a random real number s v ∈ [0, 1] and setting the position of c according to s v as follows: If s v α(πv(c )) πv(c )−1 , node v chooses c as its first preference (i.e., it setsπ v (c ) = 1 and shifts all the other candidates down by one position); otherwise, v maintains its original ranking. In the following we show that the two processes are equivalent, i.e., starting from any initial set A 0 each node in the network has the same probability to end up with c in first position in both processes. This allows us to compute the function F (A 0 ), for a given A 0 , by solving a reachability problem in graphs, as we will show later in this section. We first prove the next Lemma which will be used to show the equivalence between the two processes and to compute F (A 0 ). The lemma shows how to compute the probability that a node v is reachable from A 0 at the end of the LCF process by using the live-edge graphs or by using the probability of the incoming neighbors of v to be reachable from A 0 . We denote by G the set of all possible live-edge graphs sampled from G. For every G = (V, E ) ∈ G we denote by P (G ) the probability that the live edge graph is sampled, namely P G = v:(u,v)∈E b uv v: ∃(u,v)∈E   1 − w:(w,v)∈E b wv   . We denote by R(A 0 ) the set of nodes reachable from A 0 at the end of the LCF process and by R G (A 0 ) the set of nodes reachable from A 0 in a fixed live-edge graph G and by 1 (G ,v) the indicator function that is 1 if v ∈ R G (A 0 ) and 0 otherwise. Lemma 1. Given a set of initially active nodes A 0 , let R(A 0 ) be the set of nodes reachable from A 0 at the end of the LCF process. Then P (v ∈ R(A 0 )) = G ∈G P G · 1 (G ,v) = U ⊆Nv u∈U b uv · P ((R(A 0 ) ∩ N v ) = U ) . Proof. By the law of total probability P (v ∈ R(A 0 )) = G ∈G P v ∈ R(A 0 ) G · P G . Given a live-edge graph G sampled form G, the value of P (v ∈ R \| G ) is equal to 1 if v is reachable from A 0 in G , and it is 0 otherwise. Then G ∈G P v ∈ R(A 0 ) G · P G = G ∈G P G · 1 (G ,v) . which shows the first part of the lemma. We now show the following equality: G ∈G P G · 1 (G ,v) = U ⊆Nv u∈U b uv · P ((R(A 0 ) ∩ N v ) = U ) .(5) We can re-write the left hand side as G ∈G P G · 1 (G ,v) = U ⊆Nv G ∈G s.t. R G (A 0 )∩Nv=U P G · 1 (G ,v) . In each live-edge graph G for which P (G ) · 1 (G ,v) = 0 node v selected one of its incoming edges and then P (v selected u in LCF ) = b uv , for each u ∈ N v . Therefore, the above value is equal to U ⊆Nv G ∈G s.t. R G (A 0 )∩Nv=U u∈U P G v selected u in LCF b uv = U ⊆Nv u∈U b uv G ∈G s.t. R G (A 0 )∩Nv=U P G v selected u in LCF , where the first equality is due to the law of total probability and the last one is just reordering of the terms of the sums. In each live-edge G that does not contain the edge (u, v), the probability P (G \| v selected u in LCF ) is equal to zero. Then, G ∈G s.t. R G (A 0 )∩Nv=U P G v selected u in LCF = G ∈G s.t. R G (A 0 )∩Nv=U (u,v)∈E P G v selected u in LCF . By definition of conditional probability we have that the above sum is equal to: G ∈G s.t. R G (A 0 )∩Nv=U (u,v)∈E P (G ∩ (v selected u in LCF )) b uv . Since, in each G considered in the sum, edge (u, v) belongs to G , this is equal to: G ∈G s.t. R G (A 0 )∩Nv=U (u,v)∈E P (G ) b uv .(6) Let us now consider the right hand side of Equality (5). For each U ⊆ N v , the probability that (R(A 0 ) ∩ N v ) = U is given by the sum of the probabilities of all the live-edge graphs that satisfy this property, since these graphs represent disjoint events, we have: P ((R(A 0 ) ∩ N v ) = U ) = G ∈G s.t. R G (A 0 )∩Nv=U P G . Let us fix a node u ∈ U . For each G ∈ G such that (R G (A 0 ) ∩ N v ) = U , there exists a live-edge graph G that has the same edges as G but has edge (u, v) as incoming edge of v. Since all the other edges of G are equal to those of G , then (R G (A 0 ) ∩ N v ) = U . We have that P G =            P (G ) b uv · b u i v if ∃(u i , v) in G , P (G ) b uv ·   1 − u i ∈Nv b u i v   otherwise. Therefore, G ∈G s.t. R G (A0)∩Nv=U P (G ) = G ∈G s.t. R G (A0)∩Nv=U (u,v)∈E ui∈Nv b uiv P (G ) b uv + 1 − ui∈Nv b uiv P (G ) b uv = G ∈G s.t. R G (A0)∩Nv=U (u,v)∈E P (G ) b uv . Equality (5) follows since the above expression is equal to (6). The next theorem shows the equivalence between LT R and LCF . Theorem 2. Given a set of initially active nodes A 0 , let A LT R and A LCF be the set of nodes such thatπ v (c ) = 1 at the end of LT R and LCF , respectively, both starting from A 0 . Then, for each v ∈ V , P (v ∈ A LT R ) = P (v ∈ A LCF ). Proof. We exclude from the analysis nodes v with π v (c ) = 1 since they keep their original ranking in both models. Let us start by analyzing the LT R process. Let A be the set of active nodes at the end of the LT R process that starts from A 0 . If U is the maximal subset of active neighbors of v (i.e. U = A ∩ N v ), then we can write the probability that v ∈ A LT R given U , as P v ∈ A LT R (A ∩ N v ) = U = P t v α(π v (c )) π v (c ) − 1 u∈U b uv = α(π v (c )) π v (c ) − 1 u∈U b uv . The overall probability that v ∈ A LT R is P v ∈ A LT R = U ⊆Nv P v ∈ A LT R (A ∩ N v ) = U · P (U = (A ∩ N v )) = α(π v (c )) π v (c ) − 1 U ⊆Nv u∈U b uv · P ((A ∩ N v ) = U ) . Let us now analyze the LCF process. In order for v to be in A LCF it must hold that the coin toss has a positive outcome and that v ∈ R. Thus, P v ∈ A LCF = α(π v (c )) π v (c ) − 1 P (v ∈ R(A 0 )) .(7) By Lemma 1, we have P v ∈ A LCF = α(π v (c )) π v (c ) − 1 U ⊆Nv u∈U b uv · P ((R(A 0 ) ∩ N v ) = U ) . By Theorem 1, P ((R(A 0 ) ∩ N v ) = U ) = P ((A ∩ N v ) = U ) , and hence the theorem follows. We now exploit Theorem 2 to show how to compute the value of F (A 0 ). For each positive integer r m, we denote by V r c i the set of nodes that have candidate c i in position r. In the case of plurality rule, F (A 0 ) is the expected cardinality of A LT R , that is F (A 0 ) = E \|A LT R \| = v∈V P v ∈ A LT R . By Theorem 2 and Equality (7), this is equal to v∈V P v ∈ A LCF = F (∅) + v∈V, πv(c )>1 α(π v (c )) π v (c ) − 1 P (v ∈ R(A 0 )) . By Lemma 1, it follows that F (A 0 ) = F (∅) + v∈V,πv(c )>1 α(π v (c )) π v (c ) − 1 G ∈G P G · 1 (G ,v) . We can rewrite the above formula as follows: F (A 0 ) − F (∅) = m r=2 v:πv(c )=r α(r) r − 1 G ∈G P G · 1 (G ,v) = m r=2 α(r) r − 1 G ∈G P G v:πv(c )=r 1 (G ,v) = m r=2 α(r) r − 1 G ∈G P G · \|{v : v ∈ R G (A 0 ) ∧ π v (c ) = r}\| = m r=2 α(r) r − 1 G ∈G P G \|R G (A 0 , V r c )\|, where, for a graph G ∈ G and a positive integer r m, we denoted by R G (A 0 , V r c ) the subset of V r c of nodes reachable from a set of nodes A 0 in G , R G (A 0 , V r c ) = {v : v ∈ R G (A 0 ) ∧ π v (c ) = r}. It follows that the function F (A 0 ) is a non-negative linear combination of functions \|R G (A 0 , V r c )\|. In the next lemma, we show that R G (A 0 , V r c ) in G is a monotone submodular 3 function of the initial set of nodes A 0 . This implies that also F (A 0 ) is monotone and submodular w.r.t. A 0 and the same holds for F (A 0 ) − F (∅). Therefore, we can use Greedy (Algorithm 1) to find a set A 0 whose value F (A 0 ) − F (∅) is at least 1−1/e times the one of an optimal solution for election control problem [NWF78]. Note that, we can use the same algorithm to approximate F (A 0 ) within the same approximation bound. Lemma 2. Given a graph G ∈ G and a positive integer r m , the size of R G (A 0 , V r c ) in G is a monotone submodular function of the initial set of nodes A 0 . Proof. Given A 0 ⊆ V , for any v ∈ V \ A 0 , the nodes in V r c that are reachable from A 0 in G are reachable also from A 0 ∪ {v}. Therefore, \|R G (A 0 ∪ {v}, V r c )\| \|R G (A 0 , V r c )\|. Let us consider two sets of nodes S, T such that S ⊆ T ⊆ V and a node v ∈ V \ T . We show that \|R G (S ∪ {v}, V r c )\| − \|R G (S, V r c )\| \|R G (T ∪ {v}, V r c )\| − \|R G (T, V r c )\|. Since v ∈ S ∪ {v}, we have that \|R G (S ∪ {v}, V r c )\| − \|R G (S, V r c )\| = \|R G (S ∪ {v}, V r c ) \ R G (S, V r c )\|. Moreover, for any two sets of nodes B, C we have that R G (B∪C, V r c ) = R G (B, V r c )∪ R G (C, V r c ). Hence R G (S ∪ {v}, V r c ) \ R G (S, V r c ) = [R G (S, V r c ) ∪ R G ({v}, V r c )] \ R G (S, V r c ) = R G ({v}, V r c ) \ R G (S, V r c ). Similarly, \|R G (T ∪ {v}, V r c )\| − \|R G (T, V r c )\| = \|R G ({v}, V r c ) \ R G (T, V r c )\|. Since S ⊆ T , then R G (S, V r c ) ⊆ R G (T, V r c ) and then R G ({v}, V r c ) \ R G (S, V r c ) ⊇ R G ({v}, V r c ) \ R G (T, V r c ) , which implies the statement. Maximizing the Score: Scoring Rule In this section we extend the results of Section 4 to the general case of the scoring rule, in which a scoring function f assigns a score to each candidate according to the positions he was ranked in the voters' lists. The overall approach is similar, but more general: We first define an alternative random process to LT R and show its equivalence to LT R; then we use this model to compute F (A 0 ) and show that it is a monotone submodular function of the initial set of active nodes A 0 . This latter result allows us to compute a set A 0 that has an approximation guarantee of 1 − 1/e on the maximization of the score of the target candidate. The alternative random process, called Live-edge Dice Roll (LDR), is defined as follows. Definition 2. (Live-edge Dice Roll process) 1. Each node v ∈ V selects at most one of its incoming edges with probability proportional to the weight of that edge, i.e., edge (u, v) is selected with probability b uv , and no edge is selected with probability 1 − u∈Nv b uv . 2. Each node v with π v (c ) > 1 that is reachable from A 0 in the live-edge graph rolls a biased π v (c )-sided dice and changes its list according to the outcome. This is equivalent to picking a random real number s v in [0, 1] and setting the position of c according to s v as follows: π v (c ) =          1 if s v α(πv(c )) πv(c )−1 , if α(πv(c )) πv(c )− +1 < s v α(πv(c )) πv(c )− , for = 2, . . . , π v (c ) − 1, π v (c ) if s v > α(π v (c )). Ifπ v (c ) = π v (c ), all candidates betweenπ v (c ) and π v (c ) − 1 are shifted down by one position. In the next theorem we show that processes LT R and LDR have the same distribution. Proof. Let A be the set of active nodes at the end of the LT R process that starts from A 0 . The probability that an active node moves candidate c to position is given by the following function. Definition 3. For each r, ∈ {1, . . . , m} we define: P (r, ) =      α(r) r−1 if = 1, α(r) r− − α(r) r− +1 if = 2, . . . , r − 1, 1 − α(r) if = r. In particular, for a node v, the probability that the second step of LDR yields π v (c ) = , for = 1, . . . , π v (c ), is P(π v (c ), ). Thus, P π LT R v (c ) = = U ⊆Nv P π LT R v (c ) = (A ∩ N v ) = U · P ((A ∩ N v ) = U ) . If U is is the maximal subset of active neighbors of v (i.e., U = A ∩ N v ), then we can write the probability thatπ LT R v (c ) = given U as follows: P π LT R v (c ) = (A ∩ N v ) = U = P t v α(π v (c )) π v (c ) − 1 u∈U b uv if = 1; P π LT R v (c ) = (A ∩ N v ) = U = P α(π v (c )) π v (c ) − + 1 u∈U b uv < t v α(π v (c )) π v (c ) − u∈U b uv if = 2, . . . , π v (c ) − 1; P π LT R v (c ) = (A ∩ N v ) = U = P t v > α(π v (c )) u∈U b uv if = π v (c ). In other words, P π LT R v (c ) = (A ∩ N v ) = U = P(r, ) u∈U b uv . Therefore, P π LT R v (c ) = = P(π v (c ), ) U ⊆Nv u∈U b uv P ((A ∩ N v ) = U ) . In LDR, P π LDR v (c ) = = P(v ∈ R(A 0 )) · P(π v (c ), ). By Lemma 1, it follows that By definition, the value of F (A 0 ) is F (A 0 ) = E v∈V f (π v (c )) = v∈V πv(c ) =1 f ( )P (π v (c ) = ) . In LDR, P π LDR v (c ) = = P(v ∈ R(A 0 )) · P(π v (c ), ), moreover, by Lemma 1, P(v ∈ R(A 0 )) = G ∈G P (G ) 1 (G ,v) . Then, F (A 0 ) = v∈V πv(c ) =1 f ( )P(π v (c ), ) G ∈G P G 1 (G ,v) , which can be rewritten as F (A 0 ) = m r=1 r =1 f ( )P(π v (c ), ) G ∈G P G v:πv(c )=r 1 (G ,v) = m r=1 r =1 f ( )P(π v (c ), ) G ∈G P G \|{v v ∈ R(G ) ∧ π v (c ) = r}\| = m r=1 r =1 f ( )P(π v (c ), ) G ∈G P G \|R G (A 0 , V r c )\|. Thus, F (A 0 ) is a non-negative linear combination of the monotone submodular function \|R G (A 0 , V r c )\| (see Lemma 2), and hence F (A 0 ) − F (∅) is also monotone and submodular. Thus, we can use Greedy (Algorithm 1) to find a (1 − 1/e)-approximation to the problem of maximizing the score of the target candidate [NWF78]. Approximating Margin of Victory We have seen in previous sections that we can map the problem of maximizing the score of the target candidate to that of influence maximization both in the plurality (Section 4) and in the scoring rules (Section 5); we also defined two alternative processes (Definitions 1 and 2) and showed their equivalence to LT R for both rules (Theorems 2 and 3). By showing that the objective function is monotone and submodular w.r.t. the initial set of seed nodes (Lemma 2) it follows that Greedy (Algorithm 1) finds a (1 − 1/e)-approximation of the optimum [NWF78]. In the following we show how to achieve a constant factor approximation to the original problem of maximizing the MoV by only maximizing the score of the target candidate. Given the equivalence of the processes with LT R, we can formulate our original objective function as the average MoV G computed on a sampled live-edge graph G , namely E [MoV(A 0 )] = E [MoV G (A 0 )], where MoV G (A 0 ) = µ G (∅) − µ G (A 0 ), and µ G is the change in margin on a fixed G . We formulate the margin on the live-edge graphs in a way that is akin to that of [WV18b]: We can exploit such formulation to prove our constant factor approximation with the same proof structure since also in our case the objective function is monotone and submodular (Lemma 2). In particular in the simple case of the plurality rule we have that E [MoV G (A 0 )] := m r=2 α(r) r − 1 \|R G (A 0 , V r c )\| + min cz max c i \|V 1 c i \| − \|V 1 cz \| + m r=2 α(r) r − 1 \|R G (A 0 , V r c ∩ V 1 cz )\| , where: the first term is the number of points gained by the target candidate after LT R; the second term (the first inside the minimum) is the number of points of the most voted opponent before LT R; the third is the total number of points that the most voted opponent after LT R had before the process; the fourth term is the number of points that the most voted opponent after LT R lost because of the shifting of candidate c . Similarly, in the general case of arbitrary scoring rules, we have E [MoV G (A 0 )] := m r=2 r−1 =1 P(r, ) \|R G (A 0 , V r c )\| (f ( ) − f (r)) + min cz max c i m r=1 f (r)\|V r c i \| − m r=1 f (r)\|V r cz \| + m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V h cz )\| (f (h) − f (h + 1)) , where the meaning of the terms is similar to above. This latter formulation is just a generalization of the plurality case whenever we choose f such that f (1) = 1 and f (r) = 0, for each r ∈ {2, . . . , m}. In this way we would have that the gain in score would be just 1 and that α(r) r−1 = P (r, 1). In the following we prove that, up to the loss of a constant-factor in the approximation ratio, it suffices to concentrate only on the score of the target candidate c and not on the margin w.r.t. the most voted opponent. Proof. Let A 0 be the solution found by Greedy (Algorithm 1) in the election control problem and let A 0 be the optimal solution. Letc andĉ respectively be the candidates that minimize the second term of E [MoV G (A 0 )] and E [MoV G (A 0 )]. Note that E [MoV G (A 0 )] = F (A 0 ) − F (∅) + \|V 1 c \| − \|V 1 c \| + m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h) − f (h + 1)), where c is the most voted candidate before the process. Since F (A 0 ) − F (∅) = m r=2 r−1 =1 P(r, ) \|R G (A 0 , V r c )\| (f ( )−f (r)) and F (A 0 )−F (∅) (1−1/e)(F (A 0 )− F (∅)), we get E [MoV G (A 0 )] F (A 0 ) − F (∅) + \|V 1 c \| − \|V 1 c \| (1 − 1/e) m r=2 r−1 =1 P(r, ) \|R G (A 0 , V r c )\| (f ( ) − f (r)) + \|V 1 c \| − \|V 1 c \| 1 3 (1 − 1/e) m r=2 r−1 =1 P(r, ) \|R G (A 0 , V r c )\| (f ( ) − f (r)) + m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h) − f (h + 1)) + m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V ĥ c )\| (f (h) − f (h + 1)) + \|V 1 c \| − \|V 1 c \| . Note that this is possible thanks to Theorem 3 and because the last two terms in the last inequality are smaller than the first term for any solution A 0 and candidate c i since the solution A 0 can only increase the score of c . Therefore, for any other candidate c i the score can only decrease. With some additional algebra we get that E [MoV G (A 0 )] 1 3 (1 − 1/e)MoV G (A * 0 ) + \|V 1 c \| − \|V 1 c \| + m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h) − f (h + 1)). By definition ofĉ we have that \|V 1 c \| − \|V 1 c \| m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V ĥ c )\| (f (h) − f (h + 1)) − m r=2 r−1 =1 r−1 h= P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h) − f (h + 1)) and therefore E [MoV G (A 0 )] 1 3 (1 − 1/e)MoV G (A * 0 ). Destructive Election Control In this section we focus on the destructive election control problem. The model is similar to the constructive one (see Section 3): Here we define, for each node v ∈ V , the number of positions of which c shifts down after the LT R process as π ↓ v (c ) := min   m − π v (c ),     α(π v (c )) t v u∈A, (u,v)∈E b uv       . The final position of c in v will beπ v (c ) := π v (c ) + π ↓ v (c ) and the overall score that c gets is F D (A 0 ) := E v∈V f (π v (c ) + π ↓ v (c )) . Formally, the problem can be defined as that of finding an initial set of seed nodes A 0 such that max A 0 E [MoV D (A 0 )] := E [µ(A 0 ) − µ(∅)] s.t. \|A 0 \| B, namely to find an initial set of seed nodes of at most size B that maximizes the expected MoV D , i.e., minimizes the expected MoV. Similarly to the constructive case, we aim at decreasing the overall score of a target candidate c as much as possible since, as before, in this way we can achieve a constant factor approximation. To do that we provide a reduction from the destructive to the constructive case. Given an instance of destructive control, we build an instance of constructive control in which we simply reverse the rankings of each node and complement the scoring function to its maximum value. Roughly speaking, this reduction maintains invariant the absolute value of the change in margin of the score of any candidate between the two cases. Formally, for each v ∈ V , the new instance has a preference list defined as π v (c) := m − π v (c) + 1 for each candidate c ∈ C, and, for each position r ∈ {1, . . . , m}, has a scoring function defined as f (r) := f max − f (m − r + 1), where f max := max r∈{1,...,m} f (r). For each v ∈ V , the ranking of c in the new instance is π v (c ) := m − π v (c ) + 1. For each solution A 0 found in the new instance, i.e., a constructive one, the overall score of c after the process is F (A 0 ) := E v∈V f (π v (c ) − π ↑ v (c )) , where π ↑ v (c ) := min π v (c ) − 1, α(πv(c )) tv u∈A, (u,v)∈E b uv . Let F D (∅) = F (∅) and F (∅) := v∈V f (π v (c )). Then the following lemma holds. Lemma 3. F D (∅) − F D (A 0 ) = F (A 0 ) − F (∅), for every A 0 . Proof. Observe that π ↑ v (c ) = π ↓ v (c ) and that π v (c ) = m − π v (c ) + 1. It follows that F (A 0 ) − F (∅) = E v∈V [f max − f (m − (π v (c ) − π ↑ v (c )) + 1)] − E v∈V [f max − f (m − π v (c ) + 1)] = E v∈V [f (m − π v (c ) + 1) − f (m − (π v (c ) − π ↑ v (c )) + 1)] = E v∈V [f (m − π v (c ) + 1) − f (m − π v (c ) + 1 + π ↑ v (c ))] = E v∈V [f (π v (c )) − f (π v (c ) + π ↓ v (c ))] = F (∅) − F D (A 0 ). The reduction, together with Lemma 3, allows us to maximize the score of the target candidate in the constructive case and then to map it back to destructive case. Theorem 5. Greedy (Algorithm 1) is a 1 2 (1 − 1/e)-approximation algorithm for the problem of destructive election control in arbitrary scoring rule voting systems. Proof. Let A 0 be the solution found by Greedy (Algorithm 1) in the election control problem and let A 0 be the optimal solution. Letc andĉ respectively be the candidates that minimize the first term of E [MoV D (A 0 )] and E [MoV D (A 0 )]. By Lemma 3 we have that E [MoV D (A 0 )] = E [µ(A 0 ) − µ(∅)] = F (∅) − F D (A 0 ) − \|V 1 c \| + \|V 1 c \| + m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h + 1) − f (h + 1)) = F (A 0 ) − F (∅) − \|V 1 c \| + \|V 1 c \| + m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h + 1) − f (h + 1)) where c is the most voted candidate before the process. Since F (A 0 ) − F (∅) is an instance of the score in the constructive case we able to approximate this value, thus we get E [MoV D (A 0 )] 1 − 1 e F (A * 0 ) − F (∅) − \|V 1 c \| + \|V 1 c \| + m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h + 1) − f (h + 1)) 1 2 1 − 1 e F (∅) − F D (A * 0 ) − \|V 1 c \| + \|V 1 c \| + m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h + 1) − f (h + 1)) + \|V 1 c \| − \|V 1 c \| 1 2 1 − 1 e MoV D (A * ) + \|V 1 c \| + m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h + 1) − f (h + 1)) − \|V 1 c \| By definition ofc we have that \|V 1 c \| − \|V 1 c \| m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V h c )\| (f (h + 1) − f (h + 1)) − m−1 r=1 m h=r+1 m =r+1 P(r, ) \|R G (A 0 , V r c ∩ V ĥ c )\| (f (h + 1) − f (h + 1)) and therefore MoV D (A 0 ) 1 2 1 − 1 e MoV D (A * 0 ). Simulations In this section we present some experimental results that show how our approximation algorithm performs on real-world networks. We chose four heterogeneous social and communication networks on which political campaigning messages could spread, namely: 1. facebook, 4 an undirected network of 10 Facebook users, with 2,888 nodes and 2,981 edges; 2. irvine, 4 a directed network of instant messages exchanged between students at U.C. Irvine, with 1,899 nodes and 20,296 edges; 3. netscience, 5 an undirected network of research collaborations in network science, with 1,461 nodes and 2,742 edges; 4. polblogs, 5 a directed network of hyperlinks between web blogs on US politics, with 1,224 nodes and 19,025 edges. We made the two undirected networks directed, by doubling the edges and orienting them; moreover, to adhere to the Linear Threshold Model, we assigned random weights to edges of the graphs since they are unweighted. Recent literature in influence maximization shows that advanced techniques can be used to scale our approximation algorithm to much larger networks [TXS14]. We considered three different scenarios, each with a different number of candidates, i.e., m = 2, 5, 10. For each scenario, we assigned a random preference list to each node of the networks; this assignment was performed 10 distinct times, by randomly permuting its preference list. We separately analyzed three different initial budgets, i.e., B = 5, 10, 15, and three different values of α (the rate at which the position of candidate c changes in the preference list of each node), i.e., α = 0.1, 0.5, 1. For each combination of parameters (dataset, number of candidates, preference list assignment, budget, and α) we performed 20 experiments for each of the two considered voting systems, namely plurality rule and borda count (as example for the scoring rule), in the constructive election control scenario. We measured the Probability of Victory (PoV), i.e., the fraction of times c won out of the 20 experiments, and the Margin of Victory (MoV), average value of the difference between the score of candidate c and the score of the most voted opponent. 6 All the experiments were run in parallel on a machine with four 16-core AMD Opteron TM 6376 with 2.3 GHz CPU, 16 MB L2 cache, 64 GB RAM, running Ubuntu 16.04.5 LTS. Overall, we performed 43,200 distinct runs for an average running time of approximately 15 minutes per run. Figure 1 shows the effectiveness of the algorithm in the scenario with m = 10 candidates running for the elections and a fixed budget B = 5, 10, 15 using as voting system the plurality rule and borda count. We study the impact of α on the behavior of our algorithm (Figure 1). The algorithm succeeds for α = 0.5, 1.0, making candidate c win all or most of the times on all datasets but netscience: We believe this is due to the topology of the dataset, with 267 connected components, that limitates the influence diffusion process. Instead, when α = 0.1 it is difficult for the initially targeted voters to influence their friends in all datasets but facebook, where most of the nodes have few friends and can be easily influenced by the 10 users on which the network is centered. In fact the lower is the value of α the higher is the rate at which c shifts in the preference lists of the voters. Tables 1, 2, and 3 report detailed results of the experiments. Table 1 reports unified results for plurality rule and borda count voting systems, given their equivalence in the scenario with only m = 2 candidates running for the elections. Figure 1 gives a visual interpretation of the results, considering the scenario with m = 10, which is the "hardest" among the considered ones, since it has the maximum number of candidates and the minimum budget. Each boxplot considers 200 observations, i.e., the results obtained by permuting the preference list of each voter 10 times and repeating 20 experiments on each of them, and shows the results for all considered values of B. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 500 1000 1500 2000 facebook irvine netscience polblogs (k) MoV, borda count, B=10 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q µ and σ are, respectively, the mean and the standard deviation of the observations averaged over the 10 preference list permutations. Discussion of Results The results in our paper are very significant. Nowadays social media are are significant sources of information for voters and the massive usage of these channels for political campaigning is a turning point. Potential attackers can manipulate the outcome of elections through the spread of targeted ads and/or fake news. Being able to control the information spread can have a great impact, but it is not easy to achieve given that traditional media sources are relatively transparent. Therefore, it is essential to protect the integrity of electoral processes to ensure the proper operation of democratic institutions. Our results indicate that social influence is a salient threat to election integrity: We provide an approximation algorithm to maximize the MoV of a target candidate, that can be used by an attacker to control the election results and is of fundamental importance to protect their fairness. There is only another paper that focus on the problem of election control through social influence [WV18b]. Compared to it, we consider a more realistic model (LTM instead of ICM) that takes into account the amount of influence that voters exercise on each other. We believe that our algorithm could be used in real-life scenarios to predict election results and to understand what degree of control has been exercised. Our results assume the knowledge of election data that are not available (degrees of influences and preferences of voters), but that can be estimated. Even if such estimation is not easy, experimental results show that greedy has good performances even on real-world datasets where this data are uncertain. With this respect, we are aware of recent studies that analyze the robustness of greedy w.r.t. inaccurate estimations of the degrees of influence; again, ground truth for such quantities is not available and good estimates are hard to get. Nevertheless, experimental results on greedy algorithm for Influence Maximization showed that the worst case hardness theoretical results do not necessarily translate into bad performance on real-world datasets [HK18]. Conclusion and Future Work Online social networks are increasingly utilized for political campaigning since specific users can be targeted by advertisement and/or fake news. We focused on the problem of controlling election through social influence: Given a social network of people willing to vote, we aim at selecting a fixed-size subset of voters such that their influence on the others will change the outcome of the elections, making some specific candidate win or lose. We described a powerful extension of the Linear Threshold Model, which describes the change of opinions taking into account the amount of exercised influence. We provided a constant factor approximation algorithm to the problems of constructive and destructive election control, considering arbitrary scoring rule voting systems, including plurality and borda count. Greedy (Algorithm 1) achieves a 1 3 (1 − 1/e) approximation ratio in the constructive scenario, since we showed that any scoring function is monotone submodular w.r.t. the initial set of active nodes. Similarly, we get a 1 2 (1 − 1/e) approximation ratio for the destructive scenario. We performed a simulation of our algorithm in our model, examining it on realworld networks using synthetic election data, i.e., random degrees of influences of voters on each other and random preference lists for each voter. We ran the simulation with different combinations of parameters, varying B, α, \|C\|, and π v for each v ∈ V on 4 networks exhibiting heterogeneous topologies, namely facebook, polblogs, irvine, netscience. We observed that Greedy is able to find a solution that makes the target candidate win the election in the plurality rule with 10 candidates between 50% and 88% of the times, depending on the value of α that changes the degree of influence, using only 5 seed nodes. As future research directions we would like to further study our model in a wider range of scenarios which are not currently captured, including multi-winner and proportional representation systems. We also believe that approaches that mix constructive and destructive control could be analyzed to get better approximation ratios. Moreover, we would like to extend our model in order to consider a more uncertain scenario, in which the preferences of voters are not known. Finally, it would be interesting to study how to prevent election control for the integrity of voting processes, e.g., through the placement of monitors in the network [ZATN15,AAAF17] or by considering strategic settings [YVAH18,WV18a]. Theorem 3 . 3Given a set of initially active nodes A 0 and a node v ∈ V , letπ LT ) be the position of node v at the end of LT R and LDR, respectively, both starting from A 0 . Then, P π LT R v (c ) = = P π LDR v (c ) = , for each = 1, . . . , π v (c ). u∈U b uv P ((R(A 0 ) ∩ N v ) = U ) . By Theorem 1, P ((R(A 0 ) ∩ N v ) = U ) = P ((A ∩ N v ) = U ), which shows the statement. Theorem 4 . 4Greedy (Algorithm 1) is a 1 3 (1 − 1/e)-approximation algorithm for the problem of election control in arbitrary scoring rule voting systems. PoV, plurality rule, B=5 PoV, plurality rule, B=10 PoV, plurality rule, B=15 ) MoV, borda count, B=5 Figure 1 : 1PoV and MoV with m = 10. Each plot compares the results on different datasets (facebook, irvine, netscience, polblogs) and for different values of α. For each dataset, from left to right: α = 0.1 (red ), α = 0.5 (green), α = 1.0 (blue). Table 1 : 1PoV and MoV values relative to the experiments with m = 2. .0 1.00 0.00 1.00 0.00 1.00 0.00 1234.91 31.05 1472.39 28.42 1476.18 28.37PoV MoV B = 5 B = 10 B = 15 B = 5 B = 10 B = 15 α µ σ µ σ µ σ µ σ µ σ µ σ facebook 0.1 1.00 0.00 1.00 0.00 1.00 0.00 149.78 46.10 165.63 47.32 168.22 46.14 0.5 1.00 0.00 1.00 0.00 1.00 0.00 631.90 39.11 743.61 35.70 751.42 38.55 1irvine 0.1 0.95 0.15 0.96 0.13 0.99 0.03 66.93 39.44 82.04 40.21 89.84 38.75 0.5 1.00 0.00 1.00 0.00 1.00 0.00 230.48 36.68 313.38 23.76 350.47 19.72 1.0 1.00 0.00 1.00 0.00 1.00 0.00 441.67 40.74 624.90 39.16 693.16 59.91 netscience 0.1 0.74 0.42 0.78 0.39 0.81 0.36 29.95 40.73 32.55 40.80 34.71 40.86 0.5 0.99 0.04 1.00 0.00 1.00 0.00 55.38 39.76 71.31 40.03 86.40 40.77 1.0 1.00 0.00 1.00 0.00 1.00 0.00 85.98 39.57 122.36 40.12 151.88 41.31 polblogs 0.1 0.91 0.29 0.92 0.25 0.93 0.22 47.16 30.39 52.69 30.57 55.24 30.40 0.5 1.00 0.00 1.00 0.00 1.00 0.00 150.11 29.34 178.11 28.84 196.28 27.95 1.0 1.00 0.00 1.00 0.00 1.00 0.00 280.13 27.29 344.81 23.40 387.12 17.19 Table 2 : 2PoV and MoV values relative to the experiments with m = 5. .0 1.00 0.00 1.00 0.00 1.00 0.00 1314.98 103.33 1572.05 98.62 1588.18 99.25PoV Table 3 : 3PoV and MoV values relative to the experiments with m = 10. .0 1.00 0.00 1.00 0.00 1.00 0.00 1701.58 133.16 2036.01 131.07 2055.55 131.70PoV We actually study the change in the margin -not just the margin -to have well defined approximation ratios also when the margin is negative.2 Note that MoV is always positive since the scoring function f is nonincreasing. 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Youze Tang, Xiaokui Xiao, Yanchen Shi, Proceedings of the 2014 ACM SIGMOD International Conference on Management of Data, SIGMOD '14. the 2014 ACM SIGMOD International Conference on Management of Data, SIGMOD '14New York, NY, USAACMYouze Tang, Xiaokui Xiao, and Yanchen Shi. Influence maximization: Near-optimal time complexity meets practical efficiency. In Proceedings of the 2014 ACM SIGMOD International Conference on Management of Data, SIGMOD '14, pages 75-86, New York, NY, USA, 2014. ACM. Defending Elections Against Malicious Spread of Misinformation. B Wilder, Y Vorobeychik, ArXiv e-printsB. Wilder and Y. Vorobeychik. Defending Elections Against Malicious Spread of Misinformation. ArXiv e-prints, September 2018. International Foundation for Autonomous Agents and Multiagent Systems. 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[ "Noncollinear magnetism and spin-orbit coupling in 5d pyrochlore oxide Cd 2 Os 2 O 7", "Noncollinear magnetism and spin-orbit coupling in 5d pyrochlore oxide Cd 2 Os 2 O 7" ]	[ "Hiroshi Shinaoka \nNanosystem Research Institute \"RICS\"\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568UmezonoTsukubaJapan\n\nScience and Technology Agency (JST)\nCREST\n332-0012Honcho, KawaguchiSaitamaJapan, Japan\n", "Takashi Miyake \nNanosystem Research Institute \"RICS\"\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568UmezonoTsukubaJapan\n\nScience and Technology Agency (JST)\nCREST\n332-0012Honcho, KawaguchiSaitamaJapan, Japan\n", "Shoji Ishibashi \nNanosystem Research Institute \"RICS\"\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568UmezonoTsukubaJapan\n\nScience and Technology Agency (JST)\nCREST\n332-0012Honcho, KawaguchiSaitamaJapan, Japan\n" ]	[ "Nanosystem Research Institute \"RICS\"\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568UmezonoTsukubaJapan", "Science and Technology Agency (JST)\nCREST\n332-0012Honcho, KawaguchiSaitamaJapan, Japan", "Nanosystem Research Institute \"RICS\"\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568UmezonoTsukubaJapan", "Science and Technology Agency (JST)\nCREST\n332-0012Honcho, KawaguchiSaitamaJapan, Japan", "Nanosystem Research Institute \"RICS\"\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568UmezonoTsukubaJapan", "Science and Technology Agency (JST)\nCREST\n332-0012Honcho, KawaguchiSaitamaJapan, Japan" ]	[]	We investigated the electronic and magnetic properties of the pyrochlore oxide Cd2Os2O7 using the density-functional theory plus on-site repulsion (U ) method, and depict the ground-state phase diagram with respect to U . We conclude that the all-in/all-out non-collinear magnetic order is stable in a wide range of U . We also show that the easy-axis anisotropy arising from the spin-orbit (SO) coupling plays a significant role in stabilizing the all-in/all-out magnetic order. A pseudo gap was observed near the transition between the antiferromagnetic metallic and insulating phases. Finally, we discuss possible origins of the peculiar low-temperature(T ) properties observed in experiments. PACS numbers: 75.47.Lx,71.15.Mb,71.70.Ej Pyrochlore transition-metal oxides A 2 B 2 O 7 are a class of materials that have been extensively investigated for several decades[1]. Recently, particular attention has been paid to 5d transition-metal oxides, such as iridates (A 2 Ir 2 O 7 ), in search for unconventional phenomena that are induced by the competing spin-orbit (SO) coupling and electron correlation.Cd 2 Os 2 O 7 is one of the few compounds whose low-T magnetic structure has been experimentally determined among a number of magnetic 5d pyrochlore oxides. The Cd 2+ ion is non-magnetic, whereas the Os 5+ ion with a 5d 3 configuration can be magnetic. This compound exhibits a purely electronic continuous metal-insulator transition (MIT) concurrently with a Néel ordering at T MIT ≃ 227 K [2-5]. Magnetic-susceptibility [2, 3] and µSR measurements [6] suggested the Néel ordering, whereas powder neutron diffraction did not confirm the abovementioned observation[7]. Recently, Yamaura et al. have successfully detected a magnetic reflection at the wave vector of q = 0 using resonant X-ray scattering on high-quality single crystals[5]. They showed that only the so-called all-in/all-out magnetic order [seeFig. 1(a)] is compatible with the cubic symmetry of the crystal among the possible q = 0 magnetic orders. The all-in/all-out magnetic ordering has been suggested also in other 5d pyrochlore oxides, i.e, Nd 2 Ir 2 O 7 in experimental work [8] and Y 2 Ir 2 O 7 in theoretical work[9].Despite the recent experimental progress, we are still far from fully understanding the electronic properties of this compound. In particular, several puzzling electronic properties have been reported experimentally: (1) In contrast to the opening of an optical gap of the order of 800 cm −1 (≃ 1100 K) [10], the semiconducting gap continuously vanishes toward low T 's[2,3]. This indicates the absence of a clear charge gap at low T 's which seemingly contradicts the semiconducting behavior of the resistivity up to T MIT . (2) The Néel transition temperature of this compound (= T MIT ) is, to the beset of our knowledge, one of the highest among the magnetic pyrochlore (a) 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 Gap (eV) U eff (eV) (c) 0.0 0.4 0.8 1.2 Os moment (b) NMM AFM AFI FIG. 1: (color online). (a) Cubic unit cell of Cd2Os2O7 contains 8 formula units. Only the corner sharing tetrahedron network of Os atoms is shown for clarity. The arrows represent the all-in/all-out magnetic order with the ordering vector of q = 0. (b)/(c) U eff dependences of the Os magnetic moment mOs, the charge gap ∆C, and the direct gap ∆D. The phase diagram consists of a non-magnetic metal (NMM), an antiferromagnetic metal (AFM), and an antiferromagnetic insulator (AFI).oxides[1]. This observation is quite surprising because geometrical frustration tends to prevent long-range magnetic ordering. These puzzling characteristics and the similarity with the Ir oxides urged us to investigate this compound as a prototype of 5d pyrochlore oxides.In this Letter, we investigate the ground-state properties of Cd 2 Os 2 O 7 using extensive LSDA+SO+U calculations (LSDA denotes the local spin density approximation). We explore the ground-state phase diagram as a function of U eff (≡ U − J). Here, U eff is an empirical parameter, and U eff = 1-2 eV is expected to be appropriate for spatially extended 5d orbitals[9]. We show that the all-in/all-out magnetic order is stable in an antiferromagnetic metal (AFM) and in an antiferromagnetic insulator (AFI). We observe that strong magnetic anisotropy arising from the SO coupling stabilizes the all-in/all-out magnetic order. Finally, we discuss possible origins of the experimentally-observed puzzling characteristics of	10.1103/physrevlett.108.247204	[ "https://arxiv.org/pdf/1111.6347v2.pdf" ]	32,326,827	1111.6347	f4e3743520db72a562ca1da24d8f927b9f4821c9	Noncollinear magnetism and spin-orbit coupling in 5d pyrochlore oxide Cd 2 Os 2 O 7 15 Apr 2012 (Dated: April 17, 2012) Hiroshi Shinaoka Nanosystem Research Institute "RICS" National Institute of Advanced Industrial Science and Technology (AIST) 305-8568UmezonoTsukubaJapan Science and Technology Agency (JST) CREST 332-0012Honcho, KawaguchiSaitamaJapan, Japan Takashi Miyake Nanosystem Research Institute "RICS" National Institute of Advanced Industrial Science and Technology (AIST) 305-8568UmezonoTsukubaJapan Science and Technology Agency (JST) CREST 332-0012Honcho, KawaguchiSaitamaJapan, Japan Shoji Ishibashi Nanosystem Research Institute "RICS" National Institute of Advanced Industrial Science and Technology (AIST) 305-8568UmezonoTsukubaJapan Science and Technology Agency (JST) CREST 332-0012Honcho, KawaguchiSaitamaJapan, Japan Noncollinear magnetism and spin-orbit coupling in 5d pyrochlore oxide Cd 2 Os 2 O 7 15 Apr 2012 (Dated: April 17, 2012)arXiv:1111.6347v2 [cond-mat.str-el] We investigated the electronic and magnetic properties of the pyrochlore oxide Cd2Os2O7 using the density-functional theory plus on-site repulsion (U ) method, and depict the ground-state phase diagram with respect to U . We conclude that the all-in/all-out non-collinear magnetic order is stable in a wide range of U . We also show that the easy-axis anisotropy arising from the spin-orbit (SO) coupling plays a significant role in stabilizing the all-in/all-out magnetic order. A pseudo gap was observed near the transition between the antiferromagnetic metallic and insulating phases. Finally, we discuss possible origins of the peculiar low-temperature(T ) properties observed in experiments. PACS numbers: 75.47.Lx,71.15.Mb,71.70.Ej Pyrochlore transition-metal oxides A 2 B 2 O 7 are a class of materials that have been extensively investigated for several decades[1]. Recently, particular attention has been paid to 5d transition-metal oxides, such as iridates (A 2 Ir 2 O 7 ), in search for unconventional phenomena that are induced by the competing spin-orbit (SO) coupling and electron correlation.Cd 2 Os 2 O 7 is one of the few compounds whose low-T magnetic structure has been experimentally determined among a number of magnetic 5d pyrochlore oxides. The Cd 2+ ion is non-magnetic, whereas the Os 5+ ion with a 5d 3 configuration can be magnetic. This compound exhibits a purely electronic continuous metal-insulator transition (MIT) concurrently with a Néel ordering at T MIT ≃ 227 K [2-5]. Magnetic-susceptibility [2, 3] and µSR measurements [6] suggested the Néel ordering, whereas powder neutron diffraction did not confirm the abovementioned observation[7]. Recently, Yamaura et al. have successfully detected a magnetic reflection at the wave vector of q = 0 using resonant X-ray scattering on high-quality single crystals[5]. They showed that only the so-called all-in/all-out magnetic order [seeFig. 1(a)] is compatible with the cubic symmetry of the crystal among the possible q = 0 magnetic orders. The all-in/all-out magnetic ordering has been suggested also in other 5d pyrochlore oxides, i.e, Nd 2 Ir 2 O 7 in experimental work [8] and Y 2 Ir 2 O 7 in theoretical work[9].Despite the recent experimental progress, we are still far from fully understanding the electronic properties of this compound. In particular, several puzzling electronic properties have been reported experimentally: (1) In contrast to the opening of an optical gap of the order of 800 cm −1 (≃ 1100 K) [10], the semiconducting gap continuously vanishes toward low T 's[2,3]. This indicates the absence of a clear charge gap at low T 's which seemingly contradicts the semiconducting behavior of the resistivity up to T MIT . (2) The Néel transition temperature of this compound (= T MIT ) is, to the beset of our knowledge, one of the highest among the magnetic pyrochlore (a) 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 Gap (eV) U eff (eV) (c) 0.0 0.4 0.8 1.2 Os moment (b) NMM AFM AFI FIG. 1: (color online). (a) Cubic unit cell of Cd2Os2O7 contains 8 formula units. Only the corner sharing tetrahedron network of Os atoms is shown for clarity. The arrows represent the all-in/all-out magnetic order with the ordering vector of q = 0. (b)/(c) U eff dependences of the Os magnetic moment mOs, the charge gap ∆C, and the direct gap ∆D. The phase diagram consists of a non-magnetic metal (NMM), an antiferromagnetic metal (AFM), and an antiferromagnetic insulator (AFI).oxides[1]. This observation is quite surprising because geometrical frustration tends to prevent long-range magnetic ordering. These puzzling characteristics and the similarity with the Ir oxides urged us to investigate this compound as a prototype of 5d pyrochlore oxides.In this Letter, we investigate the ground-state properties of Cd 2 Os 2 O 7 using extensive LSDA+SO+U calculations (LSDA denotes the local spin density approximation). We explore the ground-state phase diagram as a function of U eff (≡ U − J). Here, U eff is an empirical parameter, and U eff = 1-2 eV is expected to be appropriate for spatially extended 5d orbitals[9]. We show that the all-in/all-out magnetic order is stable in an antiferromagnetic metal (AFM) and in an antiferromagnetic insulator (AFI). We observe that strong magnetic anisotropy arising from the SO coupling stabilizes the all-in/all-out magnetic order. Finally, we discuss possible origins of the experimentally-observed puzzling characteristics of We investigated the electronic and magnetic properties of the pyrochlore oxide Cd2Os2O7 using the density-functional theory plus on-site repulsion (U ) method, and depict the ground-state phase diagram with respect to U . We conclude that the all-in/all-out non-collinear magnetic order is stable in a wide range of U . We also show that the easy-axis anisotropy arising from the spin-orbit (SO) coupling plays a significant role in stabilizing the all-in/all-out magnetic order. A pseudo gap was observed near the transition between the antiferromagnetic metallic and insulating phases. Finally, we discuss possible origins of the peculiar low-temperature(T ) properties observed in experiments. Pyrochlore transition-metal oxides A 2 B 2 O 7 are a class of materials that have been extensively investigated for several decades [1]. Recently, particular attention has been paid to 5d transition-metal oxides, such as iridates (A 2 Ir 2 O 7 ), in search for unconventional phenomena that are induced by the competing spin-orbit (SO) coupling and electron correlation. Cd 2 Os 2 O 7 is one of the few compounds whose low-T magnetic structure has been experimentally determined among a number of magnetic 5d pyrochlore oxides. The Cd 2+ ion is non-magnetic, whereas the Os 5+ ion with a 5d 3 configuration can be magnetic. This compound exhibits a purely electronic continuous metal-insulator transition (MIT) concurrently with a Néel ordering at T MIT ≃ 227 K [2][3][4][5]. Magnetic-susceptibility [2,3] and µSR measurements [6] suggested the Néel ordering, whereas powder neutron diffraction did not confirm the abovementioned observation [7]. Recently, Yamaura et al. have successfully detected a magnetic reflection at the wave vector of q = 0 using resonant X-ray scattering on high-quality single crystals [5]. They showed that only the so-called all-in/all-out magnetic order [see Fig. 1(a)] is compatible with the cubic symmetry of the crystal among the possible q = 0 magnetic orders. The all-in/all-out magnetic ordering has been suggested also in other 5d pyrochlore oxides, i.e, Nd 2 Ir 2 O 7 in experimental work [8] and Y 2 Ir 2 O 7 in theoretical work [9]. Despite the recent experimental progress, we are still far from fully understanding the electronic properties of this compound. In particular, several puzzling electronic properties have been reported experimentally: (1) In contrast to the opening of an optical gap of the order of 800 cm −1 (≃ 1100 K) [10], the semiconducting gap continuously vanishes toward low T 's [2,3]. This indicates the absence of a clear charge gap at low T 's which seemingly contradicts the semiconducting behavior of the resistivity up to T MIT . (2) The Néel transition temperature of this compound (= T MIT ) is, to the beset of our knowledge, one of the highest among the magnetic pyrochlore oxides [1]. This observation is quite surprising because geometrical frustration tends to prevent long-range magnetic ordering. These puzzling characteristics and the similarity with the Ir oxides urged us to investigate this compound as a prototype of 5d pyrochlore oxides. In this Letter, we investigate the ground-state properties of Cd 2 Os 2 O 7 using extensive LSDA+SO+U calculations (LSDA denotes the local spin density approximation). We explore the ground-state phase diagram as a function of U eff (≡ U − J). Here, U eff is an empirical parameter, and U eff = 1-2 eV is expected to be appropriate for spatially extended 5d orbitals [9]. We show that the all-in/all-out magnetic order is stable in an antiferromagnetic metal (AFM) and in an antiferromagnetic insulator (AFI). We observe that strong magnetic anisotropy arising from the SO coupling stabilizes the all-in/all-out magnetic order. Finally, we discuss possible origins of the experimentally-observed puzzling characteristics of Cd 2 Os 2 O 7 . In the following calculations, we use a fullyrelativistic two-component first-principles computational code, QMAS (Quantum MAterials Simulator) [11]. We employ the projector augmented-wave method [12] and the LSDA+SO+U method [13][14][15][16][17]. The relativistic effect including the SO coupling is fully considered in solving the relativistic Kohn-Sham equation [16,17]. In the following calculations, we adopt a face-centered cubic primitive unit cell for Cd 2 Os 2 O 7 containing two formula units, and restrict consideration to q = 0 magnetic ordering. Brillouin-zone integrations were performed using up to 12×12×12 k-point samplings using the improved tetrahedron method [18]. We used a planewave cutoff energy of 40 Ry. The following calculations were done with the experimental lattice structure at 180 K: a = 10.1598Å and x(O 1 ) = 0.319 [3]. Every Os atom is located at the center of an OsO 6 octahedron. For x(O 1 ) > 0.3125, each oxygen octahedron is slightly compressed along the local 111 axis that connects the centers of the two neighboring Os tetrahedra. To identify the antiferromagnetic ordering, we calculate the local magnetic moment projected on an Os atom, m Os , by integrating the magnetic moment within a radius of 2.5 a.u. (= 1.323Å). The direct gap, ∆ D , is defined as the minimum gap between the conduction and valence bands, which approximately corresponds to the optical gap. In the following discussion, ∆ C denotes the charge gap, which identifies the MIT. Figure 1(b) shows the computed ground-state phase diagram with respect to U eff . At small U eff 's, the ground state is non-magnetic metal (NMM). By increasing U eff , the ground state turns into the AFM phase at U eff ≃ 0.75 eV, and further into the AFI phase at U eff ≃ 1.2 eV. We found that the all-in/all-out magnetic order is the most stable in the entire parameter region of the AFM and AFI phases: Os moments with the same magnitude point toward or away from the centers of the tetrahedra along the local 111 axes as illustrated in Fig. 1(a). We obtained m Os ≃ 0.8-1.1 µ B /Os in the AFI phase, which is considerably smaller than 3 µ B /Os for the high-spin state. Note that the effect of the distortion of the octahedral crystal field is not strong enough to split the t 2g manifold. As suggested by the previous LSDA studies [19,20], the band structure looks semi-metallic on the high-symmetry lines. In fact, we obtained ∆ D ≃ 0.035 eV at U eff = 0.0 eV. Although ∆ D decreases with increasing U eff , ∆ D remains non zero within the NMM phase as shown in Fig. 1(c). inate in low-lying flat bands. The low-energy structure of the density of states remains essentially unchanged within the NMM phase [see Fig. 3 1.0 Γ L W X Γ E (eV) Γ L W X Γ (a) U eff =0.0 eV (b) ef =0.5 eV U f (d) U eff =0.9 eV (c) U eff =0.8 eV (e) U eff =1.0 eV (f) U eff =1.1 eV (h) U eff =1.5 eV (g) U eff =1.25 eV(b)]. With increasing U eff , the ground state turns into the AFM phase at U eff ≃ 0.8 eV. We found that the allin/all-out magnetic order is the most stable in the AFM and AFI phases. We used various initial spin and charge densities for the iterative scheme, but we could not find any other stable magnetic solutions. In the AFM phase, the time-reversal symmetry breaking lifts the Kramers degeneracy in the NM band structure. Note that the AF transition is associated with no Brillouin-zone folding because the all-in/all-out order preserves the translational symmetry of the lattice. The direct gap, ∆ D , closes at U eff ≃ 0.9 eV with a band inversion between the valence and conduction bands [see Fig. 1(c) and the circle in Fig. 2(d)] [33]. As m Os develops with increasing U eff , ∆ D opens again and becomes increasingly larger, finally resulting in the continuous MIT at U eff ≃ 1.2 eV [ Fig. 1(c)]. The MIT is characterized by vanishing elec- tron and hole Fermi surfaces (Lifshitz transition). This is clearly distinguished from the Slater transition in which a charge gap appears because of a magnetic superlattice structure [21]. Another notable observation near the MIT is that the density of states is considerably suppressed near the Fermi level up to about 0.2 eV, which is remarkably higher than ∆ C [Figs. 3(d)-(f)]. As seen in Fig. 3(d), this pseudo gap starts to develop in the AFM phase, suggesting that it comes from the modification of the band structure near the Fermi level with the emergent magnetic order. Next, we discuss the magnetic properties of the AFI phase. It is well known that pyrochlore antiferromagnets tend to be frustrated when the spins are isotropic. This is clearly seen in the nearest-neighbor classical antiferromagnet which exhibits no phase transition and remains paramagnetic down to zero T with a macroscopic degeneracy [22]. The ground-state manifold consists of degenerate states in each of which the summation of four spin moments vanishes on every tetrahedron. The degeneracy can be lifted by magnetic anisotropy which arises from the SO coupling. In particular, the all-in/all-out order is selected as the unique ground state by the local 111 easy-axis anisotropy [23][24][25]. Figure 4(a) shows magnetic anisotropy energies calculated for U eff = 1.25 and 2.0 eV, respectively. Here E g (θ) is the energy of the self-consistent solution that is obtained under the constraint that every Os moment is rotated from that of the ground state around the [001] axis, and its magnitude is equal to m Os in the ground state. One can clearly see that E g remarkably increases with the rotation by as large as about 40 meV/Os and 70 meV/Os for U eff = 1.25 and 2.0 eV, respectively. This proves the existence of strong magnetic anisotropy in this compound even near the MIT. To gain deeper insight into the nature of the magnetic anisotropy, we extend the analysis using a phenomenological model for the energy change associated with the rotations of the Os moments: H = J i,j ,i<j m i · m j − A sia i ( m i · α i ) 2 +A DM i,j ,i<j d ij · ( m i × m j ) ,(1) where m i is a unit direction vector of the magnetic moment at Os site i. Here J, A sia , and A DM are the nearestneighbor exchange interaction, the single-ion anisotropy, and the DM interaction, respectively. Note that m Os weakly depends on U eff in the AFI phase, and this effect is renormalized into the interaction parameters. The unit vector α i is along the local 111 axis at site i. The case where A sia > 0 corresponds to the easy-axis anisotropy. The unit vectors d ij are the direction vectors of the DM interaction, which are the only one symmetry-allowed form [26] [34]. In Ref. 26, the cases where A DM > 0 and A DM < 0 are distinctly referred to as the "direct case" and "indirect case"; the former favors the all-in/all-out ordering. The model parameters can be extracted by fitting energies obtained in electronic structure calculations by Eq. (1). Hereafter, the energy is always measured per Os atom. In the following calculations, the magnitudes of the Os moments are fixed to m Os of the ground state. On the basis of Eq. (1), the energy difference between the all-in/all-out and 3-in/1-out states is given by E e (0) − E g (0) = J + 2 √ 2A DM .(2) Here, the 3-in/1-out state is the lowest excited state for J > 0 and A sia > 0, which is obtained by flipping one of four non-equivalent spins in the all-in/all-out state [see are obtained respectively as follows: E g (θ) − E g (0) = E e (θ) − E e (0) + 4 √ 2 3 A DM (1 − cos θ) ,(3)E e (θ) − E e (0) = A sia 1 − 1 9 (2 cos θ + 1) 2 ,(4) where θ denotes the rotation angle around the [001] axis. Table I summarizes the model parameters extracted for U eff = 1.25 and 2.0 eV. The values of A sia and A DM were obtained by fitting the magnetic anisotropy energies at small θ's, i.e., in the range of 0 • ≤ θ ≤ 60 • [35]. We found that the values of J and A sia are positive for both the abovementioned values of U eff 's. The antiferromagnetic J might originate in the superexchange coupling via the O site. The easy-axis anisotropy A sia is found to be even larger than J for both U eff 's. On the other hand, A DM is considerably smaller than A sia , and vanishes as U eff increases. This might be because the DM interaction appears as a perturbation with respect to hopping between the nearest-neighboring Os atoms. These results suggest that the all-in/all-out order is mainly stabilized by the large easy-axis anisotropy rather than the DM interaction. In the case of Y 2 Ir 2 O 7 [9], the allin/all-out ordering is ascribed to the direct DM interaction because Ir 4+ has an effective total angular momentum of J eff = 1/2 [9,27,28] [36]. The observation of the large A sia indicates that Os 5+ has a larger value of J eff (J eff > 1/2). Assuming that the real material is located near the MIT, e.g, U eff ≃ 1.25 eV, the calculated results provide a natural explanation for the peculiar low-T properties of this compound. (2) The strong easy-axis anisotropy on the order of several tens meV stabilizes the all-in/all-out order cooperatively with the antiferromagnetic exchange interaction. This can account for the high antiferromagnetic transition temperature of this compound. The present result indicates that this compound has a magnetic gap of several tens meV. Further experiments are needed to detect the magnetic gap and confirm the present observation. Before closing this Letter, we would like to comment on future work. Contrary to the recent theoretical proposals [29,30], all the three phases in the phase diagram have a trivial Z 2 topological invariant. The band inversion in the AFM phase might suggest the presence of Dirac cones. A detailed analysis of the nature of the band inversion is a part of our future study. In contrast to Cd 2 Os 2 O 7 , Hg 2 Os 2 O 7 remains metallic below the AF transition temperature [31]. This material might be located in the AFM phase. The absence of a clear charge gap below the continuous MIT associated with the Néel ordering is also observed in NaOsO 3 perovskite [32]. Clearly, first-principle studies on this related compound will be interesting. In summary, we performed LSDA+SO+U calculations to explore the ground-state properties of Cd 2 Os 2 O 7 . We also found the all-in/all-out magnetic order as the ground state in a wide region of the phase diagram, supporting recent X-ray experimental results. We also showed that this magnetic order is stabilized by strong easy-axis anisotropy originating in spin-orbit coupling. Furthermore, we found a pseudo gap in the density of states near the metal-insulator transition. These numerical results provide a natural explanation for the puzzling lowtemperature properties of this compound. The present result might open up new possibilities to explore Os compounds as a stimulating playground for the interplay of electron correlation and spin-orbit coupling. PACS numbers: 75.47.Lx,71.15.Mb,71.70.Ej FIG . 1: (color online). (a) Cubic unit cell of Cd2Os2O7 contains 8 formula units. Only the corner sharing tetrahedron network of Os atoms is shown for clarity. The arrows represent the all-in/all-out magnetic order with the ordering vector of q = 0. (b)/(c) U eff dependences of the Os magnetic moment mOs, the charge gap ∆C, and the direct gap ∆D. The phase diagram consists of a non-magnetic metal (NMM), an antiferromagnetic metal (AFM), and an antiferromagnetic insulator (AFI). Figures 2(a)-(b) show the calculated electronic band structures near the Fermi level. At U eff = 0.0 eV [Fig. 2(a)], the Fermi level lies in the half-filled t 2g bands consisting of twelve Kramers(doubly)-degenerate bands. Figures 3(a)-(f) show the calculated electronic density of states. At U eff = 0.0 eV [Fig. 3(a)], one clearly see a sharp peak near the Fermi level, which may orig- FIG. 2 : 2(color online). Band structures on high-symmetry lines computed at U eff = 0.0, 0.5, 0.8, 0.9, 1.0, 1.1, 1.25, and 1.5 eV. Energy, E, is measured from the Fermi level (EF) [(a)-(f)] or from the center of the charge gap [(g), (h)]. Conduction and valence bands are denoted by green and red lines, respectively. The direct gap closes near the L point at U eff ≃ 0.9 eV [see the circle in (d)]. FIG. 3 : 3(color online). Density of states computed at U eff = 0.0, 0.5, 0.9, 1.1, 1.25, and 1.5 eV for the same data as inFig. 2. The inset of (e) is an enlarged plot of (e) near the charge gap. The density of states is measured per primitive unit cell. The shaded areas denote a pseudo gap, which appears near the MIT. Figs. 4 4(b) and (c)]. Changes in the energies of the allin/all-out [E g (θ)] and 3-in/1-out ordered states [E e (θ)] ( 1 ) 1The large direct gap, ∆ D , and small charge gap, ∆ C , near the MIT are consistent with the experimental observation of a large optical gap and the absence of a clear charge gap at low T 's. In fact, ∆ D ≃ 0.17 eV estimated at U eff = 1.25 eV is comparable to the optical gap of about 0.1 eV observed below T MIT [10]. The pseudo gap is expected to develop below T MIT concurrently with the onset of the Néel ordering. The density of states vanishes toward low T 's in the wide energy range. This observation explains the semiconducting behavior of resistivity up to the high T MIT . FIG. 4 : 4(color online). (a) Magnetic anisotropy energies estimated by the rotation of Os magnetic moments around the [001] axis. (b)/(c) all-in/all-out and 3-in/1-out ordered states rotated by θ = 0 • and θ = 120 • around the [001] axis. Every spin is perpendicular to the local 111 axis at θ = 120 • . 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The product of the parity eigenvalues for the occupied bands changes its sign between U eff = 0.8 and 0.9 eV at the L point. The middle points of nearest-neighboring Os sites are not inversion centers of the pyrochlore lattice. The middle points of nearest-neighboring Os sites are not inversion centers of the pyrochlore lattice. The deviations of the data from the fitting curves for θ > 60 • might come from higher order terms omitted in the simple model. The deviations of the data from the fitting curves for θ > 60 • might come from higher order terms omitted in the simple model (1). No uniaxial single-ion anisotropy is present in J eff = 1/2 or S = 1/2 systems. No uniaxial single-ion anisotropy is present in J eff = 1/2 or S = 1/2 systems.	[]
[ "Proximity effect with noncentrosymmetric superconductors", "Proximity effect with noncentrosymmetric superconductors" ]	[ "Gaetano Annunziata \nMax-Planck-Institut für Festkörperforschung\nDepartment of Physics\nNorwegian University of Science and Technology\nHeisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway\n", "Dirk Manske \nMax-Planck-Institut für Festkörperforschung\nDepartment of Physics\nNorwegian University of Science and Technology\nHeisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway\n", "Jacob Linder \nMax-Planck-Institut für Festkörperforschung\nDepartment of Physics\nNorwegian University of Science and Technology\nHeisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway\n" ]	[ "Max-Planck-Institut für Festkörperforschung\nDepartment of Physics\nNorwegian University of Science and Technology\nHeisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway", "Max-Planck-Institut für Festkörperforschung\nDepartment of Physics\nNorwegian University of Science and Technology\nHeisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway", "Max-Planck-Institut für Festkörperforschung\nDepartment of Physics\nNorwegian University of Science and Technology\nHeisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway" ]	[]	We describe the superconducting proximity effect taking place in a contact between a noncentrosymmetric superconductor and a diffusive normal/ferromagnetic metal within the quasiclassical theory of superconductivity. By solving numerically the Usadel equation with boundary conditions valid for arbitrary interface transparency, we show that the analysis of the proximity-modified local density of states in the normal side can be used to obtain information about the exotic superconductivity of noncentrosymmetric materials. We point out the signatures of triplet pairing, the coexistence of triplet and singlet pairing, and particular orbital symmetries of the pair potential. Exploiting the directional dependence of the spin polarization pair breaking effect on the triplet correlations, we show how the order relation between triplet and singlet gaps can be discriminated and that an estimation of the specific gap ratio is possible in some cases. PACS numbers: arXiv:1211.5138v1 [cond-mat.supr-con]	10.1103/physrevb.86.174514	[ "https://arxiv.org/pdf/1211.5138v1.pdf" ]	118,588,437	1211.5138	c5168be2854eb49c750c21ea8406c6f2d6ac6f1b	Proximity effect with noncentrosymmetric superconductors Gaetano Annunziata Max-Planck-Institut für Festkörperforschung Department of Physics Norwegian University of Science and Technology Heisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway Dirk Manske Max-Planck-Institut für Festkörperforschung Department of Physics Norwegian University of Science and Technology Heisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway Jacob Linder Max-Planck-Institut für Festkörperforschung Department of Physics Norwegian University of Science and Technology Heisenbergstrasse 1D-70569, N-7491Stuttgart, TrondheimGermany, Norway Proximity effect with noncentrosymmetric superconductors (Dated: Received May 10, 2014)PACS numbers: We describe the superconducting proximity effect taking place in a contact between a noncentrosymmetric superconductor and a diffusive normal/ferromagnetic metal within the quasiclassical theory of superconductivity. By solving numerically the Usadel equation with boundary conditions valid for arbitrary interface transparency, we show that the analysis of the proximity-modified local density of states in the normal side can be used to obtain information about the exotic superconductivity of noncentrosymmetric materials. We point out the signatures of triplet pairing, the coexistence of triplet and singlet pairing, and particular orbital symmetries of the pair potential. Exploiting the directional dependence of the spin polarization pair breaking effect on the triplet correlations, we show how the order relation between triplet and singlet gaps can be discriminated and that an estimation of the specific gap ratio is possible in some cases. PACS numbers: arXiv:1211.5138v1 [cond-mat.supr-con] I. INTRODUCTION Unconventional superconductors are characterized by a pairing state breaking not only the U(1) gauge symmetry. The most famous class of such materials is high-temperature cuprates in which anisotropic d-wave pairing is likely realized. 1 It is also known that electrons forming Cooper pairs need not to be bound in a spin-singlet state: Spin-triplet states with total spin quantum number S = 1 are also possible. 2 Several materials have been discovered showing strong signatures of a p-wave spin-triplet pairing state. 3 A full understanding of pairing states in unconventional superconductors is still lacking but it is widely accepted that their origin cannot be explained within the Bardeen-Cooper-Schrieffer theory of conventional s-wave superconductors. 4 The determination of the symmetry of the superconducting order parameter is extremely important since it is often the first step to understand the mechanisms generating the superconducting phase. Among unconventional superconductors a special place is occupied by noncentrosymmetric superconductors (NCSs), an intriguing class of compounds which has recently generated much attention due to its unique properties. 5 In these materials the superconducting phase develops in a low-symmetry environment with a missing inversion center. This broken symmetry generates a Rashba-type 6 spin-orbit coupling (SOC) and prevents the usual even/odd classification of Cooper pairs according to orbital parity, allowing a mixed-parity superconducting state. Since according to the Pauli principle even (odd) orbital parity states are associated with spin-singlet (triplet) states, disregarding for the moment the possibility of odd-frequency pairing, 7 singlet and triplet Cooper pairs are allowed to coexist in NCSs. The interest in these materials does not only stem from the lacking of a clear understanding of pairing-state symmetry and pairing mechanism. Indeed it has been recently recognized that NCSs can possibly be in a topologically nontrivial phase. This phase is believed to be characterized by line nodes, nontrivial topology of Bogoliubov-quasiparticles wavefunctions, zero-energy Majorana modes in vortex cores, and gapless edge states, all being stable and topologically protected against Hamiltonian parameters small variations. [8][9][10][11][12] Moreover they are predicted to be suitable materials to generate and host spin polarized supercurrents. [13][14][15] Since the discovery of noncentrosymmetric CePt 3 Si, exhibiting a superconducting phase below T c = 0.75K at ambient pressure, 16 the family of NCSs has largely grown in number. Relevant examples are CeRhSi 3 , 17 CeIrSi 3 , 18 CeCoGe 3 , 19 CeIrGe 3 , 20 UIr, 21 Li 2 Pd x Pt 3−x B, 22 Mo 3 Al 2 C, 23 and Y 2 C 3 . 24 It is worth noting that superconductivity developing at oxide interfaces such as LaAlO 3 /SrTiO 3 is intrinsically noncentrosymmetric. 25 We will focus on the cerium compounds in what follows since within our formalism they will admit a common description. They share the same generating point group C 4v lacking a mirror plane normal to the c axis and have a qualitatively similar T − P phase diagram, with a superconducting dome partially intersecting an antiferromagnetic phase. 20,[26][27][28][29] They differ in the amplitudes and patterns of the ordered magnetic moments, they have a different electronic specific heat Sommerfeld coefficient and effective masses. CePt 3 Si is the compound with the largest Sommerfeld coefficient, a signature of strong electronic correlations, 30,31 and it is the only one to become superconducting at ambient pressure. A detailed collection of available data on these material can be found in Ref. 5 and references therein. The determination of superconducting pairing state effectively realized is still an open issue. Several experiments point toward an unconventional state with the possibility of lines nodes. [32][33][34][35][36] The interpretation of experimental results can nevertheless be controversial since strong electronic correlations and SOC have been shown 30,31 to influence the experimental findings. 37 Indeed a triplet pairing state seems to be suggested by a critical magnetic field H c2 larger than the condensation energy, e.g. beyond the Pauli-Clogston limit. 38 There is still controversy about this point since a superconducting phase developing close to an antiferromagnetic one would naturally correspond to singlet pairing state. In this scenario the large critical magnetic field could be explained taking into account the SOC. 39 The standard route to identify a triplet pairing state is to exploit the interplay of superconductivity and magnetism, namely the magnetic field direction dependence of its pair breaking effect on Cooper pairs. While the pair breaking exist for any direction of the field for singlet superconductors, for triplet superconductors if the field is oriented parallel to the triplet equal spin pairing direction its effect is null. So measuring the spin susceptibility across superconducting transition can help identify triplet pairing states. However, the same experimental techniques used to ascertain that the prototypical triplet superconductor SrRuO 4 40 has an equal spin pairing state with spin oriented in the ab plane 41 fail when applied to CePt 3 Si. 37 While theoretical calculations including SOC predict a zero-temperature susceptibility value which is half the value of the one in the normal state when the field is in a pair-breaking configuration, 42 experiments 37 find a temperature-independent susceptibility for any field direction. The impossibility to analyze the triplet pairing states with the common techniques is a consequence of the strong electronic correlations which enlarge the almost temperatureindependent van Vleck term over the Pauli term in the spin susceptibility. 30,31 Another complication is that a magnetic field shifts the system in the mixed phase which in NCSs is predicted to be a helical vortex phase. 43 We believe that these complications can be overcome by separating in space superconductivity and magnetism, and that an analysis of pairing states realized in NCSs can be put forth by considering hybrid contacts with ferromagnets. In particular analyzing the superconducting proximity effect 44 in such hybrids, the superconducting correlations propagating in the proximate region will be specific of the pairing state in NCSs but will not suffer from the complications described above. The strategy of using hybrid contacts, junctions, and tunneling spectroscopy 45 to probe the pairing state of superconductors has been very fruitful. These techniques can be exploited to determine the gap amplitude in conventional superconductors and even the phase change or the existence of nodes in pairing states of unconventional superconductors. 46,47 Many theoretical studies exist on tunnel and Josephson junctions with NCSs. 10,12,14,[48][49][50][51][52][53][54] Here we address the problem of proximity effect in hybrid contacts of NCSs with normal and ferromagnetic metals. The most direct way to probe the proximity effect in experiments is to measure the local density of states (LDOS) in the nonsuperconducting region and to look at how it deviates as a function of position and energy from the normal-state LDOS for the influence of superconducting correlations. This can be done by placing contacts at different points of the normal region 55 or with a STM tip capable of sweeping it. 56 Our aim is to show that this kind of measurement can be useful to determine both the orbital and spin pairing state of NCSs. In particular we point out the signatures of triplet pairing, triplet and singlet pairing coexistence, and particular orbital pair potentials realizations. We show how the order relation between triplet and singlet gaps can be discriminated and that an estimation of their specific gap ratio is possible in some cases. The remainder of this paper is organized as follows. In Sec. II the system is described and the formalism and equations employed are introduced. In Sec. III we report our results for the proximity effect both in normal metals and ferromagnets. We compare different types of NCSs with different types of gaps. We also consider ferromagnets with varying exchange field amplitude and direction. Our conclusions are given in Sec. IV. FIG. 1: (Color online) In (a) the system under consideration is sketched. It is a contact between a NCS and a diffusive ferromagnet (DF) separated by an infinitely thin insulator. In such a setup the proximity effect features can be probed by looking at how the normal LDOS in DF is modified by the presence of the superconductor with, e.g., a position and energy resolved STM measurement. In the particular case of NCS both even-frequency spin-singlet and odd-frequency spin-triplet superconducting correlations are expected to propagate in the normal part of the contact; the latter can be long ranged depending on the relative orientation of exchange field in DF and d-vector in NCS. In (b) the amplitude and phase of the pair potentials matrices in two different types of NCSs are considered. Numerical simulations suggest that in cerium family NCSs d f potentials could be stabilized rather than the more commonly assumed sp ones. 66 II. MODEL AND FORMALISM We consider an effective 2D contact between a clean noncentrosymmetric superconductor and a diffusive ferromagnet (NCS/DF) separated by a thin insulating barrier (see Fig. 1). The contact is at equilibrium at T = 0. The system lies in the xy plane with x being the direction orthogonal to the interface. The NCS side is considered as a reservoir while the DF side is assumed to have a finite width L. We describe the contact within the quasiclassical theory of superconductivity, a formalism widely used in the analysis of superconducting heterostructures (see Ref. 57 for a general introduction and Refs. 58 for its application to superconducting hybrids). The approximations behind this model have revealed being able to grasp the main low-energy experimental features of heterostructures involving conventional and unconventional superconductors, both for ballistic and diffusive systems. The key consideration of the theory is that whenever the Fermi energy is much larger than other energy scales in the system, only a small error is carried out by considering only physical processes taking place close to the Fermi energy. Within this formalism, all the physical information is coded in the quasiclassical Green's function defined aŝ g(r, k F , ε) = i π ∞ −∞ dξ kĜ (r, k, ε),(1) whereĜ is the position, momentum, and energy dependent Fourier-transformed Gor'kov Green's function in the Wigner representation and ξ k = k 2 /2m. We have chosen units such thath = c = 1 and we use. . . to denote 4 × 4 matrices in spin ⊗ Nambu space and . . . for 2×2 matrices in spin space. While the momentum amplitude dependence has been integrated out, the quasiclassical Green's function still depends on Fermi momentum direction, i.e., k F = (cos ϕ, sin ϕ, 0), where ϕ is the azimuthal angle in the xy plane (see Fig. 1). From now on we omit the subscript in the momenta since it is clear that in the quasiclassical theory they are always fixed on the Fermi surface. In nonequilibrium conditions a larger number of Green's functions together with proper distribution functions are necessary. This theory, known as Keldysh formalism, is redundant in our case and the retarded or advanced 4 × 4 Green's functions matrices are sufficient. They have the general structureĝ R,A = g R,A f R,A −f R,A −g R,A ,(2) where g R,A and f R,A are the normal and anomalous propagator blocks, the latter being non zero only when superconducting correlations exist. Each block has spin structure g R,A = g R,A g R,A ↑↓ g R,A ↓↑ g R,A ,(3) and the tilde operation is defined as f R,A (r, k F , ε) = f R,A (r, −k F , −ε) * .(4) Moreover the matrices are normalized such that g R,A ·ĝ R,A =1.(5) Since retarded and advanced matrices are not independent in what follows we will use only the retarded matrix dropping the suffix R. Once the latter is known, the LDOS can be calculated as N(ε)/N 0 = 1 2 Re Tr g ,(6) where N 0 is the density of states in the absence of superconductivity. Measuring the LDOS in the nonsuperconducting side of superconducting hybrids is the most direct way to probe the proximity effect so we will focus on this quantity in what follows. The quasiclassical Green's function can be calculated by solving the Eilenberger equation, 59 valid in systems with arbitrary impurity scattering, and the Usadel equation, 60 valid in diffusive systems with strong impurity scattering. Since we are considering a contact between a clean superconductor and a diffusive ferromagnet, we have to solve different equations in the x ≷ 0 regions and their solutions have to be connected at x = 0 through a proper boundary condition. Our choice to consider different impurity scattering regimes follows from the consideration that superconductivity in noncentrosymmetric materials (and more generally unconventional non s−wave superconductivity 61 ) is known to occur only in rather clean samples. 16,62 In NCSs the absence of a center of inversion generates an electric field which, according to special relativity, in the rest frame of electrons is felt as a magnetic field proportional to their orbital momenta, resulting in an effective antisymmetric Rashba spin-orbit coupling g k = −g −k . The particular form of this coupling for a noncentrosymmetric material can be determined by symmetry considerations. Moreover the broken inversion symmetry prevents the classification of Cooper pairs' wave function orbital parity which is allowed to be an admixture of even and odd parts. As a consequence in NCSs an admixture of triplet and singlet Cooper pairs can exist. It is worth noting that the Pauli principle allows rather than dictates the existence of a triplet pair potential in NCSs and that ideas to identify it are highly desirable. When considering the proximity effect in a NCS contact, both singlet and triplet correlations can manifest in the non superconducting side. Since these become isotropic in a diffusive medium, the triplet correlations have to be odd in frequency in order to respect the overall antisymmetry dictated by the Pauli principle. This exotic type of correlation can decay in a magnetic medium on a scale typical of a non magnetic medium, at odds with standard even-frequency singlet correlations which are strongly suppressed by an exchange field. 63,64 In a NCS the gap matrix in spin space has the general form ∆ k = ∆ s f s (k)iσ y + ∆ t f t (k)d k · σiσ y ,(7) where ∆ (s,t) is the singlet (triplet) component amplitude, f (s,t) (k) are structure factors, and the well-known d-vector d k 65 is used to parametrize the triplet pair potentials. The usual notation is used for Pauli matrices with σ the vector of them. It is convenient to normalize Eq. (7) introducing the total gap amplitude ∆ 0 ≡ max k Tr ∆ † k ∆ k /2 and the triplet/singlet gap amplitudes ratio r ≡ ∆ t /∆ s ∈ [0, ∞), result- ing in ∆ k = ∆ 0 f s (k) √ 1 + r 2 iσ y + r f t (k) √ 1 + r 2 d k · σiσ y .(8) With the parametrization employed the relative weight of triplet and singlet gaps amplitudes r can be tuned without af-fecting the total gap amplitude ∆ 0 = √ ∆ s 2 + ∆ t 2 and the extreme cases of pure singlet and triplet superconductors can be recovered for r = 0 and r → ∞, respectively. The value assumed by the ratio r in NCSs is extremely important since Andreev bound states can develop at the surface if r > 1 (∆ t > ∆ s ), depending on the orbital symmetry of pair potentials. Indeed it has been recognized that NCSs manifest a rich and peculiar bound-state structure. [10][11][12] So it would be highly desirable to estimate experimentally the value of r or at least discriminate between the r ≷ 1 regimes. Another important point is to probe the structure factors and orbital symmetries of pair potentials. Several theoretical studies have pointed out the possible singlet and triplet pair potentials that are supposed to be realized in NCSs. 31,66,67 Part of our subsequent analysis is aimed toward these points. Similar ideas have been developed analyzing Raman scattering, tunneling conductance (without proximity effect) and Josephson effect features. 10,12,14,[48][49][50][51][52][53][54]68 It is widely accepted that the spin structure of triplet Cooper pair condensate is such that d k g k since this condition maximizes the superconducting critical temperature. 39 The form of the spin-orbit coupling vector g k depends on crystallographic point-group symmetries of the material. 69 We focus here on the most common family of NCSs: the cerium compounds. Their generating point group is the tetragonal point group C 4v corresponding to g k ∝ (−k y , k x , 0). 69 The resulting triplet pair potential has finite S = 1, m = ±1 and null S = 1, m = 0 components. Moreover ∆ and ∆ have equal magnitudes resulting in an unitary triplet state. They occupy the diagonal elements of the gap matrix while the singlet elements are the off-diagonal ones. The k dependence of the pair potentials in NCSs is a matter of debate. The most common assumption is that they are the minimal spherical harmonics consistent with their spin structure, i.e., s−wave for singlet and p−wave for triplet pair potentials. Several studies suggest that this scenario is realized in NCSs. 31 This sp case is obtained in our formulation by choosing f s (k) = f t (k) = 1. The associated gap matrix ∆ sp is depicted in panel (b) of Fig. 1. The singlet off diagonal functions have opposite sign while the triplet diagonal ones have opposite chirality p x ± ip y . Theoretical studies point out that higher spherical harmonics pair potentials are possible. 66 The singlet pair potential can be d x 2 −y 2 −wave and the triplet can Fig. 1 and is obtained in our formulation by be f −wave. The gap matrix in the d f case ∆ d f is depicted in panel (b) ofchoosing f s (k) = f t (k) = k 2 x − k 2 y . Other possibilities have been explored in the literature such as f s (k) = f t (k) = 2k x k y which is related to the LaAlO 3 /SrTiO 3 heterointerface 12 and f s (k) = 2k x k y , f t (k) = 1 which is not related to any material but has been considered for theoretical interest. 53 Part of the subsequent analysis is devoted to the comparison of the sp and d f scenarios, the two most probable pair potential realization in NCSs. Several studies of NCSs within the quasiclassical theory of superconductivity exist in the literature. 13,54,64,70 One often employed simplification is to neglect the spin-orbit coupling splitting of the Fermi surface since the Fermi energy is always much larger than the spin-orbit coupling. It has been pointed out how this simplification gives results qualitatively similar to the ones obtained without it. 10,12 The application of quasiclassical theory to a system with SOC depends on the strength of this interaction with respect to Fermi energy: For strong SOC one has to calculate two quasiclassical propagators in the two bands diagonalizing the SOC while for weak SOC the band splitting can be neglected and SOC can be considered as a self-energy. 5 While SOC can appear a strong interaction with respect to the superconducing gap, it is certainly weak with respect to the Fermi energy in the materials under examination 71 so we will adopt the second strategy. Since we are considering a NCS/DF contact where the NCS is in the clean limit, in the region x < 0 its asymptotic quasiclassical Green's functionĝ S can be obtained from the Eilenberger equation [ερ 3 −Σ k +∆ k ,ĝ S ] =0,(9) whereρ 3 = diag(1, 1, −1, −1) and Σ k = g k · σ 0 0 [g −k · σ] T .(10) After a proper rotation in spin space 70 Eq. 9 can be solved analytically. Its solution for generic ratios of triplet and singlet gap amplitudes r and for generic structure factors readŝ g S = 1 2 (ĝ + +ĝ − ) ,(11) wherê g ± =     N ± ∓ie −iϕ N ± ±ie −iϕ A ± A ± ±ie iϕ N ± N ± −A ± ±ie iϕ A ± ±ie iϕ A ± A ± −N ± ±ie iϕ N ± −A ± ±ie −iϕ A ± ∓ie −iϕ N ± −N ±     ,(12) with normal and anomalous elements N ± = ε ε 2 − ∆ 2 0 f 2 ± 1+r 2 ,(13)A ± = ∆ 0 f ± (1 + r 2 )ε 2 − ∆ 2 0 f 2 ± ,(14) where f ± = f s (k) + f t (k)r.(15) The calculated Green's function does not depend explicitly on SOC magnitude but its structure is dictated by its existence. This is a well-known result of the application of quasiclassical theory to NCSs. 70 However, the absence of SOC is compensated in our formalism by the parameter r, since the amplitude of SOC has been shown to be related to the triplet gap amplitude in extended Hubbard model studies. 66 The quasiclassical Green's functionĝ for x > 0 in the DF can be obtained from the Usadel equation D∂ x (ĝ∂ xĝ ) + i[(ε + iδ)ρ 3 + diag[h · σ, (h · σ) T ],ĝ] =0,(16) where D = v 2 F τ/2, τ being the relaxation time associated with elastic impurity scattering, δ is an energy associated with inelastic scattering, e.g., quasiparticle damping, "diag" indicates a diagonal block matrix, and h is the exchange field vector in DF. The characteristic energy scale of the diffusive side is the Thouless energy E Th = D/L 2 . Equation (16) can be solved with two boundary conditions at x = 0 and x = L. While at x = L the simple Neumann condition ∂ xĝ \| L =0(17) can be applied, the situation at x = 0, where the solution of the Usadel equation has to be connected with the asymptotic solution of the Eilenberger equation [Eq. (11)], is far more complicated. For contacts with conventional superconductors this problem has been solved and a boundary condition valid for arbitrary interface scattering between the sides of the contact has been derived by Nazarov. 72 A simplified version valid in the tunneling case had been derived by Kuprianov and Lukichev. 73 The case of magnetic interface has also been treated. 74 In the case of unconventional superconductors a serious complication arises since the Green's function in the superconductor has a k (or ϕ) dependence which is absent in the diffusive normal side since the strong impurity scattering isotropizes the Green's function. This problem has been solved by Tanaka et al. 75 who have generalized the Nazarov boundary condition for unconventional superconductors, both in a singlet and a triplet state. An infinitely thin insulating barrier located at x = 0 gives rise to a finite scattering potential Hδ(x) which lowers the transmission probability T as T (ϕ) = 4 cos 2 (ϕ) 4 cos 2 (ϕ) + Z 2 ,(18) where ϕ is the angle with respect to the normal to the interface for a given trajectory (see Fig. 1), and Z = 2mH/k is the dimensionless barrier strength. The barrier resistance R B is then defined as R B = 2 R 0 π/2 −π/2 dϕ T (ϕ) cos(ϕ) ,(19) where R 0 = 4π 2 /k 2 A is the contact Sharvin resistance, A being the constriction area. Introducing Γ = R B /R F , with R F the nonsuperconducting side resistance, the boundary condition at the interface can be written as Γ Lĝ\| 0 ∂ xĝ \| 0 = 2 ĝ\| 0 ,B(ϕ) ,(20) where . . . represents a transmission-mediated angular average on a half-Fermi surface, f (ϕ) = π/2 −π/2 dϕ f (ϕ) cos(ϕ) π/2 −π/2 dϕ T (ϕ) cos(ϕ) ,(21) andB(ϕ) depends on the asymptotic Green's function in the NCS. Defining T (ϕ) = T (ϕ)/ 2 − T (ϕ) + 2 1 − T (ϕ) ,(22) andĤ ± (ϕ) = 1 2 [ĝ S (ϕ) ±ĝ S (π − ϕ)] ,(23) B(ϕ) can be written aŝ B(ϕ) = −T 1 +Ĥ −1 − + T 2ĝ \| 0Ĥ −1 −Ĥ+ −T ĝ\| 0 ,Ĥ −1 − +Ĥ −1 −Ĥ+ − T 2ĝ \| 0Ĥ −1 −Ĥ+ĝ \| 0 .(24) In Eq. (24) the fraction between matrices is only symbolic and should be understood as a matrix product between the inverse of the denominator and the numerator. The Usadel equation [Eq. (16)] and the boundary conditions [Eqs. (17), (20)] form a boundary value problem which cannot be solved analytically. In the following section we report results obtained by solving numerically the boundary value problem with a finite difference code that implements the three-stage Lobatto IIIa formula and an initial guess for the Green's function is adapted self-consistently toward the final solution. III. RESULTS In order to look at the features of the proximity effect in the NCS/DF contact, once the quasiclassical Green's function has been obtained we calculate the LDOS in the non superconducting side with Eq. (6) as a function of energy and position. Notice that our LDOS is normalized with the normal state one such that when N(ε)/N 0 = 1 there is no proximity effect. Our main target is to find signatures in the proximity effect that can give hints on the nature of superconductivity in NCSsnamely, if there is triplet superconductivity, what is the ratio of triplet and singlet gap amplitudes?-distinguishing between sp and d f pair potential symmetries. We use ∆ 0 as energy unit and the superconducting coherence length ξ as the length unit. The energy is measured from the Fermi energy. The imaginary energy iδ in the Usadel equation [Eq. (16)], besides describing finite lifetime effects which are always present in real systems, has the advantage of stabilizing the code and facilitating the achievement of convergence. It is set to δ = 0.01∆ 0 while other parameters will take several values. The formalism developed in the previous section allows us to explore limiting cases. The asymptotic solution for the Green's function in NCSs [Eq. (11)] describes pure singlet and pure triplet superconductors for r = 0 and r → ∞, respectively. Moreover by neglecting the exchange field in the Usadel equation, the quasiclassical Green's function in a diffusive normal metal (DN) can be computed. We start exploring this case before considering the effect of ferromagnetism on the proximity effect. A. Proximity effect in NCS/DN In Fig. 2 we show the LDOS as a function of energy comparing sp and d f NCSs for several r = ∆ t /∆ s values (r = 0, 0.5, 1, 2, ∞ from bottom to top at ε = 0 in both panels). The LDOS is calculated in the normal side at the border opposite to the interface, i.e., at x = L (see Fig. 1), and its width is fixed to L = ξ. In a typical setup L ∼ 10 nm. The interface between the NCS and DN is assumed to be in an intermediate transparency regime with Γ = 0.1 and Z = 2. A change in the values of these parameters modifies quantitatively the LDOS but not its general features, except for L since the proximity effect eventually disappears farthest from the interface. In the sp case for r = 0, i.e., without a triplet gap, the typical proximity effect between an s−wave superconductor and a normal metal is recovered. Its signature is the opening of a minigap 80 in the LDOS of the normal metal. Its width depends on junction parameters such as interface transparency and normal layer length. 81 In a NCS with 0 < r < 1, e.g., r = 0.5, the minigap is still there but its width is smaller with respect to the pure s−wave case. Assuming equal amplitude for triplet and singlet gaps, i.e., r = 1, the LDOS is almost flattened at its normal-state value and a weak proximity effect exists only close to the Fermi energy ε = 0. This peculiar behavior can be regarded as a signature of a transition to a topologically nontrivial phase. Indeed the features of the LDOS change abruptly for r > 1. The main feature is a well-defined zeroenergy peak (ZEP) which is associated with the proximity effect from triplet superconductors. 76 In the d f NCS case the LDOS is quantitatively less modified by the proximity effect since the pair potentials are not fully gapped. The r dependence of the LDOS in this case is similar to the sp case but the energy dependence in the r ≷ 1 regimes differs. Indeed a dip rather than a minigap exists in the LDOS at low energy for r < 1. This is the typical situation in a [100] contact between a high T c cuprate and a normal metal while a [110] interface, i.e., an effective d xy rather than a d x 2 −y 2 pair potential, would not generate any proximity effect. [77][78][79] The dip becomes narrower increasing r as long as r < 1. Again there is almost no proximity effect for r = 1 and a ZEP develops for r > 1. This peak is different from the one in the sp case since shoulders are present at finite energy giving it a bell shape. The shoulders are a signature of additional peaks at finite energies reflecting the nonmonotonic dispersion of bound states in the d f case. 12 This peculiar type of peak is clearly distinguishable from a standard ZEP in tunneling experiments. 82 The LDOS features described depend quantitatively but not qualitatively on the contact parameters; i.e., the width of the minigaps and the width and height of the peaks depend on the particular values of L, Z, and Γ, but do not disappear as long as proximity effects take place. We have chosen an intermediate transparency interface even if this is not the usual experimental situation only for the sake of clarity. The main difference in considering a tunneling interface is that the peaks and minigaps appear more narrow, but their features and the possibility of their distinction remain unaltered. At the end of this section the more proper case of tunneling interface will be considered when we report on the penetration length of superconducting correlations in the normal side. From the analysis of the proximity effect in NCS/DN several considerations can be made. For both the sp and d f cases the regimes r ≷ 1 are clearly distinguishable by the low-energy LDOS, i.e., minigap for r < 1 and ZEP for r > 1 in the sp case and dip for r < 1 and ZEP for r > 1. The same differences can be exploited to distinguish be-tween sp and d f NCSs. Even for r > 1 where they both show a ZEP in the LDOS, a distinction is possible since only in the d f case the peak has shoulders at finite energies due to the nonmonotonic dispersion of the bound states. The determination of the particular value of r rather than the regime r ≷ 1 appears to be more difficult. Indeed it is clear that the features of the LDOS in NCS/DN are mostly dominated by the larger gap. In other words a NCS with singlet gap larger than the triplet one gives rise to a proximity effect very similar to the one from a pure singlet superconductor and vice versa. This is a general behavior of NCSs not only manifested in the proximity effect. However, there are quantitative dependencies on the particular value of r assumed in both regimes r ≷ 1 such as the width of the minigap and the height of the ZEP such that in principle a fitting procedure can be employed to determine its value from the experimental data. 55 A simpler estimation is nevertheless possible by looking at the proximity effect in NCS/DF. B. Proximity effect in NCS/DF The analysis of the proximity effect in a NCS/DF junction can ease the identification of triplet pairing in NCSs, the distinction between different types of NCSs, and the estimation of the triplet/singlet gap ratio r. Even if the simpler NCS/DN analysis can give sufficient information in this sense, loopholes can exist. The point is that observing ZEPs in LDOS is not sufficient to claim the existence of a triplet pairing in the superconductor since the peak could be generated by unconventional singlet superconductivity or could indeed be a signature of triplet correlation but generated from spin-sensitive processes at the interface or magnetic impurities rather than by a real triplet pair potential in the superconductor. The idea is to exploit the pair-breaking effect of the exchange field on the Cooper pairs. While a spin-polarized environment tends always to destroy singlet Cooper pairs, this is not true for triplet Cooper pairs where a polarization parallel to the equal spin pairing direction has no effect at all. 63,64,83 Considering that the d-vector of triplet superconductivity is built such that the equal spin pairing direction is in the plane orthogonal to it, this condition reads h⊥d k . If h has some components parallel to d k pair breaking takes place even for triplet Cooper pairs. Indeed the standard route to identify triplet pairing is to measure the polarization direction dependence of spin susceptibility across the superconducting transition. While this technique has been proven successful for Sr 2 RuO 4 , the prototypical triplet superconductor, it fails when applied to NCSs due to strong electronic correlations. A possible way to overcome this difficulty is to extract the superconducting correlations by the complicated environment of the NCS and analyze how they react to a pair-breaking field in a much simpler environment such as a ferromagnet, that is, study the proximity effect in NCS/DF. In our case the d-vector lies in the xy plane so an exchange field out of the plane of the junction h ẑ is not pair breaking while any finite component in the xy plane is pair breaking for triplet correlations. For singlet correlations the exchange field is pair breaking for any orientation. . This can be exploited as follows. Reporting the NCS/DN case we have concluded that a NCS with singlet gap larger than the triplet one gives rise to a proximity effect very similar to the one from a pure singlet superconductor and vice versa, and that a distinction between the extreme cases of pure triplet/singlet and mixed cases in the regimes r ≷ 1 requires a quantitative rather than qualitative analysis. In NCS/DF the situation is different. The fact that the exchange field is pair breaking for every direction for singlet correlations but not for triplet correlations gives the possibility to distinguish between a pure singlet case (conventional or not) and a real mixed case where singlet and triplet gaps exist. Indeed the presence of a subdominant triplet gap can be ascertained by a residual proximity effect in strong ferromagnets with exchange field in a nonpair-breaking configuration for the triplet correlations [compare the solid lines of panels (a), (b), (e), and (f)]. Of course in order to observe the pair-breaking effect it is not necessary to have a field totally in plane but a finite in-plane component is enough. Panels (c), (d), (g), and (h) show how the exchange field modifies the LDOS when the NCSs are mostly triplet superconductors; i.e., ∆ t = 2∆ s . In this case when the field is not in a pair-breaking direction [see panels (c) and (g)], its effect on the LDOS is minimal. At low energies there is no difference from the NCS/DN case, while small quantitative deviations exist at larger energy. The reason for the insensitivity of the LDOS to the subdominant component in the r > 1 regime lies in the fact that the NCS is in a topologically non trivial phase with Andreev bound states at the interface. These states totally dominate the physics of the contact. Once they have been triggered by the condition r > 1, they become only weakly dependent on the magnitude of the subdominant singlet gaps rendering the LDOS almost insensitive to an exchange field which is pair breaking only for the singlet correlations. 53 In particular the zero-energy bound state with momentum orthogonal to the interface does not depend at all on the subdominant singlet component and only the bound states associated with large-angle trajectories have an appreciable dependence on it. Moreover the larger contribution to the proximity effect comes from almost orthogonal trajectories since they are associated with a larger transmission probability [see Eq. (18)]. When the exchange field is pair breaking for triplet correlations [see panels (d) and (h)], the peaks observed in the NCS/DN case split and lower until eventually the proximity effect disappears. The different behavior for in-plane and out-of-plane fields can be exploited to ascertain whether the peaks observed in the LDOS of NCS/DN are really generated by triplet correlations. Indeed only in this case a sensitivity on the exchange field direction in NCS/DF should be present. C. Penetration length of superconducting correlations The spatial dependence of the LDOS is worth exploring since the penetration depth of superconducting correlations in the normal side is also a source of information about the pairing state. Figure 4 shows the zero-energy LDOS in the nonsuperconducting side of the contact as a function of the normalized distance from the interface L/ξ. Only the sp symmetry is plotted since at zero energy the d f symmetry results differ only quantitatively. In each panel Γ = 10, Z = 10 are fixed and a dashed line representing the absence of proximity effect is plotted. In panel (a) the case of NCS/DN is shown for several r values, r = ∆ t /∆ s = 0, 0.5, 2, ∞ from bottom to top (the first two curves are superposed). It is clear that the range of proximity effect in NCS/DN does not depend on the triplet/singlet gaps ratio, but when the triplet gap is dominant the magnitude of the proximity effect is much more sensitive to the distance from the interface. The range of the proximity effect can have an r dependence in NCS/DF. Panels fect. Since the penetration length is easy to measure and the quasiclassical theory is able to give a good estimation for it from the contact parameters in a conventional superconductor/ferromagnet structure, 84 the range of the proximity effect could be a precious source of information for NCS superconductivity since an unexpectedly long range proximity effect would be a strong signature of the existence of a subdominant triplet component. This could also be ascertained in the NCS's Josephson junction showing a coupling on distances larger than expected. When the NCS is mostly a triplet superconductor this sensitivity disappears as a consequence of Andreev bound state formation as explained before. Panels (d) and (e) shows this case with an exchange field out of plane and in plane and h/∆ 0 = 5. The curves plotted are for r = 1.5, 2, ∞ bottom to top close to the interface. In this case the proximity effect is long ranged for out of plane field. Indeed it decays on the same scale of a NCS/DN contact [see panel (a)]. In a pairbreaking field configuration oscillations around the normalstate LDOS appear and the penetration length is reduced as in a NCS/DF contact with r < 1 [see panel (c)]. The r dependence in the r > 1 regime is weak and only small quantitative differences emerge close to the interface. One could try to overcome this by analyzing the range of correlations at finite energies rather than zero energy; however the r dependence that emerges is always small compared to the one in the r < 1 regime. This could be seen by the small deviation induced by h in panel (c) and (g) of Fig. 3. From the analysis of the proximity effect in NCS/DF several considerations can be made. The interplay of ferromagnetism and superconductivity can give extra information with respect to the NCS/DN case. The sp and d f types can be distinguished; real equal spin pairing correlations generated by a triplet gap in NCS can be discriminated by directional dependence of the pair-breaking exchange field in both energy and space resolved analysis; in NCSs with a dominant singlet gap the existence of a subdominant triplet gap and an estimation of triplet/singlet gap ratios are possible. IV. CONCLUSIONS We have studied the proximity effect in a 2D contact between a clean noncentrosymmetric superconductor and a diffusive metal and ferromagnet within the quasiclassical theory of superconductivity. To look at the features of the proximity effect we have analyzed how the presence of the noncentrosymmetric superconductor influences the LDOS in the nonsuperconducting side by numerically solving the boundary problem determined by the quasiclassical Usadel and Eilenberger equations. We have found unique signatures in the proximity effect of the exotic superconductivity expected in NCSs and our findings can be easily exploited in tunneling spectroscopy experiments, overcoming the difficulties encountered in spin susceptibility measurements. Focusing on the most common family of noncentrosymmetric superconductors, the cerium compounds, we have found that the existence of a dominant triplet gap can be clearly ascertained by a zero-energy peak in the LDOS measured in a proximate normal metal. We have also shown that the features of the LDOS are mostly dominated by the larger gap, a noncentrosymmetric superconductor with singlet gap larger than the triplet one giving rise to a proximity effect very similar to the one from a pure singlet superconductor and vice versa. However, quantitative differences exist and the particular value of the triplet/singlet gap specific ratio could be determined by a fitting procedure of experimental data. Unique signatures of orbital symmetries exist for both triplet and singlet dominating gaps. So the proximity effect can distinguish between sp and d f orbital symmetries. For a dominant singlet gap the sp symmetry induces a minigap and the d f a dip in the normal metal. For a dominant triplet gap both induce a zero-energy peak but in the d f case it is accompanied by weaker peaks at finite energies. The range of superconducting correlations penetration in the normal metal does not depend on the orbital symmetries and on the gap amplitude ratio. The analysis of proximity effect in a ferromagnet gives additional insights. We have shown that it can help to ascertain whether the peaks seen in the normal metal case are really a signature of a dominant triplet gap in the noncentrosymmetric superconductor and not a spurious effect generated by magnetic impurities or spin-sensitive processes at the interface since a genuine triplet zero-energy peak is insensitive to an exchange field oriented out of the junction plane while it is destroyed by a sufficiently strong in-plane field. In order to change the orientation of the field, two separate samples where the exchange field is locked to different orientations via, e.g., antiferromagnetic coupling, or with different crystallographic orientations can be grown to effectively change the orientation of the exchange field. We have shown that another signature of triplet dominant gap is that the zero-energy peak survives far inside the non superconducting side despite the exchange field when it is in a nonpair-breaking configuration. The exchange field can be exploited again to distinguish between sp and d f types of noncentrosymmetric superconductors since for a field of the order of the superconducting gap they show different low-energy behaviors, sp showing a dome and d f a peak in the regime where the singlet gap is dominant. We have shown that in this regime the presence of a subdominant triplet component can be ascertained by the existence of a residual proximity effect in strong ferromagnets with exchange field in a non-pairbreaking configuration for the triplet correlations. This effect can be easier to observe by looking at the penetration length of superconducting correlations in the ferromagnet since the larger the subdominant triplet component the larger the range of the proximity effect. Since this quantity is easy to measure and the quasiclassical theory is able to give a good estimation for it from the contact parameters in a conventional superconductor/ferromagnet structure, the range of the proximity effect could be a precious source of information for NCS superconductivity since an unexpectedly long range proximity effect would be a strong signature of the existence of a subdominant triplet component. The specific value of r can be deduced by the enhancement in the penetration length in this case. The identification of a subdominant singlet component in a mostly triplet noncentrosymmetric superconductor appears more subtle instead since the the Andreev bound states existing in this regime are only weakly dependent on the magnitude of the subdominant singlet gaps rendering the LDOS almost insensitive to an exchange field pair breaking only for the singlet correlations. The influence of interface scattering on the triplet pair potential of NCSs deserves a comment since it is expected to be pair breaking for anisotropic superconducting order parameters. 85 This effect cannot be easily incorporated in our model since we are solving different equations for each side of the contact and they are connected by a boundary condition containing an integration. However, it has been shown 13 that taking into account this effect the surface Andreev bound states do not disappear but merely change dispersion for large-angle trajectories. Since the contribution to the proximity effect of a trajectory is inversely proportional to its angle, that would introduce only small quantitative changes and does not affect our conclusions. With the ongoing activity of characteriz-ing exotic superconducting materials where triplet and mixed pairing are believed to be present we believe our results on proximity effect may serve as a useful tool to experimentally identify superconducting properties of NCSs. FIG. 2 : 2(Color online) LDOS calculated in the nonsuperconducting side of a NCS/DN contact (see Fig. 1) for different NCSs. Several r = ∆ t /∆ s values are plotted in both panels (r = 0, 0.5, 1, 2, ∞ from bottom to top at ε = 0). L = ξ, Γ = 0.1, Z = 2 are fixed. The LDOS is calculated at x = L. FIG. 3 : 3(Color online) LDOS calculated in the nonsuperconducting side of a NCS/DF contact (see Fig. 1) for different NCSs, gaps ratio values, exchange field amplitudes and directions in DF. The NCS/DN case (dotted lines) is plotted for comparison. Short-dashed, long-dashed, and solid lines are for h/∆ 0 = 0.5, 1, 5, respectively. L = ξ, Γ = 0.1, Z = 2 are fixed. The LDOS is calculated at x = L. Fig- ure 3 3shows the LDOS calculated in the nonsuperconducting side of a NCS/DF contact for different NCSs (the first row for the sp type and the second for the d f type). The two panels of a column differ only in the type of NCS with all other parameters being equal. Both r ≷ 1 regimes are shown for two possible orientations of the exchange field h ẑ and h⊥ẑ. The NCS/DN case (dotted lines) is plotted for comparison and in each panel short-dashed, long-dashed, and solid lines differ in the amplitude of the exchange field, h/∆ 0 = 0.5, 1, 5, respectively. L = ξ, Γ = 0.1, Z = 2 are fixed and the LDOS is calculated at x = L. Panels (a), (b), (e), ( f ), show how the exchange field modifies the LDOS when the NCSs are mostly singlet superconductors, i.e., ∆ s = 2∆ t . In this case the minigap and the dip existing in the NCS/DN case are narrowed for low fields, while for fields of the order of the superconducting gap an enhancement of LDOS is possible and ZEPs in the d f case or domes in the sp case can develop. These different shapes can be used to distinguish between the two types of NCSs. The enhancement rather than the suppression of LDOS is indeed found in measurement of contacts between singlet superconductors and ferromagnets.84 For larger exchange field values such as to overcome the Pauli-Clogston limit, the pairbreaking effect eventually becomes so strong as to destroy completely the proximity effect [see solid curves of panels (b) and (f)]. However in a situation of absence of pair breaking for the triplet correlations a residual weak proximity effect can exist [see solid curves of panels (a) and (e)] (b) and (c) are for a NCS/DF contact with a mostly singlet NCS and with an exchange field out of plane and in plane and h/∆ 0 = 5. The cases r = 0, 0.4, 0.6, 0.8, 0.9 are plotted from bottom to top close to the interface. The oscillation of LDOS around the normal-state value and the faster decay with respect to the nonmagnetic case typical of singlet superconductor/ferromagnet contacts are present for both orientations of the field. However when the field is not pair breaking on the subdominant triplet correlations an r dependence of superconducting penetration length clearly appears. The larger the triplet component the larger the range of the proximity ef-FIG. 4: (Color online) Zero-energy LDOS calculated in the non superconducting side of a NCS/DN contact (a) and of a NCS/DF contact [(b)-(e)] as a function of the normalized distance from the interface L/ξ. Here only the sp symmetry is plotted since at zero energy the d f symmetry results differ only quantitatively. Everywhere Γ = 10, Z = 10 are fixed and a dashed line representing the absence of proximity effect is plotted. In (a) r = ∆ t /∆ s = 0, 0.5, 2, ∞ are considered from bottom to top (the first two are superposed and practically indistinguishable). In (b) and (c) the case ∆ t < ∆ s is considered for two orientations of the exchange field in the non superconducting side and h/∆ 0 = 5. The cases r = 0, 0.4, 0.6, 0.8, 0.9 are plotted from bottom to top close to the interface. 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[ "THE Q-TENSOR SQUARE OF FINITELY GENERATED NILPOTENT GROUPS, q ≥ 0", "THE Q-TENSOR SQUARE OF FINITELY GENERATED NILPOTENT GROUPS, q ≥ 0" ]	[ "Noraí R Rocco ", "Eunice C P Rodrigues " ]	[]	[]	In the present paper the authors extend to the q−tensor square G ⊗ q G of a group G, q a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square G⊗G (q = 0). The results are applied to the computation of G⊗ q G for finitely generated nilpotent groups G, specially for free nilpotent groups of finite rank. We also generalize to all q ≥ 0 results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square G ⊗ G when G is a n−generator nilpotent group of class 2. We end by computing the q−tensor squares of the free n−generator nilpotent group of class 2, n ≥ 2, for all q ≥ 0. This shows that the above mentioned upper bound is also achieved for these groups when q > 1.	null	[ "https://arxiv.org/pdf/1603.05424v1.pdf" ]	119,170,486	1603.05424	34b60d08709f16fb419ce7b8341b88607593088a	THE Q-TENSOR SQUARE OF FINITELY GENERATED NILPOTENT GROUPS, q ≥ 0 Noraí R Rocco Eunice C P Rodrigues THE Q-TENSOR SQUARE OF FINITELY GENERATED NILPOTENT GROUPS, q ≥ 0 In the present paper the authors extend to the q−tensor square G ⊗ q G of a group G, q a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square G⊗G (q = 0). The results are applied to the computation of G⊗ q G for finitely generated nilpotent groups G, specially for free nilpotent groups of finite rank. We also generalize to all q ≥ 0 results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square G ⊗ G when G is a n−generator nilpotent group of class 2. We end by computing the q−tensor squares of the free n−generator nilpotent group of class 2, n ≥ 2, for all q ≥ 0. This shows that the above mentioned upper bound is also achieved for these groups when q > 1. Introduction Let G and G ϕ be groups, isomorphic via ϕ : g → g ϕ for all g ∈ G. Consider the group ν(G), introduced in [23] as (1) ν (G) = G ∪ G ϕ \| [g, h ϕ ] k = [g k , (h k ) ϕ ] = [g, h ϕ ] k ϕ , ∀g, h, k ∈ G . It is a well known fact (see [23]) that the subgroup Υ(G) = [G, G ϕ ] of ν(G) is isomorphic to the non-abelian tensor square G ⊗ G, as defined by Brown and Loday in their seminal paper [8]. A modular version of the operator ν was considered in [10], where for any non-negative integer q the authors introduced and studied a group ν q (G), which in turn is an extension of the so called q-tensor square of G, G⊗ q G, first defined by Conduché and Rodriguez-Fernandez in [11] (see also [14], [7]). In order to describe the group ν q (G), if q ≥ 1 then let G = { k \| k ∈ G} be a set of symbols, one for each element of G (for q = 0 we set G = ∅, the empty set). Let F ( G) be the free group over G and ν(G) F ( G) be the free product of ν(G) and F ( G). As G and G ϕ are embedded into ν(G) we shall identify the elements of G (respectively of G ϕ ) with their respective images in ν(G) * F ( G). Let J denote the normal closure in ν(G) * F ( G) of the following elements, for all k, k 1 ∈ G and g, h ∈ G : g −1 k g (k g ) −1 ; (2) (g ϕ ) −1 k g ϕ (k g ) −1 ; (3) ( k) −1 [g, h ϕ ] k [g k q , (h k q ) ϕ ] −1 ; (4) ( k) −1 kk 1 ( k 1 ) −1 ( q−1 i=1 [k, (k −i 1 ) ϕ ] k q−1−i ) −1 ; (5) [ k, k 1 ] [k q , (k q 1 ) ϕ ] −1 ; (6) [g, h] [g, h ϕ ] −q .(7) Definition 1.1. The group ν q (G) is defined to be the factor group (8) ν q (G) := (ν(G) * F ( G))/J. Note that for q = 0 the sets of relations (2) to (7) are empty; in this case we have ν 0 (G) = ν(G) * F ( G))/J ∼ = ν(G). Let R 1 , . . . , R 6 be the sets of relations corresponding to (2), . . . , (7), respectively, and let R be their union, R = 6 i=1 R i . Therefore, ν q (G) has the presentation: ν q (G) = G, G ϕ , G \| R, [g, h ϕ ] k [g k , (h k ) ϕ ] −1 , [g, h ϕ ] k ϕ [g k , (h k ) ϕ ] −1 , ∀g, h, k ∈ G . There is an epimorphism ρ : η q (G) G, g → g, h ϕ → h, k → k q . On the other hand the inclusion of G into ν(G) induces a homomorphism ı : G → ν q (G). We have g ıρ = g and thus ı is injective. Similarly the inclusion of G ϕ into ν(G) induces a monomorphism  : G ϕ → ν q (G). Thus we shall identify the elements g ∈ G and g ϕ ∈ G ϕ with their respective images g ı and (g ϕ )  in ν q (G). Now let G denote the subgroup of ν q (G) generated by the images of G. By relations (4), G normalizes the subgroup T = [G, G ϕ ] in ν q (G) and hence Υ q (G) = T G = [G, G ϕ ]G is a normal subgroup of ν q (G). Thus we obtain ν q (G) = G ϕ · (G · Υ q (G)), where the dots mean internal semidirect products. It should be noted that the actions of G and G ϕ on Υ q (G) are those induced by the defining relations of ν q (G): for any elements g, x ∈ G, h ϕ , y ϕ ∈ G ϕ and k ∈ G we have [g, h ϕ ] x = [g x , (h x ) ϕ ] and ( k) x = (k x ); similarly, [g, h ϕ ] y ϕ = [g y , (h y ) ϕ ] and ( k) y ϕ = (k y ). In addition, for any τ ∈ Υ q (G), (gτ ) y ϕ = g[g, y ϕ ]τ y ϕ ∈ GΥ q (G). By [10, Proposition 2.9] Υ q (G) is isomorphic to the q-tensor square G ⊗ q G, for all q ≥ 0. We then get a result (see [10,Corollary 2.11]) analogous to one due to Ellis in [14]: ν q (G) ∼ = G (G (G ⊗ q G)); this generalizes a similar result found in [23] for q = 0. The commutator approach to G⊗G for the case q = 0, provided by the isomorphism between G ⊗ G and the subgroup [G, G ϕ ] of ν(G) (see [23], and also [15]), has proven suitable to treat of non-abelian tensor products of groups, Schur multipliers and many other relevant invariants involving covering questions in groups; see for instance, references [15], [24], [20], [4], [12], [21] and the GAP Package "POLYCYCLIC" in [13]. The extension of the existing theory from q = 0 to all non-negative integers q, as addressed for instance in [10], broadens the scope of these connections, now in a hat ("power") and commutator approach to the q-tensor square, q ≥ 0. In section 2 we extend to G ⊗ q G, q ≥ 0, some structural results found in [5] and [24] concerning G ⊗ G. In section 3 it is established an upper bound for the minimal number of generators of G ⊗ q G when G is a finitely generated nilpotent group of class 2, thus generalizing a result of Bacon found in [2]. We end by computing the q-tensor square of the free nilpotent group of rank n ≥ 2 and class 2, N n,2 , q ≥ 0; this will show, as in the case q = 0 (see [2,Theorem 3.2]), that the cited upper bound is also attained for these groups when q > 1, although in this case N n,2 ⊗ q N n,2 is a non-abelian group. Notation is fairly standard (see for instance [22]). If x and y are elements of a group G then we write y x for the conjugate x −1 yx and [x, y] for the commutator x −1 y −1 xy. Our commutators are left normed: [x, y, z] = [[x, y], z] for all x, y, z ∈ G, and so on, recursively, for commutators of higher weights. The order of x (resp. of G) is written o(x) (resp. \|G\|). As usual, γ i (G) denotes the i th term of the lower central series of G. For future reference we recall the well known Hall-Witt identity: (9) [x, y −1 , z] y [y, z −1 , x] z [z, x −1 , y] x = 1, ∀x, y, z ∈ G. In view of the isomorphism given by [10,Proposition 2.9], from now on we identify G ⊗ q G with the subgroup Υ q (G) = [G, G ϕ ]G ≤ ν q (G) and write [g, h ϕ ] in place of g ⊗ h, for all g, h ∈ G. Following [10] we write ∆ q (G) for the subgroup [g, g ϕ ]\|g ∈ G ≤ Υ q (G), which by Lemma 2.1 (vii) is a central subgroup of ν q (G). We write τ q (G) for the factor group ν q (G)/∆ q (G). The subgroup Υ q (G)/∆ q (G) of τ q (G) is isomorphic to the q-exterior square G ∧ q G. In order to avoid any confusion we usually write [G, G ϕ ] τ ( G) to identify the q-exterior square G ∧ q G with the image of [G, G ϕ ] in τ q (G). We shall eventually write T to denote the subgroup [G, G ϕ ] of ν q (G) in order to distinguish it from the nonabelian tensor square G ⊗ G ∼ = [G, G ϕ ] ≤ ν(G) in the case q = 0. The material presented here incorporates part of the doctoral thesis [25] of the second named author, written under the supervision of the first. Some Structural Results In this section we extend results found in [5] and [24] related to the non-abelian tensor square, from G ⊗ G to G ⊗ q G, q ≥ 0. We begin by including some previous, technical results for future references. The following basic properties are consequences of the defining relations of ν q (G). ν q (G), for all g, h, x, y ∈ G. (i) [g, h ϕ ] [x,y ϕ ] = [g, h ϕ ] [x,y] ; (ii) [g, h ϕ , x ϕ ] = [g, h, x ϕ ] = [g, h ϕ , x] = [g ϕ , h, x ϕ ] = [g ϕ , h ϕ , x] = [g ϕ , h, x]; (iii) If h ∈ G (or if g ∈ G ) then [g, h ϕ ][h, g ϕ ] = 1; (iv) [ x, [g, h ϕ ]] = [ x, [g, h]]; (v) ( x) g = x[x q , g ϕ ]; (vi) If [g, h] = 1 then [g, h ϕ ] and [h, g ϕ ] are central elements of ν q (G), of the same finite order dividing q. If in addition g, h are torsion elements of orders o(g), o(h), respectively, then the order of [g, h ϕ ] divides the gcd(q, o(g), o(h)). (vii) [g, g ϕ ] is central in ν q (G), for all g ∈ G; (viii) [g, h ϕ ][h, g ϕ ] is central in ν q (G); (ix) [g, g ϕ ] = 1, for all g ∈ G ; (x) If [x, g] = 1 = [x, h], then [g, h, x ϕ ] = 1 = [[g, h] ϕ , x]. Corollary 2.2. Let G be any group and let g, h be arbitrary elements in G. Then (i) [G , G ϕ ] = [G, G ϕ ]; (ii) [G , Z(G) ϕ ] = 1; (iii) If gG = hG then [g, g ϕ ] = [h, h ϕ ]; (iv) If o (x) denotes the order of a coset xG ∈ G/G , then [g, h ϕ ][h, g ϕ ] has order dividing the gcd(q, o (g), o (h)); (v) The order of [h, h ϕ ] divides the gcd(q, o (h) 2 , 2o (h)). Proof. Part (i) follows directly from Lemma 2.1 (iii) (see also [5, (7) and Lemma 2.1 (vi). For our purposes we establish the following proposition, which may have its own interest. Proposition 2.3. Let G be a nilpotent group of class 2. Then the following hold in ν q (G): (i) G centralizes [G, G ϕ ]; (ii) [G , G ϕ ] (= [G, G ϕ ]) is a central subgroup of ν q (G); (iii) Υ q (G) ( ∼ = G ⊗ q G ) is nilpotent of class at most 2. Proof. (i) follows straightforward from 2.1 (iv) and relation (2), once G has nilpotency class 2. (ii). For all g, h ∈ G and c ∈ G we have: [c, g ϕ ] h = [c, g ϕ ][c, g ϕ , h] = [c, g ϕ ][c, g, h ϕ ] (by Lemma 2.1, (i)) = [c, g ϕ ] (since [c, g] = 1, as G ≤ Z(G)) = [c, g ϕ ] h ϕ (by definition of ν q (G)). In addition, for all k ∈ G, by Lemma 2.1 (iv) and relations (2) we have that k [c,g ϕ ] = k [c,g] = (k [c,g] ) = k, since [c, g] = 1. This proves part (ii) (using the definition of ν q (G)), because [G , G ϕ ] is generated by all those [c, g ϕ ] above. (iii). That Υ q (G) is nilpotent and has nilpotency class at most 3 follows from [10, Proposition 2.7, (i)]. Now, Υ q (G) = [G, G ϕ ]G and thus, once G centralizes [G, G ϕ ], we have (Υ q (G)) = [[G, G ϕ ]G, [G, G ϕ ]G] ≤ [G, G ϕ ] [G, G] . Induction arguments can be used, together with Lemma 2.1 (i), (ii), (iv) and (v) and defining relations (6) -(7) to get: (a) [G, G ϕ ] = [G , (G ) ϕ ] (see also [5, Proposition 1.3 (i)] or [23, Theorem 3.3]); (b) [G, G] ≤ G [G , G ϕ ] ≤ [G , G ϕ ]. Consequently, (Υ q (G)) ≤ [G , G ϕ ], which by part (ii) is central in ν q (G). This com- pletes the proof. For a finitely generated abelian group A, its q-tensor square Υ q (A) can be computed by repeated applications of the following two results from [10]. Lemma 2.4. [10, Corollary 2.16] Let G = N × H be a direct product and set N = N/N N q , H = H/H H q . Then (i) Υ q (G) = Υ q (N ) × [N, H ϕ ][H, N ϕ ] × Υ q (H); (ii) [N, H ϕ ] ∼ = (N ⊗ Zq H) ∼ = [H, N ϕ ]. Lemma 2.5. [10, Theorem 3.1] Let C n (resp. C ∞ ) be the cyclic group of order n (resp. ∞), q a non-negative integer and d = gcd(n, q). Then C ∞ ⊗ q C ∞ ∼ = C ∞ × C q ; C n ⊗ q C n ∼ =      C n × C d , if d is odd; C n × C d , if d is even and either 4\|n or 4\|q; C 2n × C d/2 , otherwise. Thus, if A = r i=1 C i is a direct product of the cyclic groups C i , i = 1, . . . , r, where C i = x i , then Υ q (A) = r i=1 Υ q (C i ) × 1≤i<j≤r [C i , C ϕ j ][C j , C ϕ i ]. Here we have Υ q (C i ) = [x i , x ϕ i ], x i and [C i , C ϕ j ][C j , C ϕ i ] = [x i , x ϕ j ][x j , x ϕ i ], [x i , x ϕ j ] . Since ∆ q (A) = [a, a ϕ ] \| a ∈ A , we observe, like in [24, Proposition 3.3], that ∆ q (A) = [x i , x ϕ i ], [x j , x ϕ k ][x k , x ϕ j ] \| 1 ≤ i ≤ r, 1 ≤ j < k ≤ r and thus it does not depend on the particular set X = {x 1 , . . . , x r } of generators of A. Consequently, we can write Υ q (A) = ∆ q (A)E q X (A), where E q X (A) = x i , [x j , x ϕ k ] \| 1 ≤ i ≤ r, 1 ≤ j < k ≤ r . Remark 2.6. If x and y are commuting elements in any group G then by relations (5)-(7) and Lemma 2.1 (vi) we get xy = x y [x, y ϕ ] −( q 2 ) = x y [y, x ϕ ] −( q 2 ) = yx, and hence [x, y ϕ ] −( q 2 ) = [y, x ϕ ] −( q 2 ) . In particular, if q = 2 then [x, y ϕ ] = [y, x ϕ ]. This means for instance that in the decomposition of Υ q (A) found above, the groups [C i , C ϕ j ] and [C j , C ϕ i ] are not necessarily independent. Moreover, the identity (x n ) = ( x) n [x, x ϕ ] −( n 2 )( q 2 ) shows that the subgroups x i and [x i , x ϕ i ] of Υ q (C i ) may have non trivial intersection. Consequently, unlike the case q = 0, the subgroup E q X (A) is not necessarily a complement of ∆ q (A) (see also [5,Section 2]). Now let G be any group and write G ab = G/G . The natural projection G G ab induces an epimorphism π : ν q (G) → ν q (G ab ). We denote by π 0 the restriction of π to Υ q (G). By [10, Lemma 2.14 (iii)] we have that Ker(π 0 ) = [G , G ϕ ][G, G ϕ ] G , which reduces to [G , G ϕ ] G , by force of Corollary 2.2 (i). In addition, using relations (5) and (7), an induction argument as in the proof of the Proposition2.3 (ii) shows that (G ) ≤ [G , G ϕ ] and, consequently, Ker(π 0 ) = [G , G ϕ ]. The next Lemma extends [5, Lemma 2.1] to all q ≥ 0 (see also [24,Proposition 3.3]). We shall omit the proof. Lemma 2.7. Let q be a non negative integer and G be a group such that G ab is finitely generated. Assume that G ab is a direct product of the cyclic groups C i = x i G , for i = 1, . . . , r and set E q (G) = x i , [x j , x ϕ k ] \| 1 ≤ i ≤ r, 1 ≤ j < k ≤ r rangle [G , G ϕ ]. Then, (i) ∆ q (G) = [x i , x ϕ i ], [x j , x ϕ k ][x k , x ϕ j ] \| 1 ≤ i ≤ r, 1 ≤ j < k ≤ r ; (ii) Υ q (G) = ∆ q (G)E q (G). With the above notation, let π 1 denote the restriction of π 0 to ∆ q (G), π 1 : ∆ q (G) ∆ q (G ab ), and let N = Ker(π 1 ). Therefore, N = ∆ q (G)∩[G , G ϕ ] (= ∆ q (G) ∩ E q (G)) , a central subgroup of Υ q (G). Our next theorem generalizes, to all q ≥ 0, Proposition 2.2 in [5], which in turn improves Proposition 3.3 in [24]. Theorem 2.8. Let q ≥ 0 and assume that G ab is finitely generated. Then, with the notation of Lemma 2.7, the following hold: (i) Υ q (G)/N ∼ = ∆ q (G ab ) × (G ∧ q G); (ii) If q ≥ 1 and q is odd, then N = 1 and thus ∆ q (G) ∼ = ∆ q (G ab ) and Υ q (G) ∼ = ∆ q (G ab ) × (G ∧ q G); (iii) For q = 0 or q ≥ 2 and q even, if G ab has no element of order two or if G has a complement in G, then also N = 1, ∆ q (G) ∼ = ∆ q (G ab ) and Υ q (G) ∼ = ∆ q (G ab ) × (G ∧ q G); (iv) For q ≥ 2 and q even, if G ab has no element of order 2, then ∆ q (G) is a homocyclic abelian group of exponent q, of rank t+1 2 ; (v) If G ab is free abelian of rank t, then the conclusion of the previous item holds for all q ≥ 1, while ∆ q (G) is free abelian of rank t+1 2 if q = 0. Proof. (i): By Lemma 2.7 (ii) we have Υ q (G) N = ∆ q (G)E q (G) N ∼ = ∆ q (G) N × E q (G) N . Now, ∆ q (G)/N ∼ = ∆ q (G ab ) and E q (G)/N = E q (G)/(∆ q (G) ∩ E q (G)) ∼ = Υ q (G)/∆ q (G) ∼ = G ∧ q G. This proves (i). (ii), (iii), (iv) and (v): Suppose that the torsion subgroup of G ab is the direct product of the cyclic groups x i G of order n i , 1 ≤ i ≤ s, and let the free part of G ab be the direct product of the cyclic groups y j G , 1 ≤ j ≤ t. Thus, o (x i ) = n i and o (y j ) = ∞. Set X := {x i \| 1 ≤ i ≤ s} and Y := {y j \| 1 ≤ j ≤ t}. Then G is generated by X ∪ Y ∪ G . Using Lemma 2.1 and Corollary 2.2 (see also [24, Proposition 3.3 and Remark 5]) we find that ∆ q (G) is generated by the set ∆ X ∪ ∆ Y ∪ ∆ XY , where ∆ X = {[x i , x ϕ i ], [x j , x ϕ k ][x k , x ϕ j ] \| 1 ≤ i ≤ s, 1 ≤ j < k ≤ s}, ∆ Y = {[y j , y ϕ j ], [y k , y ϕ l ][y l , y ϕ k ] \| 1 ≤ j ≤ t, 1 ≤ k < l ≤ t}, ∆ XY = {[x i , y ϕ j ][y j , x ϕ i ] \| 1 ≤ i ≤ s, 1 ≤ j ≤ t}. Set n ik = gcd(n i , n k ). Parts (iv) and (v) of Corollary 2.2 give ([x i , x ϕ k ][x k , x ϕ i ]) n ik = 1 and ([x i , y ϕ j ][y j , x ϕ i ]) n i = 1, while [x i , x ϕ i ] n i ∈ Ker(π 0 ), ∀i, k = 1, . . . , s, i < k, ∀j = 1, . . . t. Actually, Ker(π 0 ) is generated by the set {[x i , x ϕ i ] n i \| 1 ≤ i ≤ s},i (= o (x i )) is odd, then [x i , x ϕ i ] n i = 1, while [x i , x ϕ i ] 2n i = 1 if n i is even. Consequently, N = Ker(π 0 ) is an elementary abelian 2-group of rank at most r 2 (G ab ), the 2 − rank of G ab (see also [24,Corollary 3.6]). On the other hand, we should take into account that q is involved in the upper bound found in Corollary 2.2 ( v). Thus, if q ≥ 1 and q is odd, then gcd(q, 2n i ) = gcd(q, n i ) \| n i and hence [x i , x ϕ i ] n i = 1, for all i = 1, . . . , s. Therefore N = 1 in this case, proving part (ii). It should be also clear that N = 1 if r 2 (G ab ) = 0. This proves (iii) in the case where G ab has no element of order 2. Now, if G has a complement C in G, then every g ∈ G can be written as g = xh with x ∈ C and h ∈ G . Corollary 2.2 (iii) says that [g, g ϕ ] = [x, x ϕ ] and thus ∆ q (G) = [x, x ϕ ] \| x ∈ C = ∆ q (G ab ). This completes the prof of part (iii) (see also [5,Proposition 2.2]). Finally, we observe that [x i , x ϕ i ] = 1 = [x i , y ϕ j ][y j , x ϕ i ] in the case where r 2 (G ab ) = 0 and q ≥ 2, q even. Here we have ∆ q (G) = ∆ Y and [y j , y ϕ j ] q = 1 = ([y k , y ϕ l ][y l , y ϕ k ]) q , ∀j, k, l = 1, . . . , t, k < l. That each of these t+1 2 generators has order q follows immediately from Lemma 2.5, where we found C ∞ ⊗ q C ∞ ∼ = C ∞ × C q . Part (v) follows by an analogous argument , as in part (iv); the last assertion can be also found in [24,Corollary 3.6]. The proof is complete. We state the next Lemma for easy of reference, which in a certain sense extends ideas found in [19] for the case q = 0. A proof is given in [8] for q = 0 (see also [5,Proposition 3.2] for an alternative proof for this case) and in [16] for q ≥ 1. Lemma 2.9. Let F/R be a free presentation of a group G. Then G ∧ q G ∼ = F F q /[R, F ]R q . Notice that there is a map ρ : ν q (G) −→ G, g −→ g, g ϕ −→ g and k −→ k q . Let ρ = ρ\| Υ q (G) : Υ q (G) −→ G, [g 1 , g 2 ϕ ] −→ [g 1 , g 2 ], k −→ k q . Following [10] we write θ q (G) = Ker(ρ) and µ q (G) = Ker(ρ ) = Υ q (G) ∩ θ q (G). It follows that Υ q (G)/µ q (G) ∼ = G G q . If G = F/R is a free presentation of G, then (10) H 2 (G, Z q ) ∼ = R ∩ F F q /R q [R, F ] = (G ∧ q G) ∩ M q (G), where M q (G) = R/R q [R, F ] is the q-multiplier of G. From this we obtain (see for instance [10,Theorem 2.12]): (11) µ q (G)/∆ q (G) ∼ = H 2 (G, Z q ), for all q ≥ 0. Corollary 2.10. Let F n be the free group of rank n. Then (i) For q ≥ 1, F n ⊗ q F n ∼ = C ( n+1 2 ) q × (F n ) (F n ) q . (ii) ([9, Proposition 6]) For q = 0, F n ⊗ F n ∼ = C ( n+1 2 ) ∞ × (F n ) . Proof. Since F ab n is free abelian of rank n, by Theorem 2.8 (ii), (iii) and (v), we have: Υ q (F n ) ∼ = ∆ q (F ab n ) × (F n ∧ q F n ). (i): If q ≥ 1 then ∆ q (F ab n ) ∼ = C ( n+1 2 ) q and, by Lemma 2.9 (i) with G = F n and R = 1, F n ∧ q F n ∼ = (F n ) (F n ) q . This proves (i). (ii): If q = 0 then ∆ q (F ab n ) ∼ = C ( n+1 2 ) ∞ and, again by the previous Lemma, F n ∧ F n ∼ = (F n ) . This completes the proof. Corollary 2.11. Let N n,c = F n /γ c+1 (F n ) be the free nilpotent group of class c ≥ 1 and rank n > 1. Then (i) For q ≥ 1, N n,c ⊗ q N n,c ∼ = C ( n+1 2 ) q × (F n ) (F n ) q γ c+1 (F n ) q γ c+2 (F n ) . (ii) ([6, Corollary 1.7]) For q = 0, N n,c ⊗ N n,c ∼ = C ( n+1 2 ) ∞ × (N n,c+1 ) . Proof. (i) and (ii) follow by similar arguments as above, taking into account that here we have R = γ c+1 (F n ) and thus [R, F ], as in Lemma 2.9, is precisely γ c+2 (F n ). q-Tensor Squares of Nilpotent Groups of Class 2 In this section we restrict our considerations to finitely generated nilpotent groups G of class two. We begin with a general result concerning polycyclic groups found in [10]; this generalizes to all q ≥ 0 a result due to Blyth and Morse in [4] for q = 0, which in turn extends to all polycyclic groups a similar result for finite solvable groups found in [24]. (a 1 , . . . , a n ) . Then (i) [G, G ϕ ], a subgroup of ν q (G), q ≥ 0, is generated by [G, G ϕ ] = [a i , a ϕ i ], [a i , a ϕ j ][a ϕ j , a i ], [a α i , (a ϕ j ) β ], for 1 ≤ i < j ≤ n, 1 ≤ k ≤ n , (ii) Υ q (G), a subgroup of ν q (G), q ≥ 1, is generated by Υ q (G) = [a i , a ϕ i ], [a i , a ϕ j ][a ϕ j , a i ], [a α i , (a ϕ j ) β ], (a k ), for 1 ≤ i < j ≤ n, 1 ≤ k ≤ n , where α = 1 if o(a i ) < ∞ ±1 if o(a i ) = ∞ and β = 1 if o(a j ) < ∞ ±1 if o(a j ) = ∞. (iii) ∆ q (G) is generated by the set {[a i , a ϕ i ], [a i , a ϕ j ][a ϕ j , a i ], for 1 ≤ i < j ≤ n}. Now let G be a finitely generated nilpotent group of class two and assume that G is generated by g 1 , g 2 , · · · , g n . Thus, any element g ∈ G can be written as (12) g = n i=1 g m i i 1≤j<k≤n [g j , g k ] l jk , where the exponents m i and l jk are integers. Consequently, G has the following polycyclic generating set (13) {g i , 1 ≤ i ≤ n} ∪ {[g j , g k ], 1 ≤ j < k ≤ n}. The following theorem extends a result of Bacon in [2, Theorem 3.1] (see also [3]) for all q ≥ 0. We provide a proof for the case q = 0 using the commutator approach; the general case follows straightforward from this case and Lemma 3.1 (ii), but we shall prove it in this case too, for the sake of completeness. , for all q ≥ 0; (iii) In particular, if G has finite exponent e(G) and gcd(q, e(G)) = 1 , then d(G ⊗ q G) ≤ n 2 . Proof. On assuming that G is generated by g 1 , . . . , g n then we obtain the polycyclic generating set given by (13). Thus, by Lemma 3.1 (i) we have that Υ(G) = [G, G ϕ ] is generated by the following set of elements: {[g α i , (g ϕ j ) β ] : 1 ≤ i, j ≤ n}∪ {[g α i , ([g j , g k ]ϕ) β ] : 1 ≤ i ≤ n, 1 ≤ j < k ≤ n}∪ {[[g j , g k ] β , (g ϕ i ) α ] : 1 ≤ i ≤ n, 1 ≤ j < k ≤ n}∪ {[[g r , g s ] α , ([g t , g u ] ϕ ) β ] : 1 ≤ r < s ≤ n, 1 ≤ t < u ≤ n}, where α, β ∈ {−1, 1}. Now by Lemma 2.1, parts (ii), (iii), (ix), (x), and the fact that G has class 2, we can further reduce the above set to obtain (14) { [g i , g ϕ j ] : 1 ≤ i, j ≤ n} ∪ {[g i , [g j , g k ] ϕ ] : 1 ≤ i ≤ n, 1 ≤ j < k ≤ n}. This new set has n 2 generators of the form [g i , g ϕ j ] and n(n − 1) generators of the form [g i , [g j , g i ] ϕ ]. It remains to count the generators of the form [g i , [g j , g k ] ϕ ], when i, j, k are all distincts and j < k. Now by [23,Corollary 3.2] ν(G) has nilpotency class at most 3 and thus, again [g ϕ j , g ϕ k , g i ][g ϕ k , g ϕ i , g j ][g i , g j , g ϕ k ] = 1. It then follows that [g i , [g j , g k ] ϕ ] = [g j , [g i , g k ] ϕ ][g k , [g j , g i ] ϕ ]. Therefore, d([G, G ϕ ]) ≤ 1 3 n(n 2 + 3n − 1). This proves part (i). (ii): Part (i) also proves (ii) in case q = 0, giving us the better bound for d(G ⊗ G). Thus, we shall assume q ≥ 1. Since Υ q (G) = [G, G ϕ ]G it suffices to control the number of generators of both [G, G ϕ ] and G. We already know by part (i) that [G, G ϕ ] is generated by the set {[g i , g ϕ i ] : 1 ≤ i, j ≤ n} ∪ {[g i , [g j , g k ] ϕ ] : 1 ≤ i ≤ n, 1 ≤ j < k ≤ n}. Now, by definition the subgroup G is generated by G = { g, g ∈ G}. By the defining relations (5) of ν q (G) we have gh = g ( q−1 i=1 [g, (h ϕ ) −i ] g q−1−i ) h, for all g, h ∈ G, and hence gh ≡ g h (mod [)G, G ϕ ], for all g, h ∈ G. Since every element g ∈ G has a unique expression in the form (12) and given the fact that, for every commutator [g j , g k ] ∈ G , [g j , g k ] = [g j , g ϕ k ] q ∈ [G, G ϕ ] (by relations (7)), we see in later stage that G is generated, modulo [G, G ϕ ], by the n elements g 1 , . . . , g n . Therefore, we conclude that d(Υ q (G)) ≤ 1 3 (n 3 + 3n 2 + 2n). This proves part (ii). (iii): If in particular G has finite exponent e(G) and gcd(q, e(G)) = 1, then we see by Lemma 2.1 (vi) that all generators of the forms [g i , g ϕ i ] and [g i , [g j , g k ] ϕ ] are trivial. Consequently, d(G ⊗ q G) ≤ n 2 in this case. The proof is complete. In [1] Aboughazi computed the nonabelian tensor square of the Heisenberg group H = F 2 /γ 3 (F 2 ), where F 2 denotes the free group of rank 2. There, it is found that H ⊗ H ∼ = Z 6 , thus showing that the bound in Theorem 3.2 is sharp. Later, Bacon in [2, Theorem 3.2] computed N n,2 ⊗ N n,2 for all n ≥ 2, to show that the bound is also reached for the free n−generator nilpotent group of class 2, n > 1: N n,2 ⊗ N n,2 is a free abelian group of rank 1 3 n(n 2 + 3n − 1). It is not difficult to extend these results to the q−tensor square Υ q (N n,2 ), q ≥ 1, to show that the bound in Theorem 3.2 (ii) is also attained. In fact, the next proposition is but a specialization of Corollary 2.11. We write M (G) to denote the Schur multiplier H 2 (G, Z) of G. Proposition 3.3. Let N n,2 be the free nilpotent group of rank n > 1 and class 2, N n,2 = F n /γ 3 (F n ). Then, (i) ([2, Theorem 3.2]) N n,2 ⊗ N n,2 is free abelian of rank 1 3 n(n 2 + 3n − 1). More precisely, N n,2 ⊗ N n,2 ∼ = ∆(F ab n ) × M (N n,2 ) × N n,2 . (ii) N n,2 ⊗ q N n,2 ∼ = (C q ) (( n+1 2 )+Mn(3)) × N n,2 N q n,2 , where M n (3) = 1 3 (n 3 − n) is the q−rank of γ 3 (N n,2 )/γ 3 (N n,2 ) q γ 4 (N n,2 ), according to the Witt's formula. Consequently, for q > 1 d(N n,2 ⊗ q N n,2 ) = 1 3 (n 3 + 3n 2 + 2n). Proof. As in the proof of Corollary 2.11, using Lemma 2.9 we can write G ∧ q G ∼ = F n F q n /[R, F n ]R q . In view of (10), H 2 (G, Z q ) ∼ = R ∩ F n F q n /R q [R, F n ]. Taking into account that R = γ 3 (F n ) ≤ F n F q n we have the exact sequence (15) 1 → γ 3 (F n ) γ 4 (F n )γ 3 (F n ) q → F n F q n γ 4 (F n )γ 3 (F n ) q → F n F q n γ 3 (F n ) → 1. Here we find that H 2 (N n,2 , Z q ) ∼ = γ 3 (F n )/γ 4 (F n )γ 3 (F n ) q ∼ = Z Mn(3) q , for all q ≥ 0, where, by the Witt's formula for the rank of γ r (F n )/γ r+1 (F n ), N n,c ⊗ N n,c ∼ = C ( n+1 2 ) ∞ × γ 3 (F n )/γ 4 (F n ) × γ 2 (F n )/γ 3 (F n ) ∼ = C ( n+1 2 ) ∞ × C ( n 2 ) ∞ × C 1 3 (n 3 −n) ∞ . This is the result of Bacon, that d(N n,2 ⊗ N n,2 ) = 1 3 n(n 2 + n − 1). (ii): If q > 1 then we see by the generators of N n,2 ⊗ q N n,2 found in Theorem 3.2 that the image by ρ of the subgroup g i , [g j , g ϕ k ] \| 1 ≤ i ≤ n, 1 ≤ k < j ≤ n is the subgroup N n,2 N q n,2 of N n,2 , while the subgroup [gj, g i , g ϕ k ] , where [gj, g i , g k ] is a basic commutator of γ 3 (F n )/γ 4 (F n ), is isomorphic to γ 3 (F n )/γ 4 (F n )γ 3 (F n ) q ∼ = H 2 (N n,2 , Z q ), a homocyclic abelian group of exponent q and q−rank M n (3) = 1 3 (n 3 − n). Consequently, also in this case we get that the sequence (15) splits and we then find that N n,2 ⊗ q N n,2 ∼ = C ( n+1 2 )+Mn(3) q × N n,2 N q n,2 . Therefore, d(N n,2 ⊗ q N n,2 ) = 1 3 n(n 2 + 3n + 2), thus showing that the upper bound given in Theorem 3.2 (ii) is also achieved when q > 1. Lemma 3.1. ([10, Corolary 3.6]) Let G be a polycyclic group with a polycyclic generating sequence pgs(G) = Theorem 3 . 2 . 32Let G be a nilpotent group of class two with d(G) = n, then (i) ([2, Theorem 3.1]) d([G, G ϕ ]) ≤ n(n 2 +3n−1) 3 ;(ii) d(G ⊗ q G) ≤ n(n 2 +3n+2) 3 µ denotes the Möbius function. (i): If q = 0 then the exact sequence (15) splits and thus we have, by Corollary 2.11, Produit Tensoriel du Groupe D'Heisenberg. R Aboughazi, Bull. Soc. math. France. 115Aboughazi, R., Produit Tensoriel du Groupe D'Heisenberg, Bull. Soc. math. France, 115 (1987), 95-106. On the non-abelian Tensor Square of a Nilpotent Group of Class Two. M Bacon, Glasgow Math. 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Cotas Superiores para o Expoente e o número mínimo de geradores do Quadrado q-Tensorial de Grupos Nilpotentes. Eunice C P Rodrigues, Brasilia, DF, Brazil; Brasilia-DFUniversidade de Brasília ; Universidade de BrasíliaDoctoral Thesisin PortugueseRodrigues, Eunice C. P., Cotas Superiores para o Expoente e o número mínimo de geradores do Quadrado q-Tensorial de Grupos Nilpotentes, Doctoral Thesis (in Portuguese), Universidade de Brasília, Brasilia, DF, Brazil, 2011. Available at http://repositorio.unb.br/handle/10482/8717. Departamento de Matematática, Universidade de Brasília, Brasilia-DF, 70910-900 . E-Mail Brazil, Address, [email protected] E-mail address: [email protected] Rondonópolis-MT, 85735-001 Brazil E-mail address: eunicepr@hotmail. Matemática Departamento De, Universidade Federal de Mato GrossoDepartamento de Matemática, Universidade Federal de Mato Grosso, Rondonó- polis-MT, 85735-001 Brazil E-mail address: [email protected]	[]
[ "Black hole hair in generalized scalar-tensor gravity", "Black hole hair in generalized scalar-tensor gravity" ]	[ "Thomas P Sotiriou \nSchool of Mathematical Sciences\nSchool of Physics and Astronomy\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n", "Shuang-Yong Zhou \nSISSA\nINFN\nVia Bonomea 26534136TriesteItaly\n\nSezione di Trieste\nItaly\n" ]	[ "School of Mathematical Sciences\nSchool of Physics and Astronomy\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK", "SISSA\nINFN\nVia Bonomea 26534136TriesteItaly", "Sezione di Trieste\nItaly" ]	[]	The most general action for a scalar field coupled to gravity that leads to second order field equations for both the metric and the scalar -Horndeski's theory -is considered, with the extra assumption that the scalar satisfies shift symmetry. We show that in such theories the scalar field is forced to have a nontrivial configuration in black hole spacetimes, unless one carefully tunes away a linear coupling with the Gauss-Bonnet invariant. Hence, black holes for generic theories in this class will have hair. This contradicts a recent no-hair theorem, which seems to have overlooked the presence of this coupling. PACS numbers: 04.70.Bw, 04.50.Kd In general relativity, black hole spacetimes are described by the Kerr metric, so long as they are stationary, asymptotically flat, and devoid of any matter in their surroundings [1]. Stationarity is a reasonable assumption for black holes that are thought to be quiescent as endpoints of gravitational collapse. Astrophysical black holes are certainly not asymptotically flat, but one can invoke separation of scales in order to argue that the cosmological background should not seriously affect local physics and hence the structure of black holes. Finally, black holes can also carry an electromagnetic charge in the presence of an electromagnetic field. It has been conjectured that they cannot carry any other charges, which are colloquially referred to as hair [2] [27]. The no-hair conjecture was inspired by the uniqueness theorems for black hole solutions in general relativity[4][5][6][7].Hawking has proven that black holes cannot carry scalar charge, provided that the scalar couples to the metric minimally or as described by Brans-Dicke theory[8]. This result has been generalised to standard scalar-tensor theories [9] (see also earlier work by Bekenstein with the extra assumption of spherical symmetry[10,11]).All of these proofs actually demonstrate that the scalar has to be constant in a black hole spacetime, which is a stronger statement. Indeed, in principle, the scalar could have a nontrivial configuration without the black hole carrying an extra (independent) charge. This is sometime referred to as "hair of the second kind". The distinction is important if one is interested in the number of parameters that fully characterise the spacetime. But, a nontrivial configuration of the scalar is usually enough to imply that the black hole spacetime will not be a solution to Einstein's equations in vacuum, and hence it differs from the black holes of general relativity.The known proofs do not apply to theories with more general coupling between the metric and the scalar, or derivative self-interactions of the scalar. Hence, they do not cover the most general scalar-tensor theory that leads to second-order field equations, known as Horndeski theory[12]. Restricting attention to theories with second order field equations is justified, as higher order derivative models are generically plagued by the Ostrogradski instability[13]. Models that belong to this class have lately received a lot of attention in cosmology, under the name generalised Galileons [14] (see also Ref.[15] for a recent review).If black holes have hair in these theories, they could perhaps be used to indirectly detect the presence of a scalar field. The equivalence principle dictates that the matter should couple minimally to the metric and that it should not couple to the scalar field. This implies that direct detection in matter experiments is not promising. However, a non-trivial configuration for the scalar field would lead to a black hole solution that deviates from that of general relativity. The deviation could agree with the prediction of a certain model, and, in principle, accurate modelling of the spacetime could act a probe of the coupling of the scalar field to gravity and itself. The presence of scalar hair can also have bearing on the thermodynamical aspects of black holes in scalar-tensor theories. It is, hence, quite important to understand whether black holes can have nontrivial scalar configurations in the most general scalar-tensor theory.Progress in this direction was recently made in Ref.[16]. It was argued there that vacuum, static, spherically symmetric, asymptotically flat black holes have no hair in the most general scalar-tensor theory that leads to second order field equations, provided that the scalar exhibits shift symmetry, i.e., symmetry under φ → φ + constant. The most general Lagrangian with these properties is the following[17]	10.1103/physrevlett.112.251102	[ "https://arxiv.org/pdf/1312.3622v2.pdf" ]	13,637,703	1312.3622	8457df2502f46871ca152cb4ccdedca53b3dc3f8	Black hole hair in generalized scalar-tensor gravity 11 Aug 2014 Thomas P Sotiriou School of Mathematical Sciences School of Physics and Astronomy University of Nottingham NG7 2RDUniversity Park, NottinghamUK Shuang-Yong Zhou SISSA INFN Via Bonomea 26534136TriesteItaly Sezione di Trieste Italy Black hole hair in generalized scalar-tensor gravity 11 Aug 2014 The most general action for a scalar field coupled to gravity that leads to second order field equations for both the metric and the scalar -Horndeski's theory -is considered, with the extra assumption that the scalar satisfies shift symmetry. We show that in such theories the scalar field is forced to have a nontrivial configuration in black hole spacetimes, unless one carefully tunes away a linear coupling with the Gauss-Bonnet invariant. Hence, black holes for generic theories in this class will have hair. This contradicts a recent no-hair theorem, which seems to have overlooked the presence of this coupling. PACS numbers: 04.70.Bw, 04.50.Kd In general relativity, black hole spacetimes are described by the Kerr metric, so long as they are stationary, asymptotically flat, and devoid of any matter in their surroundings [1]. Stationarity is a reasonable assumption for black holes that are thought to be quiescent as endpoints of gravitational collapse. Astrophysical black holes are certainly not asymptotically flat, but one can invoke separation of scales in order to argue that the cosmological background should not seriously affect local physics and hence the structure of black holes. Finally, black holes can also carry an electromagnetic charge in the presence of an electromagnetic field. It has been conjectured that they cannot carry any other charges, which are colloquially referred to as hair [2] [27]. The no-hair conjecture was inspired by the uniqueness theorems for black hole solutions in general relativity[4][5][6][7].Hawking has proven that black holes cannot carry scalar charge, provided that the scalar couples to the metric minimally or as described by Brans-Dicke theory[8]. This result has been generalised to standard scalar-tensor theories [9] (see also earlier work by Bekenstein with the extra assumption of spherical symmetry[10,11]).All of these proofs actually demonstrate that the scalar has to be constant in a black hole spacetime, which is a stronger statement. Indeed, in principle, the scalar could have a nontrivial configuration without the black hole carrying an extra (independent) charge. This is sometime referred to as "hair of the second kind". The distinction is important if one is interested in the number of parameters that fully characterise the spacetime. But, a nontrivial configuration of the scalar is usually enough to imply that the black hole spacetime will not be a solution to Einstein's equations in vacuum, and hence it differs from the black holes of general relativity.The known proofs do not apply to theories with more general coupling between the metric and the scalar, or derivative self-interactions of the scalar. Hence, they do not cover the most general scalar-tensor theory that leads to second-order field equations, known as Horndeski theory[12]. Restricting attention to theories with second order field equations is justified, as higher order derivative models are generically plagued by the Ostrogradski instability[13]. Models that belong to this class have lately received a lot of attention in cosmology, under the name generalised Galileons [14] (see also Ref.[15] for a recent review).If black holes have hair in these theories, they could perhaps be used to indirectly detect the presence of a scalar field. The equivalence principle dictates that the matter should couple minimally to the metric and that it should not couple to the scalar field. This implies that direct detection in matter experiments is not promising. However, a non-trivial configuration for the scalar field would lead to a black hole solution that deviates from that of general relativity. The deviation could agree with the prediction of a certain model, and, in principle, accurate modelling of the spacetime could act a probe of the coupling of the scalar field to gravity and itself. The presence of scalar hair can also have bearing on the thermodynamical aspects of black holes in scalar-tensor theories. It is, hence, quite important to understand whether black holes can have nontrivial scalar configurations in the most general scalar-tensor theory.Progress in this direction was recently made in Ref.[16]. It was argued there that vacuum, static, spherically symmetric, asymptotically flat black holes have no hair in the most general scalar-tensor theory that leads to second order field equations, provided that the scalar exhibits shift symmetry, i.e., symmetry under φ → φ + constant. The most general Lagrangian with these properties is the following[17] The most general action for a scalar field coupled to gravity that leads to second order field equations for both the metric and the scalar -Horndeski's theory -is considered, with the extra assumption that the scalar satisfies shift symmetry. We show that in such theories the scalar field is forced to have a nontrivial configuration in black hole spacetimes, unless one carefully tunes away a linear coupling with the Gauss-Bonnet invariant. Hence, black holes for generic theories in this class will have hair. This contradicts a recent no-hair theorem, which seems to have overlooked the presence of this coupling. In general relativity, black hole spacetimes are described by the Kerr metric, so long as they are stationary, asymptotically flat, and devoid of any matter in their surroundings [1]. Stationarity is a reasonable assumption for black holes that are thought to be quiescent as endpoints of gravitational collapse. Astrophysical black holes are certainly not asymptotically flat, but one can invoke separation of scales in order to argue that the cosmological background should not seriously affect local physics and hence the structure of black holes. Finally, black holes can also carry an electromagnetic charge in the presence of an electromagnetic field. It has been conjectured that they cannot carry any other charges, which are colloquially referred to as hair [2][27]. The no-hair conjecture was inspired by the uniqueness theorems for black hole solutions in general relativity [4][5][6][7]. Hawking has proven that black holes cannot carry scalar charge, provided that the scalar couples to the metric minimally or as described by Brans-Dicke theory [8]. This result has been generalised to standard scalar-tensor theories [9] (see also earlier work by Bekenstein with the extra assumption of spherical symmetry [10,11]). All of these proofs actually demonstrate that the scalar has to be constant in a black hole spacetime, which is a stronger statement. Indeed, in principle, the scalar could have a nontrivial configuration without the black hole carrying an extra (independent) charge. This is sometime referred to as "hair of the second kind". The distinction is important if one is interested in the number of parameters that fully characterise the spacetime. But, a nontrivial configuration of the scalar is usually enough to imply that the black hole spacetime will not be a solution to Einstein's equations in vacuum, and hence it differs from the black holes of general relativity. The known proofs do not apply to theories with more general coupling between the metric and the scalar, or derivative self-interactions of the scalar. Hence, they do not cover the most general scalar-tensor theory that leads to second-order field equations, known as Horndeski theory [12]. Restricting attention to theories with second order field equations is justified, as higher order derivative models are generically plagued by the Ostrogradski instability [13]. Models that belong to this class have lately received a lot of attention in cosmology, under the name generalised Galileons [14] (see also Ref. [15] for a recent review). If black holes have hair in these theories, they could perhaps be used to indirectly detect the presence of a scalar field. The equivalence principle dictates that the matter should couple minimally to the metric and that it should not couple to the scalar field. This implies that direct detection in matter experiments is not promising. However, a non-trivial configuration for the scalar field would lead to a black hole solution that deviates from that of general relativity. The deviation could agree with the prediction of a certain model, and, in principle, accurate modelling of the spacetime could act a probe of the coupling of the scalar field to gravity and itself. The presence of scalar hair can also have bearing on the thermodynamical aspects of black holes in scalar-tensor theories. It is, hence, quite important to understand whether black holes can have nontrivial scalar configurations in the most general scalar-tensor theory. Progress in this direction was recently made in Ref. [16]. It was argued there that vacuum, static, spherically symmetric, asymptotically flat black holes have no hair in the most general scalar-tensor theory that leads to second order field equations, provided that the scalar exhibits shift symmetry, i.e., symmetry under φ → φ + constant. The most general Lagrangian with these properties is the following [17] L = K(X) − G 3 (X) φ + G 4 (X)R +G 4X ( φ) 2 − (∇ µ ∇ ν φ) 2 +G 5 (X)G µν ∇ µ ∇ ν φ − G 5X 6 ( φ) 3 −3 ( φ) (∇ µ ∇ ν φ) 2 + 2 (∇ µ ∇ ν φ) 3 ,(1) where K, G 3 , G 4 , and G 5 are generic functions of X := −∂ µ φ∂ µ φ/2, G iX ≡ ∂G i /∂X, ∇ µ is the co- variant derivative associated with the metric g µν , ≡ ∇ µ ∇ µ , (∇ µ ∇ ν φ) 2 ≡ ∇ µ ∇ ν φ∇ ν ∇ µ φ, (∇ µ ∇ ν φ) 3 ≡ ∇ µ ∇ ρ φ∇ ρ ∇ ν φ∇ ν ∇ µ φ, and R and G µν are the corresponding Ricci scalar and Einstein tensor respectively. The class of scalar-tensor theories in which the scalar enjoys shift symmetry is an interesting one, as the scalar is protected from acquiring a mass by radiative corrections. In what follows we will briefly review the no-hair proof of Ref. [16] and we will show that it can be straightforwardly extended to slowly rotating black holes. However, we will also scrutinise its assumptions and we will uncover a hidden assumption that is not generically satisfied by the Lagrangian in Eq. (1), unless one fine-tunes away a certain combination of terms. In fact, generic theories of this type have hairy black hole solutions. The equation of motion of the scalar in a theory described by Eq. (1) can be written as a conservation equation for the Noether current J µ associated with the shift symmetry φ → φ+constant ∇ µ J µ = 0 .(2) Assuming that the metric is static and spherically symmetric, one can make the ansatz ds 2 = −f (ρ)dt 2 + f (ρ) −1 dρ 2 + r 2 (ρ)dΩ 2(3) without loss of generality. The exact form of J µ will be discussed shortly. The proof laid out in Ref. [16] can be split into four steps. In the first step one argues that, if the scalar respects the symmetries of the metric, so that φ = φ(ρ), then the only non-vanishing component of J µ in this coordinate system should be J ρ . The angular components have to vanish because of spherical symmetry, and the J t component has to vanish because otherwise it would select a preferred time direction. The second step of the proof is to show that J ρ has to vanish on the horizon of a black hole. The Killing vector associated with time translations should become null at the horizon and in this coordinate system its norm is equal to f . So, f should vanish at the horizon. If J µ J µ = (J ρ ) 2 /f is to remain finite, then J ρ must be zero at the horizon. The third step involves Eq. (2), which can now be trivially integrated to give r 2 (ρ)J ρ = constant. But, r 2 remains finite at the horizon as a measure of the area of constant-ρ spheres. This implies that J ρ has to vanish everywhere. The fourth and final step is to argue that J ρ = 0 implies φ =constant, and, therefore, the metric will have to satisfy Einstein's equations in vacuum (assuming that G 4 (0) = 1). This last step is the trickiest one, as it relies on the actual dependence of the current on φ and its derivatives. It is argued in Ref. [16] that J ρ should be of the form J ρ = f ∂ ρ φF (∂ ρ φ ; g, ∂ ρ g, ∂ ρ ∂ ρ g) ,(4) where F is some unspecified function. It is then claimed that F will asymptote to a non-zero constant at spatial infinity if one imposes the minimal requirement that the theory will have a standard canonical kinetic term in the weak field regime. Asymptotic flatness requires f → 1 and φ ′ → 0 at infinity. But if one then tries to go to some smaller radius continuously, F and f should remain nonzero, which implies that φ ′ has to vanish everywhere. It is this last step of the proof that we will contest and, in particular, the functional dependence of J µ on φ and its derivatives. If the scalar respects the symmetries of the metric, then φ = φ(r). Adopting a more conventional coordinate system with r as the areal radius, the metric can take the form ds 2 = −R(r)dt 2 + S(r)dρ 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) . (5) Using this ansatz one can get the explicit form of the Noether current associated with shift symmetry: J r = − φ ′ S K X + rφ ′2 R ′ + 4Rφ ′2 2rRS 2 G 3X + 2Rφ ′ − 2RSφ ′ + 2rφ ′ R ′ r 2 RS 2 G 4X − 2Rφ ′3 + 2r φ ′3 R ′ r 2 RS 3 G 4XX + S φ ′2 R ′ − 3φ ′2 R ′ 2r 2 RS 3 G 5X + φ ′4 R ′ 2r 2 RS 4 G 5XX ,(6) where a prime denotes differentiation with respect to r. Every term does appear to depend at least linearly on φ ′ , as required in Ref. [16]. Additionally, assuming that K has a piece linear in X so that in the weak field limit the standard canonical kinetic term is present in the action, the current does seem to be asymptotically proportional to a constant times φ ′ : As r → ∞, asymptotic flatness requires that R, S → 1 and R ′ , S ′ → 0 and the terms that contain G i appear to vanish. So, all of the requirements on Ref. [16] seem to be justified. One potential loophole could be to consider theories where G i or their derivatives with respect to X have poles at X → 0, as X = −φ ′2 /2. However, such theories will not, in general, admit solutions in which φ = constant everywhere, as this would make the current diverge. As such, they do not fall under the purview of the proof in the first place. Moreover, in general, such theories would be unlikely to admit Lorentz-symmetric vacua, as the scalar would always be forced to be in a nontrivial configuration. There is an exception, though: suppose that the G i and their derivatives are such, so that they contain exactly the right negative powers of X in order for J r to not have a pole at X = 0 but instead have a piece that is φ-independent. In order to show that this is possible, it is actually easier to go back to the action. What we are requesting is that the field equation of the scalar contains a term that does not depend on the scalar itself. The corresponding term in the Lagrangian should then be linear in the scalar, i.e. of the form φA[g] up to a total divergence, where A[g] is a generally covariant scalar constructed from the metric and its derivatives. On the other hand, shift symmetry implies that A itself should be a total divergence. We also want the term φA in the Lagrangian to lead to a contribution to the field equations with no more than second order derivatives when varied with respect to both the scalar and the metric. There is only one choice that actually satisfies all requirements: A = G ≡ R µνλκ R µνλκ − 4R µν R µν + R 2 , i.e. φ has to have a linear coupling with the Gauss-Bonnet invariant. Indeed, consider the theory S = M 2 p 2 d 4 x √ −g R − 1 2 ∂ µ φ∂ µ φ + αφG ,(7) where α is a coupling constant and M p is the reduced Planck mass. Variation with respect to φ yields φ + αG = ∇ µ (∇ µ φ + αḠ µ ) = 0 (8) whereḠ µ is implicitly defined by G = ∇ µḠ µ . G vanishes only in flat space, which implies that φ cannot be constant everywhere for any other spacetime, including black holes. Although unlikely, it is not a priori inconceivable that black hole solutions do not exist at all in this model. This is not the case and we will provide explicit black hole solutions for this action elsewhere [17]. As a preview, we consider a perturbative treatment in the dimensionless parameterα ≡ α/l 2 , where l is the characteristic length of the system in question, e.g., the radius of the black hole horizon. Assumingα ≪ 1 (which is a reasonable assumption unless one is considering microscopic black holes) one could look for solutions that are perturbatively close to the Schwarzschild solution. At zeroth order the scalar would then be constant. This implies that the φG term will only start contributing to the field equations of the metric at order O(α 2 ). Hence, to O(α) the metric will be Schwarzschild. For the scalar, instead, one can solve eq. (8) to O(α) and obtain φ ′ = α 16m 2 − Cr 3 r 4 (r − 2m)(9) where m = l/2 is the mass of a black hole and C is an integration constant. For φ to be regular on the black hole horizon one must impose C = 2/m. This yields φ ′ = − 2α m (r 2 + 2mr + 4m 2 ) r 4 = − 8αm r 4 (r 2 + 2mr + 4m 2 )(10) Two remarkable features of the solution are already present at O(α): i) even though φ has a non-trivial profile it does not lead to an independent charge because of the regularity condition on the horizon, so the solutions will have hair of the "second kind"; ii) for fixed α the solution diverges as m → 0. The expansion parameter is in fact α ∝ α/m 2 and, hence, nonperturbative effects will be important in this regime. A more detailed analysis of these features and the full perturbative and non-perturbative solutions will be presented in Ref. [17]. The fact that the scalar field is obliged to have a nontrivial configuration in black hole spacetimes constitutes a counter-example to the statement that the most general shift-symmetric scalar-tensor theory that leads to second order field equations cannot have hairy solutions. Indeed, the theory (7) fits comfortably in the initial Lagrangian given in Eq. (1). One simply has to choose K = M 2 p X/2, G 3 = 0, G 4 = M 2 p /2, and G 5 = −2M 2 p α ln \|X\| [18]. It is straightforward to check that, for these choices, the G 5 -related terms in J r in eq. (6) become φ-independent, without the current (or any other equation of the theory) becoming divergent as φ → constant. It is crucial to point out that one does not need to restrict oneself to that choice in order to have hairy black holes. In fact, for any choice of K and G i , one could write G 5 = −2M 2 p α ln \|X\| +G 5 (X) .(11) Additionally, the coupling between φ and G cannot be done away with by going to another conformal frame, as is the case for a coupling of the type φR. Only when α is tuned to zero would φ = constant solutions be admissible. In other words, one could add to the action (7) virtually any other term that is shift symmetric and leads to a second order contribution to the field equations and the resulting theory would evade the no-hair theorem of Ref. [16]. From a classical perspective one can always choose to set α = 0. But if one is thinking of these theories as effective field theories, then one would need a symmetry that would protect α from receiving radiative corrections. For a real scalar, there are not many choices of internal symmetries. Given that the corresponding term is odd in copies of φ, one could invoke symmetry under φ → −φ. This would, however, reduce the Lagrangian of Eq. (1), and thus the applicable theory space of the no-hair theorem of Ref. [16], significantly: L = K(X) + G 4 (X)R + G 4X ( φ) 2 − (∇ µ ∇ ν φ) 2 .(12) One the other hand, it is straightforward to extend the no-hair argument of Ref. [16] to slowly rotating solutions, when it is valid in spherical symmetry. The most general stationary, axially symmetric, slowly rotating solution can take the form [19] ds 2 = −R(r)dt 2 + S(r)dρ 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) +ǫr 2 sin 2 θ Ω(r, θ)dtdϕ + O(ǫ 2 ) ,(13) where R(r) and S(r) correspond to the spherically symmetric solution, Ω(r, θ) is a function to be determined, and ǫ is the bookkeeping parameter for the slow rotation. The key argument for arriving at this metric is that the system should be invariant under reversal of the direction of rotation together with either t → −t or ϕ → −ϕ. Let us now apply the same requirement to the scalar field φ. Assuming it respects the symmetries of the metric in our coordinate system, the scalar field will not depend on t or ϕ to all orders, i.e. φ = φ(r, θ). But then the scalar cannot receive a correction which is linear in the rotation, as the linear correction would not be invariant under the combined operation mentioned above. Given that φ =constant in the spherical case, we will then have φ = constant+O(ǫ 2 ) in the slowly rotating case. The metric will then satisfy Einstein's equation to the same order. Therefore, slowly rotating black holes cannot have scalar hair. This extension to the proof of Ref. [16] is valid when a perturbative treatment in the rotation is applicable. It is a stronger result, in the sense that it demonstrates that moderate rotation cannot endow the black hole with scalar hair. Additionally, this simple argument applies to virtually any gravity theory with scalar fields, as long as the spherically symmetric solutions have constant profiles for the scalars. In summary, we have shown that in generalised scalartensor theories that are shift-symmetric and lead to second order equations the scalar field will have a nontrivial configuration in any spacetime other than flat, unless the linear coupling between the scalar and the Gauss-Bonnet invariant is suppressed. In the absence of a symmetry justifying such suppression, black holes will be endowed with scalar hair. On the other hand, we have also argued that, when it is valid to assume that static, spherically symmetric black holes will have no hair, their slowly rotating counterparts will not have hair either. Some comments are in order before closing. Firstly, we use the term "hair" loosely, to mean that the scalar has a nontrivial configuration in the black hole spacetime. This does not necessarily imply that the black hole has to carry some independent scalar charge (indeed it will not in the case of the action (7) [17]). It is, however, enough to argue that the black hole will be different than its general relativity counterpart. Secondly, our attention has been focussed on shift-symmetric theories because in specific examples of scalar-tensor theories where the scalar does not exhibit shift symmetry black holes with hair are already known, see for example Ref. [20]. It is also worth mentioning that one could contest two more of the assumptions of any no-hair theorem for scalar fields. The first one is that the scalar has to respect the symmetries of the metric. This might be particularly relevant in the context of the shift-symmetric theories considered here, because the scalar field appears in the field equations only through its derivatives. Hence, if one is interested in a spacetime where L ξ g µν = 0, where L ξ is the Lie derivative along the generator of the symmetry ξ, it suffices to impose L ξ ∇ µ φ = 0. This is a weaker condition than imposing that L ξ φ = 0, as is usually done. 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Hairy black hole solutions do exist in Einstein-Yang-Mills theory, see Ref. references therein[27] Hairy black hole solutions do exist in Einstein-Yang- Mills theory, see Ref. [3] for a review and references therein.	[]
[ "Distributed MIN-MAX Optimization Application to Time-optimal Consensus: An Alternating Projection Approach", "Distributed MIN-MAX Optimization Application to Time-optimal Consensus: An Alternating Projection Approach" ]	[ "Hu Chunhe \nNational Key Laboratory of Science and Technology on Aircraft Control\nBeihang University (BUAA)\n100191, 100191Beijing, BeijingChina, China\n", "Chen Zongji \nNational Key Laboratory of Science and Technology on Aircraft Control\nBeihang University (BUAA)\n100191, 100191Beijing, BeijingChina, China\n" ]	[ "National Key Laboratory of Science and Technology on Aircraft Control\nBeihang University (BUAA)\n100191, 100191Beijing, BeijingChina, China", "National Key Laboratory of Science and Technology on Aircraft Control\nBeihang University (BUAA)\n100191, 100191Beijing, BeijingChina, China" ]	[]	In this paper, we proposed an alternating projection based algorithm to solve a class of distributed MIN-MAX convex optimization problems. We firstly transform this MIN-MAX problem into the problem of searching for the minimum distance between some hyper-plane and the intersection of the epigraphs of convex functions. The Bregmans alternating method is employed in our algorithm to achieve the distance by iteratively projecting onto the hyper-plane and the intersection. The projection onto the intersection is obtained by cyclic Dykstras projection method. We further apply our algorithm to the minimum time multi-agent consensus problem. The attainable states set for the agent can be transformed into the epigraph of some convex functions, and the search for time-optimal state for consensus satisfies the MIN-MAX problem formulation. Finally, the numerous simulation proves the validity of our algorithm.	10.2514/6.2015-2006	[ "https://arxiv.org/pdf/1406.2459v1.pdf" ]	2,946,038	1406.2459	004ed09b20ccd18f2c87e64eb12ecbb8b8f460ae	Distributed MIN-MAX Optimization Application to Time-optimal Consensus: An Alternating Projection Approach 10 Jun 2014 Hu Chunhe National Key Laboratory of Science and Technology on Aircraft Control Beihang University (BUAA) 100191, 100191Beijing, BeijingChina, China Chen Zongji National Key Laboratory of Science and Technology on Aircraft Control Beihang University (BUAA) 100191, 100191Beijing, BeijingChina, China Distributed MIN-MAX Optimization Application to Time-optimal Consensus: An Alternating Projection Approach 10 Jun 2014 In this paper, we proposed an alternating projection based algorithm to solve a class of distributed MIN-MAX convex optimization problems. We firstly transform this MIN-MAX problem into the problem of searching for the minimum distance between some hyper-plane and the intersection of the epigraphs of convex functions. The Bregmans alternating method is employed in our algorithm to achieve the distance by iteratively projecting onto the hyper-plane and the intersection. The projection onto the intersection is obtained by cyclic Dykstras projection method. We further apply our algorithm to the minimum time multi-agent consensus problem. The attainable states set for the agent can be transformed into the epigraph of some convex functions, and the search for time-optimal state for consensus satisfies the MIN-MAX problem formulation. Finally, the numerous simulation proves the validity of our algorithm. I. Introduction The distributed convex optimization studies the problem that all agents work cooperatively to minimize some optimal function to agents' local functions. Most existing works concentrate on minimizing the sum of all local function in 1, 2, 3, 4. However, MIN-MAX optimization that can be widely applied to portfolio optimization, control system design, engineering design 5 also attract much more attentions as showed in 6, 5, 7. In this paper, we propose a cyclic alternating projection based algorithm for distributed convex optimization problem. Alternating algorithm is widely applied to optimal approximation, e.g., solving linear system, 8 linear programming, 8 signal processing 9 and even Sudoku puzzle. 8 Through the geometric interpretation, we show that the original problem can be reproduced as the problem of searching the minimum distance between the hyper-plane and the intersection of all epigraphs of local functions f i . In our algorithm, we utilize Bregman's alternating projection to obtain the distance. During the procedure, the metric projection onto the intersection is necessary, so we employ another projection algorithm−Dykstra's alternating projection as the intermediate procedure. We show that in our algorithm, agents can have the consensus on the point that achieves the minimum distance with only the information from neighbors on a cyclic interaction topology. Thus, the optimal solution of the original problem is also achieved. Further, we apply our algorithm to the time-optimal consensus problem. In the past decades, a large amount of attention has been devoted to coordinate control of multi-agent systems. 10,11,12 The principal object of coordinated control is to allow the multi-agent work together and coordinate their behaviors in a cooperative fashion to achieve a common goal efficiently. Multi-agent coordination control consists of widespread research fields, including mission assignment, formation control, rendezvous control, consensus and distributed estimation, etc. The consensus problem as the fundamental problem has received greatly development. In contrast to asymptotic, to achieve the consensus with less time cost becomes attractive and reality for practical implementation. Nowadays, numerous works solve the finite time consensus instead. Though the agents can achieve consensus within finite steps through the vanishing speed 13 or the Lyapunov function derivative larger than some positive term, 14 the time cost is still not optimal especially with input saturation constraints. To achieve consensus with minimum time cost, 15 proposed a minimal polynomial based observer to the consensus point to ensure time-optimal. Different from the work mentioned above, we directly search the time-optimal consensus state according to the ability of each agent and the agents move towards the state with optimal control. We show that if let the attainable states set, the function to time, as the local functions for the agents, the time-optimal consensus problem is the (see in convex optimization)distributed convex optimization problem. This paper is organized as follows. In Section II, the problem formulation is proposed. Section III presents some preliminary on the epigraph and the alternating projection methods. The geometric interpretation on the problem and the MIN-MAX distributed convex optimization algorithm are proposed in Section IV. The application to the time-optimal consensus problem is provided as the proof of algorithm's efficiency in Section V. Finally in Section V, conclusions are provided. A. Problem Formulation In this paper, we consider the distributed multi-agent MIN-MAX convex optimization problems which is presented by min x∈X max i∈Z+ f i (x) ,(1) where f i are convex functions that can be only known by agent i and share the same domain, X ⊆ R n is a closed and convex set known by all agents. We assume the problem (1) is well-posed such that x * achieves the minimum and the feasible solution is finite, i.e. min max f i (x) < ∞. To achieve the overall optimality of the problem, the agents should work cooperatively. The difficulty comes in the distributed formulation that every agent can only access to their own functions and communicate with neighborhood according to the interaction topology. Therefore, agents execute their own local algorithms with only limited knowledge. We make further assumption on the communication topology: Assumption 1. The communication between agents is limited in a cyclic digraph interaction topology. Without loss of generality, the interaction sequence is according to the number assignment to the agent, i.e., agent i receiving information from i − 1 ,i ∈ 2, 3, . . . , N and agent 1 receiving from N . II. Preliminary Before proposing our algorithm, the several background information are presented as followed. A. Epigraph 16 The epigraph of f : R n → R is defined as epi f = {(x, t) \|x ∈ domf, f (x) ≤ t} ,(2) which is a subset of R n+1 . A function is convex if and only if its epigraph is a convex set. Moreover, in terms of epigraphs, the point-wise maximum of convex functions corresponds to the intersection of epigraphs, which is also convex, we have epi max i f i (x) = i epi f i (x) .(3) B. Alternating projection algorithm Alternating projection algorithm is a type of geometric optimization method. Through iteratively orthogonally projecting onto finite number of Hilbert spaces successively in cyclic setting, the limit to the projection sequence provides an approximation of the initial point to those spaces. Bregman's alternating projection, known as Bregman's algorithm or Bregman's method designed for closed convex sets is always used to obtain a point in the intersection of convex sets. In considering two convex sets without intersection, Bregman's algorithm achieve the distance between the two sets. 17 Following theorem provides detailed descriptions. Assume there are two convex sets A, B ⊆ R n , and P A (·) , P B (·) denote projection on A and B, respectively. We have the following theorem on above sequences: Theorem 1. 10 Let A, B ⊆ R n be closed convex sets and {a n } ∞ n=1 ,{b n } ∞ n=1 be the sequences generated by alternating projection onto A and B from any intimal point x 0 ∈ R n : a n = P A (b n−1 ) , b n = P B (a n ) , a 1 = P A (x 0 ) . (4) 1. If A B = ∅, a n , b n → x * ∈ A B. (5) 2. if A B = ∅, a n → a * ∈ A, b n → b * ∈ B,(6) where a * − b * = dist(A, B). Ordinary alternating projection can only achieve some point arbitrarily on the intersection but not the orthogonal projection, so we employ another variant projection algorithm−Dykstra's alternating projection. This method is usually employed to the problem minimize x∈R x − r 2 subject to x ∈ n i=1 A i ,(7) which provides the best approximations to the sets. Recently, Dykstra's algorithm has been extended to solve least-squares, 18 convex optimization, 19 etc.. Dykstra's alternating projection implements correction at each projection to Bregman's method by subtracting the variable, i.e., increment. Following theorem provides detailed descriptions. Theorem 2. 10 Let A 1 , A 2 , . . . A n ⊆ R n be the closed convex sets with nonempty intersection. Given x ∈ R n iterate by x i n := P Ai x i−1 n − I i n−1 , I i n := x i n − x i−1 n − I i n−1 , x 0 n := x r n−1 ,(8) with initial values x 0 1 := x, I i 0 := 0 then x n → P n i=1 Ai (x) .(9) III. MIN-MAX distributed convex optimization algorithm In this section, we are going to interpret the original problem (1) geometrically as the distance between some hyper-plane and the intersection of all epigraphs of f i . To solve this geometric problem, we proposed our distributed alternating projection based algorithm. The proof of our algorithm guarantees the optimality of the solution. A. Geometric interpretation of the MIN-MAX Problem From the definition of epigraph in (2), the function f can be regarded as the lower boundary of its epigraph, min x∈X f (x) = min x∈X epi f (x) 0 n×1 1 .(10) Therefore, substitute the point-wise maximum max (10), i∈Z+ f i (x) intomin x∈X max i∈Z+ f i (x) = min x∈X epi max i∈Z+ f i (x) 0 n×1 1 .(11) Further, since max i∈Z+ f i (x) is the point-wise maximum of finite number of convex functions f i , apply (3) to (11) min x∈X max i∈Z+ f i (x) = min x∈X epi f i (x) 0 n×1 1 .(12) Then the original problem (1) is equivalent to min x∈X epi f i (x) 0 n×1 1(13) The geometric problem (13) can be regarded as the problem of searching the lowest point of the intersection epi f i (x). We notice that the supporting plane to the convex set epi f i (x) at the lowest point (x * , min epi f i (x * )) is (x, t) \|t = min epi f i (x * ) 0 n×1 1 ,(14) where x * also achieves the feasible solution to (1). Therefore, the point that achieves the minimum distance between the convex set epi f i (x) to any hyper-plane below and parallel to the supporting plane is exactly (x * , min epi f i (x * )). Now, we are ready to express the original problem (1) equivalently as, minimize dist (∩epif i (x) , {(x, t) \|t = t min }) subject to t min ≤ min epi f i (x * ) 0 n×1 1 .(15) If we can find the point that achieves the minimum distance, the feasible solution is also obtained. Remark 1. If the the solution to (13) is greater than zero, it is exactly the minimum distance between the epigraphs intersection epi f i (x) and the zero-time plane {(x, t) \|t = 0}. B. MIN-MAX distributed convex optimization algorithm Recall the discussions in section II that Bregmans method can obtain the distance of two convex sets. Since both the intersection and the hyper-plane are convex sets, we can directly apply Bregman's method to (15). However, the procedure in that method requires the projection onto the intersection epi f i (x), which is difficult to obtain immediately especially in the distributed setting. To achieve this goal, our distributed algorithm brings in cyclic Dykstras projection method as the intermediate part in Bregmans method which iteratively projects onto the intersection and the hyper-plane. According to Theorem 1 and 2, the algorithm for each agent performance under Assumption 1 is proposed in details with following three steps: Initialization: The hyper-plane is firstly determined by choosing a real number t min small enough. Agent i maintains its own guess on the point (x i , t i ) that achieves the minimum of the distance in (15) and the increment I i n mentioned in (8). The flag f lag called increment reset symbol that indicates whether the increment preserves or resets to zero passes over the network. Without loss of generality, Agent 1 remembers its previous guess, and after comparing with the current guess it can decide the algorithm stopping time. The communication topology is cyclic and the order is determined by the indexes assigned to the agents. 1. Agent i receives the guess of the point (x, t) i−1 n − I i n−1 and the increment reset symbol from previous agent i − 1. Agent i preforms the procedure described in Theorem 2, i.e., projecting (x, t) i−1 n − I i n−1 with the increment onto its own function epigraph epi f i (x) to get its guess (x, t) i n , and updating the increment whether reset or preserve according to the symbol, (x, t) i n = P Ai (x, t) i−1 n − I i n−1 ,(16)I i n =    (x, t) i n − (x, t) i−1 n − I i n−1 f lag = 0 0 f lag = 1 .(17) 2. According to the cyclic interaction topology,agent i passes the projection (x, t) i n and the increment reset symbol f lag to the next agent i + 1. 3. During agent 1's turn in each iteration cycle, agent 1 compares its new updated projection with its previous guess. If the error between them is acceptable, agent 1 resets the increment to 1(reset), and projects the new updated projection onto the hyper-plane {(x, t) \|t = t min }, otherwise Agent 1 receives guess from Agent N and repeats step 1, e = x 1 n − x 1 n−1 (18) f lag = 1 e < err 0 e ≥ err(19) The first two steps are the implementation of Dykstras algorithm. In step 3, once the projection of the intersection of all convex functions epigraph is obtained that is to say the error between each iteration is small enough, we implement Bregmans algorithm by projecting the result of Dykstras algorithm onto the hyper-plane. Taken the projection on the hyper-plane as the new guess of the feasible solution, the agents iteratively perform step 1-3, and finally obtained the feasible solution to problem (15). The correctness of algorithm to problem can be provided by following theorem: Theorem 3. If the functions f i (x) are convex and the hyper-plane satisfies (15), the convex set epi f i (x) has non-intersection with the hyper-plane. Give x iterate by (x p , t p ) = P epifi (x n−1 , t n−1 ) ,(20)(x n , t n ) = P s (x n , t n ) = (x p , t) ,(21) where P epifi ( ) is obtained by the procedure in theorem 2. Then x n → arg min x max f i .(22) Proof. Since epi f i (x) and the hyper-plane are convex sets, it can be immediately derived from the definition that they are separated. Apparently, (20) and (21) the implementation of theorem 1, and the projection in (21) is obtained from Theorem 2. Consequently, (x n , t n ) approaches the point that achieves the minimum distance between the intersection and the hyper-plane. As mentioned in previous subsection, the first part of that point is the feasible solution to the problem (1), therefore the limit of x n satisfies (22). IV. Application to the minimum time consensus problem The multi-agent consensus problem has attracted great attentions during the last decade, and achieved remarkable development and success. The consensus problem concentrates on the distributed negotiation between agents that can finally achieve consensus on some coordinated variables. With this fundamental concept, numerous problems can be treated by consensus such as formation, rendezvous, alignment problems, etc. However, the negotiation result and time consumption is unpredicted before the process, and influenced by communication topology and their initial state of coordinated variables. With this character, we are interested in how to achieve the minimum time consensus with admissible control input, namely timeoptimal consensus problem. Actually, if without the bounded control input constraint, the consensus can be achieved within any limited time. We firstly present this problem as an MIN-MAX distributed optimization problem, and show the consensus problems with first order agents or second order agents with zero velocity constraints to the initial and final state are can be treated as Problem 1 which can be solved by our algorithm. After that, demonstrations are provided to test the validity of our algorithm. Let the minimum reachable time be the function f i of the states in the state-space for agent i, and then the minimum reachable time for all agents is the maximum of all f i , i.e., max i f i (x) .(23) The minimum time for all agents consensus becomes min x max i f i (x) ,(24) where the state achieves above minimum is the corresponding time-optimal consensus state. If the functions are convex, the minimum time consensus problem is exactly Problem (1). The attainable set for any linear model agents with admissible control input is a continues function Ω (t), and for specific time instance t the set is closed, bounded and strictly convex. 20 Apparently, the attainable set is the epigraph of f i (t), therefore the time-optimal consensus problem can be also interpreted according to (15) as the minimizing the distance between the intersection of attainable sets and the hyper-plane. The following part will analyse the first-order and second-order systems separately. A. First order systems Consider following first order system model x = u, − u max ≤ u ≤ u max .(25) Agent i starts from the initial state x 0 to x 1 with the minimum reach-time x 0 − x 1 2 u max .(26) Its attainable state set is provided as, (x, t) \| x − x 0 u max ≤ t , t ≥ 0 ,(27) which forms a cone in state-time space. Since cones are convex sets, we can directly implement our algorithms (16)(17)(18)(19). From any initial state, the agents apply the algorithm to calculate the time-optimal state for consensus, and form their own optimal control, normally saturation control, towards the consensus state. Since the result is quite simple such that it can be covered by following subsection, and due to the page limitation, the simulation for first order agents is eliminated. B. Second-order Systems Consider the second order system model ẋ 1 = x 2 x 2 = u , −u max ≤ u ≤ u max(28) Agent i starts from an arbitrary initial states (x 11 , x 21 ) to the arbitrary states ( x 12 , x 22 ). The time optimal control should be bang-bang strategy which is to execute a max positive (negative) input for a period of time and then to reverse the input for another period of time. The control law is provided by following equation: u = sgn x 12 − x 1 − 1 2 (x 2 + x 22 ) x 2 − x 22 u max(29) The optimal time-cost is determined by the relation between initial and final state.      [t + (x 22 + x 21 )] 2 = 4 (x 12 − x 11 ) + 2x 2 21 + 2x 2 22 , x 12 − x 11 − 1 2 (x 21 + x 22 ) x 21 − x 22 < 0 [t − (x 21 + x 22 )] 2 = −4 (x 12 − x 11 ) + 2x 2 21 + 2x 2 22 , x 12 − x 11 − 1 2 (x 21 + x 22 ) x 21 − x 22 > 0(30) The attainable states set is presented as, {(x, t) \|[t + (x 22 + x 21 )] 2 ≥ 4 (x 12 − x 11 ) + 2x 2 21 + 2x 2 22 , x 12 − x 11 − 1 2 (x 21 + x 22 ) x 21 − x 22 < 0 [t − (x 21 + x 22 )] 2 ≥ −4 (x 12 − x 11 ) + 2x 2 21 + 2x 2 22 , x 12 − x 11 − 1 2 (x 21 + x 22 ) x 21 − x 22 > 0} (31) Unfortunately, the epigraph or attainable set (31) is not convex. Even when the problem has been simplified such that the agents have zero initial and final velocities, i.e. x 21 = x 22 = 0, the epigraph composed by two mirror symmetry half-parabola, t 2 ≥ 4 x 12 − x 11(32) is sill not convex, and we could not directly implement our algorithm. But we notice that if we apply following quadratic transformation to the time s = t 2 ,(33) the attainable state set becomes a convex function to time square. We assume that each agent with initial state x i , and their attainable sets s i is presented as s i ≥ 4 x − x i ,(34) where s i is the square of real time t. Therefore, we can substitute the new epigraph (34) into (15). Further, since the consensus time larger than zero, the hyper-plane can be chosen as the zero time plane {(x, t) \|t = 0}. Consider four agents, whose initial states are x 1 = (−3.542884, 0), x 2 = (3.001152, 0), x 3 = (6.924106, 0), x 4 = (−18.0296, 0), respectively. The time-optimal consensus state is (−5.5527, 0) and the optimal time is 7.0645s. We apply our algorithm to those four agents, and the demonstration result is presented as followed. The projections on the intersection and the hyper-plane is presented in Fig. 1. Through the transformation (33), the problem can be treated as the convex functions and finally achieves the minimum of intersection. We could find that the Bregman's method is only proceeded 2 times, which is quite efficient. The state-trajectory and control sequences are presented in Fig. 2 and 3. The control inputs are typically bang-bang control, and the agents achieves consensus with the minimum time. To make a further step, we can implement our algorithm without proof to the more complicated nonconvex case such that only the final velocities are assumed to be zero. When we apply different transformations, though this case does not satisfy the convex assumption, the result is quite inspiring. Let x 22 = 0 in equations (30) and (31). The optimal time-cost curves and the corresponding epigraphs are      [t + x 22 ] 2 = 4 (x 12 − x 11 ) + 2x 2 21 , x 12 − x 11 − 1 2 x 21 x 21 < 0 [t − x 21 )] 2 = −4 (x 12 − x 11 ) + 2x 2 21 , x 12 − x 11 − 1 2 x 21 x 21 > 0 ,(35) and {(x, t) \|[t + x 21 ] 2 ≥ 4 (x 12 − x 11 ) + 2x 2 21 , x 12 − x 11 − 1 2 x 21 x 21 < 0 [t − x 21 ] 2 ≥ −4 (x 12 − x 11 ) + 2x 2 21 , x 12 − x 11 − 1 2 x 21 x 21 > 0}.(36) Because the half-parabolas in (36) are with different vertexes, its hard to find a simple transformation like (33) to make the epigraph to be cones in this case. For the consideration of convex assumption, we apply different quadratic transformations to those two parts, left and right, of different agents' epigraphs. if x i1 normal > x i1 + 1 2 x i2 x i2 x i1con = x i1 normal x i2con =      1 4 (x i2 normal + x i2 ) 2 + x i2 − 1 2 x 2 i2 − x i2 x i2 x i2 normal ≥ −x i2 x i2 − 1 2 x 2 i2 − x i2 x i2 x i2 normal < −x i2 (37) where x i = (x i1 , x i2 ) is the initial state of agent i, x i normal = (x i1 normal , x i1 normal ) is the projection in normal axis and x icon = (x i1con , x i1con ) is the in transformed axis. Consider four agents, whose initial states are x 1 = (−3.542884, 5.140490), x 2 = (3.001152, 3.794066), x 3 = (6.924106, −3.281824), x 4 = (−18.0296, 1.9023), respectively. The time-optimal consensus state is (6.9366, 0) and the optimal time is 8.4467s. We apply our algorithm to those four agents, and the demonstration result is presented as followed. The projections on the intersection and the hyper-plane is presented in Fig. 4. Though we applied different transformation to the epigraph of the agent, and the transformation was distinguish for different agents, the projection still converged to the minimum of the epigraph intersection. We believe the ability of non-convex treatment is releated to the characteristics of the alternating projection. The state-trajectory and control sequences are presented in Fig. 5 and 6, which performed the same as previous example. V. Conclusion In this paper, we interpreted the distributed MIN-MAX convex optimal problems geometrically as the problem of searching the minimum distance between some hyper-plane and the epigraphs intersection of convex functions. To solve this problem, we proposed an alternating projection based algorithm which is constructed by the Bregmans and the cyclic Dykstras alternating projection method. This distributed algorithm guarantees the optimal solution to the problem with only neighbor-communication on a cyclic communication topology. Moreover, we implement our algorithm to the multi-agent minimum-time consensus problem. We have shown that the first-order systems and the second-order systems with zero initial and final velocities can be formulated into a standard distributed MIN-MAX convex optimal problems. With the finite time attainable state set as the optimize functions, the time optimal state for consensus can be obtained by our algorithm, and each agent can execute optimal control to that state. Heuristically, we also implement our algorithm to the non-convex cases such that the second-order systems achieve consensus on zero velocities state without proof. At last, the demonstrations are presented to illustrate the efficiency and validity of our algorithm, even for the non-convex cases. VI. 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[ "A surface-depth theory of the emergence of complex networks", "A surface-depth theory of the emergence of complex networks" ]	[ "Keith Malcolm Smith \nUsher Institute of Population Health Science and Informatics\nUniversity of Edinburgh\n\n" ]	[ "Usher Institute of Population Health Science and Informatics\nUniversity of Edinburgh\n" ]	[]	The broadly general characteristics of complex networks found across disciplines-such as high clustering coefficients and heavy-tailed degree distributions-has long invited the question of whether there are general generating mechanisms behind them. Here, we propose a theory of such mechanisms and undertake several experiments which validate it. This theory proposes that there are two key principles at work in the emergence of a network, constituting a 'surface' factor and a 'depth' factor making up the existence probability of network edges. The surface factor describes nodes as having tendencies for attachment which follow a log-normal distribution. The deep factor suggests that beneath these potentials for attachment there exist any number of important latent variables describing the nodes. These variables are formulated as a high-dimensional manifold and the 'distances' between pairs of nodes on this manifold constitute a similarity weighting informing on the probability that any two nodes are connected. Using standard network measures, the topology of 110 networks across a variety of disciplines shows agreement with a simple two-parameter model based on this theory. Importantly, we show that the log-normal surface factor can explain the power-law-like degree distributions of sparse networks and, more strongly, the variety of degree distributions found across networks of different densities. We also demonstrate how inverting an estimated surface factor of a complete weighted world city network provides more plausible clusters of nodes than the original network based on geometric and cultural considerations. This theory proposes a new fundamental formulation of complex networks with wide reaching consequences throughout the multidisciplinary domain of complex systems.	null	[ "https://arxiv.org/pdf/1902.00336v1.pdf" ]	59,553,539	1902.00336	26a692dda705ca4be3fba17c12d5c774fb41d7b3	A surface-depth theory of the emergence of complex networks Keith Malcolm Smith Usher Institute of Population Health Science and Informatics University of Edinburgh A surface-depth theory of the emergence of complex networks (Dated: February 4, 2019) The broadly general characteristics of complex networks found across disciplines-such as high clustering coefficients and heavy-tailed degree distributions-has long invited the question of whether there are general generating mechanisms behind them. Here, we propose a theory of such mechanisms and undertake several experiments which validate it. This theory proposes that there are two key principles at work in the emergence of a network, constituting a 'surface' factor and a 'depth' factor making up the existence probability of network edges. The surface factor describes nodes as having tendencies for attachment which follow a log-normal distribution. The deep factor suggests that beneath these potentials for attachment there exist any number of important latent variables describing the nodes. These variables are formulated as a high-dimensional manifold and the 'distances' between pairs of nodes on this manifold constitute a similarity weighting informing on the probability that any two nodes are connected. Using standard network measures, the topology of 110 networks across a variety of disciplines shows agreement with a simple two-parameter model based on this theory. Importantly, we show that the log-normal surface factor can explain the power-law-like degree distributions of sparse networks and, more strongly, the variety of degree distributions found across networks of different densities. We also demonstrate how inverting an estimated surface factor of a complete weighted world city network provides more plausible clusters of nodes than the original network based on geometric and cultural considerations. This theory proposes a new fundamental formulation of complex networks with wide reaching consequences throughout the multidisciplinary domain of complex systems. INTRODUCTION Theories of the emergence of complex networks allow us to gather insights into their potential generative mechanisms [1,2]. If such a theory is accurate, it can establish new foundations from which to understand, analyse, deconstruct and interpret network phenomena. The seminal prototype of network models is the Erdös-Rényi random graph where all edges have equal probability, p, of appearing in the graph. A realisation of this random graph is generated by assigning uniformly random values to all node pairs and substantiating the existence of only those edges whose values lie above p [3]. Alternatively, one can specify that only the edges with the m highest values should be kept [4]. For a large enough number of nodes, each graph isomorphism class (i.e. distinct graph topology) of n nodes and m edges has roughly equal probability of appearing from this model [5]. Yet, the topological characteristics of real-world networks substantially and consistently deviate from those generated by this model [6], telling us that real-world networks occupy a relatively small and highly uncommon set of graph isomorphism classes. We can classify network models either as being constructive or non-constructive. Non-constructive models such as configuration models [6,7], stochastic block models [8], and complex hierarchy models [9] are derived from observations of real world networks and are focused on practical issues for the study of specific network properties as they are found. They are not of much use for providing insights into the generative mechanisms explaining the emergence of real networks. Constructive models, on the other hand, seek to derive complex network topologies from proposed generative mechanisms, the aim of which is to provide plausible physical explanations for the non-arbitrary topological features of real world networks. The most dominant branch of constructive models derive from the theory of preferential attachment where nodes which are older in the network have a greater share of edges simply due to their age, and present with scalefree degree distributions seen in some networks [2]. More complicated formulations to fix the lack of clustering in the original model have been proposed [10,11]. Problems remain however, such as the recently shown rarity of scale-free networks [12]. Another branch of constructive models considers nodes existing in a geometrical space and connections occurring where those nodes are close together. The idea that nodes which are close together are connected together is intuitively sensible and recent evidence agrees [13]. The prototype of this approach is the random geometric graph where nodes are random samples of an n-dimensional Euclidean space [14]. This model has some interesting relevant properties to real world networks such as a high modularity and clustering, but they do not display the degree heterogeneity implicated by hub nodes typical of complex networks. Further to this, Serrano et al. proposed a hyperbolic geometric model constraining for the expected degree distribution of the network [15]. This has also recently been extended to weighted networks [16]. However, it does not provide an explanation for how these degree distributions themselves arise. From the literature to date, it would appear that there are two main aspects to be explained in the emergence arXiv:1902.00336v1 [physics.soc-ph] 1 Feb 2019 of complex networks: i) the variability in the number of edges assigned to nodes, and ii) the likelihood of any given pair of nodes to form a connection. Here, we provide a new general generative theory of complex networks which addresses these two points as factors proportional to the probability of the existence of edges. Essentially, we propose that the heterogeneous degree distributions of complex networks can be explained by assuming that the propensity for forming connections is an attribute of nodes which follows a log-normal distribution. This is deemed as a surface consideration of the network as it is a property independent of which nodes connect to which. In conjunction, we propose that the specific ways in which connections are made between nodes depends on a number of latent variables describing how similar nodes are to one another. These variables can then be regarded as dimensions of a high-dimensional manifold on which the nodes lie, and their closeness on this manifold represents the similarity of the nodes over these variables. This is deemed as a 'deep' consideration of the network as it is the property which directly influences which nodes connect to which, given their surface propensities. These surface and depth considerations are then taken as factors precisely describing the probabilities of the existence of a connections between all node pairs. Importantly, because we provide an exact theoretical formulation to describe the emergence of complex networks, it opens up a new branch of complex network theory for exploration. For example, we describe how we can estimate the surface factor and invert it to get close to the depth factor explaining more accurately the similarities between nodes. THEORY Let V = {1, . . . , n} be a set of nodes representative of individual agents. Then, suppose that these agents have individual tendencies to make connections to other agents, h i , and that these tendencies are distributed according to a log-normal distribution h ∼ LogN (µ, σ). For example, in social networks it stands to reason that the tendencies of people to make new friends is the result of a number of psychological variables, such as extroversion and charisma, while empirical evidence suggests that such variables should be modelled using a log-normal distribution [17]. We could consider whether such tendencies are additive or multiplicative for pairs of nodes, i.e. is the combined tendency of h i and h j (h i + h j ) or h i h j ? In practice, this is not of immediate importance since both the addition and product of two log-normally distributed variables are log-normal. For simplicity's sake, we shall assume that the combined tendency of h i and h j is additive. We relate to this as the surface factor of the network, since it does not really help to describe why any two nodes are connected together beyond that either or both have a strong tendency to make connections. Below this surface, however, we assume that there are similarities between agents which make it more likely for connections to occur between them. Thus, we suppose that agents are distinguishable by some number, q, of independent latent variables, x 1 , x 2 , . . . , x q . Then, the similarity of nodes i and j across these variables can be described by some distance function d ij = f (x 1 (i), x 1 (j), x 2 (i), x 2 (j), . . . , x q (i), x q (j)). (1) A very obvious and important consideration of such latent variables is simply the geometry within which the agents are set. If two agents live nearby one another, it stands to reason they are more likely to be connected to one another than to some other agent that lives far away, disregarding other variables. It is important to point out that variables could also be categorical. For instance, in a social network, people who belong to the same club, A say, are more likely to be connected than to others in another club, B. We refer to these latent variables as making up a depth factor for the network as it accurately describes the similarities of agents beyond their tendency to make connections. Combining these consideration, the probability of a connection being established between nodes i and j is proportional to node similarity (depth factor) and the combined tendency of making connections of i and j (surface factor), giving p ij ∼ d ij (h i + h j ).(2) Assuming that these as the only considerations of the probability of existence of an edge, we can take the weights of edges in our network as w ij = d ij (h i + h j )(3) up to linearity. For a complex binary network with m edges, we can then, for example, take the m largest weights as extant, as for the random graph, use a nearest neighbours connectivity approach [18], or use a combination of the two to ensure connectedness while specifiying the exact number of edges. Model Given the above, to construct a model, all we need is a description of the properties of the latent variables, x i . We know that geometry is a key consideration of networks, and thus we have up to three variables which can be approximated using a random geometric graph where coordinates are chosen uniformly at random over the interval [0, 1]. The most simple model would prescribe all variables as equivalent and independent, thus we shall simply model similarities between nodes as distances of a random geometric graph in q dimensions. Of course, it is likely that different variables will have different distributive properties in reality, but, as we shall demonstrate, this simple assumption actually works quite well in practice for modelling a diverse range of complex networks. Our model, then, has probabilistic weights for each edge proportional to w ij = d ij (h i + h j ),(4) where d ij = q k=1 (x ik − x jk ) 2(5) for each x i ∼ U ([0, 1]), and h ∼ LogN (µ, σ). Now, µ does not affect the relative values in (4), i.e. µ will not affect relationships of the form w ij ≤ w st for any i, j, s, t ∈ V, thus essentially, we only need to consider the shape parameter, σ, of the log-normal distribution. Thus, the only parameters of this model are the number of dimensions of the deep factor, q, and the shape parameter for the log-normal distribution of the surface factor, σ and, for a network, G, with n nodes and m edges, we can describe its surface-depth model as G s-d (q, σ). EXPERIMENTS Explaining topological properties of sparse binary networks We modelled 110 real world binary networks collected from two difference sources. This was done iteratively on the two model parameters and the best fit was achieved by minimising the Root Mean Squared Error (RMSE) of five important and distinct topological metrics. These were the clustering coefficient, C, global efficiency [19], E, normalised degree variance [20], V , Louvain's modularity [21], Q, and assortativity [22], r. These were chosen both so that they covered distinctly formulated topological aspects, and so that the values were all of similar magnitudes (between 0 and 1, or -1 for assortativity) and thus the minimisation was not biased to any particular index. This kind of minimisation has been previously used in e.g. [23,24]. Models were ensured to have all nodes with at least degree 1 by including the nearest neighbours for each node. The rest of the edges were then selected simply from the edges with highest weights across all model weights until the number of edges matched the real network. We studied two datasets of networks for this. The first consists of 25 networks taken from the network repository across different domains [25]. This consists of eight social networks-karate club, hi-tech firm, dolphins, wikivote, Hamsterster, Enron email, Dublin contact, and Uni email; six biological networks-mouse brain, macaque cortex, c elegans metabolic, mouse protein, plant protein, and yeast protein; three ecological networks-Everglades, Mangwet and Florida; three infrastructure networks-US airports, euroroads and power grid; and three economic networks-global city network (binarised at 20% density), US transactions commodities 1979 and US transactions industries 1979. Many of these are classic benchmark networks. The second network dataset is the corpus used in [26]. Of this dataset, we looked at the 184 static networks and, for the sake of computational time, chose to look only at those between 20 and 500 nodes in size. Further, we discarded bipartite networks as these have 0 clustering and thus obviously need a different depth factor consideration than the random geometric graph which has a large clustering coefficient. We thus ended up with 85 networks. The most accurate surface-depth model was then chosen following Algorithm 1. Compute C, E, V Q and r of each of these models and take the mean over realisations for each 5: Compute the RMSE between indices of G and mean of G s-d (q, σ) 6: Take σ as the σ parameter of minimum RMSE model Take the model with the minimum RMSE value from this step as the minimum for the model with q dimensions 9: The minimum across q of the minimum RMSEs across σ is then taken as the model of best fit to G Note, we took a maximum of q = 10 arbitrarily to save on time as we assume the topological properties of the model are asymptotic with q, as demonstrated in the supplementary material, so if it is still far away by q = 10 it is unlikely to ever get too close. For the 25 network repository networks, the minimum RMSE score for each network, alongside the corresponding q and σ of the model, is shown in Table I. For 50 model realisation, we compared the degree distributions of the best-fit model with real networks using Kolmogorov-Smirnoff (KS) two-sample statistical tests. As is standard, the null hypothesis, that the distributions were not different, was rejected in the case that p < 0.05. The results indicate that only around half of the networks have degree distributions indistinguishable from their best-fit models, Table I between minimum RSME and percentage of null hypothesis rejections. However, Fig 1 shows comparisons of degree distributions of the surface-depth model and network repository networks. The similarity between distributions across all networks of various size, density and domain is remarkable. We can see even those which failed the KS tests have very similar distributions. We thus suggest that these failures are not caused by the surface factor of the model, but rather by the simplified depth factor used. We then looked at the second network dataset using the same methods as above. The minimum RMSEs of the model are shown in Fig 3, grouped by network class. Of the larger classes, the connectomes were best fitted by our model, followed by protein interaction networks. Food webs, by comparison, were quite poorly approximated by our model, often with RMSE> 0.1. The similarity between degree distributions was assessed again using two sample KS tests. We found a greater rate of of distribution similarity in this dataset than in the network repository with 70.6% of networks having a fraction greater than 80% of successfully rejected null hypotheses. Observations were similar as for the network repository, where most of those which were not rejected still had observably similar distributions. Interestingly, even though food web networks were not well approximated by our model, their degree distributions were on the whole largely indistinguishable from those of the model. Looking more closely, it appeared there was an exceptional difference in the clustering coefficients in this case. Median differences for each index across food web networks were as follows: C model − C real = 0.2753, E model − E real = 0.0206, V model − V real = 0.0593, r model − r real = 0.0185, Q model − Q real = 0.0449. The very low relative clustering in food web networks makes sense since we can expect that it is uncommon for predators of the same prey to hunt one another as well. This suggests that better modelling of the depth factor would help to capture the information in food web networks and we thus conjecture that accurate modelling of latent variables would be enough to help explain different real world network topologies. Depth Factor Recovery of the World City Network Given our theory, it would be of great interest to see what the depth factor of a real network would look like. However, recovering the depth factor of a sparse binary network poses a very challenging problem, as we are unable to determine which edges are stronger to a given node than any other from the binary edges. What we can do, however is to apply our methods to a fully weighted network by assuming that the weights of the network are linearly proportional to the underlying surface-depth factors of the network. Just such a network is available from the Globalisation and World Cities research network [27,28], constructed using relationships of producer service firms at the forefront of economic influence within each city. First, we looked at an example of recovering the depth factor from a surface-depth model, where we could directly compare the depth factor with our estimation attempts. We considered estimating the surface factor using both the weighted degree distribution and just by tuning a log-normal distribution to get the best result. In this case, we just generated another set of log-normally distributed samples using the same parameters as our known surface factor. Fig 4, bottom row, shows the outcome. Although the weighted degree distribution worked fairly well, it was clear that tuning a log-normal distribution could achieve a more accurate result. For the world city network, we fine-tuned a log-normal distribution until it produced an observably balanced adjacency matrix, where the original node hierarchy appeared to be successfully inverted, as seen in Fig 5. This was achieved at parameters µ = 0.5, σ = 0.55. K-Nearest Neighbour (KNN) graphs with K = 5 were then computed from the global city network and its estimated depth factor. Modules were computed using Louvain's modularity method [19]. The KNN graphs were then plotted using the same force-based algorithm where connected nodes are attracted and non-connected nodes repelled from one another [29], Fig 6. Remarkably, surface inversion of the hub-centric world city network produced a highly mod- ular network with geometric qualities. On inspection, spaces within the network layout were notable by their global proximity and cultural ties. We analysed this statistically in the case of global proximity (details in the supplementary material). The supplementary material contains tables of the five nearest neighbours of each city for each approach. Of these, 63.64% were found to be proximal on the globe (either being in the same continent or observably close) for the tuned log-normal inversion compared to 50.55% for the degree-based inversion and just 37.82% for the original network. Furthermore, the five cities with greatest weighted degree (London, New York, Paris, Tokyo and Hong Kong) appeared in 76.64% of the nearest neighbours in the original network, compared to 46.18% in the degree-based inversion and just 14.91% of the tuned log-normal inversion, with 9.27% being that expected by random chance. In addition, 52 of the 55 cities were found within the 5 nearest neighbours of all cities in the tuned log-normal inversion approach, whereas this number was just 15 for the original network and 38 for the degree-based inversion. Cultural ties were assessed qualitatively, for example Barcelona and Madrid being in the same community as all Latin American cities appeals to their cultural ties, whereas Latin American cities were not all found in the same community in the original network. Also, Eastern Europe and East Asia both had clearly distinct communities in the recovered depth factor but not so in the original network. DISCUSSION Evolution and dynamics of networks can be easily accounted for in our theory by shifts occurring in shallow and deep factors. For instance, a node may take on different values of its latent variables thus changing the nodes to which it is most similar which would result in a change to the edges the node makes. Otherwise, the node may increase or decrease its surface factor value giving it a higher/lower tendency to make connections, again resulting in a dynamic change of the network. New nodes could be assumed to appear somewhere within the latent variable space but with an initially low tendency to make the connections. The proposal that a depth factor of weight similarities can be extracted has clear implications in terms of geometric deep learning [30]. Along similar lines, a recent study considered using machine learning approaches on a hyperbolic network model [31]. It seems that such methods can be fairly straightforwardly translated to the geometries of the proposed depth factor and we expect our study will open up interesting future research along these lines. FIG. 5. Weighted adjacency matrices (ordered by weighted degree) of the global city network, an estimated depth factor of the network using the weighted degree and an estimated depth factor using a tuned log-normal distribution, respectively. FIG. 6. Plot of the five-nearest neighbours graph of the world city network (left) and its recovered depth factor (right) with detected communities shown in different colours. Clusters in the depth factor are observably more distinguishable, whereas relationships between the nodes in the original network are dominated by a few nodes. Compute indices C, E, V , Q and r of network G 2: for q ∈ {1, 2, . . . , 10} do 3: Compute 20 realisations, G s-d (q, σ), of model with the same size and density as G with σ ranging from 0.05 up to 1 in steps of 0.05 4: realisations of each surface-depth model with σ within 0.05 of σ in steps of 0.01 8: FIG. 1 . 1Comparison of the degree distributions between real-world networks and their respective closest fit surface-depth model. These are log-log plots where there is a clear scaling distribution. FIG. 2. Difference in network indices between 110 real networks and their optimised models. Distributions centred around zero indicate lack of consistent difference between models and real networks. FIG. 3 . 3Minimum RMSE between models and real networks from the ICON corpus, ordered by network class. FIG. 4 . 4Example of recovering the depth factor from a surface-depth model. Adjacency matrices of the depth factor only, the surface factor only and the surface-depth model are shown in the first row, repsectively. Attempted recovery using the weighted degree distribution of the model, attempted recovery using an estimated surface factor and the comparison of the distributions between the surface factor and the models weighted degree are displayed in the bottom row, respectively. final column, with only around 46% having rejected over 80% of KS tests. This does not appear to depend strongly on the RMSE however, with a Spearman correlation of only ρ = 0.3742 TABLE I . IMinimum root mean squared error (min RMSE) among models found for each sparse network alongside the corresponding model parameters (q & σ) KS % indicates percentage of Kolmogorov-Smirnoff tests over 50 model realisations in which the null hypothesis fails to be rejectedNetwork size density RMSE q σ KS % karate club 34 0.1390 0.0697 2 0.42 66% hi-tech firm 36 0.1444 0.0279 10 0.08 100% Dolphins 62 0.0841 0.0297 6 0.07 100% wikivote 889 0.0074 0.0300 9 0.07 78% Hamsterster 2426 0.0057 0.0145 7 0.05 0% mouse brain 213 0.7160 0.0270 8 0.01 100% macaque cortex 242 0.1047 0.0253 7 0.08 94% c elegans 453 0.0198 0.0461 6 0.19 0% mouse protein 1455 0.0015 0.0160 7 0.09 0% plant protein 1745 0.0020 0.0257 5 0.04 0% Yeast protein 2114 0.0010 0.0363 10 0.09 0% Everglades 69 0.3762 0.0561 10 0.15 100% Mangwet 97 0.3106 0.0447 10 0.09 100% Florida 128 0.2553 0.0765 10 0.05 100% US airports 456 0.3658 0.0098 4 0.40 100% Euroroad 1174 0.0021 0.0549 10 0.03 100% Power grid 4941 0.0005 0.0365 6 0.04 0% Global city 55 0.2000 0.0674 10 0.36 0% US commodities 506 0.3317 0.0177 5 0.29 100% US industries 507 0.3516 0.0207 5 0.29 20% enron email 143 0.0614 0.0284 4 0.09 14% dublin contact 410 0.0330 0.0354 3 0.04 100% Uni email 1133 0.0085 0.0309 10 0.04 76% EPA hyperlink 3031 0.0014 0.0613 3 0.05 14% Techrouters 2113 0.0030 0.0119 7 0.05 0% Note- . 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[ "Nonexistence of locally but not globally supersymmetric orbifolds", "Nonexistence of locally but not globally supersymmetric orbifolds" ]	[ "Stefan Groot Nibbelink [email protected] \nInstitute of Engineering and Applied Sciences\nRotterdam University of Applied Sciences\nG.J. de Jonghweg 4 -63015 GGRotterdamThe Netherlands\n" ]	[ "Institute of Engineering and Applied Sciences\nRotterdam University of Applied Sciences\nG.J. de Jonghweg 4 -63015 GGRotterdamThe Netherlands" ]	[]	Motivated by the smallness of the cosmological constant we investigate whether it is possible to have vanishing one-loop heterotic string partition functions for six-dimensional non-supersymmetric toroidal orbifolds. A straightforward way to realize this presents itself, when each orbifold sector separately preserves some Killing spinors, but none of them survives in all sectors combined. By applying some representation theory to the abstract finite point groups underlying toroidal orbifolds it turns out, that this is never possible. This leads to a nonexistence proof of locally but not globally supersymmetric orbifolds.	10.1088/1742-6596/1586/1/012030	[ "https://arxiv.org/pdf/1903.03789v1.pdf" ]	119,392,403	1903.03789	d19999064ef01eb23e1b0f90563d7211effd2291	Nonexistence of locally but not globally supersymmetric orbifolds 9 Mar 2019 Stefan Groot Nibbelink [email protected] Institute of Engineering and Applied Sciences Rotterdam University of Applied Sciences G.J. de Jonghweg 4 -63015 GGRotterdamThe Netherlands Nonexistence of locally but not globally supersymmetric orbifolds 9 Mar 2019 Motivated by the smallness of the cosmological constant we investigate whether it is possible to have vanishing one-loop heterotic string partition functions for six-dimensional non-supersymmetric toroidal orbifolds. A straightforward way to realize this presents itself, when each orbifold sector separately preserves some Killing spinors, but none of them survives in all sectors combined. By applying some representation theory to the abstract finite point groups underlying toroidal orbifolds it turns out, that this is never possible. This leads to a nonexistence proof of locally but not globally supersymmetric orbifolds. Motivation The current status of particle physics and cosmology is one of great success in light of the experimental confirmations of their Standard Models and confusion about what could be next. The link of the Standard Model of Particle Physics that had been elusive for a long time, the Higgs particle, is now being measured with ever increasing precision. Similarly, the Standard Model of Cosmology is being confirmed with ever higher experimental scrutiny. One of its ingredients is that the universe is equipped with a tiny, yet, nonvanishing cosmological constant Λ: It is 10 −120 orders of magnitude smaller than its natural gravitational scale. Any attempt to understand this vacuum energy within the realm of field theory has failed so far. String theory comes with the promise of being a unified quantum theory of all interactions. In particular, gravity, gauge interactions and chiral particle spectra arise at the same stage in the construction of the heterotic string. Even though there are various theoretical arguments why nature at its most fundamental level should be supersymmetric and string theories are best understood with some amount of target space supersymmetry present, there has been absolutely no experimental evidence for supersymmetry so far. Especially the LHC experiments are now probing well into the region where by naive naturalness and hierarchy arguments one would have expected the first, if not many, hints for supersymmetry. Moreover, a positive cosmological constant seems to be inconsistent with supersymmetry. Hence, if string theory really describes our universe, the smallness of the vacuum energy should be investigated in the context of non-supersymmetric string theory, which is much harder to access and control theoretically. One of the main selling points of string theory is that, contrary to generic quantum field theories, string theory is free of divergences. Even the cosmological constant is a finite and calculable quantity. Given that all string theories are presumably related by dualities, in principle any question can be investigated within various string constructions. In particular, perturbative heterotic string theory is a well-studied subject, hence it is clear how the vacuum energy can be computed. The fact, that the cosmological constant is so tiny, may be taken as an indication that the cosmological constant should vanish perturbatively to all orders and only arises due to non-perturbative effects. For this to be feasible the cosmological constant should vanish at least at the one-loop level. In string theory this corresponds to the well-studied torus partition function: When it is integrated over the fundamental domain of the torus, the one-loop vacuum energy is determined. One-loop vanishing cosmological constants have been obtained for non-supersymmetric asymmetric orientifold constructions [1,2,3]. However, such models focussed on the gravitational sector alone, the real challenge is to have a very tiny cosmological constant and at the same time to realize the Standard Model particles. Nonsupersymmetric heterotic strings [4,5] have been investigated in the past [6,7,8,9,10,11,12] and have gained renewed interest [13,14,15,16,17] in light of the elusiveness of supersymmetry in accelerator experiments. The construction of non-supersymmetric heterotic models with spectra that are very close to that of the Standard Model (not its supersymmetric extension) has been quite successful [18,19,20,21]. If one starts to think about generic non-supersymmetric string constructions one quickly realizes that this is a daunting task. Since string theories tend to live in ten dimensions, a compactification of six dimensions is mandatory. As there is an infinite (over-countable) number of six dimensional manifolds this becomes a very difficult endeavour. Fortunately, the number of six dimensional toroidal orbifolds is finite and strings can be exactly quantized on them. Hence, the close to 29 million such orbifolds provide a large but trackable testing ground for such investigations. In this work we aim to achieve the vanishing of the one-loop vacuum energy by having the integrant, the one-loop partition function to vanish. We will show that the only orbifolds for which this is possible are in fact the known supersymmetric ones. This show that also in string theory obtaining a systematic solution to the cosmological constant problem in a non-supersymmetric setting is very challenging. Overview In this proceedings, based on [22], we first recall some necessary aspects of the space group description of toroidal orbifolds. After that we determine the conditions for the one-loop partition function to vanish 1 . This helps us to characterize candidate non-supersymmetric orbifold geometries. Next we make use of the classification of toroidal orbifolds to identify the candidate geometries. Using those explicitly or alternatively making use of abstract finite group theory, we show that no such non-supersymmetric orbifolds exist. Space group description of toroidal orbifolds The space group provides a convenient language to describe toroidal orbifolds [24,25]. A toroidal orbifold T 6 /G can be build as follows. The starting point is a six dimensional lattice spanned by six basis vectors e i : Γ = {e m\|m ∈ Z 6 } . This lattice is then used to define a six dimensional torus as the quotient T 6 = R 6 /Γ defined by the periodicities: X ∼ X + e m , m ∈ Z 6 , where the basis vectors e i are combined to the vielbein e = (e i ). To obtain an orbifold the torus is modded out by the action of a finite group G. Such an orbifold group G is either generated by twists: X ∼ D v (θ) X , θ = 1 or by roto-translations: X ∼ D v (θ) X + e q , q ∈ Q 6 . Here D v (θ) defines a six dimensional matrix representation of the abstract finite group element θ ∈ G and it is related to an integral matrix D v (θ) = e −1 D v (θ)e. As can be inferred from the defining relations above, roto-translations are twists accompanied by translations. In order that these translations q are physical, i.e. cannot be absorbed by the torus lattice translations, q ∈ Z 6 . The space group S combines the torus lattice and the orbifold group elements: (1; e m) ∈ S , m ∈ Z 6 and (θ; e q) ∈ S , θ = 1 , q ∈ Q 6 . The finite point group P is a projection of the space group S: S → P : (θ, e q) → θ , hence, the point group retains only the twist information of the orbifold group G. The group multiplication of the space group elements g = (θ; e q) and g = (θ ; e q ) is given by g · g = (θ; e q) · (θ ; e q ) = (θθ ; D v (θ) e q + e q) . Consequently, space group elements may not commute because their point group parts θ and θ do not commute or because their action on the lattice does not: D v (θ) − 1 q = D v (θ ) − 1 q . The action D v (θ) associated to a given space group element g = (θ, e q) can be diagonalized in a complex coordinate basis as D v (θ) =     e 2πi v 1 g 0 0 0 e 2πi v 2 g 0 0 0 e 2πi v 3 g     in terms of a local twist vector v g = 0, v 1 g , v 2 g , v 3 g . The characterization local here emphasizes that this diagonalization procedure has to be applied to each space group element g separately. We reserve the term global to emphasize properties that hold for all space group elements. Unless the corresponding point group elements commute, the local twist vectors of different space group elements are defined with respect to different bases. For string theory we should be able to define spinors on the six dimensional torus. Assuming for now that they exist, the action of a space group element on an eight-component spinor is given by D s (θ) = e 2πi v 1 g σ 3 2 ⊗ e 2πi v 2 g σ 3 2 ⊗ e 2πi v 3 g σ 3 2 . The expressions for D s (θ) and D v (θ) show that Spin(6) = SU(4) is the double cover of SO (6): Both D s (θ) and −D s (θ) are associated to the same element D v (θ), because D v (θ) is inert under v a g → v a g +1, while D s (θ) changes sign. The properties of the spinorial representation D s (θ) decide on how many supersymmetries a given space group element preserves, i.e. how many Killing spinors it admits. A space group element g = (θ; e q) admits a Killing spinor Ψ inv. , if D s (θ) Ψ inv. = Ψ inv. has non-trivial solutions Ψ inv. = 0. Given that the possible eigenvalues of D s (θ) are exp(±2πi v a g ), a = 0, 1, 2, 3, where v g = 1 2       v 1 g + v 2 g + v 3 g −v 1 g + v 2 g + v 3 g v 1 g − v 2 g + v 3 g v 1 g + v 2 g − v 3 g       for a space group element g = (θ; e q) to admit at least one Killing spinor, at least one of the entries of v g needs to vanish modulo integers. Consequently, −1 ∈ Spin(6) breaks all supersymmetries, since the corresponding local twist vector would be v g = (0, 1 2 , 1 2 , 1 2 ) so that none of the components of v g = ( 3 4 , 1 4 , 1 4 , 1 4 ) vanish modulo integers. Since, the two choices for the spinor embedding of a space group element g = (θ, e q) differ by −1, at most one choice of D s (θ) admits some Killing spinors. Vanishing of the one-loop partition function As emphasized in the introduction the cosmological constant is a finite calculable quantity in string theory. In heterotic string theory the four dimensional one-loop cosmological constant Λ 4D is computed as the integral of the full partition function Z full : Λ 4D ∼ F d 2 τ τ 2 2 Z full (τ,τ ) . The standard choice for the fundamental domain F of the one-loop Teichmueller parameter τ is depicted by the blue shaded region in the picture next to the expression for the one-loop cosmological constant. Given that the integral over the fundamental domain F is very complicated, we asked: Can we construct non-supersymmetric heterotic orbifolds which have a vanishing one-loop partition function? To investigate this possibility, let us consider what the full partition function consists of. The full partition function Z full = Z 4D Mink. Z 6D int. can be divided into a four dimensional Minkowskian noncompact partition function Z 4D Mink. , which never vanishes, and a six dimensional internal orbifold partition function [25] Z 4D Mink. = 1 τ 2 1 η 2 2 = 0 and Z 6D int. = 1 \|P\| [g,h]=0 Z X g h Z ψ g h Z Y g h , respectively. The latter itself can be divided into three parts that are associated with the 6D internal coordinate fields X, their worldsheet superpartners described by the right-moving fermions ψ, and the 16D left-moving gauge degrees of freedom Y . The sum indicated here is over all commuting space group element g, h ∈ S normalized by the finite number of point group elements \|P\|. Contrary the other parts of the internal partition function, the right-moving fermionic partition function This confirms the well-known result that all supersymmetric orbifolds have vanishing partition functions. Also any non-supersymmetric toroidal orbifold for which i. a Killing spinor exists locally in every commuting (g, h)-sector; ii. but none globally, will have a vanishing one-loop partition function. The second condition excludes globally supersymmetric orbifolds. Do such orbifolds exist? On first sight one might be inclined to say yes, since there are orbifold examples with different local and global supersymmetry breaking patterns: Consider for example the space group S of the DW(0-2) Z 2 × Z 2 orbifold [26,27] generated by the two orbifold elements: 1 2 e 5 and the torus translations g i = 1, e i . The two-tori fixed by these two orbifold elements are depicted schematically below: g θ -fixed two-tori g ω -fixed two-tori Each of these two-tori preserve N = 2 supersymmetry from a four dimensional perspective. However, the fixed tori of g θ and g ω perserve different N = 2 supersymmetries such that this orbifold only preserves N = 1 globally. Yet, nowhere in the orbifold geometry supersymmetry is broken to N = 1 locally, since the fixed two-tori of g θ and g ω are shifted with respect to each other by 1 4 e 5 and therefore never intersect. g θ = θ, 0 , g ω = ω, Classification of toroidal orbifolds In light of this example it seems very reasonable to assume, that also orbifolds exist, for which locally always some amount of Killing spinors are preserved, but none globally. To investigate this possibility, we make use of the fact that all six dimensional toroidal orbifolds have been classified [28,29]. This classification has three levels: There are 7,103 Q-classes enumerating all the inequivalent point groups P. Each of these point groups may act on various inequivalent lattices Γ leading to 85,308 Zclasses. Finally, there are 28,927,915 affine-classes, corresponding to all the inequivalent space groups S (differing by their roto-translations) that can act on these lattices. Among these close to 29 million toroidal orbifolds only 520 preserve at least N = 1 supersymmetry with 60 distinct underlying Qclasses [30,31]. This orbifold classification can be used to identify the Q-classes of the toroidal orbifolds, that admit Killing spinors in all sectors locally, but none globally [22]: # Q-classes Restriction 7,103 All inequivalent geometrical point groups P ⊂ O(6) 1,616 Orientable geometrical point groups P ⊂ SO(6) 106 No element from P rotates in a two-dimensional plane only 63 Each element θ ∈ P admits a choice with some local Killing spinors 60 Geometrical point group compatible with some global Killing spinors This table starts from all 7,103 inequivalent geometrical point groups. Out of those only 1,616 preserve orientation which is mandatory in order to admit spinors on the geometry. The next cut is made by vetoing any point group element that rotates in a single two-dimensional plane only, since such rotations necessarily break any supersymmetry in at least one orbifold sector. Out of the 106 possible Q-classes only 63 are such, that each point group element individually admits some Killing spinors. Since most of them correspond to the 60 Q-classes of the globally supersymmetric orbifolds, this leaves three candidate Q-classes for toroidal orbifolds with the desired properties: CARAT-index Point group Generator relations Order Local twist vectors 3375 Dic 3 = Z 3 Z 4 θ 4 1 = θ 3 2 = 1 , 12 1 4 , 1 4 , − 1 2 θ 2 θ 1 θ 2 = θ 1 1 3 , − 1 3 , 0 5751 Q 8 θ 4 1 = 1, θ 2 1 = θ 2 2 , 8 1 4 , 1 4 , − 1 2 θ 1 θ 2 θ 1 = θ 2 1 4 , − 1 4 , 0 6737 SL(2, 3) θ 3 1 = θ 4 2 = 1 , 24 1 3 , 1 3 , − 2 3 (θ 2 θ 1 ) 2 = θ 2 1 θ 2 1 4 , − 1 4 , 0 The first column gives their CARAT-indices to identify the relevant Q-classes. The second column provides their abstract point groups; their defining relations are given in the next column and their order is tabulated after that. The final column states the local twist vectors, obtained in two different bases. As can be easily confirmed, they indeed separately preserve some amount of supersymmetry. For these three candidate Q-classes all possible inequivalent spinor representations of their point groups were constructed. In all cases there is at least one point group element that does not preserve any Killing spinor! This leads to the conclusion, that there does not exist any non-supersymmetric orbifold for which all point group elements separately preserve some Killing spinors [22]. Nonexistence proof of locally but not globally supersymmetric orbifolds The last paragraph of the previous section sketched an argument for the nonexistence of locally but not globally supersymmetric orbifolds by explicit construction of all spin representations associated to all (candidate) point groups P. To confirm this result, we also made use of representation theory of abstract finite groups [22]. In the following we provide some of the necessary ingredients of this analysis. First of all, the 7,103 Q-classes of 6D orbifolds provided by CARAT correspond to only 1,594 different abstract point groups. The reason for this is, that for a given abstract group there may exist several inequivalent representations as integral 6 × 6-matrices. In light of the discussion above, the following representations of an abstract point group P are relevant: Geometrical point group Abstract point group Name Matrix repr. Repr. Character Trivial 1 1 χ 1 = 1 Spinor D s (θ) 4 χ 4 Vector D v (θ) 6 = [4] 2 χ 6 To ensure that the four-dimensional spinor representation 4 can be thought of as the double cover of the six-dimensional vector representation 6, the former is assumed to be obtained as the two times antisymmetrized 4 denoted by [4] 2 . Now, consider any 4-representations of P (in particular, neither 4 nor 6 need to be irreducible) that fulfils the following three properties: (i) The 4 lies inside SU(4) = Spin (6): On all conjugacy classes of P: χ [4] 4 = 1. (ii) The 6 = [4] 2 is isomorphic to a Q-class: There should be an integral 6 × 6-matrix representation D v within the CARAT Q-classes such that χ 6 = χ v = Tr D v . (iii) The 4 does not contain the trivial singlet representation. Any G-invariant Killing spinor satisfies: D 4 (θ) Ψ inv. = Ψ inv. for all G ⊂ P. Consequently, the projector on the G-invariant subspace reads: P G = 1 \|G\| θ ∈ G D 4 (θ ) Hence, the number of G-invariant Killing spinors is counted by: N G = Tr P G = 1 \|G\| θ ∈ G Tr D 4 (θ ) = 1 \|G\| θ ∈ G χ 4 (θ ) = χ 4 , χ 1 G = n G 1 The number of G-invariant Killing spinors equals the number of trivial singlets n G 1 in the branching of a 4-representation of P into irreducible representations of G. In particular, the number of local Killing spinors preserved by θ is given: N θ = n θ 1 , since any element θ ∈ P of order N θ generates a θ ∼ = Z N θ ⊂ P. Similarly, the number of global Killing spinors is given by: N = n P 1 , e.g. how many trivial singlets the 4-representation contains. In other words, it would be possible to obtain a locally but not globally supersymmetric orbifold, if there exists a point group P such that for all θ ∈ P: n θ 1 > 0, while n P 1 = 0. This explains the condition (iii) mentioned above. For each of the 1,594 different abstract groups P we considered all faithful (but in general reducible) 4-representations and required, that they satisfy the three conditions (i) -(iii) stated above. By constructing all Z N ⊂ P subgroups we showed, that for each remaining 4, there is at least one cyclic subgroup, for which the 4 does not contain the trivial Z N -singlet representation. This implies that for all non-supersymmetric six dimensional toroidal orbifolds, there is always a sector without any local Killing spinor. Hence there do not exist any locally but not globally supersymmetric toroidal orbifolds. h ∈ S share at least one Killing spinor. Using the contributions of the non-level matched proto-graviton and proto-gravitrino states in the partition function one can argue, that in non-supersymmetric heterotic string constructions it is never possible to have the partition function to vanish[16,23]. It is interesting to see how this conclusion is confirmed by a rather involved analysis of toroidal orbifolds. AcknowledgementsI would like to thank the organizers of the conference DISCRETE 2018 in Vienna and the convener Alon Faraggi, in particular, to give me the opportunity to speak at this conference. In addition, I would like to thank Johanna Knapp for the kind hospitality at the mathematical physics group at Vienna University. 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[ "Spectral Analysis, Model Theory and Applica- tions of Finite-Rank Perturbations", "Spectral Analysis, Model Theory and Applica- tions of Finite-Rank Perturbations" ]	[ "Dale Frymark ", "Constanze Liaw " ]	[]	[]	This survey focuses on two main types of finite-rank perturbations: self-adjoint and unitary. We describe both classical and more recent spectral results. We pay special attention to singular self-adjoint perturbations and model representations of unitary perturbations.(2010). Primary 47A55; Secondary 44A15, 30E20, 47A56, 47A10.Mathematics Subject Classification	10.1007/978-3-030-43380-2_9	[ "https://arxiv.org/pdf/1904.09833v1.pdf" ]	127,987,385	1904.09833	7d636bac4832c15501af7a46a05ad1d41de90d0f	Spectral Analysis, Model Theory and Applica- tions of Finite-Rank Perturbations 11 Apr 2019 Dale Frymark Constanze Liaw Spectral Analysis, Model Theory and Applica- tions of Finite-Rank Perturbations 11 Apr 2019Dedicated to the memory of R.G. Douglas, a magnificent person, administrator and mathematician.Finite-Rank PerturbationsRepresentationsSpectral The- oryModel Theory This survey focuses on two main types of finite-rank perturbations: self-adjoint and unitary. We describe both classical and more recent spectral results. We pay special attention to singular self-adjoint perturbations and model representations of unitary perturbations.(2010). Primary 47A55; Secondary 44A15, 30E20, 47A56, 47A10.Mathematics Subject Classification Introduction Let A be a self-adjoint (possibly unbounded) operator on a separable Hilbert space H. Fix a d-dimensional subspace K ≤ H. Consider all self-adjoint perturbations A + K with Ran K ⊂ K. All self-adjoint perturbations A + K are formally given by the family of self-adjoint finite-rank perturbations: A Γ = A + BΓB * (1.1) for some Hermitian d × d matrix Γ, where B : C d → K is an invertible coordinate operator that takes the standard basis {e k } d k=1 of C d into a basis Be k of K. Reducing our attention to the essence of the problem, we always assume without loss of generality that K is cyclic for A on H, that is, H = clos span{(A−zI) −1 K : z ∈ C\R}. See Section 5 for a more general definition of A Γ which applies when the functions Be k do not belong to the Hilbert space H, but are instead taken from a larger space. The family of self-adjoint rank-one perturbations represents a special case of the family of finite-rank perturbations given in equation (1.1), and can be formally given by A γ = A + γ( · , ϕ)ϕ, ϕ ∈ K (1.2) with parameter γ ∈ R. See Subsection 3.1 for the precise definition, as well as Subsection 1.1 regarding notation on A versus A. Interest in this type of perturbation problem originally arose from the theory of self-adjoint extensions [80]. Natural applications to the variation of boundary conditions of differential operators, in particular Sturm-Liouville operators, were investigated by Aronszajn and Donoghue in the 1950's. Other famous perturbation theoretic results, such as those by von Neumann and Kato-Rosenblum, apply because rank-one perturbations are trace class. The great achievements in this field furnish a rather concrete description of the spectral properties of the perturbed operators A γ . See Section 3 for a sampling of these results. The spectral theory for quantum mechanical systems (see e.g. [6]), large random matrices (see e.g. [16]) and free probability probability (see e.g. [15]), and the decoupling of CMV matrices (see e.g. [74,Section 4.5]) present other standard applications. Additional applications to quantum graph theory arise from transforming the graph to a tree by adding partition vertices to existing edges and imposing boundary conditions on the partition vertices [19,Ch. 3]. The number of partition vertices that needs to be added in order to transform a graph into a tree is equal to the first Betti or cyclomatic number of the original graph, which equals the number of edges minus the number of vertices plus the number of connected components. In the late 1980's and early 1990's, a surge of interest took place in perturbation theory following the discovery of the celebrated Simon-Wolff criterion, which was used in a proof of Anderson localization for the discrete random Schrödinger operator in dimension one. A brief discussion of the Simon-Wolff criterion is included in Subsection 9.3. Given two arbitrary operators on the same Hilbert space, it is generally not easy to find out whether they are related via a rank-one or a finiterank perturbation. The situation is different if we consider two (so-called) Anderson-type Hamiltonians. We refer the reader to Subsection 9.3 for a definition. For now it suffices to know that they are perturbation problems with a random perturbation that is almost surely non-compact. Under mild assumptions, the essential part of two realizations of an Anderson-type Hamiltonian are related by a rank-one perturbation (almost surely with respect to the product of the probability measures), see [51]. Unitary perturbation theory is the other main topic of this survey. Let U be a unitary operator on a Hilbert space H. Fix a d-dimensional subspace R ≤ H. Then the set of operators K with Ran K ⊂ R that make U + K a unitary operator can be parametrized by unitary d × d matrices. Specifically, there is a bijective coordinate operator J : C d → R so that K = J(α − I)J * U for a unitary d×d matrix α. The created family of unitary finite-rank perturbations of U is given by U α = U + J(α − I)J * U, (1.3) with α taken from the unitary d × d matrices. Without loss of generality, we focus on the domain altered by assuming that R is a * -cyclic subspace for U, i.e. we assume that H = clos span{U k R : k ∈ Z}. The special case when d = 1 is closely related to Aleksandrov-Clark theory, and is described in Subsection 4.1. In this setting, the family of perturbations in equation (1.3) reduce to the well-known family of unitary rank-one perturbations U α = U + α( · , U * ϕ) H ϕ,(1.4) with α ∈ T and ϕ ∈ R. Again, see Subsection 1.1 for notation. While self-adjoint and unitary operators are intimately connected via the Cayley transform, it is well-known (see e.g. [21,Theorem 4.3.1]) that this correspondence is not a bijection between the two operator classes. In fact, even when the mappings are well-defined, the Cayley transform does not explicitly take (1.1) to its analog (1.3). This can be seen for the rank-one setting in Liaw-Treil [55, pp. 124-128]. Also notice that we encounter some inconveniences arising from unbounded operators in the self-adjoint setting. Of course, the unbounded case is exactly what occurs when dealing with boundary conditions of differential operators and several other applications. The unitary setting, on the other hand, is always restricted to bounded operators (see Remark 3.2). It is therefore surprising that, in spite of these differences, many results on self-adjoint finite-rank perturbations have analogs in the unitary setting. It is also common to find that the problems raise similar questions, e.g. about the boundary behavior of analytic functions. Families of rank-one and finite-rank perturbations seem rather elementary, yet their study has revealed a quite subtle nature. Their complexity is verified by connections to several deep fields of analysis: Nehari interpolation problem, holomorphic composition operators, rigid functions, existence of the limit of the Julia-Carathéodory quotient, Carleson embedding, and functional models. Some of these connections are the topic of existing books and surveys, including [23,55,68,70]. While writing this survey, it became evident that a complete account of the subject of finite-rank perturbations is worthy of a whole book due to the connections to many other fields of mathematics. We decided to focus on a few aspects, while only briefly mentioning others. For example, some deserving topics such as related function theoretic nuances are not surveyed in detail. We also often refer to existing surveys and books on the topic such as, e.g. [6,23,55,68,70,75], in order to not overlap excessively. It should be noted that some central objects of perturbation theory, such as Aleksandrov Spectral Averaging and Poltoratski's Theorem, appear in the Appendix (Section 9) for convenience. Section 2 contains highlights of classical perturbation theory that provide additional context for the more specific results to come. In particular, we focus on aspects of the spectrum that are invariant under different types of perturbations. Sections 3 and 4 present well-known features of rank-one perturbation theory in the self-adjoint and unitary settings respectively. Section 3 includes a discussion of singular perturbations, some spectral results (including Aronszajn-Donoghue theory) and Nevanlinna-Herglotz functions, which form the backbone of the theory. The unitary setting of Section 4 is built upon Aleksandrov-Clark theory and features the Sz.-Nagy-Foiaş and de Branges-Rovnyak approach, as well as the overarching Nikolski-Vasyunin transcription free model theory. The latter reduces to the ones by Sz.-Nagy-Foiaş and de Branges-Rovnyak by choosing a specific weight. These model representations form rather concrete applications of model theory. Sections 5 through 8 focus on finite-rank perturbations. Where possible, the presentation runs in analogy to Sections 3 and 4. For finite-rank self-adjoint perturbations the setup (Section 5) is a bit more involved, and we include information on extension theory, as well as a summary of some mathematical physics applications. In Section 6 we present known results regarding the spectral analysis of finite-rank perturbations and compare them to Aronszajn-Donoghue theory. Section 7 contains information about model spaces culminating in the Nikolski-Vasyunin model theoretic representation of unitary finite-rank perturbations. A short exposition on related Krein spaces and reproducing kernel Hilbert spaces is provided. In Section 8 relationships between the family of spectra of the perturbation problem and the characteristic function are presented. In the appendix Section 9 we take a moment to convey just the ideas behind several other well-deserving topics in the field. We refer to other literature for more information. Notation We use different notation to help the reader distinguish between the unitary the self-adjoint setting. In the self-adjoint setting, a rank-one perturbation of an operator A will be denoted as A γ , where γ ∈ R. We will use "boldface" A Γ for a finiterank perturbation that is given by a self-adjoint matrix (d × d)-matrix Γ. The real spectral measures for these cases will be referred to as µ γ and µ Γ respectively. An additional superscript will be added when the trace of the matrix-valued spectral measures is required: µ tr Γ . Also, the subscript will be entirely dropped when referring to objects corresponding to the unperturbed operator A, e.g. µ = µ 0 , F = F 0 , F = F 0 , etc. In the unitary setting, a rank-one perturbation of an operator U will be denoted as U α , where α ∈ T. A finite-rank perturbation will be given by U α with unitary (d × d)-matrix α. Notation similar to the self-adjoint setting will be used for the spectral measures, e.g. µ α and µ α . Here, the subscript α indicates that we work with unitary perturbations. Characteristic functions and model spaces will be denoted in the rank-one case by θ and K θ , and in the finite-rank case by θ and K θ . Dropping the subscript again refers to objects that correspond to the unperturbed operator U, except this operator arises from using α = I, e.g. µ = µ I , etc. We will simply write I for the identity matrix, with the dimension inferred from context. Spaces will be written in "mathcal" notation, e.g. H and H s (A). In particular, D and D * refer to the deficiency spaces on the unitary side. Perturbation-theoretic background We begin by presenting some central ideas from classical perturbation theory of self-adjoint operators, in order to better frame later discussions. A linear operator A from a Banach space X to a Banach space Y is said to be compact if the image A(X 1 ) of any bounded subset X 1 ⊂ X is relatively compact in Y. Consider linear operators acting on a Hilbert space H. The class of compact operators S is then obtained by taking the closure of the set of finite-rank operators with respect to the operator norm topology. A characterization of the spectrum of self-adjoint operators that differ by a compact perturbation is available. Recall that the spectrum of an operator A, denoted by σ(A), is the closure of the set of all λ ∈ C for which operator A − λI is not invertible. The essential spectrum is the spectrum minus the isolated eigenvalues of finite (algebraic) multiplicity. In the self-adjoint setting, we can view compact operators as compact perturbations of the zero operator to see that compact operators are characterized as those whose only (possible) accumulation point of eigenvalues is the origin. A more refined standard definition restricts the speed at which the eigenvalues tend to 0. Namely, the von Neumann-Schatten classes, S p , consist of compact operators whose sequence of singular values {s k } belongs to ℓ p . Here, the singular values of an operator T are defined as the eigenvalues of \|T \| = (T * T ) 1/2 . Self-adjoint operators thus have the property that s k = \|λ k \|, where λ is the sequence of eigenvalues. Carey and Pincus [22] characterized trace class, S 1 , perturbations A+B of A. Apart from leaving the absolutely continuous spectrum invariant, it must be possible to split the isolated eigenvalues of A and A + B as follows into three categories. The first and second categories are comprised of the eigenvalues of A and of A + B, respectively, that have summable distance from the essential spectrum of A. The third category contains all remaining eigenvalues of A and A+B. And there must exist a bijection ϕ mapping those eigenvalues of A in this category to those remaining eigenvalues of A + B so that the sum of \|λ − ϕ(λ)\| over all eigenvalues λ of A in this category is finite. In other words the remaining eigenvalues of A have trace class distance to the remaining ones of A + B. To emphasize a dichotomy, we mention that absolutely continuous spectrum can be destroyed by a Hilbert-Schmidt operator of arbitrarily small Hilbert-Schmidt norm: Theorem 2.3 (Weyl-von Neumann, see e.g. [43, p. 525]). Let A be a selfadjoint operator. For every η > 0, there exists a self-adjoint operator B with Hilbert-Schmidt norm less than η so that A + B has pure point spectrum. Since the Hilbert-Schmidt norm dominates the standard operator norm, this means that the absolutely continuous spectrum may be unstable under arbitrarily small perturbations. Theorem 2.3 was first proved by Weyl [80] for compact perturbations and then for the smaller class of Hilbert-Schmidt perturbations by von Neumann [79]. Extensions to normal operators and perturbations were proved by Berg [17] for compact operators and by Voiculescu [77,78] for Hilbert-Schmidt perturbations. These results form the basis of K-homology theory, which studies the homology of the category consisting of locally compact Hausdorff spaces. On the side, we mention Baranov [12] where a model representation and a spectral synthesis for rank-one perturbations of normal operators is achieved. In order to avoid possible confusion, we spell out that we are not (at least not explicitly) reaching for a spectral synthesis, or other questions usually related to K-homology. Instead, we are primarily interested in spectral invariants and describing the spectral measure under perturbations. Aspects of self-adjoint rank-one perturbations Scales of Hilbert Spaces When considering perturbations like Equation (1.2), it is sometimes convenient to loosen our restrictions on the perturbation vector ϕ to expand our possible applications, e.g. to changing boundary conditions of differential operators. We say that the perturbation is bounded when the vector ϕ is from the Hilbert space H. The previous sections have dealt exclusively with bounded perturbations. If ϕ / ∈ H, we say the perturbation is singular. These perturbations are significantly more complicated; it is imperative to ensure that the perturbation is well-defined in order to extend the tools that are presented in Subsection 3.2. The description here roughly follows that of [6]. Let A be a self-adjoint (possibly unbounded) operator on a separable Hilbert space H. Consider the non-negative operator \|A\| = (A * A) 1/2 , whose domain coincides with the domain of A. Alternatively, if A is bounded from below, the shifted operator A + kI, k ∈ R sufficiently large, will provide a non-negative operator. We introduce a scale of Hilbert spaces. It is not difficult to see that the spaces satisfy the nesting properties . . . ⊂ H 2 (A) ⊂ H 1 (A) ⊂ H = H 0 (A) ⊂ H −1 (A) ⊂ H −2 (A) ⊂ . . . , and that for every two s, t with s < t, the space H t (A) is dense in H s (A) in the norm · s . Indeed, the operator (A+1) t/2 defines an isometry from H s (A) to H s−t (A). In the rest of the subsection, we will use the brackets · , · to denote both the scalar product in the Hilbert space H and the action of the functionals. For instance, if ϕ ∈ H −s (A), ψ ∈ H s (A), then ϕ, ψ := (\|A\| + I) −s/2 ϕ, (\|A\| + I) s/2 ψ , where the brackets on the right hand side denote the scalar product. Throughout the literature of other fields similar constructions occur under different names. For instance, the pairing of H 1 (A), H, and H −1 (A) is sometimes referred to as a Gelfand triple or rigged Hilbert space. Also, when A is the derivative operator, these scales are simply Sobolev spaces (with p = 2). More details about Hilbert scales can be found in [47]. It is worth noting that these Hilbert scale are related to those generated by so-called left-definite theory [57]. This theory employs powers of a semibounded self-adjoint differential operator to create a continuum of operators whereupon spectral properties can be studied. The theory can be applied to self-adjoint extensions of self-adjoint operators, which can be viewed as finite-rank perturbations, see e.g. [29,30] and the references therein. Rank-one perturbations of a given operator A arise most commonly when the vectors ϕ are bounded linear functionals on the domain of the operator A, so many applications are focused on H −2 (A). Here, we only discuss the case ϕ ∈ H −1 (A) for the sake of simplicity. However, references usually contain information on extensions to ϕ ∈ H −2 (A), and information on the case when ϕ / ∈ H −2 (A) can be found in [25,48]. Spectral Theory of Rank-One Perturbations A nice overview of what is now known as Aronszajn-Donoghue theory was given in [75]. Extensions of Aronszajn-Donoghue theory to the case when the spectral measure is associated with a perturbation vector ϕ ∈ H −2 (A) can found in [5] and [45], but here we take ϕ ∈ H −1 (A) unless otherwise mentioned. The results compare the spectral measures µ and µ γ of the unperturbed and the perturbed operators and are expressed through the scalarvalued Borel transform F γ (z) := R dµ γ (t) t − z for z ∈ C\R, (3.2) which is abbreviated F for γ = 0. One of the standard identities at the heart of the theory is often referred to as the Aronszajn-Krein formula F γ (z) = F (z)/(1+γF (z)). The distinction of whether or not a point has mass is encrypted in the functions F and G(x) := dµ(t) (x − t) 2 .S γ = x ∈ R lim y→0 F (x + iy) = −1/γ; G(x) = ∞ , P γ = x ∈ R lim y→0 F (x + iy) = −1/γ; G(x) < ∞ , and C = x ∈ R lim y→0 Im F (x + iy) = 0 , contain spectral information of the perturbed operator A γ as follows: (i) For fixed γ = 0, the sets S γ , P γ and C are mutually disjoint. (ii) Set P γ is the set of eigenvalues, and set C (S γ ) is a carrier for the absolutely (singular) continuous measure, respectively. (iii) For γ = β the singular parts of A γ and A β are mutually singular. Remark 3.4. Set X being a carrier for a measure τ means that τ (R\X) = 0. Any (measurable) set that contains the support of a measure is also a carrier. Since we do not require a carrier to be closed, there may be carrier sets that are strictly contained in the support of a measure. The density function of the absolutely continuous measure and the pure point masses of A γ are completely described by the following result. Proposition 3.5. Assume that γ = 0. (i) For λ ∈ P γ we have µ γ ({λ}) = 1 γ 2 G(λ) . (ii) The density function of the absolutely continuous part of A γ is given by dµ γ (x) dx = 1 π lim y→0 + Im F (x + iy) \|1 + γF (x + iy)\| 2 , with respect to Lebesgue a.e. x ∈ R. We mention that the limit in part (ii) of the proposition exists with respect to Lebesgue a.e. x. Indeed, by the Aronszajn-Krein formula Im F \|1+γF \| 2 = Im F γ , and F γ is analytic on the upper half-plane. A characterization of the singular continuous part of A γ has been sought after but is still outstanding. Only partial results have been established. Instead of elaborating on the details here, we refer the reader to [23,54,75] and the references therein. We also point the reader to [55] for a discussion of, and references for, the question: "How unstable can the singular spectrum become?" The measures µ and µ γ , which are the spectral measures associated with rank-one perturbations of self-adjoint operators, are associated with scalar Nevanlinna-Herglotz functions. These functions are analytic self-maps of the upper half plane C + and possess the Nevanlinna-Riesz-Herglotz representation F (z) = c + dz + R 1 t − z − t 1 + t 2 dµ(t), and µ is a measures which satisfies the decay condition R (1+t 2 ) −1 dµ(t) < ∞.Eq. (5) F (z) = −1/z δ {0} (t)dt Eq. (6) F (z) = ln(z) χ (−∞,0) (t)dt Eq. (7) F (z) = ln(−1/z) χ (0,∞) (t)dt Eq. (8) F (z) = z r − cos( rπ 2 ) \|t\| r π −1 sin(rπ)χ (−∞,0) dt, r ∈ (0, 1) Eq. (10) F (z) = tan(z) n∈Z δ {nπ} (t)dt Eq. (17) F (z) = ln z−t1 z−t2 χ [t1,t2] (t)dt with t 1 < t 2 Aspects of Unitary Rank-One Perturbations and Model Theory Consider the unitary rank-one perturbation problem given by equation (1.4). Let µ α be the spectral measure of U α with respect to the * -cyclic vector ϕ, which is simultaneously also * -cyclic for U α for all α ∈ T. Then, the Spectral Theorem says U α can be represented by the operator that acts via multiplication by the independent variable on the space L 2 (µ α ). The operator U 0 is well-known to be a completely non-unitary contraction, i.e. it is not unitary on any of its invariant subspaces. Therefore, it (and hence the family of measures {µ α }) corresponds to the compression of the shift operator in a model representation associated with a characteristic function θ. Studying the intricacies of these model representations emerges as one of the main strategies in this field. Model spaces are subspaces of a weighted L 2 -space, of which we discuss several: the one by Clark, which resembles a simplified Sz.-Nagy-Foiaş model; the one by de Branges-Rovnyak which was e.g. studied by the Sarason school; and an overarching description of model theory developed by Nikolski-Vasyunin. This final formulation essentially incorporates the former ones by choosing an appropriate weight function. Aleksandrov-Clark Theory and Sz.-Nagy-Foiaş Model for Perturbations with Purely Singular Spectrum A seminal paper by Clark [24] laid the foundation that connects rank-one perturbations with reproducing kernel Hilbert spaces. The field has since grown into what is now known as Aleksandrov-Clark theory, honoring the deep insights gained by Aleksandrov about Clark measures -especially in the presence of an absolutely continuous component. A nice exposition of Aleksandrov-Clark theory can be found in [23], which we mostly follow along with in this section. We refer readers interested in a more general exposition of the Sz.-Nagy-Foiaş model spaces to [76]. For roughly the second half of this subsection, we work with characteristic functions that are inner, or equivalently, within the Clark setting of purely singular spectral measures. For an analytic function θ : D → D and a point α ∈ T, the function u α (z) := ℜ α + θ(z) α − θ(z) = 1 − \|θ(z)\| 2 \|α − θ(z)\| 2 ,(4.1) is positive and harmonic on D. For each α, a theorem by Herglotz [34] says this function corresponds uniquely to a positive measure µ α with u α = P µ α . Here, P µ α = T 1−\|z\| 2 \|ζ−z\| 2 dµ α (ζ) is the Poisson integral of µ α . We let A θ := {µ α : α ∈ T} denote the family of measures associated with the function θ. We will call A θ the family of Clark measures of θ when θ is an inner function, i.e. a bounded analytic function with unit modulus a.e. on T. Note that when θ is a general analytic self-map of the disk, the family A θ is usually referred to as the Aleksandrov-Clark measures of θ. With the Herglotz transformation (Hµ)(z) = T ζ+z ζ−z dµ(ζ) of a measure µ = µ 1 , it can easily be verified that the function θ(z) := (Hµ)(z) − 1 (Hµ)(z) + 1 , (4.2) is an analytic self map of the disk. The condition θ(0) = 0 is equivalent to each µ α ∈ A θ being a probability measure [23, Proposition 9.1.8]. These Clark measures can be used to describe the unitary perturbations of an important operator. To do so, we define the shift operator S : H 2 → H 2 by (Sf )(z) = zf (z), where H 2 = H 2 (D) denotes the Hardy space. Likewise, for later, we define the backward shift operator to be ( S * f )(z) = f (z)−f (0) z . Beurling's Theorem [20] then says that the S-invariant subspaces of H 2 are exactly those that can be written as θH 2 for some inner function θ. In order to take advantage of this relationship, we now assume θ is an inner function with θ(0) = 0. Assuming that θ(0) = 0 is not essential, but rather a convenience. Sometimes we will refer to such functions as characteristic functions. Given such a θ, the Sz.-Nagy-Foiaş model space [76] can then be defined as K θ := H 2 ⊖ θH 2 . (4.3) Beurling's Theorem further implies that S * -invariant subspaces of H 2 are simply model spaces K θ corresponding to some inner θ. On the side, we mention two major advances in complex analysis: (i) Douglas-Shapiro-Shields [27] have shown that for f ∈ H 2 , f ∈ K θ if and only if the meromorphic function f /θ on D has a pseudo-continuation to a function f θ ∈ H 2 (C \ D) with f θ (∞) = 0 (also see [23,Theorem 8.2.5]). The analogous result was also shown there to hold for conjugate pairs of H p spaces. (ii) A milestone has been achieved with the Ahern-Clark Theorems [2,3] with respect to understanding when the Julia-Carathéodory angular derivative exists. This result was generalized by Fricain-Mashreghi [31] to a characterization based on the existence of radial limits for higher derivatives. Also see the survey by Garcia-Ross [32, Theorem 6.11] for a summary. Moving on with our program, let P θ be the orthogonal projection of H 2 onto K θ . The compression of the shift operator is thus defined as S θ = P θ S\| K θ . This allows us to write the family of rank-one perturbations on K θ : V α f = S θ f + α f, θ z 1, with α ∈ T. (4.4) In particular, the following theorem of Clark says that these are the only unitary rank-one perturbations of S θ . . Any operator X that is both unitary and a rank-one perturbation of S θ can be written as X = V α for some α ∈ T. Let µ α be the Clark measure associated with the inner function θ and the point α ∈ T. Since V α is a cyclic unitary operator, the spectral theorem says that V α can be represented as multiplication by the independent variable on some L 2 (ν) space. It turns out that the space L 2 (ν) can be canonically identified with L 2 (µ α ). Let M be the operator on the space L 2 (µ α ) acting via multiplication by the independent variable. Then, the unitary operator that intertwines, C α M = V α C α , and maps the constant function 1 ∈ L 2 (µ α ) to some vector in the defect space Ran (I − S * θ S θ ) 1/2 is called the adjoint Clark operator. It is given by the normalized Cauchy transform C α : L 2 (µ α ) → Hol(D) with (C α g)(z) := K(gdµ α ) Kµ α , where K is the Cauchy transform (Kν)(z) = T dν(ζ) 1−zζ . The Clark operator is often denoted by Φ in literature, so that C α = Φ * . These representations gives us access to spectral information regarding the Clark family {µ α }, α ∈ T. The Sz.-Nagy-Foiaş representation simplifies to the setting described in this subsection precisely when operator V 1 has no absolutely continuous part (or, equivalently, when the characteristic function θ is inner). This poses a significant restriction. The de Branges-Rovnyak model is an alternative representation of the situation under weaker conditions. In the most general Aleksandrov-Clark situation, one is required to deal with the full two-storied Sz.-Nagy-Foiaş model space K θ = H 2 clos ∆L 2 ⊖ θ ∆ H 2 , instead of just the first component as in (4.3). The defect function ∆ is ∆(z) = (1 − θ(z) * θ(z)) 1/2 for z ∈ T. For further reference, see [62, Section 1.3.5]. de Branges-Rovnyak Model and Perturbations in the Extreme Case In this subsection, we assume that the characteristic function θ is an extreme point, i.e. that T ln(1 − \|θ(z)\|)dm(z) = −∞. It is well-known that θ extreme if and only L 2 (µ) = H 2 (µ) for the corresponding Aleksandrov-Clark measure µ = µ 1 . This situation is ideal for the de Branges-Rovnyak model space, as it now reduces from two components K θ = g + g − : g + ∈ H 2 , g − ∈ H 2 − , g − − θ * g + ∈ ∆L 2 to a one component space. Here we used the notation H 2 − := L 2 ⊖ H 2 . We describe the reduced one-component de Branges-Rovnyak model space: So, assume θ : H 2 → H 2 is extreme. Then the de Branges-Rovnyak model space H(θ) ⊂ H 2 consists of functions in the range space of the defect operator, i.e. H(θ) = (I − \|θ\| 2 ) 1/2 H 2 . The canonical norm on this space is the range norm which arises by taking the minimal norm of the pre-image of an element from H(θ). Much of the success of this approach is based upon the fact that H(θ) is a reproducing kernel Hilbert space with reproducing kernel k θ w (z) = 1−θ(w)θ(z) 1−wz . The deep structure of this space is the focus of [71]. Here we only mention a few items relevant to perturbation theory. We will omit other interesting topics such as multipliers of H(θ), the theory regarding the Julia-Carathéodory angular derivatives and Denjoy-Wolff points -all of which are detailed in [71]. The connection with the corresponding Aleksandrov-Clark measure µ is made through equation (4.2), see e.g. [71,Chapter III]. Much of the development in this area is attributed to the dissertation of Ball [11]. For instance, it was shown there that the measure µ has an atom at a point z 0 ∈ T if and only if the function θ(z)−1 z−z0 belongs to H(θ), see e.g. [71, Section (III-12)]. General perturbations and Nikolski-Vasyunin Model Theory Not all rank-one perturbations satisfy any of the conditions under which we can use the representations detailed in Subsections 4.1 and 4.2. Model theory for unitary perturbations in the general setting is much more complicated. Instead of a one-story model space, the general setting requires a two-story model space. While this description is superior in abstraction and admits more general settings, the models discussed in Subsections 4.1 and 4.2 have provided many deep insights over the years. An overarching treatise of the Sz.-Nagy-Foiaş, the de Branges-Rovnyak model space and other model spaces (e.g. the one studied by Pavlov) was achieved by Nikolski-Vasyunin [62,63,64,65]. There, a general so-called transcription free model space was introduced as a subspace of a (possibly) two-storied weighted space L 2 (D * ⊕ D, W ) on the unit circle. Here, the defect spaces of contraction V 0 are given by D = clos Ran (I − S * θ S θ ) 1/2 and D * = clos Ran (I − S θ S * θ ) 1/2 . We also note that the defect spaces D and D * were identified with T in Subsections 4.1 and 4.2. This L 2 space then reduces to the Sz.-Nagy-Foiaş, the de Branges-Rovnyak, the Pavlov model spaces, and other transcriptions by making specific choices of the weight W . The connection to rank-one perturbations comes from the dependence of this W on the characteristic function θ. General rank-one perturbations were studied in Liaw-Treil [52]. This subject is included in the lecture notes by Liaw-Treil [55] on the relationship between rank-one perturbations and singular integral operators. Instead of repeating large chunks of information here, we refer the reader to those lecture notes. Self-Adjoint Finite-Rank Perturbations We adapt the self-adjoint finite-rank setup given in (1.1) to account for singular perturbation vectors, which are useful in many applications. We mostly follow along with [6,Ch. 3]. We begin by defining the coordinate operator B : C d → H −2 (A) that takes the standard basis {e k } d k=1 ⊂ C d to {ϕ k } d k=1 ⊂ H −2 (A) . Note that we are changing notation slightly from Section 1, as we used to think of B as an operator B : C d → Ran B that was invertible. As before, we assume without loss of generality the invertibility of B on its range. Consider finite-rank perturbations of a self-adjoint operator A on the separable Hilbert space H given by A Γ = A + BΓB * ,(5.1) where Γ is a Hermitian d × d matrix and the operator BΓB * is an operator of rank d from the Hilbert space H 2 (A) to the Hilbert space H −2 (A). Note that we can assume without loss of generality that the matrix Γ is invertible. If Γ is not invertible, then the orthogonal complement to the kernel of the operator Γ yields a finite-rank operator of rank strictly less than d determined by a non-degenerate Hermitian matrix. The vectors ϕ k can be thought of as modifying the domain of A by d dimensions that are in H −2 (A). However, to ensure that each of these vectors are non-degenerate and adding new dimensions, we will call the set of vectors ϕ k ∈ H −2 (A)\H, k = 1, . . . , d, H-independent if and only if the equality d k=1 c k ϕ k ∈ H, c k ∈ C, implies c 1 = c 2 = · · · = c d = 0. If a desired set is not H-independent, then the matrix BΓB * will not be invertible and define a degenerate perturbation of rank strictly less than d. For this reason, we consider only H-independent perturbations. On the side we mention that Ker(BΓB * ) = Ker(B * ), because we are assuming Γ to be invertible and H-independence of ϕ k . Note that the vectors ϕ k having unit norm in H −2 (A) is not a restriction, as every H-independent system {ϕ k } can be orthonormalized. We assume unit norm in the following discussions and results. Singular Finite-Rank Perturbations If we let the vectors ϕ k , k = 1, . . . , d be H-independent, all vectors ψ ∈ Dom(A 0 * ) can be represented as: ψ = ψ + d k=1 a +k (ψ)(A − iI) −1 ϕ k + a −k (ψ)(A + iI) −1 ϕ k , (5.2) where ψ ∈ Dom(A 0 ), a ± (ψ) ∈ C. The theory of self-adjoint extensions of symmetric differential operators, commonly referred to as Glazman-Krein-Naimark theory [4,61], should be compared to this setup. The Dom(A 0 ) should be thought of as a "minimal" domain for the operator A, as the domain is unaffected by the perturbation BΓB * and will be contained in the domains of all extensions. Likewise, the "maximal" domain is represented by Dom(A 0 * ) and equation (5.2) Self-Adjoint Extensions The self-adjoint finite-rank perturbation given by (5.1) can be adapted as an application to self-adjoint extension theory. Namely, self-adjoint extensions of the operator A 0 are parametrized by d × d unitary matrices by the classical Glazman-Krein-Naimark theory [4,61]. Let V be such a matrix and the vector notation a ± ≡ {a ± } d k=1 denote the coefficients from equation ( . Let ϕ k ∈ H −1 (A)\H be an H-independent basis such that (A − iI) −1 ϕ j , (A + iI) −1 ϕ k = δ jk , and let Γ be a Hermitian invertible matrix. Then the self-adjoint operator A Γ = A + BΓB * is the self-adjoint restriction of the operator A 0 * to the following domain Dom(A Γ ) ={ψ ∈ Dom(A 0 * ) : a + (ψ) = −(Γ −1 + F(i)) −1 (Γ −1 − F * (i)) a − (ψ)}, where F(i) = B(A Γ − iI) −1 B * . The notation F(i) comes from the Borel transform, which we focus on in Section 6. We have A 0 = A when Γ = 0. Further note that the matrix V = (Γ −1 + F(i)) −1 (Γ −1 − F * (i)) is unitary. Hence, the theorem says that if the vectors ϕ j and the desired perturbation Γ are known, then the domain of the self-adjoint extension can be written via the explicit unitary matrix V , as in the classical theory. However, this leads to the natural question: Given the domain of a selfadjoint extension in terms of V , can we recover the perturbation Γ responsible for this domain? The answer is given by the following result. is a finite dimensional additive perturbation of the operator A. In particular, the Hermitian invertible matrix Γ is given by Γ = −Re(F(i)) + i(I − V ) −1 (I + V ) −1 . The last formula necessitates the analysis of whether I − V is invertible. This distinction is handled in the proof, where it is determined that if I −V is not invertible, then there is a degeneracy in the choice of the vectors ϕ k . This means that the set of vectors {ϕ k } d k=1 contains extra elements because we can find a new set of elements {ϕ * k } d * k=1 , d * < d, such that the corresponding matrix V * has a trivial eigensubspace. The description of domains of self-adjoint extensions resulting from finite-rank perturbations with vectors from H −2 (A) are much more involved (see e.g. [7]), and while very interesting in their own right, fall outside the scope of our discussion. Some Applications of Singular Finite Rank Perturbations The singular finite-rank perturbation setup employed in this section has a wide array of applications. Perhaps, their most common uses include point interactions for differential operators via connections to distribution theory and singular potentials of Schrödinger operators. This is immediately evident from the rank-one case when considering changing boundary conditions of regular Sturm-Liouville operators, see [73,Section 11.6]. Several contributions to the finite-rank case can be found in [6]. These include the analysis of operators with generalized delta interactions to achieve both spectral and scattering results. It is also possible to consider infiniterank perturbations, under some simplifying assumptions, to help approach problems given by two-body, three-body and few-body models. Finally, we should mention that singular perturbations can be transcribed into the theory of rigged Hilbert spaces, i.e. [46]. This theory places a larger emphasis on properties of singular quadratic forms, which can also describe self-adjoint extensions. Specific extensions, such as the Friedrichs or von Neumann-Krein cases, are sometimes easier to formulate in this context. Various aspects of spectral theory for singular finite-rank perturbations of self-adjoint operators are detailed in [46, Section 9]. Spectral Theory of Self-Adjoint Finite-Rank Perturbations Consider the family of finite-rank perturbations A Γ = A + BΓB * , see (1.1), with cyclic subspace Ran B. It is well-known that Ran B is then also cyclic for A Γ for all symmetric Γ. For simplicity let us focus on bounded perturbations in this section. By the Spectral Theorem, this perturbation family corresponds to a family of matrix-valued spectral measures µ Γ through B * (A Γ − zI) −1 B = R dµ Γ (t) t − z for z ∈ C \ R. The right hand side is the matrix-valued Borel transform, F Γ (z) := R (t − z) −1 dµ Γ (t) . We obtain the scalar spectral measures µ Γ by taking the trace of µ Γ . This trace is a scalar-valued measure which recovers the spectrum of A Γ via σ(A Γ ) = supp µ Γ . However, to access more subtle information, we formulate some of the results of the field we define the family of matrix-valued functions W Γ by dµ Γ (t) = W Γ (t)dµ Γ (t). Finally, we arrive at (W Γ ) ac := dµ Γ /dx by taking a component-wise Randon-Nikodym derivative. Absolutely Continuous Spectrum and Scattering Theory The unitary equivalence of the absolutely continuous spectrum of operators that differ by a finite-rank perturbations is available through simply applying the Kato-Rosenblum Theorem 2.2. In the more general setting of compact perturbations, the standard proof relies on the existence of the wave operators. Namely, let P ac denote the orthogonal projection from the Hilbert space onto the absolutely continuous part of A. For self-adjoint A and compact self-adjoint K it was shown that the strong operator topology limit of e it(A+K) e −itA P ac exists (see [69,Theorem 1.6] and [44, Theorem 1]), which in turn yields the Kato-Rosenblum theorem. For finite-rank perturbations, a quicker proof of unitary equivalence of the absolutely continuous spectrum is available. This proof uses the Aronszajn-Krein relation F Γ = (I + FΓ) −1 F = F(I + FΓ) −1 , (6.1) of the matrix-valued Borel transforms and was first discovered by Kuroda [49]. Via efficient notation, and in a slightly different language, Liaw and Treil [53, Appendix A.1] present this proof in a format appropriate for a graduate course. Of course, scattering theory is able to give us more information by relating how wave operators, e.g. s − lim t→∞ e it(A+K) e −itA P ac , and their packets are affected by the perturbation. For an interesting exposition of scattering theory for finite-rank perturbations confer, e.g. [48,Ch. 4]. Applications in Mathematical Physics can also be found in [48,. Alternatively, the scattering theory of finite-rank perturbations can be analyzed using boundary triples, see e.g. [14]. Validating the observation that the behavior of the absolutely continuous spectrum is one of the easier objects to capture, we conclude this subsection with its full perturbation theoretic characterization. The density of the matrix-valued spectral measure of the perturbed operator (W Γ ) ac is determined (see [53,Lemma A.3]) in terms of that of the unperturbed operator W ac by W Γ ac (x) = lim y→0 + (I + F(x + iy) * Γ) −1 W ac (x) lim y→0 + (I + ΓF(x + iy)) −1 , with respect to Lebesgue a.e. x ∈ R. In Equation (8.1) below, we also include a full description of the perturbed operator's matrix-density in terms of the matrix characteristic function of a corresponding model representation. Vector Mutually Singular Parts As evidenced by much research in the field, working with the singular spectrum will require a more subtle analysis than is necessary for the absolutely continuous part. From a naive perspective, the task at hand is to attempt to obtain some information about non-tangential boundary values z → λ of matrix-valued analytic functions on D for (µ Γ ) s -a.e. λ ∈ T. As we discuss in Section 9.2, Poltoratski's Theorem does not hold in the matrix-valued setting. Yet some positive results prevail. Recall the Aronszajn-Donoghue Theorem, which states the mutual singularity of the singular parts under rank-one perturbations, see item (iii) of Theorem 3.3. For finite-rank perturbations it is easy to construct examples for which two different perturbed operators have the same eigenvalue by taking direct sums of rank-one perturbations. The eigenvalues of the different components are completely independent from one another. Hence, a literal extension of this Aronszajn-Donoghue result cannot be true for the scalarvalued spectral measure. Through defining a vector-valued analog of the mutual singularity of matrix measures, Liaw-Treil [53, Theorem 6.2] achieved such a generalization of the Aronszajn-Donoghue Theorem. The scalar-valued spectral measures are also restricted: Theorem 6.1. [53, Theorem 6.3] Fix a singular scalar Radon measure ν, and d× d-matrices Γ > 0 and self-adjoint Γ 0 . Then the scalar spectral measures of A Γ0+tΓ are mutually singular with respect to ν for all except maybe countably many t ∈ R. Equivalence Classes and Spectral Multiplicity In [33], Gesztesy-Tsekanovskii obtained structural results for Nevanlinna-Herglotz functions that are applicable to finite-rank perturbations. Under the assumption that ker(I + ΓF(z)) = {0} for all z ∈ C + , some of these results resemble the Kato-Rosenblum Theorem 2.2 and Aronszajn-Donoghue Theorem 3.3. We begin by introducing the following sets, where 1 ≤ r ≤ d: S r (µ) ac = x ∈ R lim y→0 + F(x + iy) exists finitely, and lim y→0 + rank(Im(F(x + i0))) = r , S (µ) ac = d r=1 S r (µ) ac . Here, the existence of matrix limits are understood entrywise. Consider the equivalence classes of S r (µ Γ ) ac and S(µ Γ ) ac associated with F Γ (z); and denote them by E r (µ Γ ) ac and E(µ Γ ) ac , respectively. In this setting, Gesztesy-Tsekanovskii [33, Theorem 6.6] 1 have shown that: 1. For 1 ≤ r ≤ d, the classes E r (µ Γ ) ac , and E(µ Γ ) ac are independent of Γ. 2. Suppose µ Γ1 is a discrete point measure for some Γ 1 . Then µ Γ is a discrete point measure for all Γ. 3. The set of those x ∈ R for which, simultaneously, there is no Γ such that lim y→0 + Im F Γ (x + iy) exists and lim y→0 + det Im F Γ (x + iy) = 0, is a subset of E d (µ Γ ) ac . Model Theory of Finite-Rank Unitary Perturbations Taking a different route than Clark theory, we follow [52] to set up the problem. This perspective is more natural here, since we are interested in perturbation theory. It allows us to bypass some minor technical road blocks that arise for finite-rank perturbations (when connecting the family measures with the family of operators). Some of the model theory of rank-one perturbations carries over to model theory of finite-rank setting with the added complication that one has to keep track of the order of matrix products. For example, the description of the absolutely continuous part in terms of the characteristic function has an analog for finite-rank perturbations. Other results such as identifying when the extreme situation occurs (when the de Branges-Rovnyak transcription simplifies) need to be slightly adjusted. For this particular question, taking the trace will be appropriate. In Subsection 7.2 we briefly mention some other representations using Krein spaces and reproducing kernel Hilbert spaces. Setup and Model Spaces Recall the setting for unitary finite-rank perturbations U α = U+J(α−I)J * U with unitary α, as detailed in and around (1.3). It is well-known that Ran J also forms a * -cyclic subspace for the perturbed operators U α . Let µ α be the family of matrix-valued spectral measures on T given by the Spectral Theorem through J * (I − zU * α ) −1 J = T dµ α (ζ) 1 − zζ for z ∈ C \ T. (7.1) It is not hard to see that the operator U α is a completely non-unitary contraction for matrices α with α < 1. This provides us access to the associated model theory. Referring the reader to [ It is worth mentioning that in starting with (1.3) we do not really make a hidden assumption. We would recover the general starting point of [59] by taking U α with strict contraction α instead of U 0 with α = 0. As when dealing with rank-one perturbations, operator U 0 is unitarily equivalent to the compressed shift operator on a transcription free model space. Similar to the rank-one setting, here, the Sz.-Nagy-Foiaş model space reduces to H 2 (C d )⊖θH 2 (C d ), if and only if θ is inner (i.e. has non-tangential boundary values that are unitary with respect to Lebesgue measure a.e. on T), if and only if U has purely singular spectrum. See e.g. [52,Corollary 5.8] for a reference of the second equivalence. Also see [26]. The de Branges-Rovnyak model space reduces to one-story if and only if θ is an extreme point, and if and only if T tr(ln(I − \|θ(z)\|)dm(z) = −∞ (see [59,Theorem 4.3.1]). There seems to be no immediate description of the extreme property in terms of the operator U or the perturbation family U α . In any case, the de Branges-Rovnyak model space reduces at times when Sz.-Nagy-Foiaş model does not. When dealing with the general case of finite-rank unitary perturbations, no such reduction can be assumed a priori. This general case is the subject of Liaw-Treil [52] and some of Martin [59] holds in this generality. In [52] Liaw-Treil study the general Nikolski-Vasyunin model of finiterank Aleksandrov-Clark perturbations. Determining the unitary operator realizing this representation yields a generalization of the Clark-type operator and its adjoint. For the adjoint, the transcription choice leading to the full Sz.-Nagy-Foiaş model features a generalization of the normalized Cauchy transform. Krein Spaces and Reproducing Kernel Hilbert Spaces in Applications Krein spaces are indefinite inner product spaces; spaces which possess a Hermitian sesquilinear form that allows elements to have positive or negative values for their "norm." A Hilbert inner product can be canonically defined on Krein spaces, so they can be viewed as a direct sum of Hilbert spaces [18]. In particular, Krein spaces are naturally defined as extension spaces for symmetric operators with equal deficiency indices and have their own tools to determine spectral properties. Applications to the spectral analysis of direct sums of indefinite Sturm-Liouville operators is possible because so-called definitizable operators in Krein spaces are stable under finite-rank perturbations [13]. Furthermore, compact perturbations of self-adjoint operators in Krein spaces also preserve certain spectral points [10], and the spectral subspaces corresponding to sufficiently small surrounding neighborhoods of these points are actually Pontryagin spaces (simpler versions of Krein spaces). Representations of symmetric operators with equal deficiency indices are also possible in reproducing kernel Hilbert spaces; Hilbert spaces of functions where point evaluation is a continuous linear functional. Among other results, Aleman-Martin-Ross [9] carried out representations for Sturm-Liouville and Schrödinger (differential) operators, Toeplitz operators and infinite Jacobi matrices. The idea becomes that for each such example, the structure of the model space hosts the full information (including spectral properties) of the symmetric operator. In [9,Section 5], the characteristic functions corrresponding to these examples are computed explicitly; so that the de Branges-Rovnyak model space (which is a reproducing kernel Hilbert spaces) is completely determined. Representations in the Herglotz space constitute another interesting topic in [9]. Spectral Theory of Finite-Rank Unitary Perturbations Consider the setting of Subsection 7.1. Recall that U α for unitary α is a unitary rank d perturbation of a unitary operator U, and recall that (7.1) defines the family of associated matrix-valued spectral measures µ α . In analogy to the self-adoint setting, we define the family of matrix-valued functions W α by dµ α (t) = W α (t)dµ α (t). Taking a component-wise Randon-Nikodym derivative, we arrive at (W α ) ac := dµ α /dx. Further, recall that θ is the matrixvalued characteristic function of the completely non-unitary contraction U 0 , and that ∆(z) = (I − θ * (z)θ(z)) 1/2 . A complete explicit description of the matrix-valued spectral measures of U α in terms of the characteristic function is currently not available. In fact, the theory for finite-rank perturbations is lagging behind what is known for rank-one perturbations, see Theorem 4.2. This problem has been in recent years and continues to be a field of active study. Here we explain some results in this direction. Spectral Properties in Terms of the Characteristic Function The location of the spectrum of the perturbed operator is captured by: [60,Corollary 4.4]). The spectrum of U α consists of those points λ ∈ T at which either θ cannot be analytically continued across λ, or θ(λ) is analytically continuable with θ(λ) − α not invertible. Theorem 8.1 (see Mitkovski In combination with von Neumann's theorem, Theorem 2.1, a characterization by Lifshitz [56,Theorem 4] of the essential spectrum of U 0 says that it consists of those points λ ∈ T for which (at least) one of the following conditions fails: • θ is analytic on some open neighborhood of λ, • there is a neighborhood N λ of λ so that θ is unitary for all λ ∈ N λ ∩ T. For the absolutely continuous part of the perturbed operator's spectral measure, a full matrix-version becomes available upon combination of Liaw-Treil [52, Theorem 5.6] with the Herglotz formula (7.3) for U α , which is obtained from that for U by simultaneously replacing µ by µ α and θ by θα * . Namely, we have (I n − αb(λ) * )W α (λ)(I n − b(λ)α * ) = (∆(λ)) 2 for Lebesgue a.e. λ ∈ T, (8.1) in the sense of non-tangential boundary limits. In particular, for the absolutely continuous part, the multiplicity function is given by a non-tangential limit rank (W α (λ)) ac = lim z→λ rank ∆(z). Slightly weaker results are contained in Douglas-Liaw [26]. Singular Part in Terms of the Characteristic Function As for rank-one perturbations, capturing the singular part is a more difficult venture. The main problem here is that Poltoratski's Theorem requires a major adjustment (see Subsection 9.2 for a discussion). As a result, a description of the singular part in terms of the characteristic function is still outstanding. For regular points (i.e. those that lie in the complement of the essential spectrum), both eigenvalues and eigenvectors of U α are described in Martin [59,Proposition 5.2.2]. Namely, a regular point λ ∈ T is an eigenvalue of U α if and only if lim z→λ (αθ * (z) − U * ) exists and is not invertible. Eigenvectors are those functions χ {λ} x with x ∈ C d ∩ ker(αθ * (λ) − U * ) and where χ denotes the characteristic function. In that same proposition, a necessary and sufficient condition is provided for a point to not be an eigenvalue of U α for any unitary α. There are many open questions remaining in this area. Some of them are currently being investigated. Appendix: Brief Summaries of Other Closely Related Topics We discuss Aleksandrov Spectral Averaging and Poltoratski's Theorem. These are both central tools in the field. Thereafter, we briefly illuminate the Simon-Wolff Theorem to which we attribute some of the popularity of the topic among mathematical physicists. We wrap up with a promising direction connecting the field to modern function theoretic operator theory. Aleksandrov Spectral Averaging Undoubtedly one of the most celebrated results of the field is the following averaging formula. On the side we mention that we can retrieve restrictions on the Aleksandrov-Clark family of spectral measures, e.g., by choosing the function g to be the characteristic function of a set of Lebesgue measure zero. We being by considering the rank-one setting and will then turn to finite-rank. For part of this subsection we follow [23]. Theorem 9.1 (Aleksandrov dinsintegration theorem, see [8] and [23,Theorem 9.4.11]). For g ∈ L 1 (T) we have g(ζ)dµ α (ζ) dm(α) = g(ζ)dm(ζ). It is one of the main aspects of Theorem 9.1 that for f ∈ L 1 (T), the function Gf makes sense for Lebesgue a.e. α ∈ T and that it is integrable. It turns out that G satisfies even more subtle mapping properties. We briefly summarize those before we explain what is known for finite-rank perturbations. Due to the assumption that θ(0) = 0, we see from Subsection 4.1 that µ α = 1 and so Gf ∞ ≤ f ∞ . Note also that the function Gf is continuous whenever f is continuous. Aleksandrov Spectral Averaging for the Drury-Arveson space in the setting for inner characteristic functions was achieved by Jury [37,Theorem 2.9]. For more on Aleksandrov-Clark theory for the Drury-Arveson space see Subsection 9.4 and the references therein. Poltoratski's Theorem Deep at the heart of many results in Aleksandrov-Clark theory lies the celebrated result (proved by Poltoratski in [66]) stating that for a Radon measure τ on T and f ∈ L 2 (τ ) the normalized Cauchy transform Cf τ (z) Cτ (z) possesses nontangential boundary values z → λ for τ s -a.e. λ ∈ T. This result is so important, because it empowers us to study the behavior of the spectral measure on sets that are of Lebesgue measure zero. In particular, one can sometimes use Poltoratski's Theorem to retrieve information about the singular parts of the spectral measures. Direct sum examples of scalar characteristic functions immediately show that a literal extension of the statement of Poltoratski's Theorem is not possible to the finite-rank setting. Nonetheless, Theorem 3] have proved a finite-rank analog which features a matrix-valued numerator alongside a scalar-valued denominator as well as an multiplication by a left inverse of the coordinate map J in (1.3). This left inverse 'automatically' annihilates directions in which the limit of the ratio does not exist. Simon-Wolff Criterion In [72, Theorem 3 of Section 2] Simon and Wolff provided a characterizationformulated in terms of the spectral measure µ-of when rank-one perturbation problems A γ are pure point for Lebesgue a.e. parameters γ ∈ R. They applied their result to showing that the one-dimensional discrete random Schrödinger operator exhibits so-called Anderson localization, see [35,75]. The idea of the Simon-Wolff localization proof was to sweep through the parameter domain for the perturbed operators' random coupling constants. In Poltoratski [67] the Simon-Wolff Theorem was extended to from the rank-one to the finite-rank setting. Simon-Wolff's celebrated work initiated further applications of rankone perturbations to a generalization of random Schrödinger operators called Anderson-type Hamiltonians, see e.g. [35]. Anderson-type Hamiltonians are obtained from perturbing a self-adjoint operator by countably infintely many rank-one perturbations, each coupled by a random variable. More concretely, they are of the form A ω = A + ω i · , ϕ i ϕ i , where {ϕ i } forms an orthonormal basis of H, and ω i are independent random variables that are chosen in accordance with an identical probability distribution. In view of Section 2, the fact that the discrete random Schrödinger operator features an almost surely non-compact perturbation operator underlines the level of difficulty in dealing with such objects. In Jaksiç-Last [35,36], these methods are utilized to prove the almost sure cyclicity of the singular spectrum of the Anderson-type Hamiltonian. And in Abakumov-Liaw-Poltoratski [1], it is shown that under some condition any non-trivial vector is cyclic. In Liaw [50] these results are applied to numerically support a delocalization conjecture for the 2-dimensional discrete random Schrödinger operator. Functions of Several Variables Recall that the Drury-Arveson space H 2 (B n ) is the reproducing kernel Hilbert space of functions on the open unit ball B n of C n , n ∈ N, that arises from the reproducing kernel k(z, w) = (1 − z, w C n ) −1 with z, w ∈ B n . Jury [37] extended much of the de Branges-Rovnyak construction of Clark theory to H 2 (B n ). As before, the de Branges-Rovnyak model spaces H(θ) are contractively contained in H 2 (B n ). The family of Clark measure is replaced by a family of states on some noncommutative operator system. The backward shift is replaced by a canonical solution to the Gleason problem in H(θ). An extension of some of Jury's work to non-inner but so-called quasi-extreme characteristic functions was carried out in Jury-Martin [38]. There, the Aleksandrov-Clark measures are necessarily generalized to certain positive linear functionals. For related work on function analytic noncommutative operator theory, we refer the reader to a series of papers by Jury and Martin [39,40,41]. On the side we mention that it is not immediately clear whether a perturbation problem corresponds to this Aleksandrov-Clark theory for functions of several variables. The state of affairs for self-adjoint finite-rank perturbation problems is similar. The conditions and explicit formulas necessary to pose a well-defined problem in this area have not been investigated, to the best knowledge of the authors. However, there exists a generalization of Nevanlinna-Herglotz functions to several variables (see e.g. [58]) whose integral representation should form a framework for the analog of the Borel transform in (3.2). Theorem 2.1 (von Neumann, see e.g. [21, Theorems 3 and 6 of Ch. 9]). Let A and B be bounded self-adjoint operators. Then B is compact if and only if the essential spectra of A and A + B are the same. Theorem 2. 2 ( 2[44, Theorem 1] and [69, Theorem 1.6]). Let A and B be self-adjoint operators and assume B ∈ S 1 . Then the absolutely continuous parts of A and A + B are unitarily equivalent. s ≥ 0, define the space H s (A) to consist of ϕ from H for which the s-norm ϕ s := (\|A\| + I) s/2 ϕ H , (3.1) is bounded. The space H s (A) equipped with the norm · s is complete. The adjoint spaces, formed by taking the linear bounded functionals on H s (A), are used to define these spaces for negative indices, i.e. H −s (A) := H * s (A). The corresponding norm in the space H −s (A) is thus defined by (3.1) as well. The collection of these H s (A) spaces will be called the scale of Hilbert spaces associated with the self-adjoint operator A. Remark 3. 2 . 2The case H −1 (A) for the self-adjoint setting most closely aligns with unitary perturbations, see [55, pp. 124-128]. It is not immediately clear how the more singular perturbations, H −n (A) for n > 1, translate to the unitary side. Theorem 3 . 3 ( 33Aronszajn-Donoghue theory, e.g. [75, Theorem 12.2]). When γ = 0, the sets Examples of Nevanlinna-Herglotz functions and corresponding spectral measures.The examples Eq. (6)-(8) use the principal value of the logarithm. The integration variable is λ. The first column contains references to equations in[33, App. A]. Other examples and their sources can also be found there. Theorem 4. 1 ( 1Clark [24, Remark 2.3]) Theorem 4.2 (see e.g.[23, Proposition 9.1.14] and[32, Proposition 8.3]). In the above setting we have:1. (dµ α ) ac = u α dm (with u α (z) = (1 − \|θ(z)\| 2 )\|α − θ(z)\| −2 as in (4.1)). 2. µ α ⊥ µ β for all α = β, β ∈ T.3. µ α has a point mass at ζ ∈ T if and only if θ(ζ) = α and \|θ ′ (ζ)\| < ∞.In that case this point mass is given byµ α ({ζ}) = \|θ ′ (ζ)\| −1 . 4. The set {ζ ∈ T : lim r→1 − θ(rζ) =α} is a carrier for µ α . (Recall that µ α is purely singular in the Clark setting.) This result is in direct correspondence with the Aronszajn-Donoghue Theorem 3.3 above. Also, observe that a point mass equals the reciprocal magnitude of the derivative of the Borel transform in the self-adjoint setting, and of the Cauchy transform in the unitary setting. In fact, [23, Item (1) of Corollary 9.1.24] offers a finer carrier of the singular spectrum in terms of the lower Dini derivative of µ α . The operator A Γ on the domain Dom(A) is symmetric as an operator acting from H 2 (A) = Dom(A) to H −2 (A). The self-adjoint operator given by equation (5.1) coincides with one of the self-adjoint extensions of the operator A 0 equal to the operator A restricted to the domain Dom(A 0 ) = Dom(A) ∩ Ker(BΓB * ). Lemma 5 . 1 ([ 6 , 516Lemma 3.1.1]). Suppose that the vectors ϕ k ∈ H −2 (A)\H, k = 1, . . . , d, are H-independent and form an orthonormal system in H −2 (A). Then the restriction A 0 of the operator A to the domain Dom(A 0 ) is a densely defined symmetric operator with the deficiency indices (d, d). is a modified version of the classical von Neumann's formula (the maximal domain is the direct sum of the minimal domain and the defect spaces). The key space Dom(A 0 * ) should thus be considered as a finite dimensional extension of the spaceH 2 (A) in the sense that Dom(A 0 * ) is isomorphic to the direct sum of H 2 (A) and C d .We also emphasize that the spaces A 0 , defined via Lemma 5.1, and A 0 * , are dependent on the choice of the vectors {ϕ k } d k=1 . We can thus formulate a second scale of Hilbert spacesDom(A) = H 2 (A) ⊂ Dom(A 0 * ) ⊂ H ⊂ Dom(A 0 * ) * ⊂ H −2 (A) = Dom(A) * ,which is constructed using both the operators A and BΓB * . The norms in H −2 (A) and H 2 (A) are the standard norms from Definition 3.1. We avoid most of the specific properties of these spaces and operators, but point out that the norm in the space Dom(A 0 * ) * is listed in[6, Equation (3.11)], near other pertinent facts. 5.2). The corresponding self-adjoint operator A(V ) coincides with the restriction of the operator A 0 * to the domainDom(A(V )) = {ψ ∈ Dom(A 0 * ) : −V a − (ψ) = a + (ψ)}. (5.3)We present an explicit connection between V and Γ in Lemma 5.3 below. The extension given by the matrix V = I coincides with the original operator A. This case is handled by classical self-adjoint extension theory. However, when the perturbing vectors {ϕ k } d k=1 belong to H −1 (A), descriptions of the corresponding domains become more difficult. , Lemma 3.1.2]). Let ϕ k ∈ H −1 (A)\H, k = 1, . . . , d, be an H-independent orthogonal system. If det V + [iI + Re(F(i))] −1 [iI − Re(F(i))] = 0 then the operator A 0 * restricted to the domain of functions {ψ ∈ Dom(A 0 * ) : −V a − (ψ) = a + (ψ)} 52 , 52Sections 3 and 4], we omit the details of showing that operator U 0 corresponds to the matrixvalued characteristic function θ(z) = (Kµ(z) − I)(Kµ(z)) −1 . (7.2) Here, the identity I maps D → D and K is the Cauchy transform of a matrix-valued measure (Kν)(z) = T dν(ζ) 1−zζ . It is not hard to see that the relation in (7.2) is equivalent to the Herglotz formula (Hµ)(z) = (I + θ(z))(I − θ(z)) −1 , (7.3) with the Herglotz transformation of a matrix-valued measure (Hν)(z) = T ζ+z ζ−z dν(ζ). Now, one can reason that replacing µ by µ α in (7.3) will result in replacing θ by θα * . And we arrive at the starting point of Aleksandrov-Clark Theory, see e.g. [59, Eq. (2.5)] when θ(0) = 0. bounded Borel function f on T, let (Gf )(α) := f (ζ)dµ α (ζ). (9.2) The Monotone Class Theorem (see i.e.[23, Theorem 9.4.3]) can be used to show that if f is a bounded Borel function, then Gf is also a bounded Borel function. Hence, the integral T (Gf )(α)dm(α), makes sense. In fact, the transformation G in (9.2) can be extended to many classes of functions. Not only do we have GC ⊂ C, CL ∞ ⊂ L ∞ , and GL 1 ⊂L 1 , but also GL p ⊂ L p (1 ≤ p ≤ ∞), G(BM O) ⊂ BM O, G(V M O) ⊂ V M O,and GB s pq ⊂ B s pq , where B s pq are the Besov classes, see [8]. Now, let us turn to what is known about Aleksandrov Spectral Averaging for finite-rank perturbations. In the unitary setting, a generalization of the Aleksandrov Spectral Averaging formula for continuous functions was obtained in Elliot [28] under extra conditions and in Martin [59, Theorem 3.2.3]. For self-adjoint operators a Aleksandrov-type Spectral Averaging formula was proved, Liaw-Treil [53, Theorems 4.1, 4.6]. These formulas imply restrictions on the singular parts of families of Aleksandrov-Clark measures. The examples that use F are more singular H −2 (A) perturbations. In order to give the reader additional intuition about these measures, we include some examples from [33, App. A].[33, App. A] Borel transform Spectral Measure dµ(t) Gesztesy-Tsekanovskii present these results for a slightly more general setting, when F and F Γ are related by a certain linear fractional transformation. Their presentation reduces to ours upon making the choices Γ 1,1 = Γ 2,2 = I and Γ 2,1 = 0 and Γ 1,2 = Γ. AcknowledgmentThe authors would like to thank Hari Bercovici and Tracy Weyand for pointing out references and suggesting phrases for some of the wording referring to their field of research. Special thanks to Alan Sola for reading large parts of this survey and for making useful suggestions. Cyclic vectors in rank-one perturbation problems. E Abakumov, C Liaw, A Poltoratski, J. Lond. Math. Soc. 882E. 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[ "RECOGNITION OF SYMMETRIES IN REVERSIBLE MAPS", "RECOGNITION OF SYMMETRIES IN REVERSIBLE MAPS" ]	[ "Patrícia H Baptistelli ", "Isabel S Labouriau ", "Miriam Manoel ", "Patrícia H Baptistelli ", "Isabel S Labouriau ", "Miriam Manoel ", "\nDepartamento de Matemática\nDepartamento de Matemática\nCentro de Matemática Universidade do Porto Rua do\nCCE Universidade Estadual de Maringá Av. Colombo\nCampo Alegre, 6875790, 87020-900, 4169-007Maringá, PortoBrazil, Portugal\n", "\nICMC Universidade de São Paulo\n13560-970 Caixa Postal 668São CarlosBrazil\n" ]	[ "Departamento de Matemática\nDepartamento de Matemática\nCentro de Matemática Universidade do Porto Rua do\nCCE Universidade Estadual de Maringá Av. Colombo\nCampo Alegre, 6875790, 87020-900, 4169-007Maringá, PortoBrazil, Portugal", "ICMC Universidade de São Paulo\n13560-970 Caixa Postal 668São CarlosBrazil" ]	[]	We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, when the diffeomorphism is the composition of transversal linear involutions.	10.1016/j.jmaa.2020.124348	[ "https://arxiv.org/pdf/1812.08727v1.pdf" ]	119,156,964	1812.08727	735a8f1b81bc8b4868cf22cf54761c9308c2d82f	RECOGNITION OF SYMMETRIES IN REVERSIBLE MAPS Patrícia H Baptistelli Isabel S Labouriau Miriam Manoel Patrícia H Baptistelli Isabel S Labouriau Miriam Manoel Departamento de Matemática Departamento de Matemática Centro de Matemática Universidade do Porto Rua do CCE Universidade Estadual de Maringá Av. Colombo Campo Alegre, 6875790, 87020-900, 4169-007Maringá, PortoBrazil, Portugal ICMC Universidade de São Paulo 13560-970 Caixa Postal 668São CarlosBrazil RECOGNITION OF SYMMETRIES IN REVERSIBLE MAPS 3780reversible mapinvolutionssymmetryfixed-point subspace We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, when the diffeomorphism is the composition of transversal linear involutions. Introduction Symmetry occurs in many different contexts. It has driven attention in many fields of Mathematics and related areas whenever existence and analysis of patterns become relevant. Symmetric objects have characteristic features that are not present in generic objects. Symmetry carries geometric information that facilitates the study of such objects. One particularly important kind of symmetry -or reversing symmetry, as we shall say -is given by an involution. For a local study in R n , this is defined as the germ of a diffeomorphism ϕ : (R n , 0) → (R n , 0) satisfying ϕ • ϕ = I n , namely a germ of diffeomorphism which is its own inverse. We point out that many results in the present work are algebraic, so it shall be clear that those also hold for general invertible maps. For the same reason, although formulated for diffeomorphisms, our results often do not require differentiability and hold for homeomorphisms. One branch of applications of reversing symmetries comes from dynamical systems. We refer to [5] for references on reversibility and related problems. For discrete dynamics governed by the iteration of a (germ of) diffeomorphism F : (R n , 0) → (R n , 0), we recall that this is called reversible under a (germ of) diffeomorphism R : (R n , 0) → (R n , 0), or simply R-reversible, if F • R = R • F −1 . In other words, the symmetric copy of a trajectory is also a trajectory with time reversed. If F happens to be reversible under an involution ϕ 1 , it follows that it can be decomposed as F = ϕ 1 • ϕ 2 , for some involution ϕ 2 . Yet, F is also ϕ 2 -reversible. As a consequence, contrary to the continuous case, reversible discrete dynamical systems always have more than one reversing involution. We remark that there exist reversing symmetries that are not involutions, even for linear isomorphisms; for example, 2 × 2 symmetric matrices are reversible under the rotation of π/2. Here we assume throughout that F has an involutory reversibility. In this case, interesting dynamics resides in the class governed by diffeomorphisms that possess an infinite number of involutions. In algebraic terms, if ϕ 1 and ϕ 2 generate an Abelian group, then these are simultaneously linearizable, as a consequence of the Bochner-Montgomery theorem about linearization of a compact group of transformations around a fixed point (see [8]). In our setting, the two involutions generate a discrete group which may be non-Abelian and is generally noncompact. Then, a natural question raised in this set-up regards the local linearization around a fixed point. The simultaneous linearization and transversality of ϕ 1 and ϕ 2 is a sufficient condition for the linearization of F (see [6] for more on this topic). Also, pairs of involutions are linearizable provided F is a hyperbolic germ of diffeomorphism [10]. The work in [7] extends this result to F normally hyperbolic. In the first case, the fixed-point set of F reduces to a point; in the second case, it can be a local submanifold with positive dimension. Normally hyperbolic examples are treated in [3]. In the present work, simultaneously linearizable involutions are treated as a particular case of our results. Sets of linear involutions are treated in Section 3 for the planar case and in Section 4 for higher dimensions. Another branch of applications comes from Singularity theory in the study of divergent diagrams of folds (see, for example, [6,9,11]). We recall that given a fold f : (R n , 0) → (R n , 0), there exists a unique nontrivial involution ϕ associated with f , that is, such that f • ϕ = f . In [6] the authors prove that equivalent classes of s-tuples of divergent diagram of folds, for s ≥ 1 finite, are described through the simultaneous equivalence of the associated s-tuples of involutions. Here we remark that the same holds for an infinite sequence of folds. In addition, it follows that the study of singular sets of folds is deduced directlly from the description of fixed-point subspaces of involutions we present here, once the singular set of f is precisely the fixed-point submanifold of the associated involution ϕ. Further investigation on this topic is carried out in [2]. Here is what we shall encounter in the next sections. In Section 2 we recall some basic definitions and properties; we also present the main general results regarding symmetries and reversing symmetries of a reversible germ of diffeomorphism F . We establish in Theorem 2.8 the existence of a chain of fixed-point subspaces of the (infinite) sequence of reversing involutions which must be tracked by the iterates of F . In the particular case when the fixed-point submanifolds of ϕ 1 and ϕ 2 have codimension 1, Theorem 2.13 describes the orbits of points in the complement in R n of these fixed-point submanifolds. The connected components of this complement are interchanged by F . We also use the chain of fixed-point subspaces to find periodic points. In Section 3 the results of the Section 2 are applied to the class of pairs of transversal linear involutions on the plane. This is done using the normal forms obtained in [6] under the equivalence given by simultaneous conjugacy. We also relate the reversible dynamics to the geometry of the chain of fixed-point subspaces of the reversing involutions. In Section 4 we use the results of Section 3 to extend the analysis to dimension greater or equal to three. Here we use again normal forms of transversal linear involutions given in [6] which, for almost all cases, are suspensions of the normal forms on the plane. In short, Section 2 contains general results on reversible diffeomorphisms. These results are illustrated by detailed descriptions of the geometry in two dimensions (Section 3) and in higher dimensions (Section 4). Reversibility and equivariance Let Ω be the group under composition of the invertible maps on (R n , 0) acting on (R n , 0) by the standard action of application, Ω × (R n , 0) → (R n , 0), (ϕ, x) → ϕx = ϕ(x). Definition 2.1. Let ϕ ∈ Ω. A germ of a diffeomorphism F : (R n , 0) → (R n , 0) is ϕ-equivariant if F • ϕ = ϕ • F . It is ϕ-reversible if F • ϕ = ϕ • F −1 . In the first case, ϕ is a symmetry of F and in the second case ϕ is a reversing symmetry of F . Note that if F is ϕ-reversible then so is F −1 . For a given germ of diffeomorphism F of (R n , 0) we denote by Γ + the group formed by the symmetries of F and by Γ − the set of reversing symmetries of F, that is, (1) Γ + = {ϕ ∈ Ω : F • ϕ = ϕ • F } and Γ − = {ϕ ∈ Ω : F • ϕ = ϕ • F −1 }. The set Γ − doesn't have a group structure, it is closed under inversion but composition of two resersing symmetries belongs to Γ + . Yet, we can write Γ − = δΓ + for any δ ∈ Γ − fixed (and arbitrary). In fact, for ϕ ∈ Γ + we have F • (δ • ϕ) = δ • F −1 • ϕ = (δ • ϕ) • F −1 , hence δ • ϕ ∈ Γ − . Conversely, if ϕ ∈ Γ − then ϕ = δ • (δ −1 • ϕ) ∈ δΓ + . Moreover, Γ + ∩ Γ − = ∅ if, and only if, Γ + = Γ − , which are equivalent to F 2 = I n , the germ of the identity map in R n . Definition 2.2. An involution is a germ of a diffeomorphism ϕ : (R n , 0) → (R n , 0) satisfying ϕ • ϕ = I n . From now, we assume that there exists an involution in Γ − . In [6] it has been recognized that, in this case, reversible diffeomorphisms are in 1-1 correspondence with pairs of involutions. In fact, if ϕ 1 is an involution then F is ϕ 1 -reversible if, and only if, F = ϕ 1 • ϕ 2 for the involution ϕ 2 = ϕ 1 • F . We now also remark that, in this case, F is ϕ 2 -reversible too, for F • ϕ 2 = ϕ 1 = ϕ −1 1 = ϕ 2 • ϕ −1 2 • ϕ −1 1 = ϕ 2 • F −1 . Hence F corresponds to the pair of reversing symmetries (ϕ 1 , ϕ 2 ). In what follows we show that, more than one pair, there are actually two infinite sequences of involutions ϕ k and ϕ k such that F is ϕ k -and ϕ k -reversible if F m = I n , for all m ∈ Z. We take (2) ϕ k = ϕ 2 • F k−2 , ϕ k = F k−1 • ϕ 1 , k ∈ N, k ≥ 1. This definition is consistent for k = 1 and k = 2, with ϕ 1 = ϕ 1 . Proposition 2.3. Let ϕ 1 : (R n , 0) → (R n , 0) be an involution and let F : (R n , 0) → (R n , 0) be a ϕ 1 -reversible germ of diffeomorphism such that F m = I n , for all m ∈ Z. Then F has an infinite group of symmetries and an infinite set of reversing symmetries. Proof. F is clearly a symmetry of F itself, so the subgroup generated by F , F = {F k : k ∈ Z}, is formed by symmetries, and then Γ + is infinite. Furthermore, all ϕ k and ϕ k defined in (2) are different elements in Γ − , so Γ − is infinite. Examples are given in Sections 3 and 4. Definition 2.4. Given a germ of a diffeomorphism F : (R n , 0) → (R n , 0), the F -orbit of a point x ∈ (R n , 0) is the ordered set {x k = F k (x) : k ∈ Z}. When F is clear from the context we just call this set the orbit of x. If F : (R n , 0) → (R n , 0) is the germ of a ϕ-reversible diffeomorphism, then ϕ maps the F -orbit of x ∈ (R n , 0) into the F −1 -orbit of ϕ(x) , preserving the order. 2.0.1. The fixed-point sets. Definition 2.5. The fixed-point set of a map-germ ϕ ∈ Ω is Fix(ϕ) = {x ∈ (R n , 0) : ϕ(x) = x} and the fixed-point set of a subgroup Σ ≤ Ω is Fix(Σ) = {x ∈ (R n , 0) : γx = x, ∀ γ ∈ Σ} . If ϕ is linear and Σ is a subgroup of the linear group GL(n) then Fix(ϕ) and Fix(Σ) are naturally extended to the whole R n as linear subspaces of R n . It follows that the fixed-point set of any involution ϕ : (R n , 0) → (R n , 0) is a smooth submanifold in (R n , 0), since ϕ is conjugate to the germ of its linear part dϕ(0) at the origin (see [7,Lemma 2.2]). Denote by a 1 , . . . , a the linear subspace generated by a 1 , . . . , a . The following direct result is classical and extensively used in equivariant continuous-time dynamics: Lemma 2.6. Let F : (R n , 0) → (R n , 0) be a germ of an equivariant diffeomorphism with symmetry group Γ + . If Σ ≤ Γ + is a subgroup, then Fix(Σ) is F -invariant. Hence, fixed-point sets of symmetries of F are invariant under the discrete dynamics ruled by F . We now turn to reversing symmetries: Lemma 2.7. Let ϕ 1 : (R n , 0) → (R n , 0) be an involution and let F : (R n , 0) → (R n , 0) be a ϕ 1 -reversible germ of diffeomorphism. Consider the two sequences of reversing symmetries of F given in (2). The following equalities hold: F (Fix(ϕ k+2 )) = Fix(ϕ k ), F (Fix(ϕ k )) = Fix(ϕ k+2 ), k ∈ N, k ≥ 1. Proof. Consider x ∈ Fix(ϕ k+2 ), i.e., x = ϕ 2 (F k (x)). If y = F (x), then y = F (ϕ 2 (F k (x))) = ϕ 2 (F −1 (F k (x))) = ϕ 2 (F k−2 (F (x))) = ϕ k (y), so F (Fix(ϕ k+2 )) ⊆ Fix(ϕ k ). For the other inclusion, let y ∈ Fix(ϕ k ). Then y = ϕ k (y) = ϕ 2 • F k−2 (y) = F (ϕ 2 • F k−1 (y)) and, therefore, y = F (x) for x = ϕ 2 • F k−1 (y). Also, ϕ k+2 (x) = ϕ 2 • F k (x) = ϕ 2 • F k−1 (y) = x. The equalities for the other sequence are obtained analogously. Theorem 2.8. Applying F to the fixed-point submanifolds of the involutions of (2) the following chains are obtained: · · · −→ Fix(ϕ 2k ) −→ · · · −→ Fix(ϕ 2 ) −→ Fix(ϕ 2 ) −→ · · · −→ Fix(ϕ 2k ) −→ · · · ,(3)· · · −→ Fix(ϕ 2k+1 ) −→ · · · −→ Fix(ϕ 1 ) −→ Fix(ϕ 3 ) −→ · · · −→ Fix(ϕ 2k+1 ) −→ · · · , for k ≥ 1. Proof. Similar calculations to those of Lemma 2.7 give F (Fix(ϕ 1 )) = Fix(ϕ 3 ) and F (Fix(ϕ 2 )) = Fix(ϕ 2 ). We now use Lemma 2.7 to get the result. If for any k ≥ 1 and ∈ N we have either Fix(ϕ k ) = Fix(ϕ k+2 ) or Fix(ϕ k ) = Fix(ϕ k+2 ) or Fix(ϕ k ) = Fix(ϕ k+2 ) then the whole chain in (3) containing one of these fixed-point manifolds is finite. The following is also a direct consequence of Lemma 2.7: Corollary 2.9. All fixed-point submanifolds of the involutions of (2) with odd index have dimension equal to dim Fix(ϕ 1 ), and the ones for even index have dimension equal to dim Fix(ϕ 2 ). It should be stressed that dynamically and geometrically relevant results should not depend on the choice of coordinates. Proposition 2.11 below establishes this point. Before stating it we define the equivalence of two sets of involutions which is given by simultaneous conjugacy: Definition 2.10. Two pairs (ϕ 1 , ϕ 2 ) and (ψ 1 , ψ 2 ) of involutions on (R n , 0) are equivalent if there exists a germ of diffeomorphism h : (R n , 0) → (R n , 0) such that ψ i = h • ϕ i • h −1 , for i = 1, 2. In this case, if (ϕ 1 , ϕ 2 ) and (ψ 1 , ψ 2 ) are linear, then h is a linear isomorphism and we say that (ϕ 1 , ϕ 2 ) and (ψ 1 , ψ 2 ) are linearly equivalent. For two equivalent pairs of involutions (ϕ 1 , ϕ 2 ) and (ψ 1 , ψ 2 ), it follows that for each k ∈ N, k ≥ 2, the two pairs (ϕ k , ϕ k ) and (ψ k , ψ k ), constructed in (2) for F = ϕ 1 • ϕ 2 and G = ψ 1 • ψ 2 respectively, are equivalent. In addition, F and G are also conjugate, so they generate equivalent dynamics. These equivalences are clearly governed by the same h, which directly implies the following: The next result gives a sufficient condition for an orbit starting at a fixed-point submanifold to be periodic: Proposition 2.12. For any reversing symmetry ϕ of F , if x ∈ Fix(ϕ) and if there exists ∈ Z such that F (x) ∈ Fix(ϕ), then the orbit of x is periodic. In addition, for the sequences of reversing symmetries of F in (2) and > k, we have: (A) If x ∈ Fix(ϕ k ) ∩ Fix(ϕ ) , then the orbit of x is a periodic orbit with period that divides − k. Also, if x is a periodic point with period that divides − k and x ∈ Fix(ϕ k ), then x ∈ Fix(ϕ ). (B) If x ∈ Fix(ϕ k ) ∩ Fix(ϕ ) , then the orbit of x is a periodic orbit with period that divides − k. Also, if x is a periodic point with period that divides − k and x ∈ Fix(ϕ k ), then x ∈ Fix(ϕ ). (C) If x ∈ Fix(ϕ k ) ∩ Fix(ϕ ) , then the orbit of x is a periodic orbit with period that divides k + − 2. Also, if x is a periodic point with period that divides k + − 2 and x ∈ Fix(ϕ k ), then x ∈ Fix(ϕ ). Proof. The first part is straightforward, just noticing that x, F (x) ∈ Fix(ϕ) implies that F − (x) = F − (ϕ(x)) = ϕ(F (x)) = F (x). For the statements (A)-(C) we use (2) to get ϕ k • ϕ = F −k , ϕ k • ϕ = F k− , ϕ k • ϕ = F +k−2 . The periodicity then follows from Fix(ϕ k ) ∩ Fix(ϕ ) ⊆ Fix(ϕ k • ϕ ) = Fix(F −k ) for (A) and similarly for (B) and for (C). A particular case of the proposition above can be found in [4], namely when the whole space has even dimension and the fixed-point subspaces are n/2-dimensional submanifolds of R n . From Corollary 2.9, if dim Fix(ϕ 1 ) = dim Fix(ϕ 2 ) = n − 1, then the fixed-point submanifolds split (R n , 0) into connected regions. The result below describes how the dynamics by F = ϕ 1 • ϕ 2 behaves with respect to these regions. (2). Then F = ϕ 1 •ϕ 2 interchanges the connected components of the germ at the origin of Theorem 2.13. Let ϕ 1 , ϕ 2 : (R n , 0) → (R n , 0) be two involutions with dim Fix(ϕ 1 ) = dim Fix(ϕ 2 ) = n−1. Let ϕ k , ϕ k ∈ Γ − , k ∈ N, k ≥ 1 be as inC = R n \ ∞ k=1 (Fix(ϕ k ) ∪ Fix(ϕ k )) , determined by these fixed-point submanifolds. Proof. Take V a region whose boundary is determined by the fixed-point manifolds of involutions ψ p , ψ q ∈ ∆ = {ϕ k , ϕ k : k ≥ 1}. By path connectedness of F (V ) and Lemma 2.7, the boundary of F (V ) is determined by Fix( ψ p ) ∪ Fix( ψ q ), with ψ p , ψ q ∈ ∆ where each pair (ψ p , ψ p ) and (ψ q , ψ q ) consists of consecutive elements in one of the chains in (3), that is, F (Fix(ψ p )) = Fix( ψ p ), F (Fix(ψ q )) = Fix( ψ q ). Therefore, F (V ) is another component. It remains to consider the case when part of the boundary of a connected component V of the complement is not contained in a fixed-point manifold of any involution in ∆ (see the examples of Subsections 3.3.2, 3.3.4 and 4.1). This happens when the boundary of V meets the set C of accumulation points of these fixed-point manifolds. We claim that F (C) ⊂ C. Indeed, a point x ∈ C may be written as x = lim n→∞ x n with x n ∈ Fix(ψ n ) for some ψ n ∈ ∆. Since F is a homeomorphism, then F (x) = lim n→∞ F (x n ). Lemma 2.7 implies that F (x n ) ∈ Fix( ψ n ) for some ψ n ∈ ∆, establishing the claim. It follows that if the boundary of a component V ⊂ C consists of accumulation points of fixed-point subspaces, then V is mapped by F into a component with the same type of boundary. The same is true if the boundary of V contains both elements of fixed-point submanifolds and elements of C, completing the proof. We now define transversality of two involutions. This is a generic condition we assume for the pairs of involutions treated in the next sections. Definition 2.14. Two involutions ϕ 1 , ϕ 2 on (R n , 0), n ≥ 2, are transversal if Fix(ϕ 1 ) and Fix(ϕ 2 ) are in general position at 0, i.e., R n = T 0 Fix(ϕ 1 ) + T 0 Fix(ϕ 2 ), where T 0 Fix(ϕ i ) denotes the tangent space to Fix(ϕ i ) at 0, i = 1, 2. In the next two sections we apply the previous results to analyse the behavior of a ϕ 1 -reversible germ of diffeomorphism F associated with a pair (ϕ 1 , ϕ 2 ) of transversal linear involutions on (R n , 0), for n = 2 and n ≥ 3, respectively. For this, we restrict our study to the linear case, considering the group of symmetries Γ + and the set of reversing symmetries Γ − , defined in (1), as subsets of the linear group GL(n) and keeping the notation introduced in this section. Dynamics and geometry of linear reversible maps on the plane In this section we consider a germ of diffeomorphism F = ψ 1 • ψ 2 on (R 2 , 0), where ψ 1 and ψ 2 are transversal linear involutions (Definition 2.14). We present the results up to equivalence of pairs of involutions given by simultaneous conjugacy. Hence, the pair (ψ 1 , ψ 2 ) is considered to be in normal form, which is given in [6,Theorem 6.2]. For that, we first recall the definition of the antipodal subspace of a linear involution ϕ on R n , A(ϕ) = {x ∈ R n : ϕ(x) = −x}. Notice that R n = Fix(ϕ) ⊕ A(ϕ). Denote by Λ = [ψ 1 , ψ 2 ] the group generated by ψ 1 and ψ 2 . There are three cases to be considered: (i) Λ is Abelian; (ii) Λ is non-Abelian and A(ψ 2 ) = Fix(ψ 1 ); (iii) Λ is non-Abelian and A(ψ 2 ) = Fix(ψ 1 ). Our aim is to investigate, for the cases above, the fixed-point subspaces of the reversing involutions in Λ − = Λ ∩ Γ − and their relation to the dynamics generated by F. Let us denote by Λ + = Λ ∩ Γ + the group generated by F and by Z 2 (ϕ) the 2-element group generated by ϕ. Λ is Abelian. This is a trivial case. By [6, Theorem 6.2], (ψ 1 , ψ 2 ) is equivalent to (ϕ 1 , ϕ 2 ), where ϕ 1 (x, y) = (−x, y) and ϕ 2 (x, y) = (x, −y), so Λ = Z 2 (ϕ 1 ) ⊕ Z 2 (ϕ 2 ). Since F = −I 2 , the dynamics is rather degenerate because all the F -orbits (except the origin) are periodic of period 2. Moreover, ϕ 2k+1 = ϕ 2k+1 = ϕ 1 and ϕ 2k = ϕ 2k = ϕ 2 for all k ≥ 1. Therefore, Fix(F ) = {(0, 0)} and the fixed-point subspaces Fix(ϕ 1 ) = (0, 1) and Fix(ϕ 2 ) = (1, 0) divide the plane into four connected components that are interchanged by F (Thereom 2.13). Moreover, Γ + = Γ − = GL(2), while Λ + = Z 2 (−I 2 ) and Λ − = {ϕ 1 , ϕ 2 }. Λ is non-Abelian and X k = F k (X 0 ). we have F k = (−1) k (I 2 − kN ) = I 2 for all k ∈ Z, which implies that ϕ k (x, y) = (−1) k (x, (k − 2)x − y) and ϕ k (x, y) = (−1) k (x, −kx − y). Yet, Fix(F ) = {(0, 0)} = Fix(F 2k+1 ), Fix(ϕ 2k ) = (1, k − 1) , Fix(ϕ 2k ) = (1, −k) , Fix(F 2k ) = Fix(ϕ 2k+1 ) = Fix(ϕ 2k+1 ) = (0, 1) , for all k ≥ 1. Therefore, the fixed-point subspaces of ϕ 2k and ϕ 2k approach the y-axis as k tends to infinity (see Figure 1). For the dynamics, we use the expression of F k to conclude that the y-axis is F -invariant and that the orbits of all its points, except the origin, have period 2. For the other points, by linearity, it suffices to look at the orbits of points (1, y) given by F k (1, y) = (−1) k (1, y − k), as illustrated in Figure 1, on the right. This case provides an interesting illustration of Theorem 2.13. For instance, the sector {(x, y) : x > 0, 0 < y < x} between Fix(ϕ 2 ) and Fix(ϕ 4 ) is mapped onto the sector {(x, y) : x < 0, 0 < y < −x} between Fix(ϕ 2 ) and Fix(ϕ 2 ). Also, the sector {(x, y) : x < 0, −x < y < −2x} between Fix(ϕ 2 ) and Fix(ϕ 4 ) is mapped onto the sector {(x, y) : x > 0, −3x < y < −2x} between Fix(ϕ 6 ) and Fix(ϕ 4 ). See Figure 1 on the left. Symmetries and reversing symmetries are as follows: Λ + = (−1) k (I 2 − kN ) : k ∈ Z , Λ − = {ϕ k , ϕ k : k ∈ N, k ≥ 1} , Γ + = a 0 c a : a, c ∈ R, a = 0 and Γ − = a 0 c −a : a, c ∈ R, a = 0 , both Γ ± manifolds of dimension 2. 3.3. Λ is non-Abelian and A(ψ 2 ) = Fix(ψ 1 ). Let us denote t = tr(ψ 1 •ψ 2 ). From [6, Theorem 6.2], (ψ 1 , ψ 2 ) is equivalent to (ϕ 1 , ϕ 2 ), where (4) ϕ 1 (x, y) = (−x, y + (2 + t)x) and ϕ 2 (x, y) = (x + y, −y). The analysis here considers all possibilities for the parameter t, which is an invariant under linear simultaneous conjugacy. We have F = −1 −1 2 + t 1 + t , whose eigenvalues λ + = t + √ t 2 − 4 2 and λ − = t − √ t 2 − 4 2 satisfy λ + λ − = 1. Notice that λ + = λ − = ±1 if, and only if, t = ±2, respectively. A direct calculation gives the group of symmetries of F Γ + = a + (2 + t)b b −(2 + t)b a : a, b ∈ R and the set of reversing symmetries of F Γ − = 1 1 0 −1 Γ + = a a + b (2 + t)b −a : a, b ∈ R , both manifolds of dimension 2. Since the normal form of F depends only on the parameter t, we have subdivided this subsection in four cases. In all of them the group of symmetries and the set of reversing symmetries generated by ϕ 1 and ϕ 2 are given respectively by Λ + = F k : k ∈ Z and Λ − = {ϕ k , ϕ k : k ∈ N, k ≥ 1}. 3.3.1. Normal form (4) with t = −2. In this case, F (x, y) = (−x − y, −y), whose eigenvalues are λ 1 = λ 2 = −1 with geometric multiplicity 1. Writing F = −I 2 + N, where N is a nilpotent matrix of index 2, we have F k (x, y) = (−1) k (x + ky, y) for all k ∈ Z, which implies that ϕ k (x, y) = (−1) k (x + (k − 1)y, −y) and ϕ k (x, y) = (−1) k (x − (k − 1)y, −y). Therefore Fix(F ) = {(0, 0)} = Fix(F 2k+1 ), Fix(ϕ 2k ) = Fix(ϕ 2k ) = Fix(F 2k ) = (1, 0) , Fix(ϕ 2k+1 ) = (−k, 1) and Fix(ϕ 2k+1 ) = (k, 1) for all k ≥ 1. The fixed-point subspaces of ϕ 2k+1 and ϕ 2k+1 approach the x-axis as k tends to infinity, which is an F -invariant line (see Figure 2). 3.3.2. Normal form (4) with t = 2. In this case, F (x, y) = (−x − y, 4x + 3y), whose eigenvalues are λ 1 = λ 2 = 1 with geometric multiplicity 1. Writing F = I 2 + N for a nilpotent matrix N of index 2, we have F k (x, y) = (1 − 2k)x − ky, 4kx + (1 + 2k)y , ∀ k ∈ Z, which implies that ϕ k (x, y) = ((2k − 3)x + (k − 1)y, −4(k − 2)x − (2k − 3)y) and ϕ k (x, y) = ((1 − 2k)x − (k − 1)y, 4kx − (1 − 2k)y). Therefore Fix(F ) = Fix(F k ) = (1, −2) is the eigenspace of F associated with λ = 1 for all k ∈ Z. Moreover, Fix(ϕ k ) = (k − 1, 4 − 2k) and Fix(ϕ k ) = (k − 1, −2k) for all k ≥ 1. The fixed-point subspaces of ϕ k and ϕ k approach the F -invariant line y = −2x as k tends to infinity (see Figure 2). Here the two half-lines Fix(F ) ∩ {(x, y) ∈ R 2 : x > 0} and Fix(F ) ∩ {(x, y) ∈ R 2 : x < 0} are components of R 2 \ ∞ k=1 (Fix(ϕ k ) ∪ Fix(ϕ k )) . 3.3.3. Normal form (4) with \|t\| < 2. When −2 < t < 2, the map F has complex eigenvalues λ ± = t ± i √ 4 − t 2 2 , with \|λ ± \| = 1. Hence F is diagonalizable over C with λ ± = e ±iθ , where θ = arccos(t/2). This means that there is a change of coordinates that conjugates F to a rotation of θ radians around the origin. The complex eigenvectors associated to λ ± have the form R + iI where R = 1, − t 2 − 1 I = 0, − √ 4 − t 2 2 . From now on we use coordinates in the basis β = {R, I} of R 2 for which, taking α k = (k − 1)θ, with k ≥ 1, we have ϕ k \| β = − cos α k sin α k sin α k cos α k and ϕ k \| β = − cos α k − sin α k − sin α k cos α k , whence Fix(ϕ k ) = (sin α k , 1 + cos α k )\| β and Fix(ϕ k ) = (− sin α k , 1 + cos α k )\| β . Hence the coordinates in the basis β of the generators of Fix(ϕ k ) and Fix(ϕ k ) lie on a circle of center (0, 1) and radius 1 (see Figure 3). When θ/2π = p/q, p, q ∈ Z, q = 0, then F q = I 2 and all the F -orbits are periodic of period q. Also, Λ − is finite and their fixed-point subspaces form a finite set of lines through the origin. When θ/2π / ∈ Q, Λ − is infinite and a set of generators of Fix(ϕ k ) and of Fix(ϕ k ) can be taken to form each a dense set in the circle, and hence the union of fixed-point subspaces is dense in the plane. In the original coordinates, there is a family of concentric F -invariant ellipses and each F -orbit is dense on the ellipse that contains it. Figure 3. Fixed-point subspaces in the R × I coordinates for the involutions ϕ k in case (iii), when \|t\| < 2 for θ = 2π/5. In this case Fix(ϕ 1 ) is the I-axis, Fix(ϕ k ) = Fix(ϕ l ) = Fix(ϕ r ) when k ≡ l (mod 5) and r + k ≡ 2 (mod 5). (4) with \|t\| > 2. When t > 2 the map F has real eigenvalues λ + > 1 and 0 < λ − < 1, whereas for t < −2 the eigenvalues of F satisfy λ − < −1 < λ + < 0. Hence, F is hyperbolic and F k = I 2 for k = 0. The eigenvectors of F associated to λ ± are generated by (1, −1 − λ ± ), respectively. From now on we use coordinates in the basis Normal form β = {(1, −1 − λ + ), (1, −1 − λ − )} of R 2 for which we have F diagonal, ϕ k \| β = 0 −λ k−1 − −λ k−1 + 0 and ϕ k \| β = 0 −λ k−1 + −λ k−1 − 0 . Therefore, Fix(ϕ k ) = (1, −λ k−1 + )\| β and Fix(ϕ k ) = (1, −λ k−1 − )\| β . These fixed-point subspaces do not coincide with the eigenspaces of F . Hence, powers of F and F −1 map Fix(ϕ k ), Fix(ϕ k ) into distinct subspaces, according to Theorem 2.8. Moreover, it follows that the Fix(ϕ k )'s accumulate, when k → ∞, on the expanding eigenspace of F : if t > 2 the Fix(ϕ k )'s accumulate on the eigenspace of λ + > 1 and for t < −2 they accumulate on the eigenspace of λ − < −1 (see Figure 4). Similarly, the subspaces Fix(ϕ k )'s accumulate, when k → ∞, on the contracting eigenspace of F (the expanding eigenspace of F −1 ). Dynamics and geometry of linear reversible maps for n ≥ 3 In this section we obtain a generalization of the results of Section 3 for n ≥ 3. As we shall see, the planar case leads to a similar analysis of the dynamics of a germ of diffeomorphism F = ψ 1 • ψ 2 on (R n , 0), for n ≥ 3, where ψ 1 and ψ 2 are transversal linear involutions. Again, we denote by Λ = [ψ 1 , ψ 2 ] the group generated by the involutions and use the normal forms for the pairs of transversal linear involutions on (R n , 0) now given in [6,Theorem 7.3]. There are five cases to consider: (a) Λ is Abelian; (b) Λ is non-Abelian, tr(F ) = n and A(ψ 2 ) ⊂ Fix(ψ 1 ); (c) Λ is non-Abelian, tr(F ) = n and A(ψ 2 ) ⊂ Fix(ψ 1 ); (d) Λ is non-Abelian, tr(F ) = n and A(ψ 1 ) = A(ψ 2 ); (e) Λ is non-Abelian, tr(F ) = n and A(ψ 1 ) = A(ψ 2 ). We investigate the relation between the fixed-point subspaces of the involutions in Λ − and the dynamics generated by F. Let us denote by e i the vector with 1 in the i-th coordinate and 0 elsewhere, for i = 1, . . . , n. The following definition will be useful: Definition 4.1. The map-germ f : (R m+ , 0) → (R m+ , 0) is a suspension of f : (R m , 0) → (R m , 0) if f (x, y) = (f (x), y) , where x ∈ R m and y ∈ R . The pair ( ϕ 1 , ϕ 2 ) is a suspension of (ϕ 1 , ϕ 2 ) if each ϕ i is a suspension of ϕ i in the same system of coordinates. A consequence of the results of [6] is that a pair of linear transversal involutions (ψ 1 , ψ 2 ) is equivalent to a suspension of a pair of planar involutions, except in case (e), with n ≥ 4, for which the pair is equivalent to a suspension of a pair of involutions in R 3 . The following trivial proposition summarises the properties of suspensions. Proposition 4.2. For linear involutions ϕ i : (R m+ , 0) → (R m+ , 0), i = 1, 2, if ( ϕ 1 , ϕ 2 ) is a suspension of the involutions (ϕ 1 , ϕ 2 ), with ϕ i : (R m , 0) → (R m , 0), then: • F = ϕ 1 • ϕ 2 is a suspension of F = ϕ 1 • ϕ 2 ; • ϕ k and ϕ k , k ≥ 1 integer, are suspensions of ϕ k and ϕ k respectively; • Fix( ϕ k ) = Fix(ϕ k ) × R and Fix( ϕ k ) = Fix(ϕ k ) × R ; • The group Γ + of symmetries of F consists of matrices of the form A 0 0 B , where A ∈ Γ + is a symmetry of F and B ∈ GL( ), with a similar result for the set Γ − of reversing symmetries of F . Using [6,Theorem 7.3], the relation of cases (a)-(d) above to (i)-(iii) of the previous section is the following: (a) if Λ is Abelian, then (ψ 1 , ψ 2 ) is equivalent to ( ϕ 1 , ϕ 2 ), where ( ϕ 1 , ϕ 2 ) is a suspension of the normal forms in Subsection 3.1; (b) if Λ is non-Abelian, tr(F ) = n and A(ψ 2 ) ⊂ Fix(ψ 1 ), then (ψ 1 , ψ 2 ) is equivalent to ( ϕ 1 , ϕ 2 ), where ( ϕ 1 , ϕ 2 ) is a suspension of the normal forms in Subsection 3.2; (c) if Λ is non-Abelian, tr(F ) = n and A(ψ 2 ) ⊂ Fix(ψ 1 ), then (ψ 1 , ψ 2 ) is equivalent to ( ϕ 1 , ϕ 2 ), where ( ϕ 1 , ϕ 2 ) is a suspension of the normal forms in Subsection 3.3 with t = 2; (d) if Λ is non-Abelian, tr(F ) = n and A(ψ 1 ) = A(ψ 2 ), then (ψ 1 , ψ 2 ) is equivalent to ( ϕ 1 , ϕ 2 ), where ( ϕ 1 , ϕ 2 ) is a suspension of the normal forms in Subsection 3.3 with t = 2. All these cases satisfy the hypothesis of Theorem 2.13, i.e., Fix(ψ 1 ) and Fix(ϕ 2 ) are hyperplanes. In the next subsection we discuss the remaining case (e), which does not suspend from the planar problem. is a nilpotent matrix of index 3. Therefore F has eigenvalues λ = 1 with algebraic multiplicity n and geometric multiplicity n − 2. Since . , e n for all k ≥ 2. For k = 1, we have Fix(ϕ 1 ) = Fix(ϕ 1 ) = e 2 , e 3 , . . . , e n . Thus, when k → ∞, the fixed-point subspaces of ϕ k and ϕ k approach the F -invariant subspace x 2 = −2x 1 . The subspaces Fix(ϕ k ) and Fix(ϕ k ) have codimension 1, so we can still apply Theorem 2.13. The situation is similar to the example of Subsection 3.3.2: the invariant limit hyperplane contains two connected components of R n \ Proposition 2. 11 . 11The equalities in Lemma 2.7 and the chains in Theorem 2.8 are invariant under equivalence. Figure 1 . 1A(ψ 2 ) = Fix(ψ 1 ). By [6, Theorem 6.2], (ψ 1 , ψ 2 ) is equivalent to (ϕ 1 , ϕ 2 ), where ϕ 1 (x, y) = (−x, x + y) and ϕ 2 (x, y) = (x, −y).In this case F (x, y) = (−x, x − y), whose eigenvalues are λ 1 = λ 2 = −1 with geometric multiplicity 1. Writing F = −I 2 + N, where Left: fixed-point subspaces for the involutions ϕ 1 , ϕ 2k and ϕ 2k in case (ii). Note that Fix(ϕ 2k+1 ) = Fix(ϕ 2k+1 ) = Fix(ϕ 1 ), k = 1, 2, 3, . . .. Right: orbit of the point X 0 = (1, 0), Figure 2 . 2Fixed-point subspaces for the involutions ϕ 1 , ϕ 2k and ϕ 2k in case (iii) with t = ±2. Left: t = −2. In this case Fix(ϕ 2k ) = Fix(ϕ 2k ) = Fix(ϕ 2 ), k = 1, 2, 3, . . .. Right: t = 2. The dashed red line is Fix(F ) = {(x, y) : y = −2x} where all the lines Fix(ϕ k ) and Fix(ϕ k ) accumulate as k → ∞. Figure 4 . 4Eigenspaces for F (red dotted lines) and fixed-point subspaces for the involutions ϕ 1 , ϕ 2k and ϕ 2k in case (iii) for \|t\| > 2. Left: t = −3. Right: t = 3. 4. 1 . 1Case (e): Λ is non-Abelian, tr(F ) = n and A(ψ 1 ) = A(ψ 2 ). By[6, Theorem 7.3], (ψ 1 , ψ 2 ) is equivalent to (ϕ 1 , ϕ 2 ), where ϕ 1 (x 1 , . . . , x n ) = (−x 1 , 4x 1 + x 2 , x 3 , . . . , x n ), ϕ 2 (x 1 , . . . , x n ) = (x 1 + x 2 , −x 2 , x 2 + x 3 , x 4 , . .. , x n ). In this case F = I n + N, k ∈ Z, we have Fix(F ) = Fix(F k ) = e 3 , . . . , e n , for all k ∈ Z. Moreover, Fix(ϕ k ) = (k − 1)e 1 + (4 − 2k)e 2 , e 3 , . . . , e n and Fix(ϕ k ) = (k − 1)e 1 − 2ke 2 , e 3 , . . ∞ k=1 ( k=1Fix(ϕ k ) ∪ Fix(ϕ k )) .The group formed by the linear symmetries of F is written as Γ + × GL(n − 3), where Γ + is the 3-dimensional manifold of elements , a, b, c ∈ R, a = 2b.The set of the linear reversing symmetries of F is then Γ − × GL(n − 3), where Γ − = ϕ 1 Γ + is the 3, a, b, c ∈ R, a = 2b. CAPES/FCT provided financial support for visits of the authors to the Universities of Porto and of São Paulo, whose hospitality is gratefully acknowledged. 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On topological stability of divergent diagrams of folds. M A Teixeira, Math. Zeitschrift. 180Teixeira, M. A., On topological stability of divergent diagrams of folds. Math. Zeitschrift 180 382-390, 1981. Local and simultaneous structural stability of certain diffeomorphisms. M A Teixeira, Proc. Dyn. Sys. Stab. and Turb. Warwick. 898Lect. Notes in Mat.Teixeira, M. A., Local and simultaneous structural stability of certain diffeomorphisms, Proc. Dyn. Sys. Stab. and Turb. Warwick, in: Lect. Notes in Mat. 898 382 -390, 1981. Analytic classification of pairs of involutions and its applications. S M Voronin, Funct. Anal. Appl. 162Voronin, S. M., Analytic classification of pairs of involutions and its applications. Funct. Anal. Appl. 16(2) 21-29, 1982.	[]

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